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1106.2253
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030002 2011 L. Viña R. Gordon, Department of Electrical and Computer
Engineering,
University of Victoria, British Columbia, Canada. 030002
We report the enhancement of the optical second harmonic signal in non-
centrosymmetric semiconductor CdS quantum dots, when they are placed in close
contact with isolated silver nanoparticles. The intensity enhancement is about
1000. We also show that the enhancement increases when the incoming laser
frequency $\omega$ is tuned toward the spectral position of the silver plasmon
at $2\omega$, proving that the silver nanoparticle modifies the nonlinear
emission.
# Plasmon-enhanced second harmonic generation in semiconductor quantum dots
close to metal nanoparticles
Pablo M. Jais [conicet, lec] Catalina von Bilderling E-mail: jaisp@df.uba.ar
[cma, conicet] Andrea V. Bragas[conicet, lec] E-mail: catalina@df.uba.arE-
mail: bragas@df.uba.ar
(8 March 2011; 30 May 2011)
††volume: 3
99 conicet Instituto de Física de Buenos Aires, CONICET, Argentina. lec
Laboratorio de Electrónica Cuántica, Departamento de Física, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Intendente
Güiraldes 2160, Pabellón I - Ciudad Universitaria, Buenos Aires, C1428EHA,
Argentina. cma Centro de Microscopías Avanzadas, Facultad de Ciencias Exactas
y Naturales, Universidad de Buenos Aires, Argentina.
## 1 Introduction
Second Harmonic Generation (SHG) is the second order nonlinear process for
which two photons with the same frequency $\omega$ interact simultaneously
with matter to generate a photon of frequency $2\omega$. The nonlinear
response of a material to an applied field becomes evident in the presence of
intense fields, as the one given by a focused pulsed laser.
Due to symmetry requirements for second order nonlinear processes, the leading
dipolar term of the SHG is forbidden for centrosymmetric materials. However,
this condition is relaxed at surfaces, where the symmetry is broken. On the
other hand, for a small and perfectly spherical particle of any type of
material under homogeneous illumination, the SHG from the surface is also zero
since the geometrical shape recovers the symmetry [1].
The presence of inhomogeneities in the electromagnetic field recovers the SHG
even for symmetric materials, both from bulk and surface, as pointed out by
Brudny et al. [1]. Those inhomogeneities may be, for instance, the consequence
of a strong focusing of the field, the proximity of a surface or the presence
of another particle. Besides, electronic resonances and enhancement of the
field around the particle will increase the SHG and, in many cases, would make
it detectable.
It is very well-known, as one of the main results of nano-optics, that the
field around a metal nanoparticle is confined and enhanced when the incoming
wavelength is resonant with the surface plasmons sustained by the
nanoparticle. This effect is exploited in many different applications of metal
nanoparticles: volume reducers for diffusion measurements [2], optical probes
for high resolution imaging [3, 4, 5], biosensors [6], nanoheaters [7] and
surface enhanced Raman scattering (SERS) [8, 9], among others.
In the present paper, we show the enhancement of the SHG when non-
centrosymmetric CdS quantum dots are placed in close contact with silver
nanoparticles. The plasmon of the nanoparticles is resonant at the second
harmonic (SH) frequency and enhances the emission. As far as we know, there
are few works on the enhancement of SHG (either in semiconductor QDs or in
molecules) produced by the resonant excitation of metal nanoparticles [10, 11,
12, 13].
## 2 Materials and methods
The experiment was performed in transmission, by performing spectrally-
resolved photon counting (Fig. 1). The laser was a KMLabs tunable modelocked
Ti:Sapphire with $50\,$fs pulse width, $400\,$mW average power, $80\,$MHz
repetition rate and tunable in the range 770–$805\,$nm. The resolution of the
monochromator was $2\,$nm in all experiments. Silver nanoparticles (NPs), of
average diameter $20\,$nm, were synthesized as in Marchi et al. [14], while
the CdS quantum dots (QDs), $3\,$nm in diameter, as in Frattini et al. [15].
Figure 1: Experimental setup. PMT: photomultiplier tube; m1 to m3: mirrors, l1
to l3: lenses.
The absorption spectra for both NPs and QDs in solution are shown in Fig. 2.
The arrows mark the first and second excitonic transitions in the QDs, at
$(362\pm 4)\,$nm and $(425\pm 15)\,$nm, obtained by the second derivative
method [16]. Figure 2 also shows the two-photon photoluminescence (TPPL)
spectrum of the QDs in solution, excited at $\lambda=780\,$nm and measured
with the setup shown in Fig. 1. A Stokes shift was observed in the TPPL, in
accordance with the same effect reported for the (one photon)
photoluminescence of CdS QDs [17].
Figure 2: Absorption (red line) and two-photon photoluminescence, TPPL (red
squares), of the CdS QDs in ethanol solution. The dotted blue line is the
extinction of the Ag NP in aqueous solution.
NP samples were prepared by placing a drop of the NP solution on a glass
substrate until the solvent evaporated. The average NP concentration was
$\approx 10$ particles per $\upmu\textrm{m}^{2}$. However, optical and SEM
images revealed that the metal NPs are inhomogeneously distributed on the
substrate, with a high concentration of particles in the fringe (drop
boundary).
A CdS QD sample was prepared according to the same protocol. However, the QD
solution has a very high concentration of aminosilanes. They form a thick
layer when dried, so the QDs are immersed in an aminosilane matrix. The
estimated height of this layer was $\approx 5\,\upmu\textrm{m}$, and the
average QD concentration was $\approx 10^{5}$ particles per
$\upmu\textrm{m}^{2}$ (inferred from AFM images taken at a 1:1000 diluted
concentration).
For the mixed NP-QD sample, we first dried a drop of the metal NP solution,
and then we deposited a drop of CdS QD solution on top of it. The resulting
structure is schematically shown in Fig. 3.
Figure 3: Diagram of the NP-QD sample.
## 3 Results and discussion
For the silver NP samples we did not detect SH signal above the noise level of
the experiment (of about 200 counts per second). This result agrees with the
fact that silver is a centrosymmetric material and the NPs are almost
spherical, even considering that their interaction should produce a weak SH
signal. On the other hand, Fig. 4a shows that the CdS QD sample presents a
small SH peak (as expected for a non-centrosymmetric material) and a strong
TPPL signal. The central wavelength of the excitation laser for this
measurement was $784\,$nm, and its FWHM was $26\,$nm.
Figure 4: SH and TPPL spectra for (a) the CdS QD sample (multiplied by 5) and
(b) the mixed Ag NP-CdS QD sample. Note that the SH peak was 20 times higher
in the mixed sample, and the ratio between the SH and the TPPL changes
drastically.
However, a strong SH signal was measured in the mixed NP-QD sample, as shown
in Fig. 4b. The SHG was 20 times larger in this sample, and the SH${}/{}$TPPL
ratio increased significantly. In addition, in the mixed sample, the TPPL
decayed after a few seconds of laser irradiation while the SHG remained stable
over time. In this case, the central wavelength was $799\,$nm, and the FWHM
was also $26\,$nm.
The observations can be attributed to the enhancement of the nonlinear field
around the QDs due to silver NP plasmon resonance at $2\omega$.The incoming
light at $\omega$ is not enhanced by the silver NP, but creates a weak
nonlinear signal at $2\omega$ in the QDs. This outgoing signal is enhanced by
the field enhancement factor ($\eta$), so that the SHG intensity is enhanced
by $\eta^{2}$. Note that in the analogous case of SERS, the incoming and
outgoing signals are enhanced giving an intensity enhancement of $\eta^{4}$
[18].
The magnitude of this enhancement is not easy to calculate since the mixed NP-
QD sample contains several hot-spots (usually in the concentrated NP fringe),
and their intensities were wildly different. Nevertheless, the highest
intensities recorded for the NP-QD sample were $\approx 10^{4}\,$cts/s, while
the highest ones for the QD sample were $\approx 1700\,$cts/s. To estimate the
enhancement, we used the analytical enhancement factor (AEF), defined as [19]
$\mathrm{AEF}=\frac{I_{\mathrm{\scriptscriptstyle{NP-
QD}}}/\rho_{\mathrm{\scriptscriptstyle{NP-
QD}}}}{I_{\mathrm{\scriptscriptstyle{QD}}}/\rho_{\mathrm{\scriptscriptstyle{QD}}}},$
(1)
where $I$ and $\rho$ are the SHG intensity and the surface concentration of
QDs in the corresponding samples.
We must take into consideration that there is a thick layer of QDs that are
not in the enhancement region of the NPs, i.e., the measured signal in the NP-
QD sample is the sum of the intensity coming from QDs close to the NPs that
are enhanced, and the intensity from QDs far from the NPs that are not
enhanced. If we keep only the enhanced fraction,
$\mathrm{AEF}\approx\frac{I_{\mathrm{\scriptscriptstyle{NP-
QD}}}-I_{\mathrm{\scriptscriptstyle{QD}}}}{I_{\mathrm{\scriptscriptstyle{QD}}}}\frac{\rho_{\mathrm{\scriptscriptstyle{QD}}}}{\tilde{\rho}_{\mathrm{\scriptscriptstyle{NP-
QD}}}},$ (2)
where $\tilde{\rho}_{\mathrm{\scriptscriptstyle{NP-QD}}}$ is the concentration
of QDs near the NPs. Since the NP enhancement decays one order of magnitude in
$5\,$nm [20], we can take $\tilde{\rho}_{\mathrm{\scriptscriptstyle{NP-
QD}}}\approx\rho_{\mathrm{\scriptscriptstyle{QD}}}\frac{5\,\mathrm{nm}}{5\,\upmu\textrm{m}}$.
This gives AEF${}\approx 5\,10^{3}$. This is just an estimation of the order
of magnitude of the enhancement, since the uncertainties in the thicknesses
and maximum intensities are very high, but this estimation is consistent with
the enhancements usually found in SERS experiments [18].
To study the spectral dependence of the enhancement, and gain a deeper insight
into the behavior of the system, the incoming laser light was tuned while
keeping the laser always on the same spot. This measurement is shown in Fig.
5, where the intensity is the photon flux at half the laser wavelength,
normalized to the signal provided by a $40\,$fs pulse with $275\,$mW average
power, according to the following formula:
$I_{\mathrm{norm}}=I\frac{\delta
t}{40\,\mathrm{fs}}\left(\frac{275\,\mathrm{mW}}{P}\right)^{2}.$ (3)
The signal at the SH wavelength was, for each point of Fig. 5, a local maximum
in the spectrum.
Figure 5 shows that the SH from the QD sample increases toward the excitonic
resonance in about a factor of 2 for the whole tuning range, a value similar
to the one reported by Baranov et al. [21]. It is worth noting that the
contribution of the TPPL to this increase is much smaller than the SH signal
throughout the measured range (see Fig. 4). Showing a very different spectral
response, the SH for the mixed NP-QD system sharply increases in more than a
factor of 7 close to the resonance of the silver plasmon.
Figure 5: SHG as a function of the laser excitation wavelength. The horizontal
scale is the second harmonic of the incident photon. Red symbols show the SHG
for the QD sample, while the black symbols show the SHG for the mixed NP-QD
sample. The dotted line marks the spectral position of the maximum of the
plasmon resonance.
This measurement reinforces the hypothesis that the silver NP is modifying the
nonlinear response of the QDs thanks to the resonant excitation of the silver
NP plasmons. However, it must be noted that the enhancement does not exactly
follow the plasmon resonance spectrum shown in Fig. 2. The reason for this is
still unknown, but it is possible that the plasmon resonance was shifted due
to the interaction among NPs. Unfortunately, the tuning range of the laser was
much smaller than the spectral width of the plasmon resonance.
## 4 Conclusions
We have observed a strong enhancement ($\approx 10^{3}$) of the SHG when CdS
QDs are mixed with silver NPs, compared with a CdS QD sample. The spectral
dependence of the enhancement shows that the NP plasmons are resonantly
excited by the SH emission of the QDs. These measurements are evidence that
the SH enhancement is mediated by nanoparticle plasmons. This effect can be
applied to significantly improve traditional applications of SH measurements
such as the study of surface deposition and orientation of molecules, among
others.
###### Acknowledgements.
We thank Claudia Marchi for her help with the NP synthesis and Nora Pellegri
for providing the QDs. We also thank Alejandro Fainstein for the insightful
discussions. This work was supported by the University of Buenos Aires under
Grant X010 and by ANPCYT under Grant PICT 14209.
## References
* [1] V L Brudny, B S Mendoza, W L Mochan, Second-harmonic generation from spherical particles, Phys. Rev. B 62, 11152 (2000).
* [2] L C Estrada, P F Aramendía, O E Martínez, 10000 times volume reduction for fluorescence correlation spectroscopy using nano-antennas, Opt. Express 16, 20597 (2008).
* [3] A F Scarpettini, N Pellegri, A V Bragas, Optical imaging with subnanometric vertical resolution using nanoparticle-based plasmonic probes, Opt. Commun. 282, 1032 (2009).
* [4] T Kalkbrenner, M Ramstein, J Mlynek, V Sandoghdar, A single gold particle as a probe for apertureless scanning near-field optical microscopy, J. Microsc. 202, 72 (2001).
* [5] P Anger, P Bharadwaj, L Novotny, Enhancement and quenching of single-molecule fluorescence, Phys. Rev. Lett. 96, 113002 (2006).
* [6] A J Haes, R P Van Duyne, A nanoscale optical biosensor: Sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles, J. Am. Chem. Soc. 124, 10596 (2002).
* [7] A G Skirtach, et al., Laser-induced release of encapsulated materials inside living cells, Angew. Chem. Int. Ed. 45, 4612 (2006).
* [8] P Etchegoin, et al., New limits in ultrasensitive trace detection by surface enhanced Raman scattering (SERS), Chem. Phys. Lett. 375, 84 (2003).
* [9] P G Etchegoin, P D Lacharmoise, E C Le Ru, Influence of photostability on single-molecule surface enhanced Raman scattering enhancement factors, Anal. Chem. 81, 682 (2009).
* [10] M Ishifuji, M Mitsuishi, T Miyashita, Enhanced optical second harmonic generation in hybrid polymer nanoassemblies based on coupled surface plasmon resonance of a gold nanoparticle array, Appl. Phys. Lett. 89, 011903 (2006).
* [11] H A Clark, et al., Second harmonic generation properties of fluorescent polymer-encapsulated gold nanoparticles, J. Am. Chem. Soc. 122, 10234 (2000).
* [12] S Baldelli, et al., Surface enhanced sum frequency generation of carbon monoxide adsorbed on platinum nanoparticle arrays, J. Chem. Phys 113, 5432 (2000).
* [13] I Barsegova, et al., Controlled fabrication of silver or gold nanoparticle near-field optical atomic force probes: Enhancement of second-harmonic generation, Appl. Phys. Lett. 81, 3461 (2002).
* [14] M C Marchi, S A Bilmes, G M Bilmes, Photophysics of Rhodamine B interacting with silver spheroids, J. Colloid Interface Sci. 218, 112 (1999).
* [15] A Frattini, N Pellegri, D Nicastro, O de Sanctis, Effect of amine groups in the synthesis of Ag nanoparticles using aminosilanes, Mater. Chem. Phys. 94, 148 (2005).
* [16] A I Ekimov, et al., Absorption and intensity-dependent photoluminescence measurements on CdSe quantum dots: Assignment of the first electronic transitions, J. Opt. Soc. Am. B 10, 100 (1993).
* [17] Z Yu, et al., Large resonant Stokes shift in CdS nanocrystals, J. Phys. Chem. B 107, 5670 (2003).
* [18] E C Le Ru, P G Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy, Elsevier Science Ltd, Amsterdam (2008).
* [19] E C Le Ru, E Blackie, M Meyer, P G Etchegoin, Surface enhanced raman scattering enhancement factors: A comprehensive study, J. Phys. Chem. C 111, 13794 (2007).
* [20] S A Maier, Plasmonics: Fundamentals and Applications, Chap. 5.1, Springer Science, New York (2007).
* [21] A V Baranov, et al., Resonant hyper-Raman and second-harmonic scattering in a CdS quantum-dot system, Phys. Rev. B 53, R1721 (1996).
|
arxiv-papers
| 2011-06-11T16:52:47 |
2024-09-04T02:49:19.546992
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pablo M. Jais, Catalina von Bilderling, Andrea V. Bragas",
"submitter": "Pablo M. Jais",
"url": "https://arxiv.org/abs/1106.2253"
}
|
1106.2272
|
# Soundness and completeness of the cirquent calculus system CL6 for
computability logic∗11footnotetext: This work was supported by the NNSF
(60974082) of China.
Wenyan Xu and Sanyang Liu
Department of Mathematics, Xidian University, Xi’an, 710071, PR China
###### Abstract
Computability logic is a formal theory of computability. The earlier article
“Introduction to cirquent calculus and abstract resource semantics” by
Japaridze proved soundness and completeness for the basic fragment CL5 of
computability logic. The present article extends that result to the more
expressive cirquent calculus system CL6, which is a conservative extension of
both CL5 and classical propositional logic.
Keywords: Cirquent calculus; Computability logic.
## 1 Introduction
Computability logic(CoL), introduced by G. Japaridze [1]-[3], is a semantical
and mathematical platform for redeveloping logic as a formal theory of
computability. Formulas in CoL represent interactive computational problems,
understood as games between a machine and its environment (symbolically named
as $\top$ and $\bot$, respectively); logical operators stand for operations on
such problems; “truth” of a problem/game means existence of an algorithmic
solution, i.e. $\top$’s effective winning strategy; and validity of a logical
formula is understood as such truth under every particular interpretation of
atoms. The approach induces a rich collection of (old or new) logical
operators. Among those, relevant to this paper are $\neg$ (negation), $\vee$
(parallel disjunction) and $\wedge$ (parallel conjunction). Intuitively,
$\neg$ is a role switch operator: $\neg A$ is the game $A$ with the roles of
$\top$ and $\bot$ interchanged ($\top$’s legal moves and wins become those of
$\bot$, and vice versa). Both $A\wedge B$ and $A\vee B$ are games playing
which means playing the two components $A$ and $B$ simultaneously (in
parallel). In $A\wedge B$, $\top$ is the winner if it wins in both components,
while in $A\vee B$ winning in just one component is sufficient. The symbols
$\top$ and $\bot$, together with denoting the two players, are also used to
denote two special (the simplest) sorts of games. Namely, $\top$ is a moveless
(“elementary”) game automatically won by the player $\top$, and $\bot$ is a
moveless game automatically won by $\bot$.
Cirquent calculus is a refinement of sequent calculus. Unlike the more
traditional proof theories that manipulate tree-like objects (formulas,
sequents, hypersequents, etc.), cirquent calculus deals with graph-style
structures termed cirquents, with its main characteristic feature thus being
allowing to explicitly account for sharing subcomponents between different
subcomponents. The approach was introduced by Japaridze [4] as a new deductive
tool for CoL and was developed later in [5]-[7]. The paper [4] constructed a
cirquent calculus system CL5 for the basic $(\neg,\wedge,\vee)$-fragment of
CoL, and proved its soundness and completeness with respect to the semantics
of CoL.
The atoms of CL5 represent computational problems in general, and are said to
be general atoms. The so called elementary atoms, representing computational
problems of zero degree of interactivity (such as the earlier-mentioned games
$\top$ and $\bot$) and studied in other pieces of literature on CoL, are not
among them. Thus, CL5 only describes valid computability principles for
general problems. This is a significant limitation of expressive power. For
example, the problem $A\rightarrow A{\wedge}A$ is not valid in CoL when $A$ is
a general atom, but becomes valid (as any classical tautology for that matter)
when $A$ is elementary. So the language of CL5 naturally calls for an
extension.
Japaridze [4] claimed without a proof that the soundness and completeness
result for CL5 could be extended to the more expressive cirquent calculus
system CL6 (reproduced later), which is a conservative extension of both CL5
and classical propositional logic. This article is devoted to a soundness and
completeness proof for system CL6, thus contributing to the task of extending
the cirquent-calculus approach so as to accommodate incrementally expressive
fragments of CoL.
## 2 Preliminaries
This paper primarily targets readers already familiar with Japaridze [4], and
can essentially be treated as a technical appendix to the latter. However, in
order to make it reasonably self-contained, in this section we reproduce the
basic concepts from [4] on which the later parts of the paper will rely. An
interested reader may consult [4] for additional explanations, illustrations
and examples.
The language of CL6 is more expressive than that of CL5 in that, along with
the old atoms of CL5 called general, it has an additional sort of atoms called
elementary, including non-logical elementary atoms and logical atoms $\top$
and $\bot$. On the other hand, all general atoms are non-logical. We use the
uppercase letters $P,Q,R,S$ as metavariables for general atoms, and the
lowercase $p,q,r,s$ as metavariables for non-logical elementary atoms. A
CL6-formula is built from atoms in the standard way using the connectives
$\neg$,$\vee$,$\wedge$, with $F\rightarrow G$ understood as an abbreviation
for $\neg F\vee G$ and $\neg$ limited only to non-logical atoms, where
$\neg\neg F$ is understood as $F$, $\neg(F\wedge G)$ as $\neg F\vee\neg G$,
$\neg(F\vee G)$ as $\neg F\wedge\neg G$, $\neg\top$ as $\bot$, and $\neg\bot$
as $\top$. An atom $P$ (resp. $p$) and its negation $\neg P$ (resp. $\neg p$)
is called a literal, and the two literals are said to be opposite. A
CL6-formula is said to be elementary iff it does not contain general atoms.
Throughout the rest of this paper, unless otherwise specified, by an “atom” or
a “formula” we mean one of the language of CL6.
Where $k\geq 0$, a $k-$ary pool is a sequence $\langle
F_{1},F_{2},\ldots,F_{k}\rangle$ of $k$ formulas. Since we may have
$F_{i}=F_{j}$ for some $i\neq j$ in such a sequence, we use the term oformula
to refer to a formula together with a particular occurrence of it in the pool.
For example, the pool $\langle E,F,G,E\rangle$ has three formulas but four
oformulas. Similarly, the terms “oliteral”,“oatom”, etc. will be used in this
paper to refer to the corresponding entities together with particular
occurrences. A $k-$ary structure is a finite sequence
St$=\langle\Gamma_{1},\ldots,\Gamma_{m}\rangle$, where $m\geq 0$ and each
$\Gamma_{i}$, said to be a group of St, is a subset of $\\{1,\ldots,k\\}$.
Again, to differentiate between a group as such and a particular occurrence of
a group in the structure, we use the term ogroup for the latter. For example,
the structure $\langle\\{2,3\\},\\{2,3\\},\\{1,4\\},\emptyset\rangle$ has
three groups but four ogroups.
A $k$-ary ($k\geq 0$) cirquent is a pair $C=({\bf St}^{C},{\bf Pl}^{C})$,
where ${\bf St}^{C}$, called the structure of $C$, is a $k$-ary structure, and
${\bf Pl}^{C}$, called the pool of $C$, is a $k$-ary pool. An ogroup of such a
$C$ will mean an ogroup of ${St}^{C}$, and an oformula of $C$ will mean an
oformula of ${\bf Pl}^{C}$. Usually, we understand the groups of a cirquent as
sets of its oformulas rather than sets of the corresponding ordinal numbers.
Thus, if ${\bf Pl}^{C}=\langle E,F,G,E\rangle$ and $\Gamma=\\{2,4\\}$, we
would think of $\Gamma$ simply as the set $\\{F,E\\}$, and say that $\Gamma$
contains $F$ and $E$. When both the pool and the structure of a cirquent $C$
are empty, i.e. $C=(\langle\rangle,\langle\rangle)$, we call it the empty
cirquent.
Rather than writing cirquents as ordered tuples in the above-described style,
we prefer to represent them through (and identify them with) diagrams. Below
is such a representation for the cirquent whose pool is $\langle
E,F,G,H\rangle$ and whose structure is
$\langle\\{1,2\\},\\{2\\},\\{3,4\\}\rangle$.
$E\ \ \ \ \ \ F\ \ \ \ \ G\ \ \ \ \ \ H$$\bullet$$\bullet$$\bullet$
The top level of a diagram thus indicates the oformulas of the cirquent, and
the bottom level gives its ogroups. An ogroup $\Gamma$ is represented by a
$\bullet$, and the lines connecting $\Gamma$ with oformulas, called arcs, are
pointing to the oformulas that $\Gamma$ contains. Finally, we put a horizontal
line at the top of the diagram to indicate that this is one cirquent rather
than two or more cirquents put together.
A model is a function $M$ that assigns a truth value — true (1) or false (0) —
to each atom, with $\top$ being always assigned true and $\bot$ false, and
extends to compound formulas in the standard classical way. Let $M$ be a
model, and $C$ a cirquent. We say that a group $\Gamma$ of $C$ is true in $M$
iff at least one of its oformulas is so. And $C$ is true in $M$ if every group
of $C$ is so. Otherwise, $C$ is false. Finally, $C$ or a group $\Gamma$ of it
is a tautology iff it is true in every model.
A substitution is a function $\sigma$ that sends every general atom $P$ to
some formula $\sigma(P)$, and sends every elementary atom to itself. If, (for
every general atom $P$), such a $\sigma(P)$ is an atom, then $\sigma$ is said
to be an atomic-level substitution.
Let $A$ and $B$ be cirquents. We say that $B$ is an instance of $A$ iff
$B=\sigma(A)$ for some substitution $\sigma$, where $\sigma(A)$ is the result
of replacing in all oformulas of $A$ every (general or elementary) atom
$\alpha$ by $\sigma(\alpha)$; and $B$ is an atomic-level instance of $A$ iff
$B=\sigma(A)$ for some atomic-level substitution $\sigma$.
A cirquent is said to be binary iff no general atom has more than two
occurrences in it. A binary cirquent is said to be normal iff, whenever it has
two occurrences of a general atom, one occurrence is negative and the other is
positive. A binary tautology (resp. normal binary tautology) is a binary
(resp. normal binary) cirquent that is a tautology.
The set of rules of CL6 is obtained from that of CL5 by adding to it $\top$ as
an additional axiom, plus the rule of contraction limited only to elementary
formulas. Below we reproduce those rules from [4], followed by illustrations.
Axioms (A): Axioms are “rules” with no premises. There are three sorts of
axioms in CL6. The first one is the empty cirquent. The second one is any
cirquent that has exactly two oformulas $F$ and $\neg F$, for some arbitrary
formula $F$, and an ogroup that contains $F$ and $\neg F$. In other words,
this is the cirquent $(\langle\\{1,2\\}\rangle,\langle F,\neg F\rangle)$. The
third one is a cirquent that has exactly one oformula $\top$ and one ogroup
that contains $\top$, i.e. the cirquent
$(\langle\\{1\\}\rangle,\langle\top\rangle)$.
Mix (M): According to this rule, the conclusion can be obtained by simply
putting any two cirquents (premises) together, thus creating one cirquent out
of two.
Exchange (E): This rule comes in two versions: oformula exchange and ogroup
exchange. The conclusion of oformula exchange is obtained by interchanging in
the premise two adjacent oformulas $E$ and $F$, and redirecting to $E$ (resp.
$F$) all arcs that were originally pointing to $E$ (resp. $F$). Ogroup
exchange is the same, with the only difference that the objects interchanged
are ogroups.
Weakening (W): This rule also comes in two versions: ogroup weakening and pool
weakening. A conclusion of ogroup weakening is obtained by adding in the
premise a new arc between an existing ogroup and an existing oformula. As for
pool weakening, a conclusion is obtained through inserting a new oformula
anywhere in the pool of the premise.
Duplication (D): A conclusion of this rule is obtained by replacing in the
premise some ogroup $\Gamma$ by two adjacent ogroups that, as groups, are
identical with $\Gamma$.
Contraction (C): According to this rule, if a cirquent (a premise) has two
adjacent elementary oformulas $F$ (the first), $F$ (the second) that are
identical, then a conclusion can be obtained by merging $F$,$F$ into $F$ and
redirecting to the latter all arcs that were originally pointing to the first
or the second $F$.
$\vee-$introduction ($\vee$): For the convenience of description, we explain
this rule in the bottom-up view. According to this rule, if a cirquent (the
conclusion) has an oformula $E\vee F$ that is contained by at least one
ogroup, then the premise can be obtained by splitting the original $E\vee F$
into two adjacent oformulas $E$ and $F$, and redirecting to both $E$ and $F$
all arcs that were originally pointing to $A\vee B$.
$\wedge-$introduction ($\wedge$): This rule, again, is more conveniently
described in the bottom-up view. According to this rule, if a cirquent (the
conclusion) has an oformula $E\wedge F$ that is contained by at least one
ogroup, then the premise can be obtained by splitting the original $E\wedge F$
into two adjacent oformulas $E$ and $F$, and splitting every ogroup $\Gamma$
that originally contained $E\wedge F$ into two adjacent ogroups $\Gamma^{E}$
and $\Gamma^{F}$, where $\Gamma^{E}$ contains $E$ (but not $F$), and
$\Gamma^{F}$ contains $F$ (but not $E$), with all other ($\neq E\wedge F$)
oformulas of $\Gamma$ contained by both $\Gamma^{E}$ and $\Gamma^{F}$.
Below we provide illustrations for all rules, in each case an abbreviated name
of the rule standing next to the horizontal line separating the premises from
the conclusions. Our illustrations for the axioms (the “A” labeled rules) are
specific cirquents or schemate of such; our illustrations for all other rules
are merely examples chosen arbitrarily. Unfortunately, no systematic ways for
schematically representing cirquent calculus rules have been elaborated so
far. This explains why we appeal to examples instead.
A$F\ \ \ \ \ \ \ \neg F$A$\bullet$$\top$A$\bullet$
$E$$\bullet$M$F\ \ \ \ \ G$$\bullet$$E$$\bullet$$F\ \ \ \ \
G$$\bullet$oformula exchange$E\ \ \ \ \ F\ \ \ \
G$$\bullet$$\bullet$$\bullet$E$F\ \ \ \ \ E\ \ \ \
G$$\bullet$$\bullet$$\bullet$ogroup exchange$E\ \ \ \ \ F\ \ \ \
G$$\bullet$$\bullet$$\bullet$E$E\ \ \ \ \ F\ \ \ \
G$$\bullet$$\bullet$$\bullet$
ogroup weakening$F\ \ \ \ G\ \ \ \ \ H$$\bullet$$\bullet$$\bullet$W$F\ \ \ \
G\ \ \ \ \ H$$\bullet$$\bullet$$\bullet$pool weakening$F\ \ \ \ \ \ \ \ \ \
H$$\bullet$$\bullet$W$F\ \ \ \ G\ \ \ \ H$$\bullet$$\bullet$$E\ \ \ \ \ F\ \ \
\ G$$\bullet$$\bullet$D$E\ \ \ \ \ F\ \ \ \ G$$\bullet$$\bullet$$\bullet$
$F$required to beelementary$E\ \ \ \ \ F\ \ \ \ F$$\bullet$$\bullet$C$E\ \ \ \
\ \ \ \ \ \ F$$\bullet$$\bullet$$H\ \ E\ \ F\ \ H$$\bullet$$\bullet$$\vee$$H\
\ E\vee F\ \ H$$\bullet$$\bullet$$F\ \ \ E\ \ \ F\ \
G$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\wedge$$F\ \ E\wedge F\ \
G$$\bullet$$\bullet$$\bullet$
The above are all eight rules of CL6. As a warm-up exercise, the reader may
try to verify that CL6 proves $p\rightarrow p\wedge p$ but does not prove
$P\rightarrow P\wedge P$.
As an aside, the earlier mentioned system CL5 differs from CL6 in that the
$\top-$axiom and the contraction rules are absent there. Also, as noted, the
language of CL5 does not allow elementary atoms. In next section we will see
that our proofs are carried out purely syntactically, based on the soundness
and completeness of system CL2 (introduced in Japaridze [8]) with respect to
the semantics of CoL. That is to say we do not directly use the semantics of
CoL. So, below we only explain what the language of CL2 and its rules are,
without providing any formal definitions (on top of the brief informal
explanations given in Section 1) of the underlying CoL semantics. If
necessary, such definitions can be found in [3].
The language of CL2 is more expressive than the one in which formulas of CL6
are written because, on top of $\neg$,$\vee$,$\wedge$, it has the binary
connectives $\sqcap$ and $\sqcup$, called choice operators. The CL2-formulas
are built from atoms (including general atoms and elementary atoms) in the
standard way using the connectives $\neg$,$\vee$,$\wedge$,$\sqcap$,$\sqcup$.
As in the case of CL6-formulas, the operator $\neg$ is only allowed to be
applied to non-logical atoms. A CL2-formula is said to be elementary iff it
contains neither general atoms nor $\sqcap$,$\sqcup$. A positive occurrence
(resp. negative occurrence) of an atom is one that is not (resp. is) in the
scope of $\neg$. A surface occurrence of a subformula of a CL2-formula is an
occurrence that is not in the scope of $\sqcap$,$\sqcup$. A general literal is
$P$ or $\neg P$, where $P$ is a general atom. The elementarization of a
CL2-formula $A$ is the result of replacing in $A$ every positive surface
occurrence of each general literal by $\bot$, every surface occurrence of each
$\sqcup-$subformula by $\bot$, and every surface occurrence of each
$\sqcap-$subformula by $\top$. A CL2-formula is said to be stable iff its
elementarization is a tautology of classical logic.
CL2 has the following three inference rules.
Rule (a): $\overrightarrow{H}\mapsto F$, where $F$ is stable and
$\overrightarrow{H}$ is the smallest set of formulas such that, whenever $F$
has a surface occurrence of a subformula $G_{1}\sqcap G_{2}$, for both
$i$$\in\\{$1,2$\\}$, $\overrightarrow{H}$ contains the result of replacing
that occurrence in $F$ by $G_{i}$.
Rule (b): $H\mapsto F$, where $H$ is the result of replacing in $F$ a surface
occurrence of a subformula $G_{1}\sqcup G_{2}$ by $G_{1}$ or $G_{2}$.
Rule (c): $H\mapsto F$, where $H$ is the result of replacing in $F$ two — one
positive and one negative — surface occurrences of some general atom by a non-
logical elementary atom that does not occur in $F$.
The set $\overrightarrow{H}$ of the premises of Rule (a) may be empty, in
which case the rule (its conclusion, that is) acts like an axiom. Otherwise,
the system has no (other) axioms.
## 3 Soundness and completeness of CL6
In what follows, we may use names such as (AME) to refer to the subsystem of
CL6 consisting only of the rules whose names are listed between the
parentheses. So, (AME) refers to the system that only has axioms, exchange and
mix. The same notation can be used next to the horizontal line separating two
cirquents to indicate that the lower cirquent (“conclusion”) can be obtained
from the upper cirquent (“premise”) by whatever number of applications of the
corresponding rules. The following Lemmas 1, 2, 3, 4 are precisely Lemmas 4,
5, 10 and 11 of [4], so we state them without proofs (such proofs are given in
[4]).
###### Lemma 1
All of the rules of CL6 preserve truth in the top-down direction. Taking no
premises, (the conclusion of) axioms are thus tautologies.
###### Lemma 2
The rules of mix, exchange, duplication, contraction, $\vee$-introduction and
$\wedge$-introduction preserve truth in the bottom-up direction as well.
###### Lemma 3
The rules of mix, exchange, duplication, $\vee$-introduction and
$\wedge$-introduction preserve binarity and normal binarity in both top-down
and bottom-up directions.
###### Lemma 4
Weakening preserves binarity and normal binarity in the bottom-up direction.
###### Lemma 5
If CL6 proves a cirquent $C$, then it also proves every instance of $C$.
Proof. Let $T$ be a proof tree of an arbitrary cirquent $C$, $C^{\prime}$ be
an arbitrary instance of $C$, and $\sigma$ be a substitution with
$\sigma(C)=C^{\prime}$. Replace every oformula $F$ of every cirquent of $T$ by
$\sigma(F)$. It is not hard to see that the resulting tree $T^{\prime}$, which
uses exactly the same rules as $T$ does, is a proof of $C^{\prime}$.
###### Lemma 6
Contraction preserves binarity and normal binarity in both top-down and
bottom-up directions.
Proof. This is so because contraction limited to elementary formulas can never
affect what general atoms occur in a cirquent and how many times they occur.
###### Lemma 7
A cirquent is provable in CL6 iff it is an instance of a binary tautology.
Proof. $(\Rightarrow)$ Consider an arbitrary cirquent $A$ provable in CL6. By
induction on the height of its proof tree, we want to show that $A$ is an
instance of a binary tautology.
The above is obvious when $A$ is an axiom.
Suppose now $A$ is derived by exchange from $B$. Let us just consider oformula
exchange, with ogroup exchange being similar. By the induction hypothesis, $B$
is an instance of a binary tautology $B^{\prime}$. Let $A^{\prime}$ be the
result of applying exchange to $B^{\prime}$ “at the same place” as it was
applied to $B$ when deriving $A$ from it, as illustrated in the following
example:
$P\ \ \ s\ \ \ \neg P\ \ \ \ P\vee r\ \ \ \neg P\wedge\neg r$${\bf E}$$P\ \
\neg P\ \ \ s\ \ \ \ \ P\vee r\ \ \ \neg P\wedge\neg r$$B:$$A:$$E\ \ \ s\ \ \
\neg E\ \ \ R\vee r\ \ \ \ \neg R\wedge\neg r$${\bf E}$$E\ \ \neg E\ \ \ s\ \
\ \ R\vee r\ \ \ \ \neg R\wedge\neg r$$B^{\prime}:$$A^{\prime}:$
Obviously $A$ will be an instance of $A^{\prime}$. It remains to note that, by
Lemmas 1 and 3, $A^{\prime}$ is a binary tautology.
The rules of duplication, $\vee$-introduction and $\wedge$-introduction can be
handled in a similar way.
Next, suppose $A$ is derived from $B$ and $C$ by mix. By the induction
hypothesis, $B$ and $C$ are instances of some binary tautologies $B^{\prime}$
and $C^{\prime}$, respectively. We may assume that no general atom $P$ occurs
in both $B^{\prime}$ and $C^{\prime}$, for otherwise, in one of the cirquents,
rename $P$ into another general atom $Q$ different from everything else. Let
$A^{\prime}$ be the result of applying mix to $B^{\prime}$ and $C^{\prime}$.
By Lemmas 1 and 3, $A^{\prime}$ is a binary tautology. And, as in the cases of
the other rules, it is evident that $A$ is an instance of $A^{\prime}$.
Suppose $A$ is derived from $B$ by weakening. If this is ogroup weakening, $A$
is an instance of a binary tautology for the same reason as in the case of
exchange, duplication, $\vee$-introduction or $\wedge$-introduction. Assume
now we are dealing with pool weakening, so that $A$ is the result of inserting
a new oformula $F$ into $B$. By the induction hypothesis, $B$ is an instance
of a binary tautology $B^{\prime}$. Let $P$ be a general atom not occurring in
$B^{\prime}$. And let $A^{\prime}$ be the result of applying weakening to
$B^{\prime}$ that inserts $P$ “at the same place” into $B^{\prime}$ as the
above application of weakening inserted $F$ into $B$ when deriving $A$.
Obviously $A^{\prime}$ inherits binarity from $B^{\prime}$; by Lemma 1, it
inherits from $B^{\prime}$ tautologicity as well. And, for the same reason as
in all previous cases, $A$ is an instance of $A^{\prime}$.
Finally, suppose $A$ is derived from $B$ by contraction. Then the contracted
formula $F$ should be elementary. By the induction hypothesis, $B$ is an
instance of a binary tautology $B^{\prime}$. Let $\sigma$ be a substitution
such that $B=\sigma(B^{\prime})$. And let $F^{\prime}_{1}$, $F^{\prime}_{2}$
be two oformulas in $B^{\prime}$ “at the same place” as $F$, $F$ are in $B$,
with $\sigma(F^{\prime}_{1})=F$ and $\sigma(F^{\prime}_{2})=F$. Let $\delta$
be the substitution such that, for any general atom $P$, $\delta(P)=\sigma(P)$
if $P$ occurs in $F^{\prime}_{1}$ or $F^{\prime}_{2}$, and $\delta(P)=P$
otherwise. Thus, $\delta(F^{\prime}_{1})=\delta(F^{\prime}_{2})=F$. And let
$B^{\prime\prime}=\delta(B^{\prime})$. Obviously — for the same reasons as in
classical logic — substitution does not destroy tautologicity, so
$B^{\prime\prime}$ is a tautology because $B^{\prime}$ is so. Further, the
substitution $\delta$ does not introduce any new occurrences of general atoms,
so it does not destroy the binarity of $B^{\prime}$, either. To summarize,
$B^{\prime\prime}$ is a binary tautology. Also, of course, $B$ is an instance
of $B^{\prime\prime}$. Notice that $B^{\prime\prime}$ has $F$ and $F$ where
$B$ has the contracted oformulas $F$ and $F$. So, let $A^{\prime}$ be the
result of applying contraction to $B^{\prime\prime}$ “at the same place” as it
was applied to $B$ when deriving $A$ from it, as illustrated in the following
example:
$P\ \ r\wedge s\ \ \ r\wedge s\ \ \ \ \ \neg P\ \ \ \ P\vee q\ \ \ \neg
P\wedge\neg q$${\bf C}$$P\ \ \ \ \ \ \ \ \ \ r\wedge s\ \ \ \ \ \ \neg P\ \ \
\ P\vee q\ \ \ \neg P\wedge\neg q$$B:$$A:$$R\ \ r\wedge s\ \ \ r\wedge s\ \ \
\ \neg R\ \ \ \ \ Q\vee q\ \ \ \neg Q\wedge\neg q$${\bf C}$$R\ \ \ \ \ \ \ \ \
\ r\wedge s\ \ \ \ \ \ \neg R\ \ \ \ Q\vee q\ \ \ \neg Q\wedge\neg
q$$B^{\prime\prime}:$$A^{\prime}:$
Obviously $A$ will be an instance of $A^{\prime}$. And, by Lemma 1 and Lemma
6, $A^{\prime}$ is a binary tautology.
$(\Leftarrow)$ Consider an arbitrary cirquent $A$ that is an instance of a
binary tautology $A^{\prime}$. In view of Lemma 5, it would suffice to show
that CL6 proves $A^{\prime}$. We construct a proof of $A^{\prime}$, in the
bottom-up fashion, as follows. Starting from $A^{\prime}$, we keep applying
$\vee$-introduction and $\wedge$-introduction until we hit an essentially
literal cirquent111An essentially literal cirquent, defined in [4], is one
every oformula of whose pool either is an oliteral or is homeless. $B$. As in
the proof of Theorem 6 of [4], such a cirquent $B$ is guaranteed to be a
tautology, and $A^{\prime}$ follows from it in $(\vee\wedge)$. Furthermore, in
view of Lemma 3, $B$ is in fact a binary tautology. The tautologicity of $B$
means that every ogroup of it contains either a $\top$, or at least one pair
of opposite (general or elementary) non-logical oliterals. For each ogroup of
$B$ that contains a $\top$, pick one occurrence of $\top$ and apply to $B$ a
series of weakenings to first delete all arcs but the arc pointing to the
chosen occurrence, and next delete all homeless oformulas if any such
oformulas are present. For each ogroup of the resulting cirquent that contains
a pair of opposite non-logical oliterals, pick one such pair, and continue
applying a series of weakenings, as in the proof of Theorem 6 of [4], until a
tautological cirquent $C$ is hit with no homeless oformulas, where every
ogroup only has either a $\top$ or a pair of opposite non-logical oliterals.
By Lemma 4, $C$ remains binary. Our target cirquent $A^{\prime}$ is thus
derivable from $C$ in (W$\vee\wedge$). Apply a series of contractions to $C$
to separate all shared $\top$ and all shared elementary non-logical oliterals
$p$ or $\neg p$, as illustrated below; as a result, we get a cirquent $D$
which is still a binary tautology, but whose ogroups no longer share any
elementary oformulas.
$P\ \ \ \ \neg P\ \ \ r\ \ \ \ \neg r\ \ \neg r\ \ \ \ \ r\ \ \ \ r\ \ \ \neg
Q\ \ \neg r\ \ \neg s\ \ \neg s\ \ \ Q\ \ \ s\ \ \ \ s\ \ \top\ \ \ \top\ \ \
\top$(C)$P\ \ \ \ \neg P\ \ \ r\ \ \ \ \neg r\ \ \ \ \ r\ \ \ \neg Q\ \ \neg
r\ \ \ \neg s\ \ \ \ Q\ \ \ s\ \ \ \top\ \ \ \ \ \ \top$$D:$$C:$
It is easy to see that the binarity of $D$ implies that there are no shared
general oliterals $P$ or $\neg P$ in it except the cases when they are shared
by identical-content ogroups. Applying to $D$ a series of duplications, as
illustrated below, yields a cirquent $E$ that no longer has identical-content
ogroups and hence no longer has any shared oformulas.
$P\ \ \ \ \neg P\ \ \ r\ \ \ \ \neg r\ \ \neg r\ \ \ \ \ r\ \ \ \ r\ \ \ \neg
Q\ \ \neg r\ \ \neg s\ \ \neg s\ \ \ Q\ \ \ s\ \ \ \ s\ \ \top\ \ \ \top\ \ \
\top$(D)$E:$$P\ \ \ \ \neg P\ \ \ r\ \ \ \ \neg r\ \ \neg r\ \ \ \ \ r\ \ \ \
r\ \ \ \neg Q\ \ \neg r\ \ \neg s\ \ \neg s\ \ \ Q\ \ \ s\ \ \ \ s\ \ \top\ \
\ \top\ \ \ \top$$D:$
$A^{\prime}$ is thus derivable from $E$ in (DCW$\vee\wedge$). In turn, $E$ is
obviously provable in (AME). So, CL6 proves $A^{\prime}$.
###### Lemma 8
A cirquent is an instance of a binary tautology iff it is an atomic-level
instance of some normal binary tautology.
Proof. Our proof here almost literally follows the proof of Lemma 9 of [4].
The “if” part is trivial. For the “only if” part, assume $A$ is an instance of
a binary tautology $B$. Let $P_{1},\ldots,P_{n}$ be all of the general atoms
of $B$ that have two positive or two negative occurrences in $B$. Let
$Q_{1},\ldots,Q_{n}$ be any pairwise distinct general atoms not occurring in
$B$. Let $C$ be the result of replacing in $B$ one of the two occurrences of
$P_{i}$ by $Q_{i}$, for each $i=1,\ldots,n$. Then obviously $C$ is a normal
binary cirquent, and $B$ an instance of it. By transitivity, $A$ (as an
instance of $B$) is also an instance of $C$.
We want to see that $C$ is a tautology. Deny this. Then there is a classical
model $M$ in which $C$ is false. Let $M^{\prime}$ be the model such that:
* •
$M^{\prime}$ agrees with $M$ on all atoms that are not among
$P_{1},\ldots,P_{n},Q_{1},\ldots,Q_{n}$;
* •
for each $i\in\\{1,\ldots,n\\}$, $M^{\prime}(P_{i})=M^{\prime}(Q_{i})=$false
if $P_{i}$ and $Q_{i}$ are positive in $C$; and
$M^{\prime}(P_{i})=M^{\prime}(Q_{i})=$true if $P_{i}$ and $Q_{i}$ are negative
in $C$.
By induction on complexity, it can be easily seen that, for every subformula
$F$ of a formula of $C$, whenever $F$ is false in $M$, so is it in
$M^{\prime}$. This extends from (sub)formulas to groups of $C$ and hence $C$
itself. Thus $C$ is false in $M^{\prime}$ because it is false in $M$. But
$M^{\prime}$ does not distinguish between $P_{i}$ and $Q_{i}$ (any $1\leq
i\leq n$). This clearly implies that $C$ and $B$ have the same truth value in
$M^{\prime}$. That is, $B$ is false in $M^{\prime}$, which is however
impossible because $B$ is a tautology. From this contradiction we conclude
that $C$ is a (normal binary) tautology.
Let $\sigma$ be a substitution such that $A=\sigma(C)$. Let $\sigma^{\prime}$
be a substitution such that, for each general atom $P$ of $C$,
$\sigma^{\prime}(P)$ is the result of replacing in $\sigma(P)$ each occurrence
of each general atom by a new general atom in such a way that: no general atom
occurs more than once in $\sigma^{\prime}(P)$, and whenever $P\neq Q$, no
general atom occurs in both $\sigma^{\prime}(P)$ and $\sigma^{\prime}(Q)$.
Since $C$ is a binary tautology and is its own instance, by Lemma 7, CL6
proves $C$. Then, by Lemma 5, CL6 proves $\sigma^{\prime}(C)$ (an instance of
$C$). In view of Lemma 1, we immediately get that $\sigma^{\prime}(C)$ is a
tautology. $\sigma^{\prime}(C)$ can also be easily seen to be a normal binary
cirquent, because $C$ is so. Finally, with a little thought, $A$ can be seen
to be an atomic-level instance of $\sigma^{\prime}(C)$.
###### Lemma 9
A CL6-formula $F$ is provable in CL2 iff it is an instance of a binary
tautology.
Proof. Again, it should be acknowledged that the present proof very closely
follows the proof of Lemma 27 of [4], even though there are certain
differences.
$(\Rightarrow)$ Consider an arbitrary CL6-formula $F$ provable in CL2. Fix a
CL2-proof of $F$ in the form of a sequence $\langle
F_{n},F_{n-1},\ldots,F_{1}\rangle$ of formulas, with $F_{1}=F$. We may assume
that this sequence has no repetitions or other redundancies. We claim that,
for each $i$ with $1\leq i\leq n$, the following conditions are satisfied:
Condition 1: $F_{i}$ does not contain $\sqcap,\sqcup$.
Condition 2: Whenever $F_{i}$ contains an elementary atom not occurring in
$F$, that atom is non-logical, and has exactly two — one positive and one
negative — occurrences in $F_{i}$.
Condition 3: If $i<n$, then $F_{i}$ is derived from $F_{i+1}$ by Rule (c).
Condition 4: $F_{n}$ is derived (from the empty set of premises) by Rule (a).
Condition 4 is obvious, because it is only Rule (a) that may take no premises.
That Conditions 1-3 are also satisfied can be verified by induction on $i$.
For the basis case of $i=1$, Conditions 1 and 2 are immediate. $F_{1}$ can not
be derived by Rule (b) because, by Condition 1, $F_{1}$ does not contain any
$\sqcup$. Nor can it be derived by Rule (a) unless $n=1$, for otherwise either
$F_{1}$ would have to contain a $\sqcap$ (which is not the case according to
Condition $1$), or the proof of $F$ would have redundancies as $F_{1}$ would
not really need any premises. Thus, if $1<n$, the only possibility for $F_{1}$
is to be derived from $F_{2}$ by Rule (c). For the induction step, assume
$i<n$ and the above conditions are satisfied for $F_{i}$. According to
Condition 3, $F_{i}$ is derived by Rule (c) from $F_{i+1}$ . This obviously
implies that $F_{i+1}$ inherits Conditions 1 and 2 from $F_{i}$. And that
Condition 3 also holds for $F_{i+1}$ can be shown in the same way as we did
for $F_{1}$.
As the conclusion of Rule (a) (Condition 4), $F_{n}$ is stable. Let $G$ be the
elementarization of $F_{n}$. The stability of $F_{n}$ means that $G$ is a
tautology. Let $H$ be the result of replacing in $G$ every occurrence of
$\top$ and $\bot$ (except those inherited from $F$) by a general atom, in such
a way that different occurrences of $\top$, $\bot$ are replaced by different
atoms. In view of Condition 2 (applied to $F_{n}$), we see that, on top of
these new general atoms and the elementary atoms inherited from $F$, the only
additional atoms that $H$ contains are elementary atoms with exactly two — one
positive and one negative — occurrences. Let $H^{\prime}$ be the result of
replacing in $H$ every occurrence of every such elementary atom by a general
atom not occurring in $H$, in such a way that different elementary atoms are
replaced by different general atoms. Then it is not hard to see that
$H^{\prime}$ is binary and $F_{n}$ is an instance of $H^{\prime}$. With
Condition 3 in mind, by induction, one can further see that the formulas
$F_{n-1}$, $F_{n-2}$, $\ldots$ are also instances of $H^{\prime}$. Thus, $F$
is an instance of $H^{\prime}$. It remains to show that $H^{\prime}$ is a
tautology. But this is indeed so because $H^{\prime}$ results from the
tautological $G$ by replacing positive occurrences of $\bot$ and replacing two
— one positive and one negative — occurrences of elementary atoms by general
atoms. It is known from classical logic that such replacements do not destroy
truth and hence tautologicity of formulas.
$(\Leftarrow)$ Assume $F$ is a CL6-formula which is an instance of a binary
tautology $T$. In view of Lemma 8, we may assume that $T$ is normal and $F$ is
an atomic-level instance of it. Let us call the general atoms that only have
one occurrence in $T$ single, and the general atoms that have two occurrences
married. Let $\sigma$ be the substitution with $\sigma(T)=F$. Let $G$ be the
formula resulting from $T$ by the following steps: substituting each single
atom $P$ by $\sigma(P)$; substituting each married atom $Q$ by $\sigma(Q)$ if
$\sigma(Q)$ is elementary; substituting each married atom $R$ by a non-logical
elementary atom $r$ not occurring in $F$ if $\sigma(R)$ is general. It is
clear that then $F$ can be derived from $G$ by a series of applications of
Rule (c), with each such application replacing two — a positive and a negative
— occurrences of some non-logical elementary atom $r$ by $\sigma(R)$. So, in
order to show that CL2 proves $F$, it would suffice to verify that $G$ is
stable and hence it can be derived from the empty set of premises by Rule (a).
But $G$ is indeed stable. To see this, consider the elementarization
$G^{\prime}$ of $G$. It results from $T$ by replacing the only occurrence of
each single general atom by some elementary atom, and doing the same with both
occurrences of each married general atom. In other words, $G^{\prime}$ is an
instance of $T$. Hence, as $T$ is a tautology, so is $G^{\prime}$, meaning
that $G$ is stable.
###### Theorem 10
A formula is provable in CL6 iff it is valid in computability logic.
Proof. This theorem is an immediate corollary of Lemma 9, Lemma 7 and the
known fact (proven in [8]) that CL2 is sound and complete with respect to the
semantics of computability logic.
## References
* [1] G.Japaridze, Introduction to computability logic, Annals of Pure and Applied Logic 123(2003)1-99.
* [2] G.Japaridze, Computability logic: a formal theory of interaction, In: D.Goldin, S.Smolka, P.Wegner(Eds.), Interactive Computation: The New Paradigm, Springer Verlag, Berlin, 2006, pp. 183-223.
* [3] G.Japaridze, In the beginning was game semantics, In: O.Majer, A.-V.Pietarinen, T. Tulenheimo(Eds.), Games: Unifying Logic, Language, and Philosophy, Springer Netherlands, Dordrecht, 2009, pp. 249-350.
* [4] G.Japaridze, Introduction to cirquent calculus and abstract resource semantics, Journal of Logic and Computation 16(4)(2006)489-532.
* [5] G.Japaridze, Cirquent calculus deepened, Journal of Logic and Computation 18(6)(2008)983-1028.
* [6] G.Japaridze, From formulas to cirquents in computability logic, Logical Methods in Computer Science 7(2)(2011)1-55.
* [7] Wen-yan Xu, San-yang Liu, Deduction theorem for symmetric cirquent calculus, Quantitative Logic and Soft Computing 2010, Advances in Intelligent and Soft Computing, 82(2010)121-126.
* [8] G.Japaridze, Propositional computability logic II, ACM Transactions on Computational Logic 7(2)(2006)331-362.
|
arxiv-papers
| 2011-06-12T02:20:53 |
2024-09-04T02:49:19.553384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenyan Xu and Sanyang Liu",
"submitter": "Wenyan Xu",
"url": "https://arxiv.org/abs/1106.2272"
}
|
1106.2297
|
# Qutrit: entanglement dynamics in the finite qutrit chain in the consistent
magnetic field
E. A. Ivanchenko yevgeny@kipt.kharkov.ua Institute for Theoretical Physics,
National Science Center “Institute of Physics and Technology”,
1, Akademicheskaya str., 61108 Kharkov, Ukraine
###### Abstract
Based on the Liouville-von Neumann equation, we obtain closed system of
equations for the description of a qutrit or coupled qutrits in arbitrary
time-dependent external magnetic field. The dependence of the dynamics on the
initial states and magnetic field modulation is studied analytically and
numerically. We compare the relative entanglement measure’s dynamics in the
bi-qutrit system with permutation particle symmetry. We find the magnetic
field modulation which retains the entanglement in the system of two coupled
qutrits. Analytical formulas for entanglement measures in the chain from 2 to
6 qutrits are presented.
qutrit, spin 1, entanglement, multipartite system
###### pacs:
03.67.Bg Entanglement production and manipulation
03.67.Mn Entanglement measures, witnesses, and other characterizations
## I Introduction
Multi-level quantum systems are studied intensively, since they have wide
applications. Some of the existent analytical results for spin 1 Hioe (1983)
are derived in terms of the coherent vector Hioe and Eberly (1981). The class
of exact solutions for a three-level system is given in Ref. Ishkhanyan
(2000). The application of coupled multi-level systems in quantum devices is
actively studied Zobov et al. (2008). The study of these systems is topical in
view of possible applications for useful work in microscopic systems Scully et
al. (2003). Exact solutions for two uncoupled qutrits interacting with vacuum
are obtained in Ref. Derkacz and Jakobczyk (2006). For the case of the qutrits
interacting with stochastic magnetic field exact solutions are obtained in
Ref. Ali (2009). Exact solutions for coupled qutrits in magnetic field as far
as we know were not found.
The entanglement in multi-particle coupled systems is an important resource
for many problems in quantum information science, but its quantitative value
is difficult because of different types of entanglement. Multi-dimensional
entangled states are interesting both for the study of the foundations of
quantum mechanics and for the topicality of developing new protocols for
quantum communication. For example, it was shown that for maximally entangled
states of two quantum systems, the qudits break the local realism stronger
than the qubits Kaszlikowski et al. (2000), and that the entangled qudits are
less influenced by the noise than the entangled qubits. Using entangled
qutrits or qudits instead of qubits is more protective from interception. From
the practical point of view, it is clear that generating and saving the
entanglement in the controlled manner is the primary problem for the
realization of the quantum computers. The maximally entangled states are best
suited for the protocols of quantum teleportation and quantum cryptography.
The entanglement and the symmetry are the basic notions of the quantum
mechanics. We study the dynamics of multipartite systems, which are invariant
at any subsystem permutation. The aim of our work is finding exact solutions
for the dynamics of coupled qutrits interacting with alternating magnetic
field as well as the comparative analysis of the entanglement measures in the
chain of qutrits.
The rest of the paper is organized as following. The Hamiltonian of the
anisotropic qutrit in arbitrary alternating magnetic field is described in
Sec. II. Then the system of equations for the description of the qutrit
dynamics is derived in the Bloch vector representation. We introduce the
consistent magnetic field, which describe entire class of field forms. In
section III we derive the system of equations for the description of the
dynamics of two coupled qutrits in the consistent field and find the
analytical solution for the density matrix in the case of anisotropic
interaction. Analytical formulas, which describe the entanglement in spin
chains from 2 to 6 qutrits, are presented in Sec. IV. The results are
demonstrated graphically in Sec. V for concrete parameters. The brief
conclusions are given in Sec. VI. The auxiliary analytical results are
presented in the Appendices.
## II Qutrit
### II.1 Qutrit Hamiltonian
We take the qutrit Hamiltonian (for the spin-1 particle) in the space of one
qutrit $\mathbb{C}^{3}$ in the basis
$|1>=(1,0,0),\;|0>=(0,1,0),\;|-1>=(0,0,1)$, in external magnetic field
$\overrightarrow{\bm{h}}=(h_{1},h_{2},h_{3})$ with anisotropy, in the form
$\hat{H}=h_{1}S_{1}+h_{2}S_{2}+h_{3}S_{3}+Q(S_{3}^{2}-2/3E)+d(S_{1}^{2}-S_{2}^{2}),$
(1)
where $h_{1},\;h_{2},\;h_{3}$ are the Cartesian components of the external
magnetic field in the frequency units (we assume $\hbar=1$, Bohr magneton
$\mu_{B}=1$); $S_{1},\;S_{2},\;S_{3}$ are the spin-1 matrices (see Appendix
A); $E$ stands for the $3\times 3$ unity matrix; $Q,\;d$ are the anisotropy
constants. When the constants $Q,\;d$ are zeros, then the two Hamiltonian
eigenvalues are symmetrically placed in respect to the zero level.
### II.2 Liouville-von Neumann equation
The qutrit dynamics in the magnetic field we describe in the density matrix
formalism with the Liouville-von Neumann equation
$i\partial_{t}\rho=[\hat{H},\,\rho],~{}\rho(t=0)=\rho_{0}.$ (2)
It is convenient to rewrite Eq. (2) presenting the density matrix $\rho$ in
the decomposition with the full set of orthogonal Hermitian matrices
$C_{\alpha}$ Morris (1964) (further the summation over Greek indices will be
from 0 to 8 and over the Latin ones from 1 to 8, see Appendix A)
$\rho=\frac{1}{\sqrt{6}}C_{\alpha}R_{\alpha}=\left(\begin{array}[]{ccc}\frac{1}{3}+\frac{R_{3}}{\sqrt{6}}+\frac{R_{6}}{\sqrt{18}}&\frac{R_{1}+R_{7}-i(R_{2}+R_{5})}{\sqrt{12}}&\frac{-iR_{4}+R_{8}}{\sqrt{6}}\\\
\frac{R_{1}+R_{7}+i(R_{2}+R_{5})}{\sqrt{12}}&\frac{1}{3}-\frac{2R_{6}}{\sqrt{18}}&\frac{R_{1}-R_{7}-i(R_{2}-R_{5})}{\sqrt{12}}\\\
\frac{iR_{4}+R_{8}}{\sqrt{6}}&\frac{R_{1}-R_{7}+i(R_{2}-R_{5})}{\sqrt{12}}&\frac{1}{3}-\frac{R_{3}}{\sqrt{6}}+\frac{R_{6}}{\sqrt{18}}\\\
&&\end{array}\right).$ (3)
Since $\mathrm{Tr\,}C_{i}=0$ for $1\leq i\leq 8$, then from the condition
$\mathrm{Tr\,}\rho=R_{0}$ it follows that $R_{0}=1$. And although the results
are independent of the basis choice, in this basis the functions
$R_{i}=\mathrm{Tr\,}\rho\,C_{i}$ have the concrete physical meaning Allard and
Hard (2001). The values $R_{1},R_{2},R_{3}$ are the polarization vector
Cartesian components; $R_{4}$ is the two-quantum coherence contribution in
$R_{2}$; $R_{5}$ is the one-quantum anti-phase coherence contribution in
$R_{2}$; $R_{6}$ is the contribution of the rotation between the phase and
anti-phase one-quantum coherence; $R_{7}$ is the one-quantum anti-phase
coherence contribution in $R_{1}$; $R_{8}$ is the two-quantum coherence
contribution in $R_{1}$.
Under the unitary evolution the length of the generalized Bloch vector
$b=\sqrt{R_{i}^{2}}$ (4)
is conserved. The length of the generalized vector (4) for pure states equals
to $\sqrt{2}$. Since $i\partial_{t}\rho^{n}=[\hat{H},\rho^{n}]$
$(n=1,2,3,\dots)$, then under unitary evolution there is countable number of
the conservation laws
$\mathrm{Tr\,}\rho=c_{1}=1,~{}\mathrm{Tr\,}\rho^{2}=c_{2},\dots$, from which
only $c_{2},\,c_{3}$ are algebraically independent Elgin (1980). Additional
quadric invariants of motion can be easily obtained after equating the matrix
elements in defining the pure state. For example, two of these invariants,
which follow from the expression $(\rho^{2}-\rho)_{13}=0$, have the form
$R_{1}^{2}-R_{2}^{2}+R_{5}^{2}-R_{7}^{2}-2\sqrt{\frac{2}{3}}(1-\sqrt{2}R_{6})R_{8}=0,\;R_{5}R_{7}-R_{1}R_{2}+\frac{2}{\sqrt{3}}(\frac{1}{\sqrt{2}}-R_{6})R_{4}=0.$
(5)
For numerical calculations, these invariants control also the signs of the
values $R_{i}$ and thus the using of the invariants is useful when the
analytical solutions are difficult to find. According to the Kelly-Hamilton
theorem, the density matrix $\rho$ satisfies to its characteristic equation
$\rho^{3}-\rho^{2}+\frac{2-b^{2}}{6}\rho-\det\rho\,E=0.$ (6)
From equation (6) it follows that the density matrix determinant
$\det\rho=(\mathrm{Tr\,}\rho^{3}-\mathrm{Tr\,}\rho^{2})/3+(2-b^{2})/18$ is
also the motion invariant. The Liouville-von Neumann equation in terms of the
functions $R_{i}$ takes the form of the closed system of 8 real differential
first-order equations. This system of equations in the compact form can be
written as following Elgin (1980); Ivanchenko (2009):
$\partial_{t}R_{l}=e_{ijl}h_{i}R_{j},$ (7)
where $e_{ijl}$ are the antisymmetrical structure constants,
$h_{i}=2(h_{1},h_{2},h_{3},0,0,\frac{Q}{\sqrt{3}},0,d)$ are the Hamiltonian
components (1) in the basis $C_{\alpha}$(see Appendix A).
### II.3 The consistent field
Consider the qutrit dynamics in the alternating field of the form
$\vec{h}(t)=\left(\omega_{1}\mathrm{cn}(\omega
t|k),\;\omega_{1}\mathrm{sn}(\omega t|k),\;\omega_{0}\mathrm{dn}(\omega
t|k)\right),$ (8)
where $\mathrm{cn},\mathrm{sn},\mathrm{dn}$ are the Jacobi elliptic functions
Abramovitz and Stegun (1968). Such field modulation under the changing of the
elliptic modulus $k$ from 0 to 1 describes the whole class of field forms from
trigonometric ($\mathrm{cn}(\omega t|0)=\mathrm{cos}\omega
t,\;\mathrm{sn}(\omega t|0)=\mathrm{sin}\omega t,\;\mathrm{dn}(\omega t|0)=1$
) Rabi (1937) to the exponentially impulse ones ($\mathrm{cn}(\omega
t|1)=\frac{1}{\mathrm{ch}\omega t},\;\mathrm{sn}(\omega t|1)=\mathrm{th}\omega
t,\;\mathrm{dn}(\omega t|1)=\frac{1}{\mathrm{ch}\omega t}$) Bambini and Berman
(1981). The elliptic functions $\mathrm{cn}(\omega t|k)$
and$\;\mathrm{sn}(\omega t|k)$ have the real period $\frac{4K}{\omega}$, while
the function $\mathrm{dn}(\omega t|k)$ has the two times smaller period. Here
$K$ is the full elliptic integral of the first kind Abramovitz and Stegun
(1968). In other words, even though the field is periodic with common real
period $\frac{4K}{\omega}$, but as we can see, the frequency of the
longitudinal field amplitude modulation is two times higher than the one of
the transverse field. Such field we call consistent.
Let us make use of the substitution $\rho=\alpha_{1}^{-1}r\alpha_{1}$. Then we
obtain the equation for the matrix $r$ in the form
$i\partial_{t}r=[\alpha_{1}\hat{H}\alpha_{1}^{-1}-i\alpha_{1}\partial_{t}(\alpha_{1}^{-1}),r]$
(9)
with the matrix
$\alpha_{1}=\left(\begin{array}[]{ccc}f&0&0\\\ 0&1&0\\\ 0&0&f^{-1}\\\
\end{array}\right),$ (10)
where $f(\omega t|k)=\mathrm{cn}(\omega t|k)+i\mathrm{sn}(\omega t|k).$ Since
$\alpha_{1}S_{1}\alpha_{1}^{-1}=S_{1}\mathrm{cn}(\omega
t|k)-S_{2}\mathrm{sn}(\omega
t|k),\;\alpha_{1}S_{2}\alpha_{1}^{-1}=S_{1}\mathrm{sn}(\omega
t|k)+S_{2}\mathrm{cn}(\omega t|k),\;\alpha_{1}S_{3}\alpha_{1}^{-1}=S_{3},$
(11)
then the equation for the matrix $r$ without taking into account the
anisotropy can be written as following
$i\partial_{t}r=[\omega_{1}S_{1}+\delta\,\mathrm{dn}(\omega
t|k)S_{3},r],\;r(t=0)=\rho_{0},\;\delta=\omega_{0}-\omega.$ (12)
At $k=0$ equation (12) describes the dynamics of the qutrit in the circularly
polarized field Rabi (1937); Miller et al. (2001); Grifoni and Hanggi (1998).
The exact solutions of this equation are known and at some initial conditions
the explicit formulas are given in Ref. Nath et al. (2003). At exact
resonance, $\omega=\omega_{0}$ it is straightforward to present Eq. (2) in the
deformed field ($k\neq 0$) (8) for the given initial condition
$\rho=\rho_{0}$:
$\rho(t)=\alpha_{1}^{-1}e^{-i\omega_{1}tS_{1}}\rho_{0}e^{i\omega_{1}tS_{1}}\alpha_{1}.$
(13)
Explicit solutions for some specific initial conditions are given in the
Appendix B, Eqs. (39) – (44). From the explicit exact solutions in the
deformed field at resonance $\delta=0$ one can see that the populations and
the transition probabilities do not depend on the field deformation (it is
independent of the $k$ modulus).
Consider the solution of Eq. (12) far from the resonance in the form of
$\delta$ power expansion
$r(t)=r^{(0)}(t)+r^{(1)}(t)+\cdots.$ (14)
Then we put the expansion (14) in Eq. (12) and equate the same degree terms.
As the result we obtain the system of equations for finding $r^{(l)}(t)$:
$i\partial_{t}r^{(0)}=\omega_{1}[S_{1},r^{(0)}],$ (15a)
$i\partial_{t}r^{(l)}=\omega_{1}[S_{1},r^{(l)}]+\delta\,\mathrm{dn}(\omega
t|k)[S_{3},r^{(l-1)}],\,l=1,2,\,\ldots.$ (15b) We multiply Eq. (15a) to the
left by the matrix $e^{i\omega_{1}tS_{1}}$ and to the right by the matrix
$e^{-i\omega_{1}tS_{1}}$ for formation of the integrating multiplier
Ivanchenko (2005). Now finding the terms $r^{(l)}$ in the series (14) is
defined by the previous ones $r^{(l-1)}$ as following
$r^{(l)}(t)=-i\delta\int_{0}^{t}dt^{\prime}{e^{i\omega_{1}(t^{\prime}-t)S_{1}}}\mathrm{dn}(\omega
t^{\prime}|k)[S_{3},r^{(l-1)}(t^{\prime})]{e^{-i\omega_{1}(t^{\prime}-t)S_{1}}}.$
(16)
## III Bi-qutrit
In the space $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ the two-qutrit density
matrix can be written in the Bloch representation
$\varrho=\frac{1}{6}R_{\alpha\beta}C_{\alpha}\otimes
C_{\beta},\;R_{00}=1,\;\varrho(t=0)=\varrho_{0},$ (17)
where $\otimes$ denotes the direct product. The functions $R_{m0},R_{0m}$
characterise the individual qutrits and functions $R_{mn}$ characterise their
correlations. The length of the generalized Bloch vector
$\sqrt{R_{\alpha\beta}^{2}-1}$ for pure states equals $2\sqrt{2}$.
Consider the Hamiltonian of the system of two qutrits with anisotropic and
exchange interaction in magnetic field in the following form
$\displaystyle
H_{2}=(\overrightarrow{h}\overrightarrow{S}+Q(S_{3}^{2}-2/3E)+d(S_{1}^{2}-S_{2}^{2}))\otimes
E+$ $\displaystyle
E\otimes(\overrightarrow{\bar{h}}\overrightarrow{S}+\bar{Q}(S_{3}^{2}-2/3E)+\bar{d}(S_{1}^{2}-S_{2}^{2}))+JS_{i}\otimes
S_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h_{\alpha\beta}C_{\alpha}\otimes C_{\beta},$ (18)
where $\overrightarrow{h}$ and$\;\overrightarrow{\bar{h}}$ are the magnetic
field vectors in frequency units, which operate on the first and the second
qudits respectively, and $J$ is the constant of isotropic exchange
interaction.
The system of equations for two qutrits takes the real form in terms of the
functions $R_{m0},R_{0m},R_{mn}$ as the closed system of 80 differential
equations Ivanchenko (2009), supplemented by the initial conditions
$\partial_{t}R_{m0}=\sqrt{\frac{2}{3}}e_{pim}(h_{p0}R_{i0}+h_{pl}R_{il}),\;\partial_{t}R_{0m}=\sqrt{\frac{2}{3}}e_{pim}(h_{0p}R_{0i}+h_{lp}R_{li}),$
(19a)
$\partial_{t}R_{mn}=e_{pim}\left[\sqrt{\frac{2}{3}}(h_{pn}R_{i0}+h_{p0}R_{in})+g_{rln}h_{pr}R_{il}\right]+e_{pin}\left[\sqrt{\frac{2}{3}}(h_{mp}R_{0i}+h_{0p}R_{mi})+g_{rlm}h_{rp}R_{li}\right],$
(19b) where by definition $\mathrm{Tr\,}\rho C_{\alpha}\otimes
C_{\beta}=\frac{2}{3}R_{\alpha\beta}$ (20)
and
$h_{p0}=\sqrt{6}(\overrightarrow{h},0,0,\frac{Q}{\sqrt{3}},0,d),\,h_{0p}=\sqrt{6}(\overrightarrow{\bar{h}},0,0,\frac{\bar{Q}}{\sqrt{3}},0,\bar{d}),\,h_{11}=h_{22}=h_{33}=2J$
are the Hamiltonian expansion coefficients in the basis $C_{\alpha}\otimes
C_{\beta}$ (other coefficients equal to zero). In equations (19) Latin indices
$m,\,n$ take the values from 1 to 8. Numerical values for the structure
constants $e_{pim},\;g_{rlm}$ are given in Appendix A.
The energy of the coupled qutrits in terms of the correlation functions has
the following form
$E(t)=\frac{1}{3}(h_{p0}R_{p0}+h_{0p}R_{0p}+\sum_{i=1}^{3}h_{ii}R_{ii}).$ (21)
We study the dynamics of two qutrits in the magnetic field
$\overrightarrow{h}=(\omega_{1}\mathrm{cn}(\omega
t|k)),\;\omega_{1}\mathrm{sn}(\omega t|k),\;\omega_{0}\mathrm{dn}(\omega
t|k),\overrightarrow{\bar{h}}=(\varpi_{1}\mathrm{cn}(\varpi
t|k),\;\varpi_{1}\mathrm{sn}(\omega t|k),\;\varpi_{0}\mathrm{dn}(\omega t|k))$
at the anisotropy constants equal to 0\. We transform the matrix density
$\varrho=\alpha_{2}^{-1}r_{2}\alpha_{2}$ with the matrix
$\alpha_{2}=\alpha_{1}\otimes\alpha_{1}$. Then equation for the matrix $r_{2}$
takes the form $i\partial_{t}r_{2}=[\widetilde{H},r_{2}]$ with the transformed
Hamiltonian
$\widetilde{H}=\left(\begin{smallmatrix}{}J+D\left(-2\omega+\varpi_{0}+\omega_{0}\right)&\frac{\varpi_{1}}{\sqrt{2}}&0&\frac{\omega_{1}}{\sqrt{2}}&0&0&0&0&0\\\
\frac{\varpi_{1}}{\sqrt{2}}&D\left(\omega_{0}-\omega\right)&\frac{\varpi_{1}}{\sqrt{2}}&J&\frac{\omega_{1}}{\sqrt{2}}&0&0&0&0\\\
0&\frac{\varpi_{1}}{\sqrt{2}}&D\left(\omega_{0}-\varpi_{0}\right)-J&0&J&\frac{\omega_{1}}{\sqrt{2}}&0&0&0\\\
\frac{\omega_{1}}{\sqrt{2}}&J&0&D\left(\varpi_{0}-\omega\right)&\frac{\varpi_{1}}{\sqrt{2}}&0&\frac{\omega_{1}}{\sqrt{2}}&0&0\\\
0&\frac{\omega_{1}}{\sqrt{2}}&J&\frac{\varpi_{1}}{\sqrt{2}}&0&\frac{\varpi_{1}}{\sqrt{2}}&J&\frac{\omega_{1}}{\sqrt{2}}&0\\\
0&0&\frac{\omega_{1}}{\sqrt{2}}&0&\frac{\varpi_{1}}{\sqrt{2}}&D\left(\omega-\varpi_{0}\right)&0&J&\frac{\omega_{1}}{\sqrt{2}}\\\
0&0&0&\frac{\omega_{1}}{\sqrt{2}}&J&0&D\left(\varpi_{0}-\omega_{0}\right)-J&\frac{\varpi_{1}}{\sqrt{2}}&0\\\
0&0&0&0&\frac{\omega_{1}}{\sqrt{2}}&J&\frac{\varpi_{1}}{\sqrt{2}}&D\left(\omega-\omega_{0}\right)&\frac{\varpi_{1}}{\sqrt{2}}\\\
0&0&0&0&0&\frac{\omega_{1}}{\sqrt{2}}&0&\frac{\varpi_{1}}{\sqrt{2}}&J+D\left(2\omega-\varpi_{0}-\omega_{0}\right)\end{smallmatrix}\right).$
Since $D\overset{\mathrm{def}}{\equiv}\text{dn}(\omega t|k)|_{k=0}=1$, then
the transformed Hamiltonian $\widetilde{H}$ does not depend on time and the
solution for the density matrix in the circularly polarized field has the form
$\varrho(t)=\alpha_{2}^{-1}e^{-i\widetilde{H}t}\varrho_{0}e^{i\widetilde{H}t}\alpha_{2}|_{k=0}.$
(22)
In the consistent field at resonance $\omega=\varpi_{0}=\omega_{0}=h$ at equal
$\varpi_{1}=\omega_{1}$ the Hamiltonian eigenvalues equal to
$-2J,-J,J,J-2\omega_{1},-J-\omega_{1},J-\omega_{1},-J+\omega_{1},J+\omega_{1},J+2\omega_{1}$.
This allows to find the exact solution in the closed form for any initial
condition, since the matrix exponent $e^{i\widetilde{H}t}$ in this case can be
calculated analytically.
For larger number of the qutrits with pairwise isotropic interaction, the
generalization is evident. In the case of interaction of qudits with different
dimensionality, the reduction of the original system to the system with
constant coefficients can be done by choosing, for example, the transformation
matrix for spin-3/2 and spin-2 in the form
$\mathrm{diag\,}(f^{3/2},\,f^{1/2},\,f^{-1/2},\,f^{-3/2})\otimes\mathrm{diag\,}(f^{2},\,f,\,1,\,f^{-1},\,f^{-2}).$
(23)
However, the Hamiltonian eigenvalues cannot be found in the simple analytic
form because of the lowering the system symmetry.
## IV Entanglement in the qutrits
### IV.1 Entanglement in the bi-qutrit
For the initial maximally entangled state, which is symmetrical at the
particle permutation,
$|\psi>=\frac{1}{\sqrt{3}}\sum_{i=-1}^{1}|i>\otimes|i>,$ (24)
in the consistent field at the resonance $\omega=\varpi_{0}=\omega_{0}=h$ at
equal $\varpi_{1}=\omega_{1},$ the exact solution for the correlation
functions is given in Appendix C. The correlation functions have the property
$R_{\alpha\beta}=R_{\beta\alpha}$, i.e. the symmetry is conserved during the
evolution, since the initial state and Hamiltonian are symmetric in respect to
the particle permutation.
Given the exact solution, one can find the negative eigenvalues of the partly
transposed matrix $\varrho^{pt}=(T\otimes E)\varrho$ (here $T$ denotes the
transposition):
$\epsilon_{1}=\epsilon_{2}=-\frac{1}{27}\sqrt{69+28\cos 3Jt-16\cos
6Jt},\;\epsilon_{3}=-\frac{1}{27}\left(5+4\cos 3Jt\right).$ (25)
The absolute value of the sum of these eigenvalues
$m_{VW}=|\epsilon_{1}+\epsilon_{1}+\epsilon_{3}|$ defines the entanglement
measure (negativity) between the qutrits Vidal and Werner (2002).
The entanglement between the qutrits can be described quantitatively with the
measure Schlienz and Mahler (1995)
$m_{SM}=\sqrt{\frac{1}{8}(R_{ij}-R_{i0}R_{0j})^{2}}.$ (26)
This measure equals to 0 for the separable state and to 1 for the maximally
entangled state, and it is applicable for both pure and mixed states.
That is why for the maximally entangled initial state of two qutrits, the
entanglement in the consistent field is defined by the formulae with the found
solution for the density matrix
$m_{SM}=\frac{1}{\sqrt{6561}}\sqrt{4457+2776\cos 3Jt-632\cos 6Jt-56\cos
9Jt+16\cos 12Jt}.$ (27)
This measure is numerically equivalent to the measure $m_{VW}$ Vidal and
Werner (2002); Toth and Gühne (2009), which is defined by the absolute value
of the sum of the negative eigenvalues (25) of the partly transposed matrix.
According to the definition Pan et al. (2004) for $2$-qutrit pure state, the
entanglement measure equals to
$\eta_{2}=\frac{1}{2}\sum_{i=1}^{2}S_{i},$ (28)
where $S_{i}=-\mathrm{Tr\,}\rho_{i}\log_{3}\rho_{i}$ is the reduced von
Neumann entropy, the index $i$ numerates the particles, i.e. the other
particle are traced out.
Since the qutrit reduced matrix eigenvalues equal to
$\lambda_{1}=\lambda_{2}=\frac{1}{27}(5+4\cos
3Jt),\;\lambda_{3}=\frac{1}{27}(17-8\cos 3Jt),$ then the entanglement measure
in the bi-qutrit takes the form
$\eta_{2}=-\sum_{i=1}^{3}\lambda_{i}\log_{3}\lambda_{i}.$ (29)
Normalized by the unity, the measure I-concurrence, which is easy to
calculate, is defined by the formulae Mintert et al. (2005)
$m_{I}=\frac{\sqrt{3}}{2}\sqrt{2(1-\mathrm{Tr\,\rho_{1}^{2})}}=\frac{1}{9}\sqrt{57+32\cos
3Jt-8\cos 6Jt},$ (30)
where $\rho_{1}=\frac{1}{\sqrt{6}}C_{\alpha}R_{\alpha 0}$ is the reduced
qutrit matrix.
The time-dependence of the measure $m_{SM}$ for the symmetrical initial state
$|s>=\frac{1}{\sqrt{12}}\sum_{i\neq j}(|i>\otimes|j>+|j>\otimes|i>)$ (31)
takes the form
$m_{SM}^{|s>}=\frac{1}{\sqrt{209952}}\sqrt{102679+19136\cos 3Jt+29312\cos
6Jt-1024\cos 9Jt+800\cos 12Jt};$ (32)
at $t=0$ this measure equals to $\sqrt{23/32}$.
The measures $m_{VW},\;m_{SM},\;\eta_{2},\;m_{I},\;m_{SM}^{|s>}$ do not depend
on the parameters of the consistent field, sign of the exchange constant at
zero anisotropy parameters. It should be noted that the Wooters entanglement
measure (concurrence) in the system of two qubits with the isotropic
interaction in the circularly polarized field at resonance is also independent
of the alternating field amplitude Zhang et al. (2008), but depends on the
exchange constant magnitude and the initial conditions only.
At zero external field the entanglement measure (24) takes the analytic form
at equal non-zero anisotropy parameters $Q=d=\overline{d}=\overline{Q}$
$m_{SM}(Q)=\frac{1}{\left(9J^{2}+8QJ+16Q^{2}\right)^{2}}\sqrt{\sum_{k=0}^{4}q_{k}\cos\left(k\sqrt{9J^{2}+8QJ+16Q^{2}}\,t\right)},$
(33)
where
$q_{0}=4457J^{8}+11616QJ^{7}+47392Q^{2}J^{6}+85888Q^{3}J^{5}+163072Q^{4}J^{4}+194560Q^{5}J^{3}+221184Q^{6}J^{2}+131072Q^{7}J+65536Q^{8}$;
$q_{1}=8J^{2}(J+2Q)^{2}\left(347J^{4}+518QJ^{3}+1440Q^{2}J^{2}+1504Q^{3}J+1024Q^{4}\right)$;
$q_{2}=-8J^{2}(J+2Q)^{2}\left(79J^{4}+76QJ^{3}+320Q^{2}J^{2}+448Q^{3}J+256Q^{4}\right)$;
$q_{3}=-8J^{3}(7J-4Q)(J+2Q)^{3}(J+4Q)$, $q_{4}=16J^{4}(J+2Q)^{4}$.
### IV.2 Entanglement in the chain of qutrits
Figure 1: The time-averaged populations for the initial pure state $|-1>$
versus the normalized Larmor frequency $\omega_{0}/\omega$ at the parameters
$k=0.85$ (solid line), $k=0.2$ (dashed line), $d=Q=0$,
$\omega_{1}=1/3,\;\omega=1$ ($I$ shows the upper level $|1>$ population; $II$
shows the middle level $|0>$ population).
We consider the Hamiltonian of the chain of $N$ qutrits with the pairwise
isotropic interaction in the magnetic field $\overrightarrow{\omega}$ in the
following form
$H_{N}=\sum(\overrightarrow{\omega}\overrightarrow{S}\otimes\overbrace{E\otimes\dots\otimes
E}^{N-1}+J\overrightarrow{S}\otimes\overrightarrow{S}\otimes\overbrace{E\otimes\dots\otimes
E}^{N-2}),$ (34)
where the summation is over different possible positions of
$\overrightarrow{S}$ in the direct products. Because the maximally entangled
state of $N$ qutrits
$|\phi>=\frac{1}{\sqrt{3}}\sum_{i=-1}^{1}|i>^{\otimes N}$ (35)
and the Hamiltonian (34) have the permutation symmetry, it follows that the
density matrix of $N$ qutrits has the symmetric correlation functions. The
length of the generalized Bloch vector for pure states equals
$\sqrt{3^{N}-1}$.
The entanglement measures for the many-particle multi-level quantum systems
are not studied enough and difficult to calculate in the analytic form, that
is why we will present only analytic formulas for the entropy measure
$\eta_{N}$ Pan et al. (2004), which is defined by the eigenvalues of the
reduced one-particle matrices for each qutrit. As the result of the mentioned
symmetry, the reduced matrices are equal to each other. Therefore the
entanglement measure for $N$ qutrits is defined by the formulae
$\eta_{N}=-\sum_{i=1}^{3}r_{i}\log_{3}{r_{i}}.$ (36)
The eigenvalues of the reduced matrices for 3, 4, 5, and 6 qutrits are
presented in the table below
$\begin{array}[]{ccc}N\setminus r_{i}&r_{1}=r_{2}&r_{3}\\\ &&\\\
3&\frac{29-4\cos 5Jt}{75}&\frac{17+8\cos 5Jt}{75}\\\ 4&\frac{905-98\cos
3Jt-72\cos 7Jt}{2205}&\frac{395+196\cos 3Jt+144\cos 7Jt}{2205}\\\
5&\frac{16919-1944\cos 5Jt-800\cos 9Jt}{42525}&\frac{8687+3888\cos
5Jt+1600\cos 9Jt}{42525}\\\ 6&\frac{21977-1694\cos 3Jt-1936\cos 7Jt-560\cos
11Jt}{53361}&\frac{9407+3388\cos 3Jt+3872\cos 7Jt+1120\cos 11Jt}{53361}.\\\
&&\end{array}$ (37)
The measures $\eta_{3},\;\eta_{4},\;\eta_{5},\;\eta_{6}$ do not depend on sign
of the exchange constant like the measure $\eta_{2}$.
Figure 2: Dynamics of the spin vector components $S_{y},\;S_{z}$ for the
initial pure state $|-1>$ (dashed lines) in the circularly polarized field
with the parameters: $k=0,\;\omega_{1}=0.02,\;\omega=\omega_{0}=1,\;d=Q=0$.
Solid lines demonstrate the deformation of the spin components due to the
influence of the second spin (the fluctuator) with $J=0.1$ for the initial
pure state $|-1>\otimes|-1>$.
## V Numerical results
In Fig. 1 we present the populations of the upper and middle levels in the
qutrit averaged over the time interval $\tau\rightarrow\infty$:
$P^{+}=\frac{1}{\tau}\int_{0}^{\tau}\,dt\left(\frac{1}{3}+\frac{1}{\sqrt{6}}R_{3}(t)+\frac{1}{3\sqrt{2}}R_{6}(t)\right)$
,
$P^{0}=\frac{1}{\tau}\int_{0}^{\tau}\,dt(\frac{1}{3}-\frac{1}{3}\sqrt{2}R_{6}(t))$
in dependence on the normalized Larmor frequency $\omega_{0}/\omega$. The
population of the upper level in qutrit coincides in form with the upper level
occupation in a two-level system Ivanchenko (2005), i.e. this demonstrates the
magnetic resonance position stabilization and the presence of the parametric
resonances.
In Fig. 2 we note the considerable suppression of the qutrit spin oscillations
$S_{y}=\text{cn}(\omega t|k)\sin\omega_{1}t$ and$\;S_{z}=-\cos\omega_{1}t$ by
the environment (fluctuator) in the case of the resonance $\omega=\omega_{0}$,
$\overrightarrow{\varpi}=0$.
The bi-qutrit energy (21) in the consistent field at isotropic interaction in
the case of the solution (45) is constant and equal to $\frac{2}{3}J$.
Although the analytic expressions for the measures in a bi-qutrit
$m_{VW},\;m_{SM}$ are different, but the numerical values are practically
identical. Maximal deviation in the rectangle $(1\geq J\geq
0.01)\times(100\geq t\geq 0)$ equals $0.014$, where $\times$ denotes the
Cartesian product.
Measures $\eta_{2}$ and $m_{I}$ qualitatively coincide with the measures
$m_{VW},\;m_{SM}$.
We have found that the anysotropy of the qutrits disentangles them, namely the
entanglement is decreased down to 0.0010 (see graphs 1 and 2 in Fig. 3).
In the constant longitudinal field
$\overrightarrow{\omega}=-\overrightarrow{\varpi}=(0,\,0,\,\omega_{0})$ (the
bi-qutrit Hamiltonian eigenvalues are equal to
$J,J,x_{1},x_{2},x_{3},-p,-p,p,p$, where $x_{1},x_{2},x_{3}$ are the roots of
the equation
${x^{3}+2x^{2}J-p^{2}x-2J^{3}=0,\,p=\sqrt{J^{2}+\omega_{0}^{2}}})$ the
Hamiltonian contains the antisymmetric part, thus it follows that the density
matrix for the initial symmetric state will not be symmetric because of the
breaking the symmetry of the particle permutations. The analytic solution is
cumbersome. In the constant longitudinal impulse field
$\overrightarrow{\omega}=-\overrightarrow{\varpi}=(0,\,0,\,2\,(\theta((t-17)(t-60))+\theta((40-t)(57-t)(t-60))))$
the entanglement dynamics is blocked at $\omega_{0}\gg J$ (Fig. 4 ). This
points to the possibility to control the entanglement.
In Fig. 5 we present the comparative dynamics of the entropy entanglement
measure for 2 to 6 qutrits. The disentanglement dynamics of the measures
$\eta_{3},\eta_{4},\eta_{5},\eta_{6}$ is similar to the one in the case of two
qutrits, but with smaller oscillation amplitude, i.e. larger number of the
qutrits disentangles less than two qutrits ($0.889\leq\eta_{3}\leq 1$).
Figure 3: Disentanglement dynamics of the initially maximally entangled state
in the bi-qutrit: in the zero external field with equal anisotropy constants
$Q=d=\overline{d}=\overline{Q}=0.02507,\,J=-0.1$ (curve 1) and for $J=0.1$
(curve 2); in the consistent field the curve 3 (thick line) demonstrates
complete coincidence of the measures $m_{VW}$ and $m_{SM}$ at $J$=0.1 and zero
anisotropy constants; the curve 4 demonstrates the entropy measure $\eta_{2}$;
$I$-concurrence is presented by the curve 5 at $J$=0.1. Figure 4:
Disentanglement of the maximally entangled state (24) (solid line) in the
impulse field $\omega_{1}=\varpi_{1}=0,\,\omega_{0}=-\varpi_{0}=2,\,J=0.178$.
The dashed curve presents the evolution in zero external field. Figure 5:
Disentanglement of the maximally entangled state (28) in the chain of 2, 3, 4,
5 and 6 qutrits with $J=0.1$.
## VI Conclusion
We have shown that the time-averaged upper-level occupation probability for
the qutrit in the consistent field in dependence on the normalized Larmor
frequency $\omega_{0}/\omega$ coincides in form with the upper-level
occupation probability in the two-level system and the parametric resonances
appear (Fig. 1). In the qutrit coupled to another qutrit (fluctuator), the
spin oscillations are essentially suppressed.
The comparative analysis of the bi-qutrit entanglement measures on the base of
the analytic solution for the density matrix demonstrates that, in spite of
the different approaches to the derivation of the formulas for the
entanglement, all the formulas give quite close results (Fig. 3), and the
measures $m_{VW}\ $and$\;m_{SM}$ are practically equal. This is in accordance
with the general results for the entanglement in the systems with the
permutational symmetry Toth and Gühne (2009).
The analytical formulas for the entanglement measures
$\eta_{3},\eta_{4},\eta_{5},\eta_{6}$ are similar to the disentanglement
measure for two qutrits $\eta_{2}$, but with numerically smaller oscillation
amplitude, i.e. the larger number of the qutrits disentangles less than two
qutrits.
###### Acknowledgements.
The author is grateful to A. A. Zippa for fruitful discussions and constant
invaluable support. Many thanks are due to S. N. Shevchenko for help in
editing and useful comments.
## VII Appendix A
The matrix representation of the full set of Hermitian orthogonal operators
for spin-1 has the form
$C_{1}=S_{1}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&1\\\
0&1&0\\\
&&\end{array}\right),\,~{}C_{2}=S_{2}=\frac{i}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-1&0\\\
1&0&-1\\\ 0&1&0\\\
&&\end{array}\right),\,C_{3}=S_{3}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&-1\\\ &&\end{array}\right),$ (38a)
$C_{4}=i\left(\begin{array}[]{ccc}0&0&-1\\\ 0&0&0\\\ 1&0&0\\\
&&\end{array}\right),C_{5}=\frac{i}{\sqrt{2}}\left(\begin{array}[]{ccc}0&-1&0\\\
1&0&1\\\ 0&-1&0\\\
&&\end{array}\right),\,C_{6}=\sqrt{3}(S_{3}^{2}-2/3E)=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&0&0\\\
0&-2&0\\\ 0&0&1\\\ &&\end{array}\right),$ (38b)
$C_{7}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&-1\\\
0&-1&0\\\
&&\end{array}\right),\,C_{8}=S_{1}^{2}-S_{2}^{2}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\\\ &&\end{array}\right),\,C_{0}=\sqrt{\frac{2}{3}}E,$ (38c)
where $E$ is the unity $3\times 3$ matrix. These matrices have the property of
the trace equal to zero ${\rm Tr\,}C_{a}=0$ and orthogonality ${\rm
Tr\,}C_{a}C_{b}=2\delta_{ab}$, $1\leq a,b\leq 8$. The connection between the
basis $C_{a}$ and the Gell-Mann basis ${\lambda_{a}}$ is the following:
$C_{1}=\frac{1}{\sqrt{2}}(\lambda_{1}+\lambda_{6}),\;C_{2}=\frac{1}{\sqrt{2}}(\lambda_{2}+\lambda_{7}),\;C_{3}=\frac{1}{2}\lambda_{3}+\frac{\sqrt{3}}{2}\lambda_{8},C_{4}=\lambda_{5},\;C_{5}=\frac{1}{\sqrt{2}}(\lambda_{2}-\lambda_{7}),\;C_{6}=\frac{\sqrt{3}}{2}\lambda_{3}-\frac{1}{2}\lambda_{8},\;C_{7}=\frac{1}{\sqrt{2}}(\lambda_{1}-\lambda_{6}),\;C_{8}=\lambda_{4}.$
Non-zero structure constants $e_{abc}$ ($g_{abc}$) antisymmetric (symmetric)
in respect to the permutation of any pair of indices for the commutators
$[C_{a},C_{b}]=2ie_{abc}C_{c}$ (anticommutators
$\\{C_{a},C_{b}\\}=\frac{4}{3}E\delta_{ab}+2g_{abc}C_{c}$) are respectively
equal according to the definitions $e_{abc}=\frac{1}{4i}{\rm
Tr\,}[C_{a},C_{b}]C_{c}$:
$e_{123}=e_{147}=e_{158}=-e_{245}=e_{278}=-e_{357}=\frac{1}{2},\;e_{156}=e_{267}=\frac{\sqrt{3}}{2},\;e_{348}=-1$;
$g_{abc}=\frac{1}{4}{\rm Tr\,}\\{C_{a},C_{b}\\}C_{c}$:
$g_{336}=g_{446}=-g_{666}=g_{688}=\frac{1}{\sqrt{3}},\;g_{116}=g_{226}=g_{556}=g_{677}=-\frac{1}{2\sqrt{3}},\;g_{118}=g_{124}=g_{137}=-g_{228}=g_{235}=-g_{457}=g_{558}=-g_{778}=\frac{1}{2}$.
Hence, the product of the generators is equal to
$C_{a}C_{b}=\frac{2}{3}E\delta_{ab}+(g_{abc}+ie_{abc})C_{c}$.
## VIII Appendix B
For the initial state $|-1>$ at non-zero detuning $\delta=\omega_{0}-\omega$
the density matrix elements in the circularly polarized field have the form
$\rho_{11}=\frac{\omega_{1}^{4}}{\Omega^{4}}\sin^{4}\frac{\text{$\Omega$t
}}{2},\,\rho_{12}=-\frac{\sqrt{2}\omega_{1}^{3}}{\Omega^{4}}\sin^{3}\frac{\Omega
t}{2}e^{-i\omega t}\left(i\Omega\cos\frac{\Omega t}{2}+\delta\sin\frac{\Omega
t}{2}\right),$ (39)
$\rho_{13}=-\frac{\omega_{1}^{2}}{2\Omega^{4}}\sin^{2}\frac{\Omega
t}{2}e^{-2i\omega t}\left(\omega_{1}^{2}-2i\delta\Omega\sin\Omega
t+\left(2\delta^{2}+\omega_{1}^{2}\right)\cos\Omega t\right),$ (40)
$\rho_{22}=\frac{\omega_{1}^{2}\sin^{2}\frac{\Omega
t}{2}}{\Omega^{4}}\left(2\delta^{2}+\omega_{1}^{2}(1+\cos\Omega
t)\right),\,\rho_{23}=-\frac{\omega_{1}}{\sqrt{2}\Omega^{4}}e^{-i\omega
t}\left(2\delta^{2}+\omega_{1}^{2}(1+\cos\Omega
t)\right)\left(\delta\sin^{2}\frac{\Omega t}{2}+i\frac{\Omega}{2}\sin\Omega
t\right),$ (41)
$\rho_{33}=\frac{1}{4\Omega^{4}}\left(2\delta^{2}+\omega_{1}^{2}(1+\cos\Omega
t)\right)^{2},\,\rho_{ik}=\rho_{ki}^{\ast},$ (42)
where $\Omega=\sqrt{\omega_{1}^{2}+\delta^{2}}$ is the Rabi frequency.
For the initial doubly stochastic state $\frac{1}{\sqrt{3}}(|-1>+|0>+|1>)$ and
for the states $|0>$, $\frac{1}{2}|-1>+\frac{1}{\sqrt{2}}|0>+\frac{1}{2}|1>|$
at exact resonance $\delta=0$ in the consistent field, the density matrices
are respectively equal
$\left(\begin{array}[]{lll}\frac{1}{12}\left(\cos
2\omega_{1}t+3\right)&\frac{1}{12}f^{-1}\left(i\sqrt{2}\sin
2\omega_{1}t+4\right)&\frac{1}{12}f^{-2}\left(\cos 2\omega_{1}t+3\right)\\\
\frac{1}{12}f\left(4-i\sqrt{2}\sin 2\omega_{1}t\right)&\frac{1}{6}\left(3-\cos
2\omega_{1}t\right)&\frac{1}{12}f^{-1}\left(4-i\sqrt{2}\sin
2\omega_{1}t\right)\\\ \frac{1}{12}f^{2}\left(\cos
2\omega_{1}t+3\right)&\frac{1}{12}f\left(i\sqrt{2}\sin
2\omega_{1}t+4\right)&\frac{1}{12}\left(\cos
2\omega_{1}t+3\right)\end{array}\right),$ (43)
$\left(\begin{array}[]{lll}\frac{1}{2}\sin^{2}\omega_{1}t&-\frac{if^{-1}\sin
2\omega_{1}t}{2\sqrt{2}}&\frac{1}{2}f^{-2}\sin^{2}\omega_{1}t\\\ \frac{if\sin
2\omega_{1}t}{2\sqrt{2}}&\cos^{2}\omega_{1}t&\frac{if^{-1}\sin
2\omega_{1}t}{2\sqrt{2}}\\\ \frac{1}{2}f^{2}\sin^{2}\omega_{1}t&-\frac{if\sin
2\omega_{1}t}{2\sqrt{2}}&\frac{1}{2}\sin^{2}\omega_{1}t\end{array}\right),\,\left(\begin{array}[]{lll}\frac{1}{16}\left(5-\cos
2\omega_{1}t\right)&-\frac{if^{-1}\sin
2\omega_{1}t}{8\sqrt{2}}&\frac{1}{8}f^{-2}\sin^{2}\omega_{1}t\\\ \frac{if\sin
2\omega_{1}t}{8\sqrt{2}}&\frac{1}{8}\left(\cos
2\omega_{1}t+3\right)&\frac{if^{-1}\sin 2\omega_{1}t}{8\sqrt{2}}\\\
\frac{1}{8}f^{2}\sin^{2}\omega_{1}t&-\frac{if\sin
2\omega_{1}t}{8\sqrt{2}}&\frac{1}{16}\left(5-\cos
2\omega_{1}t\right)\end{array}\right).$ (44)
For both the initial middle level and the doubly stochastic pure initial
state, the populations of the upper and bottom levels are equal. Nath et al.
(2003). This property is fulfilled for the mixed state as well.
## IX Appendix C
The exact solution for the correlation functions of the initial state (24),
which is symmetric under the particle permutation, in the consistent field at
resonance $\omega=\varpi_{0}=\omega_{0}=h$ and equal $\varpi_{1}=\omega_{1}$
takes the form
$\displaystyle
R_{0,1}=R_{0,2}=R_{0,3}=0,\,R_{0,4}=\frac{8}{3}\,\sqrt{\frac{2}{3}}\cos^{2}\omega_{1}t\,\text{cn}u\,\text{sn}u\,\sin^{2}\frac{3Jt}{2},\,R_{0,5}=-\frac{4}{3}\sqrt{\frac{2}{3}}\,\text{cn}u\,\sin^{2}\frac{3Jt}{2}\sin
2\omega_{1}t,$ $\displaystyle R_{0,6}=\frac{2}{9}\sqrt{2}\left(3\cos
2\omega_{1}t-1\right)\sin^{2}\frac{3Jt}{2},\,R_{0,7}=\frac{4}{3}\sqrt{\frac{2}{3}}\,\text{sn}u\,\sin
2\omega_{1}t\sin^{2}\frac{3Jt}{2},$ $\displaystyle
R_{0,8}=\frac{4}{3}\sqrt{\frac{2}{3}}\,\cos^{2}\omega_{1}t\left(1-2\text{sn}^{2}u\right)\sin^{2}\frac{3Jt}{2},$
(45a) $\displaystyle R_{1,1}=\frac{1}{36}\left(16+12(\cos
3Jt+2)\left(\text{cn}^{2}u-\text{sn }^{2}u\right)\cos^{2}\omega_{1}t+2\cos
3Jt-3\cos\left(3J-2\omega_{1}\right)t\right.$ $\displaystyle\left.-12\cos
2\omega_{1}t-3\cos(3J+2\omega_{1})t\right),\,R_{1,2}=\frac{2}{3}(\cos
3Jt+2)\text{cn}u\,\text{sn}u\,\cos^{2}\omega_{1}t,\,$
$\displaystyle\,R_{1,3}=\frac{1}{3}(\cos 3Jt+2)\text{sn}u\,\sin
2\omega_{1}t,\,R_{1,4}=\frac{1}{3}\text{sn}u\,\sin 3Jt\sin 2\omega_{1}t,$
$\displaystyle\,R_{1,5}=\frac{1}{6}\left(2\left(\text{cn}^{2}u-\text{sn}^{2}u\right)\cos^{2}\
\omega_{1}t+3\cos 2\omega_{1}t-1\right)\sin
3Jt,\,R_{1,6}=\frac{1}{\sqrt{3}}\,\text{cn}u\,\sin 3Jt\sin 2\omega_{1}t,$
$\displaystyle
R_{1,7}=-\frac{2}{3}\,\cos^{2}\omega_{1}t\,\text{cn}u\,\text{sn}u\,\sin
3Jt,\,R_{1,8}=\frac{1}{\sqrt{3}}R_{1,6},$ (45b) $\displaystyle R_{22}$
$\displaystyle=$ $\displaystyle\frac{1}{18}\left(6(\cos
3Jt+2)\left(\text{sn}^{2}u-\text{cn}^{2}u\right)\cos^{2}\ \omega_{1}t+\cos
3Jt-3(\cos 3Jt+2)\cos 2\omega_{1}t+8\right),$ $\displaystyle R_{23}$
$\displaystyle=$ $\displaystyle-\frac{1}{3}(\cos 3Jt+2)\text{cn}u\,\sin
2\omega_{1}t,\,R_{24}=\frac{1}{\sqrt{3}}R_{16},\,R_{25}=-R_{17},R_{26}=\sqrt{3}R_{14},$
$\displaystyle R_{27}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(2\left(\text{cn}^{2}u-\text{sn}^{2}u\right)\cos^{2}\omega_{1}t-3\cos
2\omega_{1}t+1\right)\sin 3Jt,\,R_{28}=-\frac{1}{\sqrt{3}}R_{26},$ (45c)
$\displaystyle R_{33}$ $\displaystyle=$ $\displaystyle\frac{1}{18}\left(-2\cos
3Jt+3\cos\left(3J-2\omega_{1}\right)t+12\cos
2\omega_{1}t+3\cos\left(3J+2\omega_{1}\right)t+2\right),$ $\displaystyle
R_{34}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\,\cos^{2}\omega_{1}t\left(\text{cn}^{2}u-\text{sn}^{2}u\right)\sin
3Jt,\,R_{35}=-R_{14},\,R_{36}=0,\,$ $\displaystyle R_{37}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\,\text{cn}u\sin 3Jt\sin
2\omega_{1}t,\,R_{38}=\frac{4}{3}\cos^{2}\omega_{1}t\,\text{cn}u\,\text{sn}u\sin
3Jt,$ (45d) $\displaystyle R_{44}$ $\displaystyle=$
$\displaystyle\frac{1}{72}\left(-72\left(1-8\text{cn}^{2}u\text{sn}^{2}u\right)\cos^{4}\omega_{1}t+8\cos
3Jt-12(2\cos 3Jt+1)\cos 2\omega_{1}t+9\cos 4\omega_{1}t+19\right),$
$\displaystyle R_{45}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\,\text{sn}u\left(24\left(\text{sn}^{2}u-3\text{cn}^{2}u\right)\sin\omega_{1}t\cos^{3}\omega_{1}t+2(2\cos
3Jt+1)\sin 2\omega_{1}t-3\sin 4\omega_{1}t\right),$ $\displaystyle R_{46}$
$\displaystyle=$
$\displaystyle-\frac{2}{3\sqrt{3}}\,\cos^{2}\omega_{1}t\left(2\cos 3Jt-9\cos
2\omega_{1}\ t+7\right)\text{cn}u\,\text{sn}u,$ $\displaystyle R_{47}$
$\displaystyle=$
$\displaystyle\frac{1}{12}\,\text{cn}u\left(-24\left(\text{cn}^{2}u-3\text{sn}^{2}u\right)\sin\omega_{1}t\cos^{3}\omega_{1}t-2(2\cos
3Jt+1)\sin 2\omega_{1}t+3\sin 4\omega_{1}t\right),$ $\displaystyle R_{48}$
$\displaystyle=$ $\displaystyle
4\cos^{4}\omega_{1}t\,\text{cn}u\,\text{sn}u\left(\text{cn}^{2}u-\text{sn}^{2}u\right),$
(45e) $\displaystyle R_{55}=\frac{1}{18}\left(6\left(\cos 3Jt+6\cos
2\omega_{1}t-4\right)\left(\text{sn}^{2}u-\text{cn}^{2}u\right)\cos^{2}\omega_{1}t-\cos
3Jt+3(\cos 3Jt+2)\cos 2\omega_{1}t\right.$ $\displaystyle\quad\left.-9\cos
4\omega_{1}t+1\right),\;R_{56}=\frac{1}{6\sqrt{3}}\,\text{cn}u\left(4\sin^{2}\frac{3Jt}{2}\sin
2\omega_{1}t-9\sin 4\omega_{1}t\right),$ $\displaystyle
R_{57}=\frac{1}{6}\left(2\cos 3Jt+\cos\left(3J-2\omega_{1}\right)t+4\cos
2\omega_{1}t+6\cos
4\omega_{1}t+\cos\left(3J+2\omega_{1}\right)t-2\right)\text{cn}u\,\text{sn}u,$
$\displaystyle
R_{58}=\frac{1}{12}\,\text{cn}u\left(-24\left(\text{cn}^{2}u-3\text{sn}^{2}u\right)\sin\omega_{1}t\cos^{3}\omega_{1}t+2(2\cos
3Jt+1)\sin 2\omega_{1}t-3\sin 4\omega_{1}t\right),$ (45f) $\displaystyle
R_{66}$ $\displaystyle=$ $\displaystyle\frac{1}{36}\left(-4\cos
3Jt+6\cos\left(3J-2\omega_{1}\right)t-12\cos 2\omega_{1}t+27\cos
4\omega_{1}t+6\cos\left(3J+2\omega_{1}\right)t+13\right),$ $\displaystyle
R_{67}$ $\displaystyle=$
$\displaystyle\frac{1}{6\sqrt{3}}\,\text{sn}u\left(2(\cos 3Jt-1)\sin
2\omega_{1}t+9\sin 4\omega_{1}t\right),\,$ $\displaystyle R_{68}$
$\displaystyle=$
$\displaystyle\frac{1}{3\sqrt{3}}\,\cos^{2}\omega_{1}t\left(-2\cos 3Jt+9\cos
2\omega_{1}\ t-7\right)\left(\text{cn}^{2}u-\text{sn}^{2}u\right),$ (45g)
$\displaystyle R_{77}$ $\displaystyle=$ $\displaystyle\frac{1}{18}\left((\cos
3Jt-1)\left(3\text{cn}^{2}u-3\text{sn}^{2}u-1\right)+9\cos
4\omega_{1}t\left(\text{cn}^{2}u-\text{sn}^{2}u-1\right)\right.$
$\displaystyle\quad\left.+3(\cos 3Jt+2)\cos
2\omega_{1}t\left(\text{cn}^{2}u-\text{sn}^{2}u+1\right)\right),$
$\displaystyle R_{78}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\,\text{sn}u\left(6\left(3\text{cn}^{2}u-\text{sn}^{2}u\right)\cos^{2}\omega_{1}t+2\cos
3Jt-3\cos 2\omega_{1}t+1\right)\sin 2\omega_{1}t,$ (45h)
$R_{88}=\frac{1}{72}\left(72\left(1-8\,\text{cn}^{2}u\,\text{sn}^{2}u\right)\cos^{4}\omega_{1}t+8\cos
3Jt-12(2\cos 3Jt+1)\cos 2\omega_{1}t+9\cos 4\omega_{1}t+19\right),$ (45i)
where $u=(ht|k).$ It is straightforward to find the analytic solution for
larger number of qutrits at the same conditions.
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arxiv-papers
| 2011-06-12T09:07:40 |
2024-09-04T02:49:19.561153
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. A. Ivanchenko",
"submitter": "Eugene Ivanchenko",
"url": "https://arxiv.org/abs/1106.2297"
}
|
1106.2389
|
# Renormalization Group Evolution of the Effective Potential in the Broken
Symmetry Phase
Chungku Kim Department of Physics, Keimyung University, Daegu 704-701, KOREA
###### Abstract
We investigate the renormalization group(RG) evolution for the neutral scalar
field theory in the broken symmetry phase. By using the minimum condition of
the vacuum expectation value(VEV), we show that the RG evlution of the
effective potential in the broken symmetry phase is governed by the same RG
functions in case of the symmetric phase.
###### pacs:
11.15.Bt, 12.38.Bx
The effective potential plays an important role in studies of the vacuum
instability, dynamical symmetry breaking, and the dynamics of composite
particlesSher . The scalar field of a quantum field theory with a
spontaneously broken symmetry have a non-vanishing VEV which provide a mass to
the Higgs particle, gauge bosons and fermions. The effective potential in the
broken symmetry phase is expressed in term of the Higgs particle which have a
vanishing VEV. Recently, it was shown that the effective potential in the
broken symmetry phase is scale invariant in the case of vanishing Higgs field
which implies the scale invariance of the physical cosmological constantFoot .
In this paper, we will investigate the RG evolution of effective potential in
the broken symmetry phase in case of non-vanishing Higgs field. Although we
will consider the neutral scalar field theory for simplicity, the
generalization to more complicated cases will be straightforward.
The effective potential of the neutral scalar field theory in the minimal
subtraction scheme(MS) is independent of the renormalization mass scale $\mu$
and satisfies the RG equationRG
$(D+\gamma^{MS}\phi\frac{\partial}{\partial\phi})V_{MS}(\mu,\lambda,m^{2},\phi)=0.$
(1)
where
$D\equiv\mu\frac{\partial}{\partial\mu}+\beta_{\lambda}^{MS}\frac{\partial}{\partial\lambda}+\beta_{m^{2}}^{MS}\frac{\partial}{\partial
m^{2}}$ (2)
In the case $m^{2}<0$, the scalar field $\phi$ have a non-vanishing VEV $v$
satisfying
$\left[\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=v}=0,$ (3)
from which one can determine $v$ as a function of $\mu,\lambda$ and $m^{2}.$
Then the Higgs field $\sigma$ with a vanishing VEV is defined as
$\sigma\equiv\phi-v$ and the effective potential in the broken symmetry phase
$V_{SSB}(\mu,\lambda,m^{2},\sigma)$ is given by
$V_{SSB}(\mu,\lambda,m^{2},\sigma)\equiv
V_{MS}(\mu,\lambda,m^{2},\sigma+v(\mu,\lambda,m^{2}))$ (4)
Recently, it was shown that when $\sigma=0,$ $V_{SSB}(\mu,\lambda,m^{2},0)$
satisfies
$DV_{SSB}(\mu,\lambda,m^{2},0)=0$ (5)
Foot which implies the scale invariance of the physical cosmological
constant. In order to investigate the RG evolution of
$V_{SSB}(\mu,\lambda,m^{2},\sigma)$ in the case of non-vanishing Higgs field
$(\sigma\neq 0)$, it is necessarily to obtain the RG evolution of the
$v(\mu,\lambda,m^{2})$. Actually, the RG evolution of the VEV plays an
important role in understanding the upper bound of the Higgs boson massbound
and the CKM matrixCKM and since the one-loop calculationOne , the
perturbative calculations showed that the VEV of a scalar field have a same RG
evolution as the corresponding Higgs fieldzwan . In order to obtain the RG
evolution of the VEV of a scalar field from the minimum condition given in
eq.(3), let us take a derivative of eq.(1) with respect to $\phi$ to obtain
$(D+\gamma^{MS}\phi\frac{\partial}{\partial\phi}+\gamma^{MS})\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}=0.$ (6)
By substituting $\phi=v(\mu,\lambda,m^{2})$ in this equation and by using
eq.(3), we obtain
$\left[(D+\gamma^{MS}\phi\frac{\partial}{\partial\phi})\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=v(\mu,\lambda,m^{2})}=0$
(7)
and by applying the operation $D$ to eq.(3), we obtain an
$D\left[\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=v(\mu,\lambda,m^{2})}=0$
(8)
Note that in eq.(7), the operator $D$ is inside of the bracket
$\left[...\right]_{\phi=v(\mu,\lambda,m^{2})}$ and hence does not act on
$v(\mu,\lambda,m^{2})$ whereas in eq.(8), it is outside of the bracket
$\left[...\right]_{\phi=v(\mu,\lambda,m^{2})}$ and act on
$v(\mu,\lambda,m^{2})$. Then, we can write eq.(8) as
$\left[D\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=v(\mu,\lambda,m^{2})}+\frac{\partial^{2}V_{MS}(\mu,\lambda,m^{2},v)}{\partial
v^{2}}Dv=0$ (9)
By substituting eq.(7) to first term of eq.(9), we obtain
$\left[\frac{\partial^{2}V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi^{2}}\right]_{\phi=v(\mu,\lambda,m^{2})}(Dv-\gamma^{MS}v)=0$
(10)
Since
$\left[\frac{\partial^{2}V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi^{2}}\right]_{\phi=v(\mu,\lambda,m^{2})}$
is an arbitrary quantity, we conclude that the RG evolution of the
$v(\mu,\lambda,m^{2})$ is given by
$Dv(\mu,\lambda,m^{2})=\gamma^{MS}v(\mu,\lambda,m^{2})$ (11)
Usually, this result was expected from the argument that in the broken
symmetry phase, VEV is renormalized by the wave function renormalization
constant of the Higgs field irrespective of the functional form of
$v(\mu,\lambda,m^{2})$. But it is clear that eq.(11) cannot be satisfied by
arbitrary function $v(\mu,\lambda,m^{2})$.
Now, in order to obtain the RG evolution of the effective potential in the
broken symmetry phase, let us apply
$D+\gamma^{MS}\sigma\frac{\partial}{\partial\sigma}$ to
$V_{SSB}(\mu,\lambda,m^{2},\sigma)$ :
$\displaystyle(D+\gamma^{MS}\sigma\frac{\partial}{\partial\sigma})V_{SSB}(\mu,\lambda,m^{2},\sigma)=\left[DV_{MS}(\mu,\lambda,m^{2},\phi)\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}$
$\displaystyle+\left[\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}Dv+\gamma^{MS}\sigma\frac{\partial}{\partial\sigma}\left[V_{MS}(\mu,\lambda,m^{2},\phi)\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}$
(12)
By using eq.(11), we can combine the last two terms of the above equation as
$\gamma^{MS}v\left[\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}+\gamma^{MS}\sigma\frac{\partial}{\partial\sigma}\left[V_{MS}(\mu,\lambda,m^{2},\phi)\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}=\left[\gamma^{MS}\phi\frac{\partial
V_{MS}(\mu,\lambda,m^{2},\phi)}{\partial\phi}\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}$
(13)
Then, by substituting this result into eq.(12) and by using eq.(1), we obtain
$(D+\gamma^{MS}\sigma\frac{\partial}{\partial\sigma})V_{SSB}(\mu,\lambda,m^{2},\sigma)=\left[(D+\gamma^{MS}\phi\frac{\partial}{\partial\phi})V_{MS}(\mu,\lambda,m^{2},\phi)\right]_{\phi=\sigma+v(\mu,\lambda,m^{2})}=0$
(14)
which means that the RG evolution of the effective potential in the broken
symmetry phase is governed by the same renormalization group functions in the
MS scheme. Actually, by using the two-loop effective action2loop .
$\displaystyle V_{MS}(\mu,\lambda,m^{2},\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}m^{2}\phi^{2}+\frac{1}{24}\lambda\phi^{4}+\Lambda+\frac{1}{4}\frac{\hbar}{(4\pi)^{2}}(m^{2}+\frac{1}{2}\lambda\phi^{2})^{2}(L_{MS}-\frac{3}{2})$
(15) $\displaystyle+\frac{\hbar^{2}}{(4\pi)^{4}}\\{(\frac{1}{8}\lambda
m^{4}+\frac{1}{4}\lambda^{2}m^{2}\phi^{2}+\frac{3}{32}\lambda^{3}\phi^{4})L_{MS}^{2}+(-\frac{1}{4}\lambda
m^{4}-\frac{3}{4}\lambda^{2}m^{2}\phi^{2}-\frac{5}{16}\lambda^{3}\phi^{4})L_{MS}$
$\displaystyle+(\frac{1}{8}\lambda
m^{4}+(\frac{3}{4}+\Omega(1))\lambda^{2}m^{2}\phi^{2}+(\frac{11}{32}+\frac{\Omega(1)}{2})\lambda^{3}\phi^{4})\\}+O(\hbar^{3}),$
where
$\Omega(1)=-\frac{1}{2\sqrt{3}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin(\frac{n\pi}{3})\simeq-0.293$
(16)
and
$L_{MS}\equiv\log\left(\frac{m^{2}+\frac{\lambda}{2}\phi^{2}}{\mu^{2}}\right),$
(17)
we obtain the VEV as
$v(\mu,\lambda,m^{2})=-m\sqrt{\frac{6}{\lambda}}\\{1+\frac{1}{2}\frac{\hbar}{(4\pi)^{2}}\lambda(-L+1)+\frac{\hbar^{2}}{(4\pi)^{4}}\lambda^{2}(-\frac{1}{4}L^{2}+\frac{3}{2}L-5\Omega(1)-\frac{7}{4})\\}+O(\hbar^{3})$
(18)
where
$L\equiv\log\left(\frac{2m^{2}}{\mu^{2}}\right)$ (19)
Then, by using the RG functions given byRGfn
$\displaystyle\beta_{\lambda}^{MS}$ $\displaystyle=$
$\displaystyle\mu\frac{d\lambda}{d\mu}=3\frac{\hbar}{(4\pi)^{2}}\lambda^{2}-\frac{17}{3}\frac{\hbar^{2}}{(4\pi)^{4}}\lambda^{3}+(\frac{145}{8}+12\text{
}\varsigma(3))\frac{\hbar^{3}}{(4\pi)^{6}}\lambda^{4}+\cdot\cdot\cdot,$ (20)
$\displaystyle\beta_{m^{2}}^{MS}$ $\displaystyle=$
$\displaystyle\mu\frac{dm^{2}}{d\mu}=\frac{\hbar}{(4\pi)^{2}}\lambda-\frac{5}{6}\frac{\hbar^{2}}{(4\pi)^{4}}\lambda^{2}+\frac{7}{2}\frac{\hbar^{3}}{(4\pi)^{6}}\lambda^{3}+\cdot\cdot\cdot,$
(21) $\displaystyle\beta_{\Lambda}^{MS}$ $\displaystyle=$
$\displaystyle\mu\frac{d\Lambda}{d\mu}=\frac{1}{2}\frac{\hbar}{(4\pi)^{2}}m^{4}+\frac{1}{16}\frac{\hbar^{3}}{(4\pi)^{6}}l^{2}m^{4}\cdot\cdot\cdot,$
(22)
and
$\gamma^{MS}=\frac{\mu}{\phi}\frac{d\phi}{d\mu}=-\frac{1}{12}\frac{\hbar^{2}}{(4\pi)^{4}}\lambda^{2}+\frac{1}{16}\frac{\hbar^{3}}{(4\pi)^{6}}\lambda^{3}+\cdot\cdot\cdot.$
(23)
we can check that eq.(11) is satisfied and by substituting eq.(2) to eq.(4)
and by expanding in $\hbar$, we can see that eq.(14) is also satisfied up to
$O(\hbar^{2})$. Actually, we have checked that eq.(14) is satisfied to
$\hbar^{3}$ order.
Finally, let us discuss the RG running of the Higgs mass and the coupling
constants in the broken symmetry phase. There are two typical schemes to
define the parameters in the broken symmetry phase containing $v$. One is
using only $v^{(0)}=-m\sqrt{\frac{6}{\lambda}}$ which is the tree level value
of VEV in the tree Lagrangian and the remaining terms $v-v^{(0)}$ act as a
finite counter-terms to remove the tadpole terms in the higher order Feynman
diagramsTaylor . The other is using $v$ itself in the tree Lagrangian and
includes the tadpole diagrams in the higher order Feynman diagramsJegerlehner
. For example, in case of the coupling constant for the cubic Higgs
interaction $h\sigma^{3}$ which were absent in the symmetric phase, $h$ is
defined by $-m\sqrt{\frac{\lambda}{6}}$ in the former case and
$\frac{1}{6}\lambda v$ in the latter case. Then, from eqs.(11) and (14) we can
obtain the RG functions as
$\beta_{h}=\mu\frac{\partial
h}{\partial\mu}=(\frac{1}{2m}\beta_{m^{2}}^{MS}+\frac{1}{2\lambda}\beta_{\lambda}^{MS})h$
(24)
in the former case and
$\beta_{h}=\mu\frac{\partial
h}{\partial\mu}=(\gamma^{MS}+\frac{1}{\lambda}\beta_{\lambda}^{MS})h$ (25)
in the latter case respectively. In case of the running Higgs mass term
$\frac{1}{2}m_{H}^{2}\sigma^{2}$, $m_{H}^{2}$ is defined by $2m^{2}$ in the
former case and $-m^{2}+\frac{\lambda}{2}v^{2}$ in the latter case. Then the
corresponding RG functions are given by
$\beta_{m_{H}^{2}}\equiv\mu\frac{\partial
m_{H}^{2}}{\partial\mu}=2\beta_{m^{2}}^{MS}$ (26)
in the former case and
$\beta_{m_{H}^{2}}=\mu\frac{\partial
m_{H}^{2}}{\partial\mu}=(2\gamma^{MS}+\frac{1}{\lambda}\beta_{\lambda}^{MS})\frac{\lambda}{2}v^{2}-\beta_{m^{2}}^{MS}=(2\gamma^{MS}+\frac{1}{\lambda}\beta_{\lambda}^{MS})m_{H}^{2}+(2\gamma^{MS}+\frac{1}{\lambda}\beta_{\lambda}^{MS}-\gamma_{m^{2}}^{MS})m^{2}$
(27)
in the latter case respectively.
In summary, we have investigated the RG evolution of the VEV of a scalar field
from the minimum condition of the VEV and have shown that the RG evolution of
the effective potential of the spontaneously broken symmetry is governed by
the same renormalization group functions of a theory in case of the symmetric
phase. As a result, we can determine the RG functions of the running Higgs
mass and the coupling constant for the cubic Higgs interaction. It is easy to
show that the result of this paper can be extended to the theories which have
two Higgs particles.
## References
* (1) For a review and references, see M. Sher, Phys. Rep. 89, 273 (1989).
* (2) R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, Phys. Lett.B 664,199(2008).
* (3) C. Ford, D. R. T. Jones, P. W. Stephenson and M. B. Einhorn, Nucl. Phys. B395, 17(1993).
* (4) J. A. Casas, J. R. Espinosa and M. Quiros, Phys. Lett. B382, 374(1996) and references therein.
* (5) C. Balzereit, T. Hansman, T. Mannel and B. Plumper, Eur. Phys. J. C9, 197(1999) and references therein.
* (6) E. Ma and S. Pakvasa, Phys. Rev. D20, 2899 (1979).
* (7) See for example, H. Arason, D. J. Castano, B. Keszthelyi, S. Mikaelian, E. J. Pirad, P. Ramond and B. D. Wright, Phys. Rev. D46, 3945(1992); G. Cvevic, S. S. Hwang and C. S. Kim, Phys. Rev. D58, 116003(1998); C. R. Das and M. K. Parida, Eur. Phys. J. C20, 121 (2001).
* (8) C. Ford, D. R.T. Jones, Phys. Lett. B274, 409 (1992), 285, 399(E) (1992).
* (9) H. Kleinert, J. Neu, V. Schulte-Frohlinde, K. G. Chetyrkin, S. A. Larin, Phys. Lett. B272, 39(1991); ibid. 319, 545(1993).
* (10) J. C. Taylor, Gauge Theories of Weak Interactions (Cambridge Press, Cambridge, 1976).
* (11) J. Fleischer and F. Jegerlehner, Phys. Rev. D23, 2001(1981); F. Jegerlehner, M. Yu. Kalmykov and O. veretin, Nucl. Phys. B641,285(2002).
|
arxiv-papers
| 2011-06-13T06:32:03 |
2024-09-04T02:49:19.570175
|
{
"license": "Public Domain",
"authors": "Chungku Kim",
"submitter": "Chungku Kim",
"url": "https://arxiv.org/abs/1106.2389"
}
|
1106.2440
|
# A Game Theoretic Perspective on Network Topologies
Shaun Lichter, Christopher Griffin, and Terry Friesz
(April 4, 2011)
###### Abstract
We extend the results of Goyal and Joshi (S. Goyal and S. Joshi. Networks of
collaboration in oligopoly. Games and Economic behavior, 43(1):57-85, 2003),
who first considered the problem of collaboration networks of oligopolies and
showed that under certain linear assumptions network collaboration produced a
stable complete graph through selfish competition. We show with nonlinear cost
functions and player payoff alteration that stable collaboration graphs with
an arbitrary degree sequence can result. As a by product, we prove a general
result on the formation of graphs with arbitrary degree sequences as the
result of selfish competition. Simple motivating examples are provided and we
discuss a potential relation to Network Science in our conclusions.
## 1 Introduction
In this paper we model the emergence of collaborations among players (e.g.,
firms) as a strategic network formation game [DM97], by allowing selfish
agents to choose with which other agents they would like to form a link. Each
agent has the option to deny a link to another agent, so the formation of a
link requires the cooperation of both players. A value function assigns a
value to each particular graph and this value is distributed to agents by an
allocation function (or allocation rule). This distribution of value drives a
player’s preference for particular graph structures. The primary objective of
this paper is to extend and generalize a model by Goyal and Joshi [GJ03] for
collaboration of oligopolistic firms to provide a model that, under different
conditions, admits a stable collaboration graph with symmetric or asymmetric
degree distributions. The primary contribution of this work is showing that
nonlinear pricing and collaboration incentives exist to create arbitrary
stable network structures. This extends the work of Goyal and Joshi who showed
that linear pricing schemes lead to stable graphs that are complete only. Thus
the occurrence of exotic graphs (e.g., power law graphs) in natural
collaborations may simply be the result of nonlinear incentives.
In particular circumstances a modeler may wish to predict how a network will
change if it is perturbed in some way. Specifically, a network may be changed
by a player entering or exiting the game or when a link is added or deleted.
For example, Verizon might be interested in knowing if AT&T and T-Mobile
merged, whether the new network would be stable and how the merger might
affect the market. Models that take a statistical in nature provide forecasts
for the behavior of large networks, since bulk measures are used, but have
difficulty predicting the stability of smaller network of interest.
Unfortunately, the models developed in this paper also have some limitations
in that they require some knowledge of the objective functions of players.
Further, the models developed in this paper are static models which provide
insight into the stability of particular networks, but predictive capabilities
may require a dynamic model of objective functions. These two limitations are
discussed further in Section 6 and are the driving motivation of continued
work in this area. Nonetheless, the models developed in this paper provide a
foundation to develop modeling approaches that can predict the reaction of
networks to specific changes that may take place.
The research on network formation games is extensive. We summarize a few key
results germane to our presentation. Myerson [Mye77] considers cooperation
structures, which we call graphs, but he considers only whether agents are
connected, not the structure of the graph of connections. Nonetheless, Myerson
was the first to consider the strategic formation of networks.
Jackson and Wolinsky model social relationships where each player receives
benefits at a cost from their friends and a lower benefit from friends of
friends, but with no cost. Jackson and Wolinsky investigate network formation
via the stability of particular graph structures [JW96]. An efficient graph
structure is one with the maximum value, that is, the total value of all
agents is maximized (see Section 2 for a mathematical definition). Jackson and
Wolinksy found that efficient graph structures are not always stable, but
their analysis used pairwise stability (see Section 2), which restricts the
capabilities of agents and assumes a symmetric allocation rule.
Dutta and Mutuswami construct an allocation rule that ensures the existence of
a strongly stable efficient graph where the allocation rule is symmetric on
this subset of strongly stable graphs [DM97]. Bala and Goyal [BG00] consider a
model where links allow access to a noncompetitive product such as
information. The cost of a link is attributed to the agent initiating the
formation of the link.
Furusawa and Konishi [FK07] model free trade networks in which a link between
countries represents a free trade agreement and the absence of a link results
in a tariff. The supply and demand functions then induce a value function and
allocation rules over the countries. Belleflamme and Bloch [BB04] model market
sharing between firms where a link between two firms represents an agreement
not to infringe on each other’s market and the absence of a link implies the
firms will compete in the same market. Again, supply and demand induce a value
function and an allocation rule over the firms. Labor markets [CAJ07] and co-
author networks [JW96] have also been modeled using game theoretic networks.
Networks allow a model to portray intricate relationships among players that
cannot otherwise be described. For example, non-transitive relationships may
be modeled as in the case when (e.g.), Player 1 is connected to Players 2 and
3, but Players 2 and 3 are not connected to one another. This generality leads
to another important aspect of a network model: how do relationships of one
pair of players affect a third player? In some cases, a positive benefit may
be reaped by an outside party; other times, a negative affect may be imposed
on an outside party. Networks allow relationships to be modeled as an
externality for outside players, but the nature of the externality is
dependent on the application and the model. Currarini [Cur02] investigates the
nature of externalities in networks by modeling the value of a graph as a
function of the components of the graph. Particularly, any graph is
partitioned into one or more components and the value may be a function of the
partition.
The remainder of this paper is organized as follows: In Section 2 we provide
notation used throughout the remainder of this paper. In Section 3 we state
and prove a general result on game theoretic constructions of graphs with
arbitrary degree sequences. In Section 4 we apply this result to extend the
work of Goyal and Joshi [GJ03] and provide the main results of the paper. In
Section 5 we provide conclusions and suggest a bridge between this work and
recent work in Network Science. Finally in Section 6 we discuss future
directions of this research as well as new research directions that would make
this work more applicable to the analysis of real-world organizations.
## 2 Notational Preliminaries
We follow the notational conventions common in the graph formation literature
[JW96, DM97, Jac03]. Let $N=\\{1,2,\ldots n\\}$ be the set of nodes in a
graph, which will represent the players. In the case of oligopolies, a firm is
considered a player. The set of links in the graph is a set of pairs of nodes
(subsets of $N$ of size two). A graph $g$ is a set of links (set of subsets of
$N$ of size two). The graph $g^{c}$ is the complete graph (all subsets of $N$
of size two) and $G$ is the set of all graphs over the node set $N$, that is,
$G=\\{g:g\subseteq g^{c}\\}$. For a graph $g\in G$, the graph $g+ij$ is the
graph in which link $ij$ is added to $g$ while $g-ij$ is the graph in which
link $ij$ is removed from graph $g$. Let $\eta_{i}(g):G\rightarrow\mathbb{R}$
denote the degree of node $i$ in graph $g$ and
$\eta(g):G\rightarrow\mathbb{R}^{n}$ be the degree sequence of the graph $g$;
i.e., $\eta(g)=(\eta_{1}(g),\eta_{2}(g),\ldots,\eta_{n}(g))$. Let $[g]_{\eta}$
be the equivalence class of graphs with the same degree sequence as $g$. A
degree sequence $\mathbf{d}=\\{d_{1},\dots,d_{n}\\}$ on $n$ nodes is
graphical, if there exists a graph $g$ with $n$ nodes that has degree sequence
$\mathbf{d}$. Equivalently, a degree sequence $\mathbf{d}$ is graphical if
$\eta^{-1}(\mathbf{d})\neq\emptyset$.
The value of a graph $g$ is the total value produced by agents in the graph;
we denote the value of a graph as the function $v:G\rightarrow\mathbb{R}$ and
the set of all such value functions as $V$. A graph $g$ is strongly efficient
if $v(g)\geq v(g^{\prime})\;\;\forall\;g^{\prime}\in G$. An allocation rule
$Y:V\times G\rightarrow\mathbb{R}^{N}$ distributes the value $v(g)$ among the
agents in $g$. Denote the value allocated to agent $i$ as $Y_{i}(v,g)$. Since,
the allocation rule must distribute the value of the network to all players,
it must be balanced; i.e., $\sum_{i}Y_{i}(v,g)=v(g)$ for all $(v,g)\in V\times
G$. The allocation rule governs how the value of the graph is distributed and
thus plays a significant role in the model. The allocation rule may be used to
model a free market, utilities of players, or system rules such as taxation or
the provision of subsidies to particular players. Throughout this paper, we
denote the game $\mathcal{G}=\mathcal{G}(v,Y,N)$ as the game played with value
function $v$ and allocation rule $Y$ over nodes $N$.
The strategy set of Player $i$, $S_{i}$, is the set of nodes to which Player
$i$ may request to link and $S=\prod_{i\in N}{S_{i}}$ is the strategy space
for all players. The graph $g$ induced by strategy $s\in S$ is
$g(s)=\\{ij:j\in s_{i},i\in s_{j}\\}$. Here $s_{i}\in S_{i}$ is the strategy
for Player $i$ in the overall strategy $s$.
The payoff of Player $i$ is defined as $f^{\mathcal{G}}_{i}(s)=Y_{i}(v,g(s))$.
This is interpreted as the payoff that player $i$ receives from the strategy
$s$ in game $\mathcal{G}$ and is the proportion of the value $v(g(s))$
distributed by $Y$. This model admits a Nash equilibrium in which each player
chooses to link with no other players and no network forms. Various authors
argue that this is an inadequacy and therefore it is necessary to make
refinements to the notion of equilibrium [Jac03, DM97].
Jackson and Wolinksy use pairwise stability to model stable networks without
the use of non-cooperative Nash equilibrium [JW96].
###### Definition 2.1.
A network $g$ with value function $v$ and allocation rule $Y$ is pairwise
stable if (and only if):
for all $ij\in g$, $Y_{i}(v,g)\geq Y_{i}(v,g-ij)$ and
for all $ij\not\in g$, if $Y_{i}(v,g+ij)>Y_{i}(v,g)$, then
$Y_{j}(v,g+ij)<Y_{j}(v,g)$
Pairwise stability implies that in a stable network, for each link that
exists, (1) both players must benefit from it and (2) if a link can provide
benefit to both players, then it in fact must exist. Jackson notes that
pairwise stability may be too weak because it does not allow groups of players
to add or delete links, only pairs of players [Jac03]. Deletion of multiple
links simultaneously has been considered in [BB04].
Goyal and Joshi [GJ06] provide a model that allows a player’s allocation to be
dependent on the number of links the player has. That is:
$Y_{i}(g)=\pi_{i}(g)-\gamma\eta_{i}(g)$ (1)
Here $\pi_{i}$ is a profit function, $\eta_{i}(g)$ is the number of links
player $i$ has in $g$, and $\gamma$ is a link maintenance cost. However,
$\gamma$ may be negative to model a benefit of having more links. The value
function is then defined implicitly so that $Y_{i}(g)$ is a balanced
allocation rule.
## 3 A General Result on Graph Stability
In this section, we show how to construct an allocation rule $Y_{i}(g)$ to
ensure that a graph with a given degree sequence is stable. To do this, we
present a set of properties for the objectives of players that result in a
stable graph with an arbitrary degree sequence. Recall, a stable network is
one in which no player has an incentive to drop a link or request an
additional link to a player who reciprocates such an incentive. Consider the
scenario where for $i=1,\dots,n$, Player $i$ has a desired degree $k_{i}$. For
example, some scientific authors may like to collaborate with only a few
individuals, while some like to collaborate with many others. Ultimately, when
a collaboration network is formed, each player $i$ receives the number of
links given by $\eta_{i}(g)$. In the game where each player’s objective
penalizes them for incurring a degree $\eta_{i}(g)$ not equal to their desired
degree $k_{i}$ (under some conditions), the graph with a degree distribution
of $k=(k_{1},k_{2},\ldots,k_{n})$ will be stable.
With a slight abuse of the notation from Section 2, we first define an
allocation (payoff) to each player $i$ as $Y_{i}:G\rightarrow\mathbb{R}$ which
is consistent with an allocation function $Y:G\times V\rightarrow\mathbb{R}$
and an induced value function $V:G\rightarrow\mathbb{R}$ with
$v(g)=\sum_{i}{Y_{i}(g)}$.
###### Theorem 3.1.
Let $\mathbf{d}=(k_{1},\dots,k_{n})$ be a desired degree sequence for $n$
players in the node set $N$. Assume that Player $i$ wishes to maximize
objective $Y_{i}(g)=-f_{i}(\eta_{i}(g))=-f(\eta_{i}(g)-k_{i})$ (or minimize
$f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})$), where
$f:\mathbb{R}\rightarrow\mathbb{R}$ is a convex function with minimum at $0$.
Let $v$ be the balanced value function induced from the allocation rule
$Y=(Y_{1},\dots,Y_{n})$. If $\eta^{-1}(\mathbf{d})$ is non-empty (i.e.,
$\mathbf{d}$ is graphic) then any graph $g$ such that $\eta(g)=\mathbf{d}$ is
pairwise stable for the the game $\mathcal{G}(v,Y,N)$.
###### Proof.
Let $g\in\eta^{-1}(\mathbf{d})$ be a graph so that $\eta_{i}(g)=k_{i}$ and let
$g-ij$ be the graph $g$ after the link $ij$ is dropped from $g$ and define
$g+ij$ to be the graph $g$ after the link $ij$ is added to graph $g$. Observe
that:
$\displaystyle\eta_{i}(g-ij)=k_{i}-1,\quad\eta_{j}(g-ij)=k_{j}-1,\quad\eta_{l}(g-ij)=k_{l},$
$\displaystyle\eta_{i}(g+ij)=k_{i}+1,\quad\eta_{j}(g+ij)=k_{j}+1,\quad\eta_{l}(g+ij)=k_{l}$
where $l\not\in\\{i,j\\}$
Since $f$ is convex with minimum at $0$ and each $f_{i}$ are simply shifts of
$f$ so that they are minimized at $k_{i}$, we know that $f_{i}(k_{i})$ is a
minimum of the function $f_{i}$. Thus $Y_{i}(g)$ is maximized when
$g\in\eta^{-1}(\mathbf{d})$. It now suffices to confirm pairwise stability:
$\displaystyle
Y_{i}(\eta_{i}(g-ij))-Y_{i}(\eta_{i}(g))=Y_{i}(k_{i}-1)-Y_{i}(k_{i})<0$
$\displaystyle
Y_{i}(\eta_{i}(g+ij))-Y_{i}(\eta_{i}(g))=Y_{i}(k_{i}+1)-Y_{i}(k_{i})<0$
Thus graph $g\in\eta(\mathbf{d})$ is stable because there is no firm $i$
willing to drop an arbitrary existing link $ij$ or add an arbitrary missing
link $ij$. ∎
###### Remark 3.2.
From Theorem 3.1 it is interesting to note that the stability of the class
$\eta^{-1}(\mathbf{d})$ is not unique. We can illustrate this fact using the
following counterexample. Let $\mathbf{d}=(1,1,1,2,3)$ with
$N=\\{1,\dots,5\\}$ and let $f(x)=x^{2}$, a simple convex function with
minimum at zero, from the theorem. The following two networks are both
pairwise stable solutions to the game $\mathcal{G}=(v,Y,N)$:
Figure 1: We illustrate two graphs that are both pairwise stable for degree
sequence $\mathbf{d}=(1,1,1,2,3)$ with $f(x)=x^{2}$ in Theorem 3.1
.
It should be clear however, that the second graph is not a Pareto optimal
solution to the problem. Pareto optimality of the class of graphs
$\eta^{-1}(\mathbf{d})$ follows a posteriori from the proof of Theorem 3.1.
###### Corollary 3.3.
Assume the conditions from Theorem 3.1. A graph $g$ is a pairwise stable and
Pareto optimal solution to the game $\mathcal{G}(v,Y,N)$ if and only if
$\eta(g)=\mathbf{d}$.
###### Proof.
The fact that $\mathbf{d}$ is graphic implies that there is some graph
$g\in\eta^{-1}(g)$ and furthermore $Y_{i}(g)$ is maximized for all
$i=1,\dots,n$. Therefore any change in strategy cannot improve any player’s
current payoff. Thus, $g$ is Pareto optimal and pairwise stable.
If $g^{\prime}\not\in\eta^{-1}(\mathbf{d})$ then while $g^{\prime}$ may be
pairwise stable, it cannot be Pareto optimal since there is some other graph
$g\in\eta^{-1}(\mathbf{d})$ so that $Y_{i}(g)>Y_{i}(g^{\prime})$ for some
$i\in N$. ∎
###### Example 3.4.
Suppose that we want the degree distribution of a stable graph that results
from playing the game described in Theorem 3.1 to have a power law degree
distribution. We embed this into the objectives of the players, so the
resulting graph has the proper distribution. Let $N=100$ players attempt to
minimize their cost function
$f(\eta_{i}(g))=(\eta_{i}(g)-k_{i})^{2}+\psi$
where $\psi=2$ and the parameters of each player ($k_{i}$) are given in the
following table:
Node(s) | $k_{i}$
---|---
1-75 | 1
76-89 | 2
90-94 | 3
95-96 | 4
97 | 5
98 | 6
99 | 7
100 | 8
The distribution of values of $k_{i}$ forms an approximate (with rounding to
integers) power law distribution. This is illustrated in Figure 2:
Figure 2: The empirical desired distribution of the degrees of the players.
This degree distribution follows an approximate power law distribution.
Figure 3 illustrates one realization of a power law graph with the desired
degree distribution.
Figure 3: Sample pairwise stable collaboration Network for 100 nodes.
We can test the stability of any graph $g$ with $\eta_{i}(g)=k_{i}$. In doing
so, we observe that any graph with this distribution must be both stable
(since there is no motivation for any player to alter her configuration within
the network) and Pareto optimal.
We can experimentally analyze this simple game to determine the nature of the
average network produced during play. For a simple experiment, 100 pairwise
stable graphs were generated and some of their properties analyzed. To build
these pairwise stable graphs, we used the following procedure:
An empty graph was initialized.
Two vertices that had degree less than their desired degree were chosen at
random without replacement.
These two vertices were joined by an edge.
The graph was checked for pairwise stability. If the graph was pairwise
stable, the graph was returned. Otherwise, we continued at Step 2. Results are
shown in Figure 4.
Figure 4: Experimental results from 100 pairwise stable graphs that result
when the given degree distribution is used with the objective
$f(\eta_{i}(g))=(\eta_{i}(g)-k_{i})^{2}+\psi$.
The left histogram shows the similarity between the degree specified degree
distribution and the mean degree distribution of the ensemble of graphs
generated. A more interesting plot is the mean objective function value on a
per-player basis (with error bars). This graph indicates that most players
successfully minimize their objective value (to zero) and that the players
with the worst ability to optimize their objective functions are players with
high degree requirements. It is interesting to note that these results are a
function of the routine used to compute pairwise stable graphs. We also tried
using a modified version of the preferential attachment algorithm:
An empty graph was initialized.
Two vertices that had degree less than their desired degree and with highest
desired degree where chosen at random.
These two vertices were joined by an edge.
The graph was checked for pairwise stability. If the graph was pairwise
stable, the graph was returned. Otherwise, we continued at Step 2. Using this
algorithm, in 100 simulation runs a graph with the proscribed degree sequence
was always returned111Maple code for these experiments is available upon
request from the authors..
## 4 Application to Collaboration in Oligopoly
We present an application of the network formation game to firm collaboration
in oligopolies, which is an extension to the firm collaboration presented by
Goyal and Joshi in [GJ03]. We show the conditions under which a firm
collaboration network with an arbitrary degree sequence will be stable.
The number of firms participating in collaborative agreements (e.g., sharing
resources such as equipment, laboratory space, office space, engineers and
scientists through separate R & D subcompanies) has significantly increased
within industries that are R & D intensive [HS90, Hag93, HS94, Hag96, HLV00,
Hag02], sparking research investigating the incentives for such collaboration.
Goyal and Joshi examine the incentives for collaboration and the interaction
of these incentives under market competition using a model of horizontal
oligopolistic firm collaboration. In this model, firms compete in the market
after choosing collaborators [GJ03]. Goyal and Joshi provide theoretical
analysis on various models with varying levels of link formation costs
relative to production costs. They assume that collaboration requires a fixed
cost from each firm. These models investigate collaboration agreements where
the collaboration reduces the cost of production. This prior work assumes a
constant return to scale cost function, which they admit is a restrictive
assumption, although it has been made by others in the collaborative R & D
literature [Blo95].
### 4.1 Previous Results on Network Stability in Oligopoly
Following [GJ03], consider an oligopoly composed of $n$ firms, each of whom
may collaborate with any of the other $n-1$ firms. Firm $i$ produces a
quantity $q_{i}$ of a product. Collaboration among firms affects the marginal
cost of production. Thus a particular (collaboration) graph $g$ induces a
marginal cost vector $c(g)\in\mathbb{R}^{n}$ in which $c_{i}(g)$ is the
marginal cost of firm $i$ under collaboration graph $g$. It is assumed that
the marginal cost decreases for a firm $i$ as the number of collaborators
increase. For many of the models in [GJ03], it is assumed that the marginal
cost of firm $i$ linearly decreases with the number of collaborators for firm
$i$:
$c_{i}(g)=\gamma_{0}-\gamma\eta_{i}(g)$ (2)
where, as before, $\eta_{i}(g)$ is the number of links for firm $i$ and
$\gamma_{0}$ is the marginal cost of production when a firm has no links.
Notice that $\gamma_{0}$ is constant for all firms. Given a network $g$, there
is an induced set of costs which, along with the demand functions, produces a
set of profit functions for each firm, $Y_{i}(g)$ (the allocation of payoff
for player $i$). These profit functions then induce a Nash equilibrium of
production, which provides the precise allocation rule (i.e., profit) for each
firm on the graph. The stability of the collaboration network can then be
analyzed as in Section 3.
One example that Goyal and Joshi [GJ03] study is that of a homogenous product
oligopoly. The market marginal price function (dependent on quantity produced)
is given by
$P(Q)=\alpha-\sum_{i\in N}q_{i}$ (3)
where $Q=\sum_{i\in N}q_{i}$. Using Equation 2, the resulting profit to Player
$i$ is:
$Y_{i}(g)=\left(\alpha-\sum_{i\in
N}q_{i}\right)q_{i}-\left(\gamma_{0}-\gamma\eta_{i}(g)\right)q_{i}=(\alpha-\gamma_{0})q_{i}-\left(\sum_{i\in
N}q_{i}\right)q_{i}-\left(-\gamma\eta_{i}(g)\right)q_{i}$ (4)
Given a collaboration network $g$ and marginal cost and demand induced by
network $g$ we can use Expression 4 to find the quantity produced by each firm
at Cournot equilibrium using the standard Cournot oligopoly production
formulation [Tri88, GJ03]:
$q_{i}=\frac{\alpha-\gamma_{0}+n\gamma\eta_{i}(g)-\gamma\sum_{j\neq
i}\eta_{j}(g)}{n+1}$ (5)
The fact that $q_{i}$ is a function of the collaboration graph $g$ implies
that the quantities: $P$, $Q$ and $c_{i}$ are all functions of $g$. We will
refer to these quantities in this way in the sequel.
There is no natural market force in this model to ensure that a firm does not
produce a negative quantity. Instead, Goyal and Joshi restrict the parameters
by looking at the case where firm $i$ has no collaborators ($\eta_{i}(g)=0$)
and all others have a maximum number of collaborators ($\eta_{j}(g)=n-2$ for
all $j\neq i$) and forcing the quantity production to remain non-negative;
i.e.:
$\alpha-\gamma_{0}-\gamma(n-1)(n-2)\geq 0$ (6)
Goyal and Joshi show that with marginal cost (2) and market demand (3), the
complete network is the unique stable network [GJ03]. They point out that the
positive contribution to profit obtained by a collaboration link is on the
order of $n\gamma$ and the negative contribution to profit is on the order of
$\gamma$, so the net profit is on the order of $(n-1)\gamma$, which leads to a
firm’s incentive to saturate all of its possible links, thus making the
complete graph stable.
###### Example 4.1.
We present a numerical example of the result from [GJ03]. Let $N=5$ firms
compete in an oligopoly with inverse demand function $P=100-Q$, fixed cost
$\gamma_{0}=5$, and $\gamma=1$. Then we can see that:
$\alpha-\gamma_{0}-\gamma(n-1)(n-2)=100-5-(4)(3)=95-12=83>0$ (7)
This enforces the non-negativity of production. Since the marginal cost of
Player $i$ ($i=1,\dots,5$) decreases as its degree increases (from Equation 2)
it is easy to see that players will prefer as many links as possible. Thus,
the complete graph must be stable.
###### Remark 4.2.
The fundamental limitation to Goyal and Joshi’s work lies in the model
complexity. Maintaining collaborations is costly and each additional
collaboration may contribute less to the marginal cost than the collaboration
before it. Additionally, Goyal and Joshi’s model admits only complete
collaboration networks, omitting any other type of collaboration network. The
remainder of the paper illustrates how to generalize this result to graphs
with arbitrary degree sequences.
### 4.2 Results of Nonlinear Cost on Stability
In this section we study the effect a nonlinear variation on the marginal cost
function has on the stability of collaboration structures. We consider a
marginal cost function:
$c_{i}(g)=\gamma_{0}+f_{i}(\eta_{i}(g))$ (8)
where $f_{i}$ is some function $f_{i}:\mathbb{R}\rightarrow\mathbb{R}$. We
present two Theorems:
Theorem 4.11 extends the results from Goyal and Joshi [GJ03] to the case where
firms receive decreasing marginal benefit from collaboration via a nonlinear
decrease in production cost. This extension allows the parameter space to be
considerably less restricted than the result in [GJ03].
Theorem 4.7 shows that if the functions $f_{i}$ have minima and the parameter
space is restricted to keep production quantities nonnegative, then the
collaboration network in the oligopolistic competition model will have a graph
corresponding to the arbitrary degree sequence in the prior section. As a
result, this is a particular application of Theorem 3.1, to a general
collaboration game. To prove our theorems, we alter the assumptions on the
functions $f_{i}$ from the marginal cost (Expression 8). Theorem 4.7 requires
convex functions $f_{i}$ where as Theorem 4.11 requires $f_{i}$ to be a
decreasing and convex function.
###### Lemma 4.3.
Suppose we have an oligopoly consisting of $n$ firms in which collaboration is
defined by the graph $g$ and the profit function (allocation rule) for Firm
$i$ in that oligopoly is given by:
$Y_{i}(g,q(g))=(\alpha-\gamma_{0})q_{i}(g)-\left(\sum_{j\in
N}q_{j}\right)q_{i}(g)-(f_{i}(\eta_{i}(g)))q_{i}(g)$ (9)
then the quantity produced for firm $i$ is:
$q_{i}(g)=\frac{\alpha-\gamma_{0}-nf_{i}(\eta_{i}(g))+\sum_{j\neq
i}f_{j}(\eta_{j}(g))}{n+1}$ (10)
###### Proof.
From [Tri88], for any oligopoly with profit function of the form:
$Y_{i}(q)=aq_{i}-\left(\sum_{j\in N}q_{j}\right)q_{i}-b_{i}q_{i}$ (11)
The resulting Cournot equilibrium point on quantities is:
$q_{i}=\frac{a-nb_{i}+\sum_{j\neq i}b_{j}}{n+1}$ (12)
In our case, we have:
$\displaystyle a=\alpha-\gamma_{0}$ $\displaystyle
b_{i}=f_{i}(\eta_{i}(g))\quad\forall i$
Substituting these definitions into Expression (12) yields Expression (10).
This completes the proof. ∎
###### Remark 4.4.
It is worth noting that when for each firm $i$, $b_{i}=-\gamma\eta_{i}(g)$
then the cost function (2) and induced equilibrium quantity (5) is retrieved
from Goyal and Joshi.
###### Corollary 4.5.
Suppose that $f$ is a convex function that has a minimum at $0$. Further,
suppose $f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})$ where
$k_{i}\in\\{1,2,\ldots,n-1\\}$. If the parameters $\alpha$ and $\gamma_{0}$
and the function $f$ are such that:
$\alpha-\gamma_{0}-n\max(f(n-1),f(1-n))-\frac{1}{2}(n-1)\max(f(1)-f(0),f(-1)-f(0))>0$
(13)
and $n\geq 2$, then the Cournot equilibrium point quantities (10) are
nonnegative for all firms and for all collaboration graphs and the two
inequalities are true:
$\displaystyle 2q_{i}(g)-\frac{n-1}{n+1}[f(1)-f(0)]$ $\displaystyle>0$ (14)
$\displaystyle 2q_{i}(g)-\frac{n-1}{n+1}[f(-1)-f(0)]$ $\displaystyle>0$ (15)
###### Proof.
Since $n\geq 2$ and $f$ is convex and has a minimum at $0$, this implies that
$\frac{n-1}{n+1}[f(1)-f(0)]$ and $\frac{n-1}{n+1}[f(-1)-f(0)]$ are non-
negative. Hence, (14) and (15) imply that $q_{i}(g)$ is non-negative and hence
it suffices to only show that (14) and (15) are implied by (13).
For all $i$, function $f_{i}$ is a convex function of the degree of node $i$
in the graph $g$; the degree of node $i$ must take an integer value between
$0$ and $n-1$, which due to the convexity of $f_{i}$ and the fact that
$k_{i}\in\\{1,2,\ldots n-1\\}$ implies that the maximum of $f_{i}$ is
equivalent to $\max(f(n-1),f(-n+1))$. That is,
$f_{i}(\eta_{i}(g))\leq\max(f(n-1),f(-n+1))$
This means that (13) implies:
$\alpha-\gamma_{0}-nf_{i}(\eta_{i}(g))-\frac{1}{2}(n-1)\max(f(1)-f(0),f(-1)-f(0))>0\quad\forall
i$ (16)
Since, all $f_{i}(\eta_{i}(g))\geq 0$, we may add $\sum_{j\neq
i}f_{j}(\eta_{j}(g))$ to the left side of (16) without harming the inequality,
implying:
$\alpha-\gamma_{0}-nf_{i}(\eta_{i}(g))+\sum_{j\neq
i}f_{j}(\eta_{j}(g))-\frac{1}{2}(n-1)\max(f(1)-f(0),f(-1)-f(0))>0\quad\forall
i$ (17)
Now we divide by $n+1$:
$\frac{\alpha-\gamma_{0}-nf_{i}(\eta_{i}(g))+\sum_{j\neq
i}f_{j}(\eta_{j}(g))}{n+1}-\frac{1}{2}\frac{n-1}{n+1}\max(f(1)-f(0),f(-1)-f(0))>0\quad\forall
i$ (18)
The term on the left of (18) is $q_{i}(g)$:
$q_{i}(g)-\frac{1}{2}\frac{n-1}{n+1}\max(f(1)-f(0),f(-1)-f(0))>0\quad\forall
i$ (19)
Multiply through by two and note:
$\displaystyle 2q_{i}(g)>\frac{n-1}{n+1}\max(f(1)-f(0),f(-1)-f(0))$
$\displaystyle>\frac{n-1}{n+1}[f(1)-f(0)]\quad\forall i$ $\displaystyle
2q_{i}(g)>\frac{n-1}{n+1}\max(f(1)-f(0),f(-1)-f(0))$
$\displaystyle>\frac{n-1}{n+1}[f(-1)-f(0)]\quad\forall i$
Now (14) and (15) immediately follow. ∎
###### Remark 4.6.
This essentially means that the steeper a function $f$ around zero and on the
interval $(-n+1,n-1)$, the greater the quantity $\alpha-\gamma_{0}$ is needed
to ensure the theorem proved later in this section. It is worth pointing out
that this bound may often not be tight (i.e., the inequalities may hold true
and production quantities may be positive even when the condition is not not
met).
###### Theorem 4.7.
Suppose that $f$ is a convex function that has a minimum at $0$. Further,
suppose $f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})$. Define the change in $f$ as
$\triangle^{-}f_{i}(k_{i})=f_{i}(k_{i}-1)-f_{i}(k_{i})=f(-1)-f(0)=\triangle^{-}f(0)$
and
$\triangle^{+}f_{i}(k_{i})=f_{i}(k_{i}+1)-f_{i}(k_{i})=f(1)-f(0)=\triangle^{+}f(0)$.
Suppose $n\geq 2$ firms compete in an oligopoly with market demand
$p=\alpha-\sum_{i\in N}q_{i}$ and marginal costs
$c_{i}(g)=\gamma_{0}+f_{i}(\eta_{i}(g))$. If the parameters $\alpha$ and
$\gamma_{0}$ and the function $f$ obey condition (13), then the equivalence
class of graphs $[g]_{\eta}$ such that $\eta_{i}(g)=k_{i}$ is an equivalence
class of stable collaboration graphs.
###### Proof.
Let $g$ be a graph $g$ such that $\eta_{i}(g)=k_{i}$ for all firms $i$.
Consider a firm $i$ who may consider dropping its link with node $j$. If node
$i$ drops its link with node $j$ leading to graph $g-ij$, then
$\eta_{i}(g-ij)=k_{i}-1$ and $\eta_{j}(g-ij)=k_{j}-1$, while
$\eta_{r}(g-ij)=k_{r}$ for $r\not\in\\{i,j\\}$. Using Lemma 4.3
$q_{i}=\frac{\alpha-\gamma_{0}-nf_{i}(\eta_{i}(g))+\sum_{j\neq i\in
N}f_{j}(\eta_{j}(g))}{n+1}$ (20)
Calculate:
$\displaystyle q_{i}(g-ij)$
$\displaystyle=q_{i}(g)-\triangle^{-}f_{i}(k_{i})\left(\frac{n}{n+1}\right)+\triangle^{-}f_{j}(k_{j})\left(\frac{1}{n+1}\right)$
$\displaystyle q_{j}(g-ij)$
$\displaystyle=q_{j}(g)-\triangle^{-}f_{j}(k_{j})\left(\frac{n}{n+1}\right)+\triangle^{-}f_{i}(k_{i})\left(\frac{1}{n+1}\right)$
$\displaystyle q_{r}(g-ij)$
$\displaystyle=q_{r}(g)+\triangle^{-}f_{i}(k_{i})\left(\frac{1}{n+1}\right)+\triangle^{-}f_{j}(k_{j})\left(\frac{1}{n+1}\right)$
It then follows that
$\displaystyle Q(g-ij)$
$\displaystyle=Q(g)-\left(\frac{1}{n+1}\right)(\triangle^{-}f_{i}(k_{i})+\triangle^{-}f_{j}(k_{j}))$
$\displaystyle P(g-ij)$
$\displaystyle=P(g)+\left(\frac{1}{n+1}\right)(\triangle^{-}f_{i}(k_{i})+\triangle^{-}f_{j}(k_{j}))$
$\displaystyle c_{i}(g-ij)$ $\displaystyle=c_{i}(g)+\triangle^{-}f_{i}(k_{i})$
Now, we can calculate $Y_{i}(g-ij)$ in terms of $Y_{i}(g)$:
$\displaystyle Y_{i}(g-ij)$ $\displaystyle=q_{i}(g-ij)[P(g-ij)-c_{i}(g-ij)]$
$\displaystyle=Y_{i}(g)+q_{i}(g)\left(\frac{2}{n+1}\right)[\triangle^{-}f_{j}(k_{j})-n\triangle^{-}f_{i}(k_{i})]+\left(\frac{[\triangle^{-}f_{j}(k_{j})-n\triangle^{-}f_{i}(k_{i})]}{n+1}\right)^{2}$
Since $f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})$ this implies that
$\triangle^{-}f_{i}(k_{i})=\triangle^{-}f_{j}(k_{j})$ leading to (21) and then
(22) and (23) through algebraic manipulation. Finally, by the assumptions of
the theorem and condition (13) each of the quantities
$\triangle^{-}f_{i}(k_{i})$, $\frac{n-1}{n+1}$, and
$2q_{i}(g)-\frac{n-1}{n+1}\triangle^{-}f_{i}(k_{i})$ are nonnegative implying
(24).
$\displaystyle Y_{i}(g-ij)-Y_{i}(g)$
$\displaystyle=2q_{i}(g)\triangle^{-}f_{i}(k_{i})\left(\frac{1-n}{n+1}\right)+(\triangle^{-}f_{i}(k_{i}))^{2}\left(\frac{1-n}{n+1}\right)^{2}$
(21)
$\displaystyle=\triangle^{-}f_{i}(k_{i})\left(\frac{1-n}{n+1}\right)\left(2q_{i}(g)+\frac{1-n}{n+1}\triangle^{-}f_{i}(k_{i})\right)$
(22)
$\displaystyle=-\triangle^{-}f_{i}(k_{i})\left(\frac{n-1}{n+1}\right)\left(2q_{i}(g)-\frac{n-1}{n+1}\triangle^{-}f_{i}(k_{i})\right)$
(23) $\displaystyle<0$ (24)
This implies that if firm $i$ attempts to drop link $ij$, then
$Y_{i}(g)>Y_{i}(g-ij)$ and thus firm $i$ decreases its profit. The same will
be true for firm $j$. Hence, no firm has an incentive to drop a link from
graph $g$. Now, we will consider the case where firm $i$ attempts to add a
link to the graph $g$, giving $g+ij$ under the assumption that the link $ij$
does not exist in graph $g$. This analysis will follow closely the analysis
for $g-ij$. First note that $\eta_{i}(g)=k_{i}$ for all firms $i$ and
$\eta_{i}(g+ij)=k_{i}+1$ and $\eta_{j}(g+ij)=k_{j}+1$, while
$\eta_{r}(g+ij)=k_{r}$ for $r\not\in\\{i,j\\}$. We define
$\triangle^{+}f_{i}(k_{i})$ as
$\triangle^{+}f_{i}(k_{i})=f_{i}(k+1)-f_{i}(k)$; note the subtle difference
from the definition of $\triangle^{-}f_{i}(k_{i})$. Again using Lemma 4.3, we
calculate the production quantity for each node in graph $g+ij$:
$\displaystyle q_{i}(g+ij)$
$\displaystyle=q_{i}(g)-\triangle^{+}f_{i}(k_{i})\left(\frac{n}{n+1}\right)+\triangle^{+}f_{j}(k_{j})\left(\frac{1}{n+1}\right)$
$\displaystyle q_{j}(g+ij)$
$\displaystyle=q_{j}(g)-\triangle^{+}f_{j}(k_{j})\left(\frac{n}{n+1}\right)+\triangle^{+}f_{i}(k_{i})\left(\frac{1}{n+1}\right)$
$\displaystyle q_{r}(g+ij)$
$\displaystyle=q_{r}(g)+\triangle^{+}f_{i}(k_{i})\left(\frac{1}{n+1}\right)+\triangle^{+}f_{j}(k_{j})\left(\frac{1}{n+1}\right)$
We can then calculate the corresponding total production quantity $Q$, the
market price $P$ and marginal costs for each player for the graph $g+ij$:
$\displaystyle Q(g+ij)$
$\displaystyle=Q(g)-\left(\frac{1}{n+1}\right)(\triangle^{+}f_{i}(k_{i})+\triangle^{+}f_{j}(k_{j}))$
$\displaystyle P(g+ij)$
$\displaystyle=P(g)+\left(\frac{1}{n+1}\right)(\triangle^{+}f_{i}(k_{i})+\triangle^{+}f_{j}(k_{j}))$
$\displaystyle c_{i}(g+ij)$ $\displaystyle=c_{i}(g)+\triangle^{+}f_{i}(k_{i})$
Now, we can calculate $Y_{i}(g+ij)$ in terms of $Y_{i}(g)$:
$\displaystyle Y_{i}(g+ij)$ $\displaystyle=q_{i}(g+ij)[P(g+ij)-c_{i}(g+ij)]$
$\displaystyle=Y_{i}(g)+q_{i}(g)\left(\frac{2}{n+1}\right)[\triangle^{+}f_{j}(k_{j})-n\triangle^{+}f_{i}(k_{i})]+\left(\frac{[\triangle^{+}f_{j}(k_{j})-n\triangle^{+}f_{i}(k_{i})]}{n+1}\right)^{2}$
Since $f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})$ this implies that
$\triangle^{+}f_{i}(k_{i})=\triangle^{+}f_{j}(k_{j})$ leading to (25) and then
(26) and (27) through algebraic manipulation. Finally, by the assumptions of
the theorem and condition (13), each of the quantities
$\triangle^{+}f_{i}(k_{i})$, $\frac{n-1}{n+1}$, and
$2q_{i}(g)-\frac{n-1}{n+1}\triangle^{+}f_{i}(k_{i})$ are positive implying
(28).
$\displaystyle Y_{i}(g+ij)-Y_{i}(g)$
$\displaystyle=2q_{i}(g)\triangle^{+}f_{i}(k_{i})\left(\frac{1-n}{n+1}\right)+(\triangle^{+}f_{i}(k_{i}))^{2}\left(\frac{1-n}{n+1}\right)^{2}$
(25)
$\displaystyle=\triangle^{+}f_{i}(k_{i})\left(\frac{1-n}{n+1}\right)\left(2q_{i}(g)+\frac{1-n}{n+1}\triangle^{+}f_{i}(k_{i})\right)$
(26)
$\displaystyle=-\triangle^{+}f_{i}(k_{i})\left(\frac{n-1}{n+1}\right)\left(2q_{i}(g)-\frac{n-1}{n+1}\triangle^{+}f_{i}(k_{i})\right)$
(27) $\displaystyle<0$ (28)
This implies that if firm $i$ attempts to add a link $ij$, then
$Y_{i}(g)>Y_{i}(g+ij)$ and the firm decreases its profit. The same will be
true for firm $j$. Hence, no firm has an incentive to add a link to graph $g$.
Since no firm has an incentive to add or drop a link to graph $g$, it is
stable. This completes the proof. ∎
###### Example 4.8.
We present a numerical example of Theorem 4.7. Let $N=5$ firms compete in an
oligopoly with inverse demand function $P=100-Q$, fixed cost $\gamma_{0}=5$,
and $f(\eta(g))=(\eta(g))^{2}+\psi$ where $\psi=2$. Each firm has the shifted
function
$f_{i}(\eta_{i}(g))=f(\eta_{i}(g)-k_{i})=(\eta_{i}(g)-k_{i})^{2}+\psi$ where
$\mathbf{k}=[2,3,4,3,2]^{T}$ and . We want to test the stability of a graph
$g$ with $\eta_{i}(g)=k_{i}$ and
$f_{i}(\eta_{i}(g))=(\eta_{i}(g)-k_{i})^{2}+\psi$ for each node $i$. In order
to apply theorem (4.7), we must ensure condition (13) is met. We calculate:
$\displaystyle f(-n+1)=f(-4)=18,\quad f(-1)=3,\quad f(0)=2,$ $\displaystyle
f(1)=3,\quad f(n-1)=f(4)=18,\quad$
Hence,
$\displaystyle\max(f(n-1),f(1-n))=18,\quad\max(f(1)-f(0),f(-1)-f(0))=1,\quad$
$\displaystyle n\max(f(n-1),f(1-n))=5\cdot
18=90,\quad(n-1)\max(f(1)-f(0),f(-1)-f(0))=4\cdot 1=4,\quad$
$\alpha-\gamma_{0}-n\max(f(n-1),f(1-n))-\frac{1}{2}(n-1)\max(f(1)-f(0),f(-1)-f(0))=\\\
100-5-90-\frac{1}{2}\cdot 4=3>0$
Two stable isomorphic graphs, shown in Figure 5, have a degree sequence
equivalent to $\mathbf{k}$.
Figure 5: Two stable and isomorphic graphs that are possible configurations
for collaboration in the example oligopoly.
However, for the given parameters, there are 33 stable graphs of which only
the two shown have degree sequence equal to $\mathbf{k}$. These graphs were
computed using Maple and are shown in Figure 6.
Figure 6: The 29 stable graphs that arise from the parameters given.
It is interesting to investigate the other stable networks with degree
sequences different from $\mathbf{k}$. The graph in Row 7, Column 2 of Figure
6 shows a stable configuration where both nodes 3 and 4 would prefer one more
link, but they are already linked together and no other node requires an
additional link. Hence, the network is stable because each node is either
satisfied or no pair of nodes can bilaterally improve themselves through the
addition or removal of a link. It is interesting to note that in the case of
the graph shown, if both nodes 1 and 5, were to give up their link with one
another and instead link to nodes 4 and 3 respectively, then each node would
have minimized its marginal costs. While it is the case that nodes 1 and 5
would not benefit from such a trade that would help nodes 3 and 4. These nodes
would indirectly hurt themselves via the decreased market price as a result of
the additional quantity produced by nodes 3 and 4. Nonetheless this analysis
brings out the fact that the manner in which nodes link to one another and the
manner in which stability is analyzed, greatly affects which networks are
deemed to be stable.
###### Example 4.9.
For small numbers of firms $<7$ generating the set of stable graphs takes
seconds (even in an interpreted language like Maple). Consider the case of a
firm (in this case Firm 3) who determines that collaboration is desired, but
recognizes there may be a sink cost (not included in the model) associated
with initializing such a collaboration. For example, the time taken to
establish industry connections will cost in terms of human labor. Assuming
Firm 3 is a selfish profit maximizer and assumes that all other firms are
selfish profit maximizers who will play to a stable configuration, then the
analysis of the potentially stable graphs will inform Firm 3 on the potential
payoffs it might receive. If the network is already in a stable configuration,
then since there are three stable configurations in which Firm 3 is not
connected to any other players, then it may not be worthwhile to even explore
collaboration. On the other hand, if there is no collaboration (an unstable
condition), then Firm 3 may hope to steer the network evolution and will
evaluate its various payoffs in each possible stable configuration. The
possible payoffs are shown in Figure 7:
Figure 7: The possible payoffs Player 3 can obtain in the various stable
configurations. From Figure 6, graphs are numbered from left-to-right, top-to-
bottom.
Note that there is a substantial variation in the payoff the Player 3 may
receive. In the empty graph, Player 3 receives a payoff of $169/4$. Notice
that some collaborative scenarios disadvantage Player 3 ($3$ of the $33$)
while most improve Player 3’s payoff.
###### Remark 4.10.
Goyal and Joshi show that in a market with homogenous products and under
quantity competition, the uniquely stable network is the complete network for
specific cost functions under some parameter restrictions [GJ03]. Theorem 4.11
extends the result in Goyal and Joshi [GJ03] by using nonlinear costs and
decreasing the restriction on parameters necessary for the model to be
feasible.
###### Theorem 4.11.
Suppose the marginal cost for firm $i$ on graph $g$ is
$c_{i}(g)=\gamma_{0}+f(\eta_{i}(g))$. Define $\triangle f(k)=f(k+1)-f(k)$.
Further suppose each firm reduces its marginal cost for each additional
collaboration (Condition (1)), the collaboration cost reduction function $f$
is convex (Condition (2)), collaboration cost reduction function $f$ is
positive (Condition (3)), the production quantity is positive (Condition (4)),
and the collaboration cost reduction function $f$ is not too steep (Condition
(5)):
$x_{1}>x_{2}$ implies $f(x_{1})<f(x_{2})$;
$f$ is convex;
$f$ is positive;
$\alpha-\gamma_{0}>nf(0)$; and
$\triangle f(k_{1})-n\triangle f(k_{2})>0$ for all
$k_{1},k_{2}\in\\{0,1,\ldots n-1\\}$ Then the complete network $g^{c}$ is a
stable graph.
###### Proof.
The production quantity and profit can be calculated as before in (10) and
(9), respectively, by using that fact that $f_{i}=f$ for all $i$.
$\displaystyle q_{i}(g)$
$\displaystyle=\frac{\alpha-\gamma_{0}-nf(\eta_{i}(g))+\sum_{j\neq i\in
N}f(\eta_{j}(g))}{n+1}$ $\displaystyle Y_{i}(g)$
$\displaystyle=q_{i}(g)\left(P(Q)-c_{i}(g)\right)=q_{i}(g)\left(\alpha-\left(\sum_{j}q_{j}(g)\right)-\gamma_{0}-f(\eta_{i}(g))\right)$
Observe through algebraic manipulation that $P(Q)-c_{i}(g)=q_{i}(g)$, which
implies that
$Y_{i}(g)=q_{i}(g)\left(P(Q)-c_{i}(g)\right)=\left(q_{i}(g)\right)^{2}$.
For a complete network $g^{c}$, $\eta_{j}(g)=n-1$ for all $j$. Given $g^{c}$:
$q_{i}(g^{c})=\frac{\alpha-\gamma_{0}-f(n-1)}{n+1}$
Now for a complete network missing a link $(i,j)$, denoted as $g=g^{c}-ij$,
for nodes $i$ and $j$, $\eta_{i}(g)=\eta_{j}(g)=n-2$ but all other nodes $k$
have $\eta_{k}(g)=n-1$. Thus:
$q_{i}(g^{c}-ij)=\frac{\alpha-\gamma_{0}+(n-2)f(n-1)-(n-1)f(n-2)}{n+1}$
It follows that:
$q_{i}(g^{c})-q_{i}(g^{c}-ij)=\frac{(n-1)[f(n-2)-f(n-1)]}{n+1}$
Since $n-1>n-2$, Condition (1) implies $f(n-1)<f(n-2)$ which implies
$f(n-2)-f(n-1)>0$. Hence, $q_{i}(g^{c})-q_{i}(g^{c}-ij)>0$. By Condition (4),
$\alpha-\gamma_{0}>nf(0)$, which implies $q_{i}>0$ by Corollary 4.5 using
$f_{i}=f$. Further, since $q_{i}(g^{c})+q_{i}(g^{c}-ij)>0$, which combined
with $q_{i}(g^{c})-q_{i}(g^{c}-ij)>0$, implies
$Y_{i}(g^{c})-Y_{i}(g^{c}-ij)>0$. This implies that any firm $i$ will decrease
its profit by dropping link $ij$, hence the graph $g^{c}$ is stable. Now to
show that it is the only stable graph, suppose there is another graph $g\neq
g^{c}$ that is also stable. This implies that there exists a pair of firms
$(i,j)$ such that $ij\not\in g$. Let us consider the graph $g+ij$ relative to
the graph $g$.
$q_{i}(g+ij)-q_{i}(g)=\frac{-n}{n+1}\triangle
f(\eta_{i}(g))+\frac{1}{n+1}\triangle f(\eta_{j}(g))$
and thus that:
$Y_{i}(g+ij)-Y_{i}(g)=q_{i}(g+ij)^{2}-q_{i}(g)^{2}=[q_{i}(g+ij)-q_{i}(g)][q_{i}(g+ij)+q_{i}(g)]\\\
=q_{i}(g)\left(\frac{2}{n+1}\right)\left[\triangle f(\eta_{j}(g))-n\triangle
f(\eta_{i}(g))\right]+\left(\frac{1}{n+1}\right)^{2}\left[\triangle
f(\eta_{j}(g))-n\triangle f(\eta_{i}(g))\right]^{2}$
By Condition (5), $\triangle f(\eta_{j}(g))-n\triangle f(\eta_{i}(g))>0$ and
hence $Y_{i}(g+ij)-Y_{i}(g)\geq 0$. Similarly, $Y_{j}(g+ij)-Y_{j}(g)\geq 0$,
which implies that both nodes $i$ and $j$ may increase their profit by linking
together and so graph $g$ is not stable. This is a contradiction and so
$g^{c}$ is the only stable graph. This completes the proof. ∎
###### Example 4.12.
We present a numerical example of Theorem 4.11. Let $N=5$ firms compete in an
oligopoly with demand function $P=30-Q$, fixed cost $\gamma_{0}=5$, and
$f(\eta_{i}(g))=\frac{1}{\eta_{i}(g)+3}$. We want to test the stability of the
complete graph $g^{c}$. Conditions (1)-(4) are easy to check:
$\tfrac{df}{d\eta}=\frac{-1}{(\eta_{i}(g)+3)^{2}}<0$, which implies $f$ is
decreasing;
$\tfrac{d^{2}f}{d\eta^{2}}=\frac{2}{(\eta_{i}(g)+3)^{3}}>0$, which implies
that $f$ is convex;
$f(\eta_{i}(g))=\frac{1}{\eta_{i}(g)+3}>0$, which implies $f$ is positive;
$\alpha-\gamma_{0}=30-5=25>nf(0)=5\left(\frac{1}{3}\right)$ Condition (5) can
be verified with the below table:
$\eta_{i}(g)$ | $\eta_{j}(g)$ | $\triangle f(\eta_{i}(g))$ | $\triangle f(\eta_{j}(g))$ | $n\triangle f(\eta_{i}(g))-\triangle f(\eta_{j}(g))$ | $n\triangle f(\eta_{j}(g))-\triangle f(\eta_{i}(g))$
---|---|---|---|---|---
0 | 4 | -0.08 | -0.02 | 0.0059 | 0.4
4 | 0 | -0.02 | -0.08 | 0.3988 | 0.01
All conditions of Theorem 4.11 are met. The stability of the complete graph
can be verified exhaustively and was done using Matlab.
###### Remark 4.13.
The functions described in Theorem 4.11 generalizes the result of Goyal and
Joshi [GJ06], since it is clear that a linear function of the vertex degrees
(as assumed in [GJ06]) will satisfy the given criteria. Furthermore, Theorem
4.7 is a further generalization since we define conditions under which a
collaboration graph with an arbitrary degree distribution will be stable. This
is not possible with Goyal and Joshi’s model.
## 5 Conclusions
In this paper we showed that networks with specific structure properties may
form as a result of game theoretic interactions. Specifically, we showed a
simple way of constructing a game whose pairwise stable solutions are those
graphs with a given degree distribution. As a particular application of
network formation via game theoretic principles, we investigated the formation
of collaboration networks in oligopolistic competition. We extended the model
of Goyal and Joshi [GJ03] with a nonlinear cost function that, under
particular conditions, admits stable collaboration graphs with an arbitrary
degree sequence as well. One limitation of this approach is that we cannot
specify an exact graph structure. The degree distribution specification
generates a class of stable graphs rather than a single graph. We do not view
this as a problem since it can help explain variation in observed situations
with the same parameters.
### 5.1 Potential Relation to Network Science
Special structure in networks has been considered in several recent papers
[BA99, New03, DM02, AB02], that have cut across various subjects including
social networks, information networks, and biological networks as well as
physical networks such as power grids and road networks. A review of networks
in these disciplines may be found in various articles (see [New03, DM02, AB02]
and the references therein).
The network science literature was largely inspired by the observation of
macroscopic structural properties (e.g., small world, power law degree
distribution) of networks that occurred in several distinct network types
(e.g., social, information, biological networks), which have diverse
microscopic properties. The network science literature has largely been
devoted to finding the mechanisms by which networks form and/or evolve in
order to generate the structural properties that are observed. The momentum in
this direction has largely been driven by the statistical physics community
[BA99, New03, DM02], who argue that the phenomena of complex networks (e.g.,
power laws) may be explained by laws that reach across all complex networks
because they are phenomena that are inherent to the complexity of the
networks.
Alternatively, there has been recent interest in the structural properties of
networks that have been designed via optimization [Ald08, Nag08, ADGW03]. This
perspective is motivated largely by the fact that many networks (in the
abstract sense) are models for physical networks that are designed by humans
to function with particular objectives (or even designed by nature to serve an
evolutionary purpose). These networks are distinctly different from social
networks, which are abstract models that describe interactions between actors
(e.g., people talking, writing scientific papers or dating). These networks
(e.g., power grids, communication networks) are not designed through central
coordination, but arise as a result of the objectives of multiple independent
actors. As a result there are structural properties that often exist in these
networks that are not explained by models that do not account for these
functioning characteristics [DAL+05].
The work presented in this paper is motivated by this new work. Our
perspective is that some networks are formed as a result of the interacting
strategies of multiple players, rather than a universal rule for network
formation. This is consistent with certain recent observations on (e.g.) power
law networks [SP12]. This occurs in collaboration networks. The formation of
the collaboration network results from the strategic decisions of the players.
The network will still have particular structural properties that are
explained via game theoretic principles. In this paper, we illustrated that an
important property to the Network Science community, namely the degree
distribution, can emerge as a result of designed game mechanism. As an
explanatory model, we suggest that certain networks with interesting degree
distributions form as a result of strategic player interactions in which each
node degree in embedded (or hidden) in the objective function of the node
player. The formulation in this paper is simply an explicit representation. In
[LGF11] we illustrate a class of games that generate specific degree
distributions in which the degree is not an explicit part of the objective
functions of the players.
## 6 Future Directions
There are several directions this research could take. In [LGF11] we begin the
investigation of graph formation games with specific link bias and identify a
game theoretic mechanism that does not explicitly encode the degree
distribution of interest into the players’ objective functions and yet is
still capable of having graphs with arbitrary degree distributions as stable
solutions. This work is extended in [LFG12]. We also investigate the graph
formation game in the presence of spatial oligopolies in [LFG11]. Clearly
investigating the mechanism design problem for additional graph properties,
such as the clustering coefficient is of interest. Recent work on generation
of graphs with specific clustering coefficients [GRKF11] may provide insight
into game theoretic mechanisms for solving the same problem.
Another equally interesting future direction lies in the extraction of game
theoretic mechanisms from real-world data. Solving such a problem may take two
forms. In the first form, small real world human networks are evaluated using
traditional psycho-social interviewing techniques in an attempt to identify
objective functions or constraints consistent with the model presented in this
paper and theoretical extensions (e.g., [LFG12]). This work is planned in
collaboration with social psychologists at the authors’ parent institution. A
second form of objective function inference would be completely observational
in which, given a network that appeared to be stable, one would attempt to
infer the set of potential objective functions (and player constraints
[LGF11]) that would generate the given stable graph. Observation of network
evolution would also be useful. In this case, statistical techniques would
have to be applied to the determination of the objective function structure.
A final direction of investigation lies in the evaluation of dynamic network
formation. That is, investigating this problem in a dynamic context in which
the objective functions of players may change. We propose that this problem
can be analyzed in discrete time using techniques from competitive Markov
decision theory [JAF97]. However, the combinatorial properties of the graph
structures (for all but the simplest networks) make a brute force analysis
approach intractable. Thus a more sophisticated analysis technique may be
necessary to obtain any useful results.
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|
arxiv-papers
| 2011-06-13T13:33:37 |
2024-09-04T02:49:19.576880
|
{
"license": "Public Domain",
"authors": "Shaun Lichter, Christopher Griffin, and Terry Friesz",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1106.2440"
}
|
1106.2469
|
# Fermi and Swift Gamma-Ray Burst Afterglow Population Studies
J. L. Racusin11affiliation: NASA, Goddard Space Flight Center, Code 661,
Greenbelt, MD, USA , S. R. Oates22affiliation: Mullard Space Sciences
Laboratory, University College London, Surrey, UK , P. Schady33affiliation:
Max-Planck-Institut fur extraterrestrische Physik, Garching, Germany , D. N.
Burrows44affiliation: The Pennsylvania State University, University Park, PA,
USA , M. de Pasquale22affiliation: Mullard Space Sciences Laboratory,
University College London, Surrey, UK , D. Donato11affiliation: NASA, Goddard
Space Flight Center, Code 661, Greenbelt, MD, USA 55affiliation: Department of
Physics and Department of Astronomy, University of Maryland, College Park, MD
20742, USA 66affiliation: Center for Research and Exploration in Space Science
and Technology and NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
, N. Gehrels11affiliation: NASA, Goddard Space Flight Center, Code 661,
Greenbelt, MD, USA , S. Koch44affiliation: The Pennsylvania State University,
University Park, PA, USA , J. McEnery11affiliation: NASA, Goddard Space Flight
Center, Code 661, Greenbelt, MD, USA , T. Piran77affiliation: The Racah
Institute of Physics, Hebrew University, Jerusalem 91904, Israel , P.
Roming44affiliation: The Pennsylvania State University, University Park, PA,
USA 88affiliation: Southwest Research Institute, Department of Space
Science,6220 Culebra Rd, San Antonio, TX 78238, USA , T.
Sakamoto11affiliation: NASA, Goddard Space Flight Center, Code 661, Greenbelt,
MD, USA , C. Swenson44affiliation: The Pennsylvania State University,
University Park, PA, USA , E. Troja11affiliation: NASA, Goddard Space Flight
Center, Code 661, Greenbelt, MD, USA 99affiliation: Oak Ridge Associate
Universities , V. Vasileiou11affiliation: NASA, Goddard Space Flight Center,
Code 661, Greenbelt, MD, USA 1010affiliation: University of Maryland,
Baltimore County, MD, USA 1111affiliation: Laboratoire de Physique Th eorique
et Astroparticules, Universit e Montpellier 2, CNRS/IN2P3, Montpellier, France
, F. Virgili 1212affiliation: University of Las Vegas, Las Vegas, NV, USA , D.
Wanderman77affiliation: The Racah Institute of Physics, Hebrew University,
Jerusalem 91904, Israel , B. Zhang1212affiliation: University of Las Vegas,
Las Vegas, NV, USA
###### Abstract
The new and extreme population of GRBs detected by Fermi-LAT shows several new
features in high energy gamma-rays that are providing interesting and
unexpected clues into GRB prompt and afterglow emission mechanisms. Over the
last 6 years, it has been Swift that has provided the robust dataset of
UV/optical and X-ray afterglow observations that opened many windows into
components of GRB emission structure. The relationship between the LAT
detected GRBs and the well studied, fainter, less energetic GRBs detected by
Swift-BAT is only beginning to be explored by multi-wavelength studies. We
explore the large sample of GRBs detected by BAT only, BAT and Fermi-GBM, and
GBM and LAT, focusing on these samples separately in order to search for
statistically significant differences between the populations, using only
those GRBs with measured redshifts in order to physically characterize these
objects. We disentangle which differences are instrumental selection effects
versus intrinsic properties, in order to better understand the nature of the
special characteristics of the LAT bursts.
###### Subject headings:
gamma rays: bursts; gamma rays: observations; X-rays: bursts; ultraviolet:
general
††slugcomment: Accepted for publication in ApJ
## 1\. Introduction
The field of gamma-ray bursts (GRBs) is undergoing dramatic changes for a
second time within the past decade, as a new observational window has opened
with the launch and success of NASA’s Fermi gamma-ray space telescope. While
both NASA’s Swift gamma-ray burst explorer mission (Gehrels et al. 2004) and
Fermi are operating simultaneously, we have the ability to potentially detect
hundreds of gamma-ray bursts per year ($\sim 1/3$ of which are triggered by
Swift). This allows prompt observations in the $15-150$ keV hard X-ray band
with the Burst Alert Telescope (BAT, Barthelmy et al. 2005) and rapid follow-
up in the $0.3-10$ keV soft X-ray band with the X-Ray Telescope (XRT, Burrows
et al. 2005) and the UV/optical band by the Ultraviolet Optical Telescope
(UVOT, Roming et al. 2005) on-board Swift. There is $\sim 40\%$ overlap
between BAT triggers and triggers from Fermi’s Gamma-ray Burst Monitor (GBM,
Meegan et al. 2009) allowing for coverage from 10 keV to 30 MeV, and a special
subset detected up to 10s of GeV with Fermi’s Large Area Telescope (LAT,
Atwood et al. 2009). This wide space-based spectral window is broadened
further by ground based optical, NIR, and radio follow-up observations.
In the last 2 years, the addition of the 30 MeV to 100 GeV window from Fermi-
LAT has led to another theoretical crisis, as we attempt to understand the
origin and relationship between these new observational components and the
ones traditionally observed from GRBs in the keV-MeV band. Just as Swift
challenged our theoretical models by demonstrating that GRBs have complex
behavior in the first few hours after the trigger (Nousek et al. 2006), Fermi-
LAT is regularly observing a new set of high energy components in a small very
energetic subset of bursts (Abdo et al. 2009a, b, c, 2010; Ackermann et al.
2010). The relationship between the $>100$ MeV emission and the well studied
keV-MeV components remains unclear (Corsi et al. 2010a, b; Kumar & Barniol
Duran 2010; Razzaque et al. 2010; Zhang & Pe’er 2009; Ghisellini et al. 2010;
Pe’er et al. 2010; Piran & Nakar 2010; Toma et al. 2010; Wang et al. 2010).
The complicated Fermi-LAT prompt emission spectra do not show simply the
extension of the lower-energy Band function (Band et al. 1993), but rather the
joint GBM-LAT spectral fits can also show the presence of an additional hard
power-law that can be detected both above and below the Band function (Abdo et
al. 2009a; Ackermann et al. 2010) in some cases. There were earlier
indications of this additional spectral component in the EGRET detected GRB
941017 (González et al. 2003). However, the rarity of EGRET GRB detections
left it unclear whether this was a common high energy feature, or if special
circumstances in that GRB were responsible. This component is too shallow to
be due to Synchrotron self-Compton (SSC) as had been predicted extensively
pre-Fermi (Zhang & Mészáros 2001; Guetta & Granot 2003; Galli & Guetta 2008;
Racusin et al. 2008; Band et al. 2009). The spectral behavior of the LAT
bursts appears to rule out the theory that the soft $\gamma$-rays are caused
by a SSC or another Inverse Compton (IC) component (Ando et al. 2008; Piran et
al. 2009).
Fermi-LAT’s $>100$ MeV temporal behavior is different from the lower-energy
counterparts observed from thousands of GRBs. The LAT emission often begins a
few seconds later than the lower-energy prompt emission, and sometimes lasts
substantially longer (up to thousands of seconds; Abdo et al. 2009a, c;
Ackermann et al. 2010). This so-called “GeV extended emission” and the extra
spectral power-law component may be the same component, but the statistics in
the extended emission are limited and detailed spectral fits are often not
possible.
Several groups (Ghisellini et al. 2010; Kumar & Barniol Duran 2010; de
Pasquale et al. 2010) suggest that the high-energy extended emission is caused
by the same forward shock mechanism (with special caveats for environmental
density and magnetic field strength) responsible for the well studied
broadband afterglows that have been observed for hundreds of other GRBs.
Alternatively, Zhang et al. (2011); Maxham et al. (2011) suggest that the
$>100$ MeV emission during the prompt emission phase is of internal origin,
and the later extended emission is of external origin. We can learn more about
the mysterious new LAT components by studying the GRBs from a broadband
perspective, for which the early broadband afterglow (specifically X-ray and
optical) behavior is well studied. However, currently there is only one case
of simultaneous X-ray/optical/GeV emission in the minutes after the GRB - the
short hard GRB 090510 which was simultaneously triggered upon the Swift-BAT
and the Fermi-GBM and LAT (de Pasquale et al. 2010).
Despite the lack of simultaneous LAT and lower energy observations in long
bursts, we can still learn about the special nature of the LAT bursts by
studying their lower energy late-time afterglow observations. In this paper,
we utilize the large database of Swift afterglow observations of BAT
discovered bursts, and compare them to the simultaneous BAT/GBM triggers, and
the Swift follow-up of the LAT/GBM detected bursts, in order to learn about
the properties of the different populations of GRBs.
This paper is organized as follows: in Section 2 we discuss the sample
selection and data analysis, in Section 3 we discuss the results and
correlations apparent in the different samples, in Section 4 we discuss the
physical implications of our analysis, and in Section 5 we conclude.
## 2\. Sample Selection and Analysis
In order to study the population differences between the BAT-triggered sample,
the GBM-triggered sample, and the LAT detected sample, we use all GRBs from
these samples with measured redshifts and well constrained XRT and UVOT light
curves (at least 4 light curve bins and enough counts to construct and fit an
X-ray spectrum).
As of December 2009, the Swift XRT and UVOT instruments have observed
afterglows of 439 bursts discovered by BAT, as well as 81 bursts discovered by
other missions. We now have enough detailed observations of X-ray and
UV/optical afterglows from Swift to study them as a statistical sample, and to
attempt to separate observational biases from physical differences in GRB
populations. In this study, we included afterglow observations of all GRBs
discovered by Swift-BAT between December 2004 and December 2009 (sample
hereafter referred to as BAT), with measured redshifts in the literature (e.g.
Fynbo et al. 2009) and well-constrained light curves and spectra.
Unfortunately, the positional errors provided by GBM are too large (several
degrees) to facilitate Swift follow-up. Therefore, the only afterglow
observations we have of GBM triggered bursts, are those that simultaneously
triggered the Swift-BAT and meet all of the same criteria as the BAT sample
(sample hereafter referred to as GBM). We treat the GBM bursts separately from
the BAT sample, and do not include them in both samples. The GBM bursts have a
much wider measured spectral range during the prompt emission than the BAT
bursts, and are therefore more likely to provide an accurate measurement of
$E_{peak}$ and the Band function parameters. From an instrumental perspective,
BAT has a much better sensitivity than GBM. Therefore, the GBM bursts are
biased towards higher fluence.
We also include the LAT detected GBM triggered GRBs that have been localized
by XRT and UVOT in follow-up observations (sample hereafter referred to as
LAT). The LAT position errors for those GRBs for which follow-up was initiated
were $\sim 3-10$ arcminutes radius. Those GRBs with initial position errors
that were significantly larger than the XRT field of view (FoV) have not been
followed-up by Swift. The LAT sample includes the one BAT/GBM/LAT simultaneous
trigger (GRB 090510), and the remaining LAT detected GRBs that BAT did not
observe (i.e. were outside the BAT FoV at trigger). There has been follow-up
of 10 out of 24 LAT detected GRBs (as of March 2011) with position errors
small enough to initiate Target of Opportunity (ToO) observations. However,
(with the exception of the joint BAT/GBM/LAT trigger) these follow-up
observations began at a minimum of 12 hours after the trigger, and in some
cases, did not begin until 1-2 days post-trigger. Despite this impediment, 8
of 10 LAT GRBs were detected by XRT, and 7 of 10 by UVOT (5 in u-band, see
Section 2.2).
The breakdown of GRBs in each of the XRT and UVOT datasets for the BAT, GBM,
and LAT samples are given in Table 1. Individual bursts are included in only
one of the BAT, GBM, and LAT samples, depending on their detection by one,
two, or all three instruments. In the following Sections, we describe the data
analysis of the follow-up X-ray and UV/optical observations, as well as the
methods and sources for obtaining $\gamma$-ray prompt emission spectral fits
and fluences.
Table 1Sample Statistics
Sample | XRT | UVOT
---|---|---
BAT | 147 | 49
GBM | 19 | 11
LAT | 8 | 5
Note. — The number of GRBs in each sample that meet our selection criteria.
For a GRB to be included in the UVOT sample, observations in the u-band are
needed for normalization.
### 2.1. X-ray
The X-ray light curves and spectral fits were obtained from the XRT team
repository (Evans et al. 2007, 2009). We fit and characterized all of the
light curves using the methods of Racusin et al. (2009). Each count rate light
curve was fit with the best fitting model of either a power law, broken power
law, double broken power law, or triple broken power law, after time periods
of significant flaring were manually removed.
Figure 1.— The X-ray (0.3-10 keV) afterglow rest-frame luminosity light curves
for all of the long (top panel) and short (bottom panel) GRBs in our samples.
The BAT, GBM, and LAT GRBs are indicated by the different colors. Note the
clustering of the LAT light curves compared to the BAT and GBM samples in the
top panel.
We convert the count-rate light curves to flux light curves based on a single
counts-to-flux conversion factor obtained from the Photon Counting (PC) mode
spectral fit. In order to physically characterize the afterglows, taking
advantage of the redshift information, we convert flux to luminosity and apply
a k-correction using the following formalism from Berger et al. (2003a):
$L_{x}(t)=4\pi D_{L}^{2}F_{x}(t)(1+z)^{\alpha_{x}-\beta_{x}-1}$ (1)
where $L_{x}(t)$ is the $0.3-10~{}\textrm{keV}$ luminosity at time $t$ seconds
after the trigger; $F_{x}$ is the $0.3-10~{}\textrm{keV}$ flux at time $t$;
$D_{L}$ is the luminosity distance assuming cosmological parameters:
$H_{0}=71~{}\textrm{km}~{}\textrm{s}^{-1}~{}\textrm{Mpc}^{-1}$,
$\Omega_{m}=0.27$, and $\Omega_{\Lambda}=0.73$; $\alpha_{x}$ is the X-ray
power-law temporal decay at time $t$; and $\beta_{x}$ is the spectral energy
index at time $t$. The X-ray luminosity light curves in rest frame time for
both the long and short bursts, for all three of our samples, are shown in
Figure 1.
The X-ray spectra were also taken from the XRT team repository. Evans et al.
(2009) describes how the spectra are extracted and fit to absorbed power-laws,
with two absorption components (Galactic and intrinsic at the GRB redshift).
The spectral power-law index ($\Gamma$) is converted to the energy index via
$\beta=\Gamma-1$, where $F_{x}=t^{-\alpha_{x}}\nu^{-\beta_{x}}$.
Figure 2.— The u-band normalized afterglow rest-frame luminosity light curves
for all of the long (top panel) and short (bottom panel) GRBs in our samples.
The BAT, GBM, and LAT GRBs are indicated by the different colors.
### 2.2. UV/Optical
The UV/Optical light curves were obtained from the second UVOT GRB catalog
(Roming et al., 2011, in-preparation) and combined such that the 5 arcsec
extraction region light curves were used at count rates
$>0.5~{}\textrm{counts~{}s}^{-1}$, and 3 arcsec extraction region at lower
count rates. We combined the UVOT 7 filter light curves (where available and
detected) using the methods of Oates et al. (2009) in order to obtain the
highest signal-to-noise light curves. This involves first normalizing the
individual filter light curves for each GRB to a single band, then combining
and rebinning. We chose to normalize all of the UVOT light curves in this
study to the UVOT u-band in order to optimize the number of LAT GRB afterglow
light curves that could be used in this comparison, because that was the most
commonly used filter for the LAT burst follow-up. Note that although 7 of the
Swift followed-up LAT bursts were detected by UVOT (compared to 8 by XRT), we
could only obtain detailed u-band light curves for five. The others (GRB
090323 and GRB 100414A) were only detected in the white filter, which cannot
be normalized to the u-band without knowing the shape of the optical spectrum,
and also cannot be easily used to extract luminosity information because of
the flat wide transmission curve of this filter. Note that GRB 080916C was
detected by XRT, but not UVOT, which is consistent with the redshift (z=4.35,
Greiner et al. 2009).
We convert the UVOT count rate light curves from observed count rate to flux
via the average GRB u-band conversion factor provided by Poole et al. (2008).
We also correct for Galactic extinction, and correct for host galaxy
extinction by fitting broadband (XRT and UVOT) Spectral Energy Distributions
(SEDs) choosing the best fit dust model (MW, LMC, SMC) for each burst using
the methods of Schady et al. (2010). The conversion from flux to luminosity is
similar to that of the X-ray luminosities given in Equation 1. Using the
optical spectral index from the SEDs, we calculate the u-band k-corrected
luminosity as:
$L_{o}(t)=4\pi D_{L}^{2}F_{o}(t)(1+z)^{\alpha_{o}-\beta_{o}-1}.$ (2)
where $F_{o}=t^{-\alpha_{o}}\nu^{-\beta_{o}}$, and $F_{o}$ is the u-band flux
at time $t$.
We fit the count rate light curves in a similar manner to that of the X-ray
light curves, except that we allow an additional constant contribution to the
power-law fits to account for the flattening occasionally observed in UVOT
light curves. This flattening can be due to either host galaxy contribution or
nearby source contamination. By simply allowing the fit to include this extra
constant, we can subtract it off and extrapolate the power-law fit to the time
of interest.
The u-band rest frame luminosity light curves for the short and long bursts
for each of our BAT, GBM, and LAT samples are shown in Figure 2. The light
curve shapes have not been altered to adjust for the extra constant.
### 2.3. $\gamma$-ray
Due to the different $\gamma$-ray instruments used to detect the GRBs in our
samples, observational biases cause much of the differences between these
samples. Likely, the only differences between the GBM and BAT samples are
related to the larger and harder energy range of the GBM, and the superior
sensitivity of the BAT. However, Swift has had 6.5 years to collect a sample
of GRBs with a wide range of spectral properties and brightness. This is
reflected in the ranges of the X-ray and UV/Optical afterglow light curves.
The LAT sample on the other hand, although a subset of the GBM sample, has
inherent differences. $50\%$ of the GBM bursts occur within the LAT field-of-
view, but only a small fraction $<5\%$ are detected. For the LAT to detect a
GRB, the prompt emission must have a very high fluence. The factors that may
contribute to the high fluence in the $30~{}\textrm{MeV}-100~{}\textrm{GeV}$
bandpass include the spectrum peaking at relatively high energies, the
spectrum having a very shallow $\beta_{Band}$ index, and the presence of an
additional hard power law component (as has been detected in several LAT
bursts).
In Section 3.5, we discuss burst energetics based upon observations and limits
from afterglow light curves. The prompt emission spectral fits used in these
calculations were obtained from several sources. The BAT spectral fits and
fluences come from the second BAT GRB catalog (Sakamoto et al. 2011). The GBM
spectral fits come from either individual burst papers in the literature, or
GCN circulars. Fluences were recalculated from these fits in several different
bandpasses for the calculation of $E_{\gamma,iso}$ and prompt emission fluence
ratios.
## 3\. Results
Using our compiled luminosity rest frame light curves and SEDs, we explored
various parameters for differences and similarities between the BAT, GBM, and
LAT burst populations. The goal was to determine whether the LAT detected GRBs
are fundamentally different from the normal BAT and GBM samples, or whether
they are simply the extreme cases.
Unfortunately, except in the case of the short GRB 090510, we do not have any
early afterglow observations of the LAT bursts, therefore we limit
measurements to times for which all data sets are available. One day after the
trigger in the rest frame is within the BAT, GBM, and LAT observations, though
we also compare some properties at 11 hours (a standard observed frame time
used in other papers including de Pasquale et al. 2006 and Gehrels et al.
2008). For the cases of Swift bursts, when no data are available at this time,
we extrapolate from the earlier power-law decay index.
### 3.1. Temporal and Spectral Indices
Using the light curve fits described in Sections 2.1 and 2.2, we collect the
power-law decay indices at a rest frame time of 1 day. Typically this is the
decay index in the normal forward shock phase (post-plateau, pre-jet break) in
the X-ray light curves (Zhang et al. 2006; Nousek et al. 2006; Racusin et al.
2009). In the cases where the light curves end prior to 1 day (rest frame), we
take the final decay index. The UV/optical light curve behavior observed by
UVOT has a different early morphology, often with an initial rise followed by
a shallow decay and occasionally later steepening (Oates et al. 2009). Oates
et al. (2009) observed that the distribution of $\alpha_{o}$ after 500 s
(post-trigger) is similar to $\alpha_{x}$ during the plateau phase, suggesting
that the optical afterglows are also affected by energy injection at early
times, or their temporal profile deviates due to a break (e.g. cooling break)
between the two bands.
Similarly, we also extract the X-ray and optical spectral indices ($\beta_{x}$
and $\beta_{o}$), and plot them against the temporal indices in Figure 3.
There is significant scatter in both the temporal and spectral properties, and
no significant correlation.
We note that $\alpha_{o}$ is systematically lower than $\alpha_{x}$ at 1 day
in the rest frame, though many of the UVOT light curves were extrapolated from
earlier observations to their expected later behavior assuming no breaks.
Figure 3.— The temporal and spectral decay indices at 1 day in the rest frame
interpolated from the X-ray (top panel) and optical (bottom panel) afterglow
fits. The scatter plots and histograms show that there are no noticeable
differences between the BAT, GBM, and LAT populations.
There is little correlation in either the X-ray or optical temporal and
spectral indices, as expected (given the variety of possible closure
relations, Racusin et al. 2009). From these distributions, one can see that
there are no noticeable differences between the BAT, GBM, and LAT populations.
Kolmogorov-Smirnov (K-S) tests confirm that there are no statistically
significant differences between the populations in these measurements (Table
2).
### 3.2. Redshift
Redshift is another physical quantity that we can evaluate for each of our GRB
populations, with separation into short and long populations. Jakobsson et al.
(2006) (later updated by Fynbo et al. 2009) showed that Swift GRBs are on
average at a higher redshift ($\sim 2.2$ versus $\sim 1.5$) than pre-Swift
populations, likely due to the superior sensitivity of the Swift-BAT and
softer energy range than previous instruments. It would therefore follow that
the GBM and LAT redshift distributions may be different. Of course these are
instrumental selection effects, and as we will discuss in Section 3.4, the LAT
bursts tend to have brighter optical afterglows than BAT bursts, therefore are
more likely to have redshift measurements of their optical transients.
With our limited statistics, there are no significant differences (measured
with K-S tests, Table 2) between the redshift distributions of the BAT, GBM,
and LAT populations as illustrated in Figure 4. However, these are not
independent samples, as the GBM bursts were all also detected by BAT and
localized by XRT/UVOT, and these are only the brightest best localized LAT
bursts. Despite these caveats, there is no evidence of any differences in
redshift distributions.
Figure 4.— Cumulative distribution functions of the long and short burst
redshift distributions of the BAT, GBM, and LAT GRB populations in our sample.
Note that no distributions are plotted for the short GBM and short LAT bursts,
because only one GRB is present in each of those samples.
### 3.3. Environment
Using the parameters from the SED fits, we can constrain measurements of the
X-ray absorption ($N_{H}$) or approximate gas content and the optical
extinction ($A_{V}$) or approximate dust content, in order to learn about the
GRB environments. After we remove the Galactic absorption and extinction
contributions, these quantities probe the environment around the GRB
progenitor and along the line of sight. Figure 5 demonstrates these extinction
and absorption measurements separated into the BAT, GBM, and LAT samples.
While a K-S test (Table 2) does not show any significant differences between
the populations in either $N_{H}$ or $A_{V}$, the small number of LAT bursts
for which we could make these measurements tend toward lower values of $A_{V}$
with moderate values of $N_{H}$.
The crude gas-to-dust ratios ($N_{H}/A_{V}$, Figure 6) for each of the
different best fit dust models (MW, LMC, SMC) show two features that might
distinguish the LAT bursts: the LAT bursts are all best fit by the SMC
extinction law, and they tend towards high values of the gas-to-dust ratio.
Since many of the $A_{V}$ measurements of the LAT bursts are upper limits,
this makes the corresponding $N_{H}/A_{V}$ ratios lower limits, which would
only further distinguish the LAT bursts. Due to the low $A_{V}$ values (even
$A_{V}=0$) of the LAT bursts, they can be fit by any of the the three
extinction laws equally well (Schady et al. 2010). The gas-to-dust ratio
intrinsically depends on several factors including the original pre-GRB
environmental ratio of gas-to-dust, and the amount of dust destruction and
photoionization during the GRB event in close proximity to the bursts.
Properties of the GRB itself such as amount and spectrum of the energy output
influence the alteration of the environment. Therefore, understanding
differences in the final gas-to-dust ratio is clouded by these factors which
are difficult to disentangle between properties of the environment and the GRB
itself. This measurement is also model dependent, and can be biased by
assumptions about the spectral model. Regardless, the LAT bursts appear to
have little to no dust along the line-of-sight compared to the GBM and BAT
bursts.
At this time, there are insufficient statistics on gas and dust content of the
LAT bursts to draw any strong conclusions from this sample. Further study with
more objects and broader band data are needed to distinguish any strong
environmental differences between these populations.
Figure 5.— Intrinsic X-ray absorption ($N_{H}$) plotted against visual dust
extinction ($A_{V}$) measured from the SED fits described in Section 2.2.
Figure 6.— $N_{H}/A_{V}$ or gas-to-dust ratio for all of the GRBs in the BAT,
GBM, and LAT samples for the three best fit dust extinctions models (MW, LMC,
SMC) for each GRB.
### 3.4. Afterglow Luminosity
The luminosity light curves in Figures 1 and 2 reveal several interesting
observational and possibly intrinsic differences between the BAT, GBM, and LAT
populations. The GBM and to a higher extent LAT X-ray afterglows are clustered
much more than the BAT afterglows. From an instrumental perspective, BAT is
more sensitive to detecting faint GRBs than GBM, therefore having a wider and
fainter distribution of X-ray afterglows (that correlates with prompt fluence,
Gehrels et al. 2008) is reasonable. However, due to the correlation between
$\gamma$-ray fluence and X-ray flux, with the LAT GRBs having comparatively
extreme fluences (Swenson et al. 2010; Cenko et al. 2011; McBreen et al. 2010;
Ghisellini et al. 2010), we would have expected the LAT GRBs to be at the
bright end of the X-ray afterglow distribution. This distribution is present
in all permutations of these light curves (count rate, flux, flux density,
luminosity), so it is not an effect of one of our count rate to flux, flux
density, or luminosity correction factors.
This unexpected distribution is shown more clearly in the histograms of Figure
7, demonstrating a cross-section of the luminosity at 11 hours and 1 day in
the rest frame of each GRB.
Figure 7.— Histograms of X-ray (0.3-10 keV, left column) and optical (u band,
right column) instantaneous luminosity at times of 11 hours (bottom row) and 1
day (top row) in the rest frame of each GRB, with the long bursts (top panel)
and short bursts (bottom panel) separated. Notice that the LAT long burst
population luminosities are larger on average than that of the other samples,
but are not at the very bright end of the distribution.
### 3.5. Energetics
With this large sample of X-ray and optical afterglows, redshifts, and simple
assumptions about the environment and physical parameters, we can estimate the
total isotropic equivalent $\gamma$-ray energy output in a systematic way over
the same energy range for all of the GRBs in our samples. Despite the fact
that we do not have accurate measurements of $E_{peak}$ for most of the BAT
bursts, we can estimate both $E_{peak}$ (using the power law index correlation
from Sakamoto et al. 2009), and either estimate the Band function or cutoff
power law parameters, use typical values, or use measurements from other
instruments with larger energy coverage (e.g. Konus-Wind, Fermi-GBM, Suzaku-
WAM) especially if they have constrained $E_{peak}$. Using the assumed
spectrum for each GRB and the measured redshift, we integrate over a common
rest frame energy range (Amati et al. 2002) of 10 keV to 10 MeV, as:
$E_{\gamma,iso}=\frac{4\pi
D_{L}^{2}}{(1+z)}\int_{10~{}keV/(1+z)}^{10~{}MeV/(1+z)}E~{}F(E)~{}dE.$ (3)
The functional forms and assumptions are described in more detail in the
appendix of Racusin et al. (2009). Using this method, we infer a reasonable
value of $E_{\gamma,iso}$ for each GRB in a systematic way.
Ghisellini et al. (2010) and Swenson et al. (2010) established that LAT GRBs
include some of the most energetic GRBs ever detected. On average, the LAT
GRBs have isotropic equivalent $\gamma$-ray energy outputs ($E_{\gamma,iso}$)
that are 1-2 orders of magnitude larger than that of the Swift bursts (Figure
8). Given the well known correlations between $E_{peak}$ and $E_{\gamma,iso}$
(Amati et al. 2002; Amati 2006), and the hardness of LAT GRB spectra required
for them to be detected by LAT at all, their large $E_{\gamma,iso}$’s are not
surprising.
Figure 8.— Distribution of $E_{\gamma,iso}$ for both long (top panel) and
short (bottom panel) bursts in the BAT, GBM, and LAT samples. The GBM and BAT
distributions are statistically similar. However, the LAT GRBs are on average
more energetic than the other samples and extend above $10^{55}$ ergs.
This suggests to us that the LAT is preferentially detecting extremely
energetic GRBs compared to previous GRB experiments. The sensitivity, large
field of view, and large energy range of the LAT make it especially sensitive
to hard bursts. While the physical origin of the Amati relation is not well
understood, the energetic LAT bursts seem to qualitatively follow the same
relationship.
Applying our characterizations of the optical and X-ray light curves and SEDs
to the energetics, we can infer jet half-opening angles and collimation-
corrected $\gamma$-ray energy outputs ($E_{\gamma}$), or limits when all
observations were either pre- or post-jet break. Again, the methods used in
these calculations and jet break determination are described in detail in
Racusin et al. (2009).
Using the XRT and UVOT data alone, most of the LAT GRB afterglow light curves
(exceptions discussed below) are best characterized by single power laws, with
relatively flat slopes ($\alpha_{o,x}\lesssim 1.8$), with the exception of the
poorly sampled GRB 100414A which may have had a break in the large gap between
observations, and the short GRB 090510 which shows an early break to a steep
decay - a behavior suggestive of a “naked” short hard burst (Kumar &
Panaitescu 2000) that indicates the turnoff of the prompt emission in a low
density environment with either an afterglow too faint to detect or no
afterglow at all. However, de Pasquale et al. (2010) discussed the possibility
that the break in the optical and X-ray light curves of GRB 090510 at $\sim
2000$ seconds is an early jet break, rather than a naked afterglow (i.e. steep
fall off is either high latitude emission or post-jet break). The following
calculations use the jet break assumption, but we recommend caution when
examining the energetics of this GRB.
The LAT optical light curves, where sampled well, show shallow behavior or
contamination at late times by the host galaxy or nearby sources. This is
consistent with the idea that most of the LAT afterglow observations are pre-
jet break (with the exceptions noted above). Several recent papers (McBreen et
al. 2010; Cenko et al. 2011; Swenson et al. 2010) suggest that when using
other broadband observations (including deep late optical/NIR observations),
some of these bursts do hint at jet breaks, but the Swift data alone are
insufficient to constrain jet breaks. We will discuss the differences in jet
breaks and energetics between this paper and those of McBreen et al. (2010);
Cenko et al. (2011); Swenson et al. (2010) further in Section 4.
If we assume all of the LAT GRBs are pre-jet break (except for GRB 100414A,
which may be post jet break, and GRB 090510, which may include a jet break),
and we determine the presence of jet breaks in the X-ray afterglows of the BAT
and GBM samples using the criteria from Racusin et al. (2009), we can evaluate
the jet opening angles and collimation-corrected energetics as a function of
these populations, as shown in Figure 9.
Figure 9.— $E_{\gamma}$ as a function of $\theta_{j}$ (top panel) for the BAT,
GBM, and LAT samples, for both long and short bursts, including, pre- and
post-jet break. Arrows indicate lower or upper limits on jet break times that
translate into limits on $\theta_{j}$ and $E_{\gamma}$. Histograms of
$\theta_{j}$ and $E_{\gamma}$ for the long bursts (middle panel) and short
bursts (bottom panel) including both measurements (upper plots) and upper
limits (lower plots) as indicated in the top panel scatter plot. Note that the
LAT burst are at the upper end of the $E_{\gamma}$ distribution in comparison
to the BAT and GBM samples, and are only lower limits.
For those GRBs with only lower limits on the jet break times, we use the time
of last detection to determine the lower limit on $\theta_{j}$, and therefore
also $E_{\gamma}$. As demonstrated in Racusin et al. (2009), there are several
different characteristic times for which one can place limits on jet breaks,
and the large error bars on late-time light curve data points can mask jet
breaks (see also Curran et al. 2008). However, for Figure 9, we simply use the
time of last detection.
Table 2K-S Test Probabilities
Parameter | BAT-GBM | BAT-LAT | GBM-LAT
---|---|---|---
Long Bursts
$\alpha_{x}$ | 0.78 | 0.14 | 0.54
$\alpha_{o}$ | 0.93 | 0.44 | 0.63
$\beta_{x}$ | 0.95 | 0.59 | 0.87
$\beta_{o}$ | 0.63 | 0.27 | 0.16
$z$ | 0.55 | 0.95 | 0.76
$A_{V}$ | 0.33 | $9.0\times 10^{-2}$ | $6.8\times 10^{-2}$
$N_{H}$ | 0.31 | 0.64 | 0.76
$N_{H}/A_{V}(\times 10^{21})$ | 0.19 | $5.3\times 10^{-3}$ | $3.0\times 10^{-2}$
$L_{x,11hr}$**Rest frame time | 0.27 | $2.2\times 10^{-3}$ | $5.3\times 10^{-2}$
$L_{x,1day}$**Rest frame time | 0.36 | $1.2\times 10^{-2}$ | 0.18
$L_{o,11hr}$**Rest frame time | 0.38 | 0.29 | $4.8\times 10^{-2}$
$L_{o,1day}$**Rest frame time | 0.44 | 0.21 | $7.8\times 10^{-2}$
$E_{\gamma,iso}$ | 0.39 | $1.3\times 10^{-3}$ | $1.4\times 10^{-2}$
$\theta_{j}$ | 0.15 | – | –
$E_{\gamma}$ | $4.7\times 10^{-2}$ | – | –
Note. — Probabilities of the two distributions being drawn from the same
parent population from K-S tests. Small values indicate significant
differences between the samples. Dashes indicate that there were not enough
($\geq 2$) bursts that fit relevant criteria to perform a K-S test. Only long
burst statistics are included here, because there were not enough short hard
bursts to perform K-S tests on the GBM and LAT populations.
## 4\. Discussion
We have showed that there are observational differences between the BAT, GBM,
and LAT samples throughout the previous sections. However, the difficulty lies
in separating out the instrumental selection effects from the physical
differences between GRBs that produce appreciable $>100~{}\textrm{MeV}$
emission, and those that do not.
The median and standard deviation of the distributions of the many
observational parameters discussed in the previous and following sections are
presented for each sample in Table 3.
In the following section, we will explore physical explanations for the
observable parameter distributions including a calculation of the radiative
efficiency, and speculate on the origin of the afterglow luminosity
clustering. We also will compare and contrast the other recent studies of the
broadband observations of the LAT bursts.
Table 3Parameter Population Characterizations
Parameter | BAT long | GBM long | LAT long | BAT short | GBM short | LAT short
---|---|---|---|---|---|---
$\alpha_{x}$ | $1.46~{}(0.56,~{}130)$ | $1.43~{}(0.35,~{}18)$ | $1.58~{}(0.38,~{}7)$ | $1.50~{}(0.82,~{}12)$ | $1.22$ | $2.19$
$\alpha_{o}$ | $1.21~{}(0.76,~{}46)$ | $1.11~{}(0.35,~{}10)$ | $1.29~{}(0.59,~{}4)$ | $0.72~{}(0.39,~{}3)$ | $1.56$ | $1.01$
$\beta_{x}$ | $1.08~{}(0.37,~{}130)$ | $1.02~{}(0.18,~{}18)$ | $0.90~{}(0.25,~{}7)$ | $0.96~{}(0.34,~{}12)$ | $1.14$ | $0.79$
$\beta_{o}$ | $0.74~{}(0.24,~{}48)$ | $0.75~{}(0.29,~{}13)$ | $1.11~{}(0.47,~{}4)$ | $0.70~{}(0.14,~{}3)$ | $0.88$ | $0.77$
$z$ | $2.21~{}(1.35,~{}130)$ | $2.06~{}(1.88,~{}18)$ | $2.12~{}(1.36,~{}7)$ | $0.71~{}(0.65,~{}17)$ | $1.37$ | $0.90$
$A_{V}~{}(\textrm{mag})$ | $0.32~{}(0.28,~{}44)$ | $0.48~{}(0.35,~{}12)$ | $0.11~{}(0.13,~{}3)$ | $0.32~{}(0.20,~{}2)$ | $0.40$ | –
$log~{}N_{H}~{}(\textrm{cm}^{-2})$ | $20.87~{}(2.95,~{}130)$ | $21.22~{}(2.53,~{}18)$ | $19.52~{}(3.27,~{}7)$ | $21.17~{}(0.45,~{}12)$ | $21.26$ | $20.95$
$log~{}L_{x,11~{}hr}~{}(\textrm{erg~{}s}^{-1})$ | $45.06~{}(0.83,~{}120)$ | $45.28~{}(0.39,~{}17)$ | $45.72~{}(0.24,~{}7)$ | $43.26~{}(2.14,~{}12)$ | $45.00$ | $42.82$
$log~{}L_{x,1~{}day}~{}(\textrm{erg~{}s}^{-1})$ | $44.62~{}(0.89,~{}106)$ | $44.87~{}(0.40,~{}15)$ | $45.19~{}(0.30,~{}7)$ | $42.72~{}(2.47,~{}11)$ | $44.59$ | $42.08$
$log~{}L_{o,11~{}hr}~{}(\textrm{erg~{}s}^{-1})$ | $43.72~{}(0.71,~{}27)$ | $43.57~{}(0.47,~{}8)$ | $44.33~{}(0.57,~{}4)$ | $43.12$ | $43.27$ | $43.08$
$log~{}L_{o,1~{}day}~{}(\textrm{erg~{}s}^{-1})$ | $43.11~{}(0.98,~{}23)$ | $43.19~{}(0.42,~{}7)$ | $43.86~{}(0.41,~{}4)$ | $42.98$ | – | –
$log~{}E_{\gamma,iso}~{}(\textrm{erg})$ | $52.56~{}(0.87,~{}105)$ | $52.66~{}(0.88,~{}17)$ | $54.01~{}(0.75,~{}7)$ | $50.81~{}(0.74,~{}12)$ | $51.83$ | $52.59$
$\theta_{j}~{}(\textrm{deg})$ | $3.53~{}(3.63,~{}55)$ | $4.73~{}(3.20,~{}9)$ | – | $4.46~{}(4.93,~{}4)$ | – | $0.99$
$\ast$$\ast$Lower limits on jet opening angles and collimation corrected $\gamma$-ray energy output as shown in Figure 9$\theta_{j,lim}~{}(\textrm{deg})$ | $9.19~{}(6.34,~{}46)$ | $10.68~{}(6.36,~{}8)$ | $7.65~{}(2.95,~{}4)$ | $12.28~{}(4.13,~{}5)$ | $11.74$ | –
$log~{}E_{\gamma}~{}(\textrm{erg})$ | $49.73~{}(0.71,~{}55)$ | $50.30~{}(0.64,~{}9)$ | – | $48.31~{}(1.03,~{}4)$ | – | $48.77$
$\ast$$\ast$Lower limits on jet opening angles and collimation corrected $\gamma$-ray energy output as shown in Figure 9$log~{}E_{\gamma,lim}~{}(\textrm{erg})$ | $50.20~{}(0.80,~{}46)$ | $50.36~{}(0.74,~{}8)$ | $52.04~{}(0.49,~{}4)$ | $48.99~{}(0.79,~{}5)$ | $50.15$ | –
$log~{}E_{k}~{}(\textrm{erg})$ | $53.41~{}(0.85,~{}47)$ | $53.52~{}(0.43,~{}12)$ | $53.69~{}(0.44,~{}6)$ | $52.40~{}(1.09,~{}3)$ | $53.05$ | –
$\eta~{}(\%)$ | $17.58~{}(23.34,~{}47)$ | $19.94~{}(19.15,~{}12)$ | $64.80~{}(22.53,~{}6)$ | $6.06~{}(4.63,~{}3)$ | $5.69$ | –
Note. — Mean of the distributions of each parameter for the BAT, GBM, and LAT
samples, separated into long and short bursts. Numbers in parentheses indicate
standard deviation and the number of objects in each parameter distribution.
There is only one object in each of the GBM and LAT short burst samples.
### 4.1. Radiative Efficiency
Our first attempt to explore the underlying physics is by calculating the
radiative efficiency of the GRBs at turning their kinetic energy into
radiation during the prompt emission. We follow the formulation of Zhang et
al. (2007), which derives the kinetic energy ($E_{k}$) from the X-ray
afterglow observations, and by comparing the $\gamma$-ray prompt emission
output, we can estimate a radiative efficiency:
$\eta=\frac{E_{\gamma,iso}}{E_{\gamma,iso}+E_{k}}$ (4)
where $E_{k}$ depends on the synchrotron spectral regime (Sari et al. 1998,
$\nu>\nu_{c}$ or $\nu<\nu_{c}$) as:
$\displaystyle E_{k,52,\nu>\nu_{c}}=\left(\frac{\nu F_{\nu}}{5.2\times
10^{-14}~{}\textrm{ergs}~{}\textrm{s}^{-1}~{}\textrm{cm}^{-2}}\right)^{4/(p+2)}$
$\displaystyle\times~{}D_{28}^{8/(p+2)}(1+z)^{-1}~{}t_{d}^{(3p-2)/(p+2)}$
$\displaystyle~{}(1+Y)^{4/(p+2)}~{}f_{p}^{-4/(p+2)}~{}\epsilon_{B,-2}^{(2-p)/(p+2)}$
$\displaystyle\times~{}\epsilon_{e,-1}^{4(1-p)/(p+2)}~{}\nu_{18}^{2(p-2)/(p+2)}$
(5) $\displaystyle E_{k,52,\nu<\nu_{c}}=\left(\frac{\nu F_{\nu}}{6.5\times
10^{-13}~{}\textrm{ergs}~{}\textrm{s}^{-1}~{}\textrm{cm}^{-2}}\right)^{4/(p+3)}$
$\displaystyle\times~{}D_{28}^{8/(p+3)}(1+z)^{-1}~{}t_{d}^{3(p-1)/(p+3)}$
$\displaystyle\times~{}f_{p}^{-4/(p+3)}~{}\epsilon_{B,-2}^{-(p+1)/(p+3)}$
$\displaystyle\times~{}\epsilon_{e,-1}^{4(1-p)/(p+3)}~{}n^{-2/(p+3)}~{}\nu_{18}^{2(p-3)/(p+2)}.$
(6)
All subscripts indicate the convention $X_{n}=X/10^{n}$ in cgs units. $D_{28}$
is the luminosity distance in units of $10^{28}$ cm, $t_{d}$ is the time of
interest in units of days, $p$ is the electron spectral index derived from the
spectral index $\beta_{x}$ where:
$p=\begin{cases}2\beta_{x}+1&\nu<\nu_{c}\\\ 2\beta_{x}&\nu>\nu_{c}\\\
\end{cases}$ (7)
and
$f_{p}=6.73\left(\frac{p-2}{p-1}\right)^{(p-1)}(3.3\times
10^{-6})^{(p-2.3)/2}.$ (8)
We make several simplifying assumptions including using a single typical value
for $\epsilon_{e}=0.1$, $\epsilon_{B}=0.01$, and $n=1~{}\textrm{cm}^{-3}$ as
well as ignoring inverse Compton emission ($Y=1$). Unlike Zhang et al. (2007),
we do not determine the kinetic energy ($E_{k}$) at the deceleration time or
the break time, because we do not have the temporal context of the shallow
decay phase (the earlier or later canonical phases) in many bursts, including
the LAT bursts. Instead, we calculated $\eta$ at observed times of 11 hours
and 1 day, but only if there was evidence from the light curve morphology and
decay slopes of those measurements being during the normal decay (forward
shock) phase (Zhang et al. 2006; Racusin et al. 2009). The efficiency did not
change significantly between these two times, but more GRB X-ray afterglows
were in the normal decay phase at 11 hours than at 1 day. Therefore, the 11
hour results are plotted in Figure 10. We determined whether a particular
X-ray afterglow was above or below the cooling frequency ($\nu_{c}$) using the
closure relations and assuming the simplest cases ($p>2$, ISM or Wind
environments, slow cooling, no energy injection), similarly to Zhang et al.
(2007).
Figure 10 demonstrates the relationship between the kinetic energy and the
$\gamma$-ray prompt emission with the range of radiative efficiencies
indicated. Only 69 GRBs are included in this plot. The rest of our sample were
excluded either due to not having sufficient information to measure
$E_{\gamma,iso}$, not satisfying the relevant closure relations, lack of a
clear normal forward shock decay, or lack of emission at time of interest (in
this case 11 hours in the observed frame). Of the 69 GRBs, 51 are from the BAT
sample, 12 from the GBM sample, and 6 from the LAT sample.
Our measured efficiencies for the BAT sample cover a similar range and roughly
agree with their statement from Zhang et al. (2007) that given the above
assumptions, most ($\sim 57\%$) of BAT bursts have $\eta<10\%$. This statement
is not true for the GBM and LAT samples. In fact, only $25\%$ of the GBM
bursts have $\eta<10\%$, and none of the LAT bursts have such low radiative
efficiencies. This suggests that GBM and LAT detected GRBs are on average more
efficient at converting kinetic energy into prompt radiation, which perhaps
explains why they are substantially brighter (higher fluence) than the BAT
bursts with similar afterglow luminosities. The LAT bursts are at the high end
of the $E_{\gamma,iso}$-$E_{k}$ distributions with $\eta>40\%$ for all LAT
bursts, and $\eta>80\%$ for three of the LAT bursts. These extreme
efficiencies may be unphysical and due to a our assumptions: an internal shock
mechanism as is described in the fireball model (Rees & Mészáros 1998;
Mészáros 2002), a single value of $\epsilon_{e}$ and $\epsilon_{B}$, no
appreciable Compton component, and a single universal surrounding medium
density.
Figure 10.— Kinetic energy ($E_{k}$) derived from the X-ray afterglow
observations at 11 hours as a function of the prompt emission isotropic
equivalent $\gamma$-ray energy output ($E_{\gamma,iso}$). The dashed lines
indicated different values of the radiative efficiency ($\eta$). The LAT
bursts tend to have high $E_{\gamma,iso}$ but average $E_{k}$, and therefore
higher values of $\eta$ than the BAT or GBM samples.
We acknowledge that these differences in the distribution of $\eta$ are
degenerate with differences in $\epsilon_{e}$, $\epsilon_{B}$, and the
presence of some inverse Compton component. We also caution that our
estimations of $E_{\gamma,iso}$ for the LAT bursts do not include the extra
spectral power law observed in several of the LAT bursts. However, that would
make $\eta$ even larger, which perhaps suggests that it is the internal shock
model framework that is not valid.
### 4.2. Bulk Lorentz Factor
The bulk Lorentz factor ($\Gamma$) is a fundamental quantity needed to
describe the GRB fireball and therefore interesting to compare for different
populations of bursts. Unfortunately, it is also a difficult quantity to
accurately measure and there are several methods for placing lower or upper
limits on this quantity depending on multiple assumptions.
The most common technique applied to the Fermi-LAT detected bursts (Abdo et
al. 2009c; Ackermann et al. 2010; Abdo et al. 2009a) was originally derived by
Lithwick & Sari (2001), using the highest energy observed photon to place a
lower limit on the $\gamma$-ray pair production attenuation, setting a lower
limit on $\Gamma$. This method assumes that the GeV and seed sub-MeV photons
are emitted from the same co-spatial region and are produced by internal
shocks. It can produce extreme values of $\Gamma\gtrsim 1000$ for the LAT
bursts. Zhao et al. (2011) and Zou et al. (2011) suggest modifications for
this calculation using a two-zone model that assumes that the sub-MeV and GeV
photons are produced at very different radii from the central engine. This
modification lowers $\Gamma$ to approximately a few hundred. Hascoët et al.
(2011, in-preparation) also demonstrate that when carefully calculating the
pair production attenuation taking into account the jet geometry and dynamics,
$\Gamma$ is reduced by a factor of $\sim 2.5$.
When no high energy (GeV) observations are available, the most common method
to limit $\Gamma$ is to derive it from the deceleration time of the forward
shock which corresponds to the peak time of the optical (Sari & Piran 1999;
Molinari et al. 2007; Oates et al. 2009; Liang et al. 2010) or X-ray (Liang et
al. 2010) afterglow light curves. Often one can only set upper limits on the
deceleration time because the peak must have occurred prior to the start of
the observations, corresponding to lower limits on $\Gamma$, or be buried
under other components.
There are additional alternative methods to determine $\Gamma$ including
putting upper limits on the forward shock contribution to the keV-MeV prompt
emission by looking for deep minima or dips down to the instrumental threshold
between peaks in the prompt emission light curves (Zou & Piran 2010). The
typical values of these upper limits on $\Gamma$ are several hundred. Pe’er et
al. (2007) describe another method that estimate $\Gamma$ from the thermal
component modeling in the prompt emission spectrum using photosphere modeling.
Zhang et al. (2003) describe yet another method to estimate $\Gamma$ using
early optical data to constrain the forward and reverse shock components. The
latter two methods are worth further study, but the application of them to the
data presented here is beyond the scope of this paper.
We apply the deceleration time of the optical light curves technique to our
sample for those GRBs with UVOT light curves and measurements of $E_{k}$
(derived in Section 4.1). Unfortunately, due to the lack of early observations
of the LAT bursts, we cannot apply this method to that sample. However, we
collect estimates of $\Gamma$ using the pair-production attenuation technique
from the literature, using the typical one-zone model from Abdo et al. (2009a,
c); Ackermann et al. (2010, 2011)), as well as the two-zone estimates from Zou
et al. (2011), and compare these limits in Figure 11.
The different methods yield a wide range of $\Gamma$ for each burst ranging
from a few 10s to more than 1000. However, many of these results are upper or
lower limits. The assumptions put into each measurement and method about the
geometry and nature of the outflow have a strong influence on the results. If
we believe that the sub-MeV and GeV photons are generated in the same co-
spatial region, and ignore the two-zone model estimates, the lower limits on
$\Gamma$ of the LAT bursts are nearly a factor of 2 higher than the BAT and
GBM bursts. Unfortunately, we do not have high energy observations of many of
the BAT bursts, and we do not have early optical observations of the LAT
bursts, therefore one should compare measurements of $\Gamma$ for these
different samples with caution.
A more careful detailed study of bulk Lorentz factor estimates for the bursts
in this sample would perhaps provide more insight into concrete differences
between the samples. This would require detailed analysis of all of the prompt
emission light curves and spectra of the bursts in our sample, and this is
beyond the scope of this study.
It is also interesting to note that the high $\Gamma$ limits on the LAT bursts
are reminiscent of a structured jet model, such as the two-component jet model
where there is a narrow bright faster core surrounded by a slow wider jet,
with the narrow jet on-axis to the observer (Berger et al. 2003b; Huang et al.
2004; Racusin et al. 2008). Liu & Wang (2011) explore this model using the
broadband data on the LAT GRB 090902B and find an acceptable fit. Additional
study of the other LAT GRBs in the context of this model would be needed to
draw any stronger conclusions about the sample as a whole.
Figure 11.— Estimates of the bulk Lorentz factors ($\Gamma$) using the four
methods described in Section 4.2 with different colors indicating method. The
four methods are referred to as $\gamma\gamma$ \- the pair production
attenuation method from Lithwick & Sari (2001); $t_{peak}$ \- the forward
shock estimation from the optical peak described by Sari & Piran (1999);
Molinari et al. (2007); ZP10 - the limit on forward shock contribution to the
sub-MeV prompt emission described in Zou & Piran (2010); 2-Zone $\gamma\gamma$
\- the pair production attenuation method assuming the sub-MeV and GeV photon
come from different sources as described in Zou et al. (2011). A right
pointing triangle indicates a lower limit, a left pointing triangle an upper
limit, and an asterisk indicates a measurement. These measurements or limits
are either from our own calculations (only $t_{peak}$ method) or from the
literature. The bursts for which we could make measurements are shown and
separated into each of the BAT, GBM, and LAT samples.
### 4.3. Afterglow Luminosity Clustering
As mentioned in Section 3.4, the X-ray and UV/optical luminosity distributions
of the LAT bursts are narrower than the GBM and BAT samples. There are several
possible causes or simply related dependencies, namely, the fact that a larger
fraction of the LAT bursts are in the synchrotron spectral regime
$\nu>\nu_{c}$ ($83\%$) compared to the BAT and GBM bursts ($50-60\%$), and
that the LAT bursts have a narrow distribution (in log space) of
$E_{\gamma,iso}$. The high radiative efficiencies of the LAT bursts may be
either another cause of the narrow luminosity distribution or a consequence.
The region of luminosity parameter space fainter than the LAT bursts could be
limited by the lower detection limits of the LAT instrument, and the ability
to accurately localize only the brightest of the LAT burst for Swift follow-
up. Nearly half of the LAT detections had position errors $>0.5$ degrees
radius, which was simply not practical to initiate follow-up observations
beginning many hours or days after the triggers. In the future, if Swift
happens to simultaneously trigger on one of these fainter long bursts with a
marginal LAT detection and a fainter afterglow, then we will know whether the
luminosity clustering is only limited on the bright end.
### 4.4. Luminosity Function
In addition to the simple redshift distribution comparison, we explore the
luminosity functions of the different populations of GRBs. We use the methods
of both Virgili et al. (2009) and Wanderman & Piran (2010), which apply
different statistical methods to constrain the luminosity function shapes for
the three samples. Simply due to instrumental selection effects, the BAT, GBM,
and LAT samples probe different regions of the luminosity function. GRBs
bright enough to trigger GBM, will be brighter on average than BAT only
bursts, because GBM is less sensitive than BAT. The LAT GRBs have the highest
fluence of the GBM bursts, and given the similar redshift distributions
(Section 3.2), are therefore also the most luminous.
The Virgili et al. (2009, 2011) method compares Monte Carlo simulations of the
full parameter space drawn from the full Swift sample to the specific sample
of interest, in order to be less biased by instrumental selection effects.
Whereas, the Wanderman & Piran (2010) method does a more tradition fit to the
observed data assuming a single value for the various instrument sensitivity
levels. Both methods find consistent results for the BAT sample, fitting to a
broken power law with slopes of $0.2$ and $1.4$ for the less and more luminous
ends, respectively, and a break luminosity of
$10^{52.5}~{}\textrm{erg}~{}\textrm{s}^{-1}$. Within the substantial error
bars, the GBM sample resembles the BAT luminosity function, but does not probe
as faint. The limited LAT sample is even smaller, and mostly brightward of the
break luminosity. Therefore, it is best fit by a single power law with slope
of 0.3, shallower than the post-break slope of the BAT and GBM functions. This
shallow slope may suggest some differences in the parent population of the LAT
burst, or may simply be due to the complicated selection effects of both LAT
burst detection and the subset with accurate enough positions to initiate
follow-up. Perhaps with a larger LAT sample, this could be studied more
thoroughly. The bright end of the luminosity function of the BAT sample may
not be well enough probed to accurately constrain the shape of the luminosity
function, and therefore the LAT bursts are essential tools to study the most
luminous GRBs ever detected.
### 4.5. Comparisons to Detailed Broadband Modeling
Several other recent papers (Cenko et al. 2011; McBreen et al. 2010; Swenson
et al. 2010) did detailed broadband modeling of individual LAT bursts. Swenson
et al. (2010) also made comparisons of the LAT bursts to prompt emission and
afterglow parameters including prompt fluence and afterglow luminosities, and
found that the LAT bursts had higher X-ray count rates than $80\%$ of BAT
bursts at 70 ks. Note that our sample is different (only those with redshifts)
and likely biased towards optically brighter bursts, and we measure
luminosities rather than count rates. However we agree (also with McBreen et
al. 2010) that the optical afterglows of the long LAT bursts are in the top
half of the brightness distribution.
We also agree with the aforementioned works, that the LAT bursts are some of
the most energetic GRBs ever observed by any instrument, and even with
collimation corrections or limits on the collimation, they remain extreme
events. We do not clearly detect any jet breaks in the LAT bursts using the
XRT and UVOT data alone, except for perhaps the short burst GRB 090510, as
discussed in Section 3.5. Both McBreen et al. (2010) and Cenko et al. (2011)
claim jet breaks for some of the LAT bursts, given their ground based deep NIR
and radio observations, but most are not well constrained. Clearly, more late
time deep broadband observations (both space and ground-based) are needed for
the LAT bursts in the future to constrain their total energetics.
## 5\. Conclusions
We have systematically characterized the X-ray and UV/optical temporal and
spectral afterglow and prompt emission characteristics, energetics, and other
properties of the GRBs detected only by Swift-BAT, those detected by both
Fermi-GBM and Swift-BAT, and those detected by both Fermi-GBM and LAT in order
to understand the observational and intrinsic differences between the burst
populations. There are no significant differences between the BAT, GBM, and
LAT populations in terms of X-ray and optical temporal power law decays at
common rest frame late times, or spectral power law indices, redshifts, or
luminosities. However, the distributions of luminosities are much more narrow
for the LAT and GBM samples compared to the BAT sample. The LAT long burst
sample is also more luminous on average than the BAT sample, but within the
same distribution. There are significant differences between the populations
in terms of isotropic equivalent $\gamma$-ray energies ($E_{\gamma,iso}$),
prompt emission hardness, and radiative efficiency.
While in many ways, the late-time ($\sim 1$ day) properties of the LAT bursts
are similar to their lower energy counterparts observed by BAT, some mechanism
fundamentally makes their prompt $\gamma$-ray production more efficient, or
conversely suppresses their afterglows. As we collect more statistics on this
exciting sub-population of GRBs detected by LAT, we can study luminous and
energetic extremes. Studying the afterglows of the LAT burst, especially at
early times, will also help us to understand the additional components (extra
spectral power-law and extended $>100$ MeV emission) observed in many of these
LAT bursts. Additional future simultaneous triggers between BAT and LAT will
provide more information on the early broadband behavior of LAT bursts.
We gratefully acknowledge A. Fruchter, A. Pe’er, K. Misra, and J. Graham for
helpful discussions. We also thank the anonymous referee for their detailed
comments.
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|
arxiv-papers
| 2011-06-13T15:23:04 |
2024-09-04T02:49:19.586895
|
{
"license": "Public Domain",
"authors": "J. L. Racusin, S. R. Oates, P. Schady, D. N. Burrows, M. de Pasquale,\n D. Donato, N. Gehrels, S. Koch, J. McEnery, T. Piran, P. Roming, T. Sakamoto,\n C. Swenson, E. Troja, V. Vasileiou, F. Virgili, D. Wanderman, B. Zhang",
"submitter": "Judith Racusin",
"url": "https://arxiv.org/abs/1106.2469"
}
|
1106.2487
|
# Spectral and temporal characterization of a fused-quartz microresonator
optical frequency comb
Scott B. Papp and Scott A. Diddams National Institute of Standards and
Technology, Boulder, Colorado 80305, USA
###### Abstract
We report on the fabrication of high-$Q$, fused-quartz microresonators and the
parametric generation of a frequency comb with 36 GHz line spacing using them.
We have characterized the intrinsic stability of the comb in both the time and
frequency domains to assess its suitability for future precision metrology
applications. Intensity autocorrelation measurements and line-by-line comb
control reveal near-transform-limited picosecond pulse trains that are
associated with good relative phase and amplitude stability of the comb lines.
The comb’s 36 GHz line spacing can be readily photodetected, which enables
measurements of its intrinsic and absolute phase fluctuations.
Femtosecond-laser optical frequency combs have revolutionized frequency
metrology and precision timekeeping by providing a dense set of absolute
reference lines spanning more than an octave. These sources exhibit sub-
femtosecond timing jitter corresponding to an ultralow phase noise of $<100$
$\mu$rad on high harmonics of their typically 100’s of MHz repetition
frequency (line spacing) Diddams _et al._ (2004). This remarkable level of
performance has enabled measurements of atomic transition frequencies at the
17th digit Rosenband _et al._ (2008) and the generation of ultralow noise
signals Fortier _et al._ (2011). Beyond their natural application as an
optical clockwork, frequency combs are used in diverse applications including
precision measurements of fundamental physics Diddams _et al._ (2004);
Rosenband _et al._ (2008), direct spectroscopy and real-time trace gas
detection Thorpe _et al._ (2006), molecular fingerprinting Diddams _et al._
(2007), astronomical spectrograph calibration Murphy _et al._ (2007), and
optical arbitrary waveform generationHuang _et al._ (2008). An even broader
range of applications may be possible if the bulk and complexity of a
femtosecond comb could be reduced without significantly increasing
fluctuations in the comb spectrum.
A new class of frequency combs has recently emerged based on monolithic
microresonators, henceforth denoted microcombs Del’Haye _et al._ (2007);
Savchenkov _et al._ (2008a); Levy _et al._ (2010); Razzari _et al._ (2010);
Kippenberg _et al._ (2011). Here the comb generation relies on parametric
conversion provided simply by third-order nonlinear optical effects and is
enabled by recent advances in the quality factor $Q$ and the small volume of
microresonators. These devices require only a single continuous-wave laser
source, but the usable frequency span of the comb depends on low dispersion,
making material properties critical. Microcombs also present a unique platform
for creating large line spacings (10’s of GHz to THz). To date microcombs
possessing 100’s of lines each spaced by 100’s of GHz have been created with
silica microtoroids Del’Haye _et al._ (2007) and silicon-nitride microrings
Levy _et al._ (2010); Razzari _et al._ (2010). Uniformity Del’Haye _et al._
(2007) and control Del’Haye _et al._ (2008) of silica microcombs have also
been demonstrated. Microcombs with mode spacings of 10 to 25 GHz have also
been realized with machined crystalline resonators Savchenkov _et al._
(2008a) and aspects of their microwave-domain spectral purities have been
analyzed Savchenkov _et al._ (2008a, b).
Figure 1: High-$Q$ fused-quartz microresonators. (a) A fused quartz rod is
rotated in a lathe (not shown) and rubbed against a fixed metal form charged
with diamond abrasive. The ground form is then flame polished with household-
grade propane and oxygen (not shown). (b) Image showing an array of fused
quartz microresonators after grinding and flame polishing. The disks of 1.9 mm
diameter are separated by 0.45 mm. (c) Frequency scan of a mode for disk four
(indicated by the arrow) demonstrating an optical $Q_{0}=5.2\times 10^{8}$.
(d) Cavity ringdown at $K=1$ to determine the optical $Q$ of disk four. The
dashed lines indicate a decay time of 260 ns, corresponding to
$Q_{0}=6.2\times 10^{8}$.
Future frequency metrology applications of microcombs will require a line
spacing (repetition rate, $f_{\rm{rep}}$) in the 10’s of GHz (millimeter-scale
resonator), comb span approaching an octave, and low absolute phase and
frequency fluctuations that can be further reduced by wideband comb control
Kippenberg _et al._ (2011). Additionally, a threshold power for comb
generation in the mW range, and the capability for integration with chip-based
photonic circuits would enable novel portable applications. Here we present a
platform towards attaining these requirements. Our fabrication approach takes
advantage of precision mechanical shaping and polishing possible with fused
quartz, a versatile amorphous material with low optical losses throughout the
visible and into the infrared. We created resonators that feature high $Q$ and
low effective optical mode area, which allowed comb generation for a $<$10 mW
pump laser. By pumping above threshold, a comb with $\sim 150$ lines spaced by
36 GHz was generated. We harnessed full line-by-line control (amplitude and
phase) of the comb to generate near-transform limited optical waveforms and to
understand the comb’s relative phase stability. In the frequency domain, we
characterized the comb’s mode-spacing fluctuations to assess prospects for
stabilizing it to optical atomic standards.
Figure 2: Microresonator lineshape and corresponding comb spectra for laser
pump power of 62 mW. (a) Thermally broadened lineshape with a mostly sawtooth
shape. Power levels L1, L2, L3 respectively correspond to the parametric
threshold, the initiation of a 36 GHz comb but with unstable behavior, and the
region of stable 36 GHz comb generation. (b) Comb spectrum at L1. (c) Comb
spectrum at L2. (d) Comb spectrum at L3.
We fabricated an array of disk-like microresonators on a common substrate by
use of a combination of diamond grinding and flame polishing. A schematic of
our procedure is shown in Fig. 1a. Fused quartz rods with $\approx 2$ mm
diameter were rotated in a ball-bearing spindle. To create the basic shape of
a disk, the glass was rubbed against a metal form with triangular cross
section. Diamond abrasive between them removed material from both. At a
diamond abrasive size of 6 $\mu$m, $\sim$30 minutes of grinding was required
to create the underlying shape shown in Fig. 1b. The resulting surface was far
too rough to support high $Q$ whispering-gallery modes. By polishing the disks
with a propane-oxygen flame we achieved an extremely smooth surface.
Substantial melting of the quartz was required to alleviate subsurface damage
caused by grinding. Figure 1b shows an image of a completed disk array. Free-
spectral range measurements of several disks revealed a diameter uniformity of
0.1 $\%$.
After processing, the fused quartz rod was placed in a temperature-stabilized
mount ($<$0.01 K stability) and whispering-gallery modes of the disks were
accessed using standard fiber tapers Knight _et al._ (1997); Cai _et al._
(2000); Spillane _et al._ (2003). The coupling between disk modes and the
taper can be characterized by the parameter $K=\tau_{0}/\tau_{e}$, where
$\tau_{0}$ is the intrinsic photon lifetime and $\tau_{e}$ is the photon loss
rate due to coupling. The frequency linewidth of the resonator-taper system is
$\gamma=\tau_{0}^{-1}+\tau_{e}^{-1}$. Figure 1c shows a frequency scan over a
disk mode using a tunable diode laser at 1560 nm. A low optical power of
$\sim$10 $\mu$W was used to minimize thermal broadening of the resonance
Carmon _et al._ (2004) and taper coupling was small ($K\approx 0.03$). The
linewidth of the Lorentzian transmission feature is 0.37 MHz, which
corresponds to an optical $Q_{0}=\omega\,\tau_{0}$ of $5.2\times 10^{8}$,
where $\omega$ is the optical carrier frequency 111We have not observed any
scattering-induced mode splitting in these devices Gorodetsky _et al._
(2000); Kippenberg _et al._ (2002).. We confirmed this $Q_{0}$ measurement by
recording the resonator energy ringdown time following a rapid frequency scan
across resonance with $K=1$; see Fig. 1d. A fit to the cavity energy decay
yields $Q_{0}=6.2\times 10^{8}$, in good agreement with our linewidth
measurement. Material absorption in the fused-quartz samples we use imposes a
limit to $Q_{0}\sim 2\times 10^{10}$. We speculate that some contamination
occurs during the flame-polishing step Kuzuu _et al._ (2003); *Kuzuu2004.
To achieve the long-term stability in $K$ needed for studies of microcomb
generation, we place the taper directly in contact with the resonator surface.
Translating the taper along the axis of the quartz rod allows us to tune the
resonator-taper coupling from under- to over-coupling. Moreover, measurements
of $K$ at various locations along the disk edge provide a qualitative picture
of the mode’s spatial profile. Specifically, the resonator mode studied in
Fig. 1 c and d has two field nodes in the axial direction and is presumably of
lowest radial order. Numerical calculations based on an ideal disk model
Borselli _et al._ (2005) suggest the effective area of this mode
($A_{\rm{eff}}$) is approximately 500 $\mu$m2. The threshold for parametric
oscillation threshold is
$P_{\rm{th}}=\frac{1}{4}\,\frac{1+K}{K}\,\frac{n}{n_{2}}\,\frac{\omega}{\Delta\nu}\,\frac{A_{\rm{eff}}}{Q^{2}}$,
where $n$ ($n_{2}$) is the refractive index (nonlinear index) of fused quartz
and $\Delta\nu$ is the free spectral range. For our disk resonators we expect
$P_{\rm{th}}=$ 4 mW for $K=0.2$.
We have generated a microcomb at, and significantly above, parametric
threshold. Figure 2a-d shows a disk resonator transmission lineshape and
typical comb optical spectra. With constant laser power, we adjusted and
stabilized the intracavity power by tuning the pump laser frequency for a
fixed $K=0.2$. Here the lineshape is thermally broadened and assumes a
characteristic “sawtooth” shape Carmon _et al._ (2004). Power level L1
indicates the observed 3.7 mW threshold for parametric oscillation, which is
in excellent agreement with our prediction. Notably, the signal and idler
modes are spaced by 828 GHz ($23\times 36$ GHz); indicating the point at which
resonator-induced phase mismatch of pump, signal, and idler balances the
mismatch created by nonlinear effects Agrawal (2007); Kippenberg _et al._
(2004); Savchenkov _et al._ (2008a). Increasing the intracavity power further
resulted in a widened comb span up to $\pm$25 nm about 1560 nm and a reduction
of the comb spacing toward the fundamental 36 GHz of the disk resonator, which
is consistent with the simulations of Chembo et al. Chembo _et al._ (2010);
Chembo and Yu (2010). However, our observations indicate that 36 GHz comb
generation is complex with competition between different modes of operation.
Specifically, we present two instances of 36 GHz combs with dramatically
different properties: (1) At power level L2 (Fig. 2c) an $\approx$36 GHz comb
spacing is initiated, but a finer analysis (see Fig. 6c) reveals numerous comb
line spacing frequencies and (2) in a range about power level L3 (Fig. 2d) a
robust comb pattern exists with a single line spacing of 36.0012 GHz.
Interestingly, when we tuned to power level L2 a deviation in the sawtooth
shape of the transmission lineshape was observed. This behavior depends on
resonator-taper coupling and laser polarization, but not the laser frequency
scan rate, which we varied from 0.005 to 1 s. Moreover, in this range stable
time-domain waveforms and low noise frequency-domain performance were not
evident under the conditions of Fig. 2 222The numerical simulation of
parametric comb generation by Chembo appears to explain the spectrum of Fig.
2d with 36 GHz lines modulated at an approximately 828 GHz period, but not the
sawtooth deviation at power level L2 or the associated instability of comb
generation here Chembo _et al._ (2010); Chembo and Yu (2010). About L3 we
observed the resumption of a sawtooth lineshape.
Figure 3: Time-domain measurements of a three line comb. (a) Schematic of our
setup to control and analyze the microcomb spectrum. The amplitude and phase
of light exciting the resonator via tapered fiber is manipulated with a
programmable filter. An intensity autocorrelator is used to record the fields’
waveforms. (b) Optical spectrum following the filter and amplifier. (c) Time-
averaged autocorrelation signal for $\Delta\phi=0$ (black circles) and
$\Delta\phi=\pi$ (blue circles), which respectively demonstrate constructive
and destructive interference of the three comb fields. The solid lines show
the results of an analytic model. (d) Visibility of constructive interference
as a function of $\Delta\phi$.
The spectral structure similar to Fig. 2 has been observed in a variety of
parametric microcombs, but little is known about the temporal structure of the
output as determined by the phase relationship between the comb elements.
Measurements and control of optical waveforms can reveal crucial information
about the internal fields of the microcomb Foster _et al._ (2011); Ferdous
_et al._ (2011). Using time-domain diagnostics, we studied the relative phase
stability of the first three comb lines (pump, signal, and idler) at
parametric threshold, and the phase stability of many lines far above
threshold. For a comb composed of only three equal-amplitude lines its time
waveform has two distinct behaviors, which depend primarily on the relative
phases of the lines: constructive interference, resulting in a rudimentary
pulse at $f_{\rm{rep}}$, or partial destructive interference, resulting in a
beat signal at $2f_{\rm{rep}}$. A comb with many lines exhibits more
complicated behavior and offers high peak power waveforms.
Figure 4: Generation of picosecond pulses utilizing line-by-line comb control.
(a) Measured time-domain waveform (black line) following amplitude and phase
optimization, and simulation (red dashed line) based on the measured
amplitudes of comb lines used and assuming phase alignment of all the fields.
The inset shows the optical spectrum after optimization. (b) Optical spectrum
following broadening in highly nonlinear fiber. The comb was amplified to 1.6
W.
Our experimental starting point was a comb spectrum similar to that shown in
Fig. 2b, except with spacing of 864 GHz 333The mode spacing at threshold can
be tuned over a small range near 864 GHz with $K$.. Using a programable
amplitude and phase filter featuring 10 GHz resolution (Fig. 3a), we selected
the central three lines of this comb, equalized their amplitudes to better
than 7 %, and adjusted their phases. The optical spectrum we obtained
following amplification with a low-dispersion erbium fiber amplifier is shown
in Fig. 3b. It was delivered via a short section of optical fiber to an
intensity autocorrelator for characterization. By varying the relative phase
($\Delta\phi$) of the lowest wavelength line, we were able to explore the
transition between constructive and deconstructive interference of the comb
lines. The relative phases between the center and the highest wavelength line
were set to zero. Figure 3c shows the time-waveform autocorrelation signal
($P(\tau)$), where $\tau$ is the differential path delay, for $\Delta\phi=0$
(black open circles) and $\Delta\phi=\pi$ (blue open circles). The solid lines
indicate the results of an analytic model based on three phase-coherent
fields, which are in excellent agreement with the data. To characterize the
interference between the three fields we introduce a visibility $V$ based on
$P(\tau)$, which is defined as
$V=|\frac{P(\pi/\Delta\omega)-P(0)}{P(\pi/\Delta\omega)+P(0)}|$. The filled
points in Fig. 3d are measurements of $V$ as a function of $\Delta\phi$, and
the red line shows our analytic model. We measured a maximum (minimum)
visibility of 0.7 (0.007), and the model predicts the visibility is 0.768
(0.002). This indicates good phase coherence of the comb lines. Fitting the
model to the data with free parameter $\Delta\phi$ yields an uncertainty in
$\Delta\phi$ of 20 mrad, which we associate with an upper limit on the
fluctuations of the relative phase of the comb elements over the measurement
time of 1700 seconds 444We note this value also includes un-characterized
fluctuations intrinsic to the programable filter and the measurement
apparatus..
We also carried out time-domain waveform generation with a microcomb far above
parametric threshold. The amplitude and phase filter was used to adjust the
central 15 lines (0.54 THz) of a 36 GHz comb similar to that of Fig. 2d. Here
a manual line-by-line procedure was used to optimize the peak waveform
intensity. This resulted in stable near-transform-limited 2.5 ps optical
pulses, as demonstrated by the time-averaged autocorrelation signal in Fig.
4a. Hence, the relative phases of the 15 comb lines were presumed to be
constant across the spectrum and were correspondingly stationary in time.
Moreover, the autocorrelation signal remained unchanged despite large
perturbations to the microcomb system, such as re-locking the pump laser
frequency or re-placing the tapered fiber in contact with the disk. These
observations indicate that the internal fields of the resonator and the pulse
generation mechanism are deterministic and repeatable. Significantly, the
creation of time-domain waveforms, such as shown in Fig. 4a, implies that such
a compact and simple device could be competitive with more conventional mode-
locked lasers a source of high repetition rate ultrashort optical pulses that
would be useful for a variety of time- and frequency-domain applications
Kippenberg _et al._ (2011). For example, nonlinear broadening in fibers could
be a route to ultra-broadband spectra for comb self-referencing and comb
spectroscopy. Using a 101 m length of highly nonlinear fiber, we have
broadened our 15 line comb by more than a factor of 10, as shown in Fig. 4b.
This demonstrates sufficient peak power to drive nonlinear processes external
to the microresonator.
Figure 5: Residual phase noise of microcomb line spacing. (a) Schematic of our
experimental setup. The microcomb spectrum is split (BS) into two paths, which
are separately filtered at $\lambda_{1}=1560$ nm and $\lambda_{2}=1553.6$ nm.
Photodectors PD1,2 convert the comb spacing of the independent paths into
electrical signals near 36 GHz. PD1 (PD2) generates 2.15 mA (0.04 mA) of
photocurrent. Double-balanced mixers M1 and M2 convert these signals to
baseband, and mixer M3 offsets one path by 11 MHz. This offset frequency was
referenced to a hydrogen maser. At mixer $M_{4}$ the two paths are interfered
yielding residual noise. (b) Single-sideband phase noise of the microcomb 36
GHz tone (black line) and the contribution from the 11 MHz offset oscillator
(red line). (c) Histogram of zero-dead-time counter measurements of the signal
exiting M4. (d) Allan deviation of the microcomb line spacing (filled points).
The solid line shows the $1/\tau$ scaling exhibited at most averaging times.
Frequency-domain techniques offer significant advantages in characterizing
microcomb fluctuations, including access to the power spectral density, wide
measurement bandwidth, and extremely high precision. A key feature of our
system is a comb line spacing of 36 GHz, which enables direct photodetection
and analysis of the resulting microwave signal in the frequency domain. Here
we present measurements of the microcomb’s residual and absolute phase
fluctuations. Residual (i.e. intrinsic) noise indicates the degree that the
frequencies of different comb lines are correlated, and absolute noise
indicates the stability of the lines with respect to a fixed frequency
reference. Moreover, we have characterized the relationship between line
spacing fluctuations and pump laser intensity noise. This information provides
a benchmark for what level of control will be required for future applications
of stabilized microcombs.
To understand the intrinsic stability of the microcomb, we directly compared
the line spacing at two independent portions of its spectrum. We created two
copies of the microcomb spectrum shown in Fig. 2d, using a fiber beamsplitter,
and separately bandpass filtered them at 1560 nm and 1553.6 nm; see Fig. 5a.
Independent high-speed photodetectors created electronic signals at 36 GHz.
The frequencies of these signals were reduced to baseband with a common 35.1
GHz oscillator at mixers M1 and M2, and the path corresponding to filter
$\lambda_{1}$ was further shifted by 11 MHz with a low-phase-noise oscillator.
A final mixer (M4) interfered the two baseband signals to reject common-mode
fluctuations. Hence, only residual phase noise of the microcomb (and noise
associated with the independent paths) appeared at the output of M4. Using a
commercial phase noise analyzer we recorded the single-sideband phase noise
spectral density $L_{\phi}$, which is shown by the black line in Fig. 5b. For
reference, the red line in Fig. 5b shows the phase noise contribution of the
11 MHz offset signal. The microcomb’s residual phase noise is extremely low.
At carrier offsets greater than 10 kHz, $L_{\phi}$ is dominated by
photodetection noise and would be reduced with greater microcomb optical
power. Close to carrier the residual phase noise is mostly below -100 dBc/Hz,
a value comparable to the absolute phase noise of the best optical and
microwave oscillators Fortier _et al._ (2011). This indicates that with
appropriate frequency control such a microcomb could be harnessed for portable
low-noise synthesis applications.
Figure 6: Microcomb fluctuations measured with respect to fixed references.
For all of these experiments the entire microcomb spectrum was delivered to
PD2. (a) Free-running microwave spectrum of the microcomb operating at power
level L3 obtained via photodetection. (b) Power spectral density (PSD) of
$L_{\phi}$ (black), line-spacing signal amplitude (red), and pump laser RIN
(blue). The RIN attains its shot noise limited value of -151 dBc/Hz beyond 100
kHz. The cyan shaded region shows a scaling of the RIN to phase noise using
the measured conversion factor $\gamma_{P}$ for our system. The top range of
this area corresponds to a constant $\gamma_{P}$ with frequency, and the
bottom range accounts for the measured frequency response of pump power to
line spacing. The green line shows the phase noise contribution of the 35.5
GHz oscillator. (c) Microwave spectrum at L2 with a 30 kHz resolution
bandwidth. Under these conditions the comb exhibits multiple line spacings
near 36 GHz. (d) $L_{\phi}$ (black) and amplitude (red) noise of the microcomb
at L2. The blue trace shows pump laser RIN.
We have assessed whether the microcomb line spacing is the same at 1560 nm and
1553.6 nm by electronically counting the 11 MHz signal exiting M4. Here an
offset from 11 MHz would indicate non-equidistance of the microcomb spectrum.
Previous work by Del’Haye et. al. studied microcomb equidistance by way of an
auxiliary fiber-laser frequency comb Del’Haye _et al._ (2007). With access to
the 36 GHz comb spacing, our measurements directly characterize microcomb
equidistance for the first time. We acquired 546 consecutive (zero dead time)
one-second long measurements; Fig. 5c shows a histogram of the frequency
difference ($\Delta f$) between the microcomb and reference oscillator. The
data exhibit a Gaussian distribution with mean of -0.65 $\mu$Hz with a 25
$\mu$Hz width. Hence, the mode spacing of the comb does not change
fractionally by more than 3$\times 10^{-17}$ across 6.4 nm. The Allan
deviation $\sigma_{A}$, shown in Fig. 5d, demonstrates the intrinsic stability
of the microcomb line spacing with $\sigma_{A}$=$1\times 10^{-17}$ after only
400 s of averaging.
Many future applications of a microcomb will require a stable spectrum with
respect to fixed frequency references, such as atomic clocks. Here we show the
microcomb line spacing’s microwave spectrum, absolute phase noise, amplitude
noise, and its dependence on pump laser power. Importantly, different behavior
of these was observed at power levels L2 and L3 that was not revealed in
coarse measurements of the optical spectrum (Fig. 2c and d). The experimental
setup for these measurements was similar to Fig. 5a, except that the
beamsplitter, optical filters, and mixers M1,3,4 were removed. The
photocurrent generated at PD2 was converted to baseband with a 35.5 GHz
oscillator for analysis. For phase noise measurements we used a digital
prescaler (divide-by-265) following M1 to reject spurious amplitude
fluctuations.
First, we present our observations at L3. Figure 6a shows a single microwave
tone 55 dB above a broad asymmetric pedestal. The width of this feature is
limited by the resolution bandwidth of 10 kHz. The amplitude and phase
fluctuations of this tone are shown by the red and black traces in Fig. 6b,
respectively. Notably, the absolute phase noise is between 20 and 120 dB above
the residual noise, depending on the carrier offset. For comparison the green
line shows the phase noise of our commercial 35.5 GHz oscillator. The pump
laser power, with the relative intensity noise (RIN) spectrum shown by the
blue trace in Fig. 6b, significantly influences the comb line spacing. We
measured a line spacing-power dependence of $|\gamma_{P}|$ = 0.3 MHz/mW with a
3 dB bandwidth of $\sim$20 Hz. The correlation between pump laser RIN and
microcomb amplitude and phase noise is evident in Fig. 6b. Scaling the RIN by
$\gamma_{P}$ results in the phase noise prediction shown by the cyan-shaded
area. Pump laser power fluctuations explain a significant fraction of the line
spacing phase noise, and either passive or active reduction of the RIN will be
crucial for future experiments. In particular, obtaining a $f$-2$f$ heterodyne
beatnote Diddams _et al._ (2000); Jones _et al._ (2000); Del’Haye _et al._
(2009), a key outstanding milestone in microcomb systems, will be challenging
in the face of significant comb noise.
Figure 6c and d shows a similar analysis of the microcomb line spacing’s
spectrum (c), amplitude (red trace in (d)) and phase fluctuations (black trace
in (d)) at power level L2. Under this operating mode, the comb exhibits
multiple line spacings in a few MHz range around 36 GHz. We observe
significantly larger (10’s of dB) amplitude and phase noise, even though the
pump laser RIN is the same as at L3. The correspondence between pump RIN and
amplitude, phase noise remains evident. We have not been able to develop a
detailed understanding of, nor any strategy to mitigate, the poor performance
at L2. However, these observations emphasize that measurements beyond those of
the optical spectra will be critical to understanding the properties of the
microcomb in all its operating regimes.
In summary, we have fabricated mm-scale, high-$Q$ optical microresonators with
fused quartz. By using them, a frequency comb with 36 GHz line spacing, 50 nm
span, and $<$10 mW threshold was created near 1560 nm. We studied the
microcomb spectrum in both the time and frequency domains. Picosecond optical
pulses were generated by way of full line-by-line control of up to 15 comb
lines. Precise frequency-domain techniques enabled a direct test of comb line
fractional equidistance at the 3$\times 10^{-17}$ level, and a
characterization of the microcomb’s absolute line spacing phase noise.
Quantifying the noise of microresonator frequency combs is crucial in
assessing whether they can distribute modern microwave and optical atomic
standards. In the future we will focus generating a wider comb span, and
stabilizing the comb to Rb transitions at 780 nm by way of higher bandwidth
comb control.
We acknowledge useful conversations with Kerry Vahala and Tobias Kippenberg,
and we thank Gabe Ycas and Nathan Lemke for a thoughtful reading of the
manuscript. This work is a contribution of NIST and is not subject to
copyright in the United States.
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* Del’Haye _et al._ (2008) P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett. 101, 053903 (2008).
* Savchenkov _et al._ (2008b) A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, Optics Express 16, 4130 (2008b).
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* Note (1) We have not observed any scattering-induced mode splitting in these devices Gorodetsky _et al._ (2000); Kippenberg _et al._ (2002).
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* Note (2) The numerical simulation of parametric comb generation by Chembo appears to explain the spectrum of Fig. 2d with 36 GHz lines modulated at an approximately 864 GHz period, but not the sawtooth deviation at power level L2 or the associated instability of comb generation here Chembo _et al._ (2010); Chembo and Yu (2010).
* Foster _et al._ (2011) M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, arXiv:1102.0326v1 (2011).
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* Note (3) The mode spacing at threshold can be tuned over a small range near 864 GHz with $K$.
* Note (4) We note this value also includes un-characterized fluctuations intrinsic to the programable filter and the measurement apparatus.
* Diddams _et al._ (2000) S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. H nsch, Phys. Rev. Lett. 84, 5102 (2000).
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* Del’Haye _et al._ (2009) P. Del’Haye, T. Herr, E. Gavartin, R. Holzwarth, and T. Kippenberg, arXiv:0912.4890v1 (2009).
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|
arxiv-papers
| 2011-06-13T16:56:21 |
2024-09-04T02:49:19.597387
|
{
"license": "Public Domain",
"authors": "Scott B. Papp and Scott A. Diddams",
"submitter": "Scott Papp",
"url": "https://arxiv.org/abs/1106.2487"
}
|
1106.2518
|
# Impact Parameter Dependence of Inelasticity in $pp$ / $p\overline{p}$
Collisions
P. C. Beggio beggio@uenf.br Laboratório de Ciências Matemáticas, Universidade
Estadual do Norte Fluminense Darcy Ribeiro - UENF,
Campos dos Goytacazes - RJ, Brazil.
###### Abstract
We study the impact parameter dependence of inelasticity in the framework of
an updated geometrical model for multiplicity distribution. A formula in which
the inelasticity is related to the eikonal is obtained. This framework permits
a calculation of the multiplicity distributions as well as the inelasticity
once the eikonal function is given. Adopting a QCD inspired parametrization
for the eikonal, in which the gluon-gluon contribution dominates at high
energy and determines the asymptotic behavior of the cross sections, we find
that the inelasticity decreases as collision energy is increased. Our results
predict the KNO scaling violation observed at LHC energies by CMS
Collaboration.
###### keywords:
Eikonal approximation $pp$ / $\bar{p}p$ Inelasticity Multiplicity distribution
13.85.Hd 12.40.Ee 13.85.Ni
## 1 Introduction
Inelasticity is defined as the fraction of the available energy relesead for
multiple particle production in inelastic hadronic interactions. The remaining
part of the incident energy is carried away by the participant’s remnants, so
called leading particles. The energy dependence of inelasticity is a problem
of great interest from both theoretical and experimental standpoints [1].
However, the experimental data are scarce and, on the theoretical side, the
existing models are largely in conflict with each other even in explaining a
simple aspect as the center-of-mass energy dependence of the inelasticity [2],
[3]. For example, the decrease in inelasticity with energy is advocated by
some authors, while others believe that the inelasticity is an increasing
function of energy [1]. Hence the problem remains unsolved. Naturally,
multiplicity distributions are connected to inelasticity ones, so one can
study multiplicity distribution features in order to derive information on the
inelasticity behavior. Following this way, we have updated an existing
phenomenological procedure [4], referred as _Simple One String Model_ , which
allows simultaneous description of several experimental data from elastic and
inelastic channels through the Unitarity Equation [4]. Thus, based on the
_Simple One String Model_ formalism, able to describe the charged multiplicity
distributions from ISR to Collider energies (30.4 - 900 GeV), we have computed
the impact parameter dependence of inelasticity at fixed center-of-mass
energies, $\sqrt{s}$. We have also inferred information on energy dependence
of inelasticity. The plan of the paper is as follows. In the section 2 we
present the basic formalism of the _Simple One String Model_ and the
predictions for $pp/p\overline{p}$ overall multiplicity distributions compared
with the experimental data. In Section 3 we apply the theoretical framework
computing the inelasticity of hadronic reactions. The final remarks are the
content of Section 4.
## 2 _Simple One String Model_ for Multiplicity Distributions
The _Simple One String Model_ has been discussed in references [4], [5] and in
order to define the notation and also update the model, we shall review the
main points here. We work in impact parameter space, b, and to guarantee
unitarity, the inelastic cross sections, $\sigma_{in}$, is calculated via the
relation:
$\sigma_{in}(s)=\int d^{2}b\,Gin(s,b)\,,$ (1)
where
$Gin(s,b)=1-e^{-2\chi_{I}(s,b)}$ (2)
is the Inelastic Overlap Function. In this work we have update the model
adopting the complex eikonal function
$\chi_{pp}^{\overline{p}p}{(s,b)}=\chi_{R}{(s,b)}+i\chi_{I}{(s,b)}$, from Ref.
[6], as will be discussed in subsection 2.1. In Ref. [4] has been adopted the
Henzi Valin parametrization for $Gin(s,b)$. The probabilities of $n$ particle
production, namely multiplicity distribution, $P_{n}$, is the most general
feature of the multiparticle production processes [7] and measurements of
charged particle multiplicity distributions have revealed intrinsic features
in $pp$ / $p\overline{p}$ interactions [8]. The multiplicity distribution is
defined by the formula [9]
$P_{n}(s)={\frac{\sigma_{n}(s)}{\sum_{n=0}^{\infty}\sigma_{n}(s)}}={\frac{\sigma_{n}(s)}{\sigma_{in}(s)}},$
(3)
where $\sigma_{n}$ is the cross section of an $n$-particle process (the so-
called topological cross section). The charged multiplicity distribution, in
the impact parameter formalism, may be constructed by summing contributions
coming from hadron-hadron collisions taking place at fixed impact parameter.
In this way, the idea of a normalized multiplicity distribution at each impact
parameter $b$ is introduced [10]. Thus the multiplicity distribution is
written as
$P_{n}(s)={\frac{\sigma_{n}(s)}{\sigma_{in}(s)}}={\frac{\int
d^{2}b\,G_{in}(s,b)\left[{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}\right]}{\int
d^{2}b\,G_{in}(s,b)}}\,,$ (4)
where the topological cross section $\sigma_{n}$ is decomposed into
contributions from each impact parameter $b$ with weight $G_{in}(s,b)$. In the
original formulation [4] the quantity em brackets scales in KNO sense, and the
Eq. (4) can be written as
$P_{n}(s)={\frac{\int
d^{2}b\,{\frac{G_{in}(s,b)}{<n(s,b)>}}\left[<n(s,b)>{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}\right]}{\int
d^{2}b\,G_{in}(s,b)}}\,,$ (5)
where $<n(s,b)>$ is the average number of particles produced at $b$ and
$\sqrt{s}$ due to the interactions among hadronic constituents involved in the
collision and, in this model, $<n(s,b)>$ factorizes as [4]
$<n(s,b)>=<N(s)>f(s,b),$ (6)
where $<N(s)>$ is the average multiplicity at $\sqrt{s}$ and $f(s,b)$ is the
so called multiplicity function. Similarly to KNO, is introduced, for each
$b$, the elementary multiplicity distribution
$\psi^{(1)}\left(\frac{n}{<n(s,b)>}\right)=<n(s,b)>{\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}}.$
(7)
Thus, with Eqs. (6) and (7), Eq. (5) becomes
$\Phi(s,z)={\frac{\int
d^{2}b\,{\frac{G_{in}(s,b)}{f(s,b)}}\,\psi^{(1)}\\!\left(\frac{z}{f(s,b)}\right)}{\int
d^{2}b\,G_{in}(s,b)}},$ (8)
where $\Phi(s,z)=<N(s)>P_{n}(s)$ and $z=n/<N(s)>$. Here $z$ represents the
usual KNO scaling variable. Now, to obtain the multiplicity function $f(s,b)$
in terms of the imaginary eikonal $\chi_{I}$, it has been assumed that
1. 1.
the fractional energy $\sqrt{s^{{}^{\prime}}}$ that is deposited for particle
production in a collision at $b$ is proportional to $\chi_{I}$:
$\sqrt{s^{{}^{\prime}}}=\beta(s)\,\chi_{I}(s,b)\,.$ (9)
The physical motivation of this equation is that eikonal may be interpreted as
an overlap, on the impact parameter plane, of two colliding matter
distributions [10];
2. 2.
the average number of produced particles depends on the energy
$\sqrt{s^{{}^{\prime}}}$ in the same way as in $e^{-}e^{+}$ annihilations,
which is approximately represented by a power law in $\sqrt{s}$ [4]
$<n(s,b)>=\gamma\,\left(\frac{s^{{}^{\prime}}}{s^{{}^{\prime}}_{0}}\right)^{A},$
(10)
where $s^{{}^{\prime}}_{0}$=1 GeV2. In Ref. [11] a power law energy dependence
of multiplicity in both $pp$ and $Pb+Pb$ collisions has been analyzed based on
the gluon saturation scenario. Now, combining Eqs. (9), (10) and (6), we have
$f(s,b)=\frac{\gamma}{<N(s)>}\left[\frac{\beta(s)}{\sqrt{s^{{}^{\prime}}_{0}}}\right]^{2A}[\chi_{I}(s,b)]^{2A}=\xi(s)[\chi_{I}(s,b)]^{2A},$
(11)
with $\xi(s)$ determined by the usual normalization conditions on $\Phi$ [4]
and that serves to determine $\xi(s)$ as an energy dependent quantity,
explicitly [4]
$\xi(s)=\frac{\int d^{2}b\,G_{in}(s,b)}{\int
d^{2}b\,G_{in}(s,b)[\chi_{I}(s,b)]^{2A}}.\ $ (12)
Thus, adopting an appropriate parametrization for $G_{in}$ and $\psi^{(1)}$,
as well as an adequate value for $A$, we can test the formalism embodied in
Eq. (8) and (12), making direct comparisons with multiplicity distribution
data. In the following, we will discuss the results obtained in the context of
our updated model.
### 2.1 Inputs and Results on Multiplicity Distributions
The _Simple One String Model_ is based on the idea of multiparticle creation
due to the interactions between hadronic constituents in collisions taking
place at $b$. It is assumed that in parton-parton collision there is formation
of a string, in which probably one $q\overline{q}$ has triggered the multitude
of the final particles. In previous analysis the authors [4] considered the
experimental data available on $e^{-}e^{+}$ annihilations as possible source
of information concerning elementary hadronic interactions and, in this work,
we borrow their results. Specifically, assuming a gamma distribution
normalized to 2,
$\psi^{(1)}(z)=2\,\frac{k^{k}}{\Gamma(k)}z^{k-1}e^{-kz},$ (13)
experimental data on $e^{-}e^{+}$ multiplicity distributions were fitted,
obtaining _k_ =10.775 $\pm$ 0.064 $(\chi^{2}/N_{DF}=2.61)$. Also, the average
multiplicity data in $e^{-}e^{+}$ annihilations were fitted by Eq. (10),
giving $A$=0.258 $\pm$ 0.001 and $\gamma$=2.09 $(\chi^{2}/N_{DF}=8.89)$ in the
interval 5.1 GeV $\leq\sqrt{s}\leq$ 183 GeV, and $A$=0.198 $\pm$ 0.004
$(\chi^{2}/N_{DF}=1.7)$ for the set in the interval 10 GeV $<\sqrt{s}\leq$ 183
GeV, respectively [4]. In the analysis done in Ref. [11] the value of $A$=0.11
was obtained within the gluon saturation picture. The difference between the
values of $A$ is, probably, associated with the different sets of experimental
data used in each analysis. We recall that the value of $A$=0.11 was obtained
by using experimental data for average multiplicity of hadrons in the gluon
and quark jets in $e^{-}e^{+}$ annihilation, in the interval of the jet energy
between 0.6 $\sim$ 32 GeV [11]. At the end of the section the _One String
Model_ formalism will be tested using the three $A$ values, above mentioned.
Now is needed a parametrization for the eikonal function, and we have adopted
the QCD-inspired complex eikonal from the work of Block _et al._ [6] in which
the eikonal function is written as a combination of an even and odd eikonal
terms related by crossing symmetry
$\chi_{pp}^{\overline{p}p}{(s,b)}=\chi^{+}{(s,b)}\pm\chi^{-}{(s,b)}$. The even
eikonal is written as the sum of gluon-gluon, quark-gluon and quark-quark
contributions:
$\chi^{+}{(s,b)}=\chi_{qq}{(s,b)}+\chi_{qg}{(s,b)}+\chi_{gg}{(s,b)},$ (14)
while the odd eikonal, that accounts for the difference between $pp$ and
$p\overline{p}$, is parametrized as
$\chi^{-}{(s,b)}=C^{-}\sum\frac{m_{g}}{\sqrt{s}}e^{i\pi/4}W(b;\mu^{-}).$ (15)
The various parameters and functions involved in last two expressions are
discussed in Ref.[6]. By fixing the value of $A$=0.258, $\psi^{(1)}$ given by
Eq. (13) with $k$=10.775, adopting $G_{in}$ from analysis by Block _et al_.
and observing that $\xi(s)$ is obtained by Eq. (12), we have computed the
overall multiplicity distributions arising from $pp/\overline{p}p$ collisions
at energies 52.6, 200, 546 and 900 GeV. The theoretical curves are shown in
Figs. 1, 2, 3 and 4 together with the experimental data. The curves at
$pp$-ISR $52.6$ GeV and CERN $p\overline{p}$ Collider $546$ GeV shows
excellent agreement with the data, Figs. 1 and 3 respectively. At energies
$\sqrt{s}$=200 and 900 GeV, also in CERN $p\overline{p}$ Collider, the
agreement with data seems reasonable since the curves agree with experimental
points for $z^{\prime}\gtrsim 1$, (high multiplicities), Figs. 2 and 4
respectively. In view of the recent results on multiplicity distributions, in
$pp$ collisions at $\sqrt{s}$=0.9, 2.36 and 7 TeV at the Large Hadron Collider
(LHC), reported by CMS Collaboration [15],
Figure 1: Overall scaled multiplicity distribution data for $pp$ at ISR energy
[12], compared to theoretical prediction using the _Simple One String Model_ ,
Eqs. (8) and (12). Figure 2: Overall scaled multiplicity distribution data for
$p\overline{p}$ at Collider energy [13], compared to theoretical prediction
using the _Simple One String Model_ , Eqs. (8) and (12).
we have also computed the overall multiplicity distributions that the _One
String Model_ predicts at LHC energies. The results are shown in Fig. 5 and we
can see violation of KNO scaling, in qualitative agreement with the result
obtained by CMS Collaboration in pseudorapidity interval of $|\eta|<$2.4, as
discussed in Ref. [15]. As mentioned, two fit values of $A$=0.258 and
$A$=0.198 have been obtained in the previous study [4], the first one giving a
better account of lower energy data whereas the second one higher energy data.
As before, we have computed the corresponding hadronic multiplicity
distribution by fixing both the gamma parametrization for $\psi^{(1)}$, Eq.
(13), and the complex eikonal, Eqs. (14) and (15), and considered the two
parametrizations for the average multiplicity, $\sim s^{0.258}$
($\xi(s)$=1.424) and $\sim s^{0.198}$ ($\xi(s)$=1.348). The results at 546 GeV
are shown in Fig. 6 and, for $A$=0.198, we can see the disagreement of the
theoretical curve when compared to the data. As pointed out in [4] the
parametrization $\sim s^{0.198}$ brings information from data at high
energies, while the parametrization $\sim s^{0.258}$ is in agreement with data
at smaller energies. However, the information from the $e^{-}e^{+}$ average
multiplicities at high energies does not reproduce the overall multiplicity
distribution. Hence, by using $A$=0.256 the output seems to be more consistent
with data. In addition, there is no evidence of gluon saturation at CERN
$p\overline{p}$ Collider 546 GeV, however and as a pedagogical exercise, we
have also computed $\Phi$ considering the value of $A$=0.11 ($\xi(s)$=1.211),
Fig. 6. We would like emphasize that when the formalism is applied,
considering the three values of $A$ (0.258, 0.198 and 0.11) at energies 52.6,
200 and 900 GeV, the results are essencially the same obtained in Fig. 6. We
have expressed $\Phi$ in terms of modified the scaling variable
$z^{\prime}=n-N_{o}/<n-N_{o}>$ with $N_{o}$=0.9 representing the average
number of leading particles [4].
## 3 Inelasticity
The concept of inelasticity is essential since it defines the energy available
for particle production in high energy hadronic and nuclear collisions.
However, the impact parameter dependence of the inelasticity is a problem
unsolved. In theoretical works, it is quite natural to assume that the
multiplicity distribution and inelasticity are connected. Indeed, some authors
has defined multiplicity distributions in terms of inelasticity as [1], [16]
$P_{n}(s)=\int_{0}^{1}P(n|K)P[K(s)]dK,$ (16)
where $P[K(s)]$ is the inelasticity distribution and $P(n|K)$ is the
probability of the production of $n$ particles at the given inelasticity $K$.
Thus, based on the connection between multiplicity and inelasticity, we have
explored the parametrization of the _One String Model_ formalism and computed
the impact parameter dependence of inelasticity, as will discuss in next two
subsections.
Figure 3: Overall scaled multiplicity distribution data for $p\overline{p}$ at
Collider energy [14], compared to theoretical prediction using the _Simple One
String Model_ , Eqs. (8) and (12). Figure 4: Overall scaled multiplicity
distribution data for $p\overline{p}$ at Colider energy [13], compared to
theoretical prediction using the S _imple One String Model_ , Eqs. (8) and
(12). Figure 5: Theoretical predictions for overall multiplicity distribution
by using the _Simple One String Model_ , Eqs. (8) and (12), at LHC energies.
Figure 6: Overall scaled multiplicity distribution data for $p\overline{p}$ at
Collider energy [14], compared to theoretical prediction using the _Simple One
String Model_ , Eqs. (8) and (12), considering three different values of $A$,
Eq. (10).
### 3.1 Similarities between $pp/p\overline{p}$ and $e^{+}e^{-}$ Collisions
The idea of a universal hadronization mechanism is not new and similarities
between both processes were indeed observed [16], [17], [18]. For example, the
average multiplicities in $pp/p\overline{p}$ and $e^{+}e^{-}$ collisions
become similar when comparisons are made at the same effective energy for
hadron production. In $pp/p\overline{p}$ collisions the effective energy for
particle production, $E_{eff}$, is the energy left behind by the two leading
protons
$E_{eff}=(\sqrt{s})_{pp}-(E_{leading,1}+E_{leading,2}),$ (17)
or
$E_{eff}=(\sqrt{s})_{pp}-2E_{leading},$ (18)
in the case of symmetric events. (We recall that $(\sqrt{s})_{pp}$ and
$\sqrt{s}$ represents both the center-of-mass energy, however, in this
subsection the notation $(\sqrt{s})_{pp}$ is helpful to differentiate that
from $(\sqrt{s})_{e^{+}e^{-}}$). In $e^{+}e^{-}$ collisions the effective
energy for hadron production coincides with total center-of-mass energy of the
beam
$E_{eff}=(\sqrt{s})_{e^{+}e^{-}}=2E_{beam}.$ (19)
Thus, the same equivalent energy for both $pp/p\overline{p}$ and $e^{+}e^{-}$
collisions can be written as
$(\sqrt{s})_{pp}-2E_{leading}=E_{eff}=(\sqrt{s})_{e^{+}e^{-}}.$ (20)
For the quantitative estimation of the inelasticity, $K$, we can use the
definition [1], [2]
$E_{eff}=K(\sqrt{s})_{pp}\Rightarrow K=\frac{E_{eff}}{(\sqrt{s})_{pp}}.$ (21)
In the following, we will explore the _One String Model_ formalism, able to
describe the multiplicity distributions in wide interval of energy (30.4 - 900
GeV), to obtain information about inelasticity.
### 3.2 Computation of Inelasticity
The Eq. (9) is a key point of the formalism. Physically, it corresponds to the
energy for hadron production deposited at $b$, due to the interactions among
hadronic constituents involved in the collision. Thus, and as discussed in
last section, the fractional energy, $\sqrt{s^{\prime}}$ (Eq. (9)), and the
effective energy for hadron production, $E_{eff}$ (Eq. (21)), represents both
the same physical quantity ($\sqrt{s^{{}^{\prime}}}=E_{eff}$). Now, by using
the Eq. (9), let us write the inelasticity, Eq. (21), as a function of
$\sqrt{s}$ and $b$ as
$2K(s,b)=\frac{\sqrt{s^{{}^{\prime}}}}{(\sqrt{s})_{pp}}=\frac{\beta(s)\chi_{I}(s,b)}{\sqrt{s}}.$
(22)
The factor 2, in the above equation, is due to the fact that the multiplicity
distributions data are normalized to 2. However, we can not calculate the
$K(s,b)$ until the value of $\beta(s)$ is known. To estimate $\beta(s)$ we
note that the parameter $\xi(s)$, which is introduced in Eq. (11), is related
with $\beta(s)$ by
$\xi(s)=\frac{\gamma}{<N(s)>}\left[\frac{\beta(s)}{\sqrt{s^{{}^{\prime}}_{0}}}\right]^{2A}.$
(23)
We recall that $<N(s)>$ is the average multiplicity at $\sqrt{s}$. By using
the values of $A=0.258$, $\gamma$=$2.09$, discussed in subsection 2.1, and the
values of $<N(s)>$ imputed from experiments [12], [13] and [14], and also
observing that $\xi(s)$ is obtained from Eq. (12), we have estimated the
values of $\beta$ at various energies. The results are displayed in Table 1.
We can see clearly that $\beta$ increases as the collision energy also
increases. $\beta$ can be parameterized as $\beta(s)=77.48\sqrt{s}+0.4168$
with $\chi^{2}/N_{DF}$=1.
Table 1: $\beta(s)$ estimated values at various energies. The values of $<N(s)>$ was imputed from Refs. [12], [13] and [14]. $\sqrt{s}$ GeV | $\xi(s)$ | $<N(s)>$ | $\beta(s)$ GeV
---|---|---|---
52.6 | $1.612$ | $11.55$ | $69.295$
200 | $1.517$ | $21.4$ | $203.531$
546 | $1.424$ | $27.5$ | $292.729$
900 | $1.377$ | $35.6$ | $452.376$
2360 | $1.286$ | $-$ | $1061.13$
7000 | $1.188$ | $-$ | $2995.08$
14000 | $1.130$ | $-$ | $5912.68$
Now we proceed to compute the impact parameter dependence of inelasticity and
infer some information on its energy dependency.
### 3.3 Results and Discussion
Based on the connection between multiplicity distribution and inelasticity, we
have update and applied the _One String Model_ formalism deriving an
expression, Eq. (22), which allows us to study the impact parameter and energy
dependence of inelasticity. Adopting the Block _et al._ QCD-inspired
parametrization for $\chi_{pp}^{\overline{p}p}{(s,b)}$ [6] and by using the
estimated values of $\beta(s)$, Table 1, we have applied the Eq. (22) by
fixing the collision energy and computed the inelasticity as a function of
$b$. We show, in Fig. 7, the results from our analysis. Naturally, the
inelasticity decreases as a function of impact parameter, $b$. The
inelasticity behavior is essentially the same at energies 546 and 900 GeV. At
energies 52.6 and 200 GeV, we can see appreciable difference just in the
region of $b\sim 0$ (central collisions). It is interesting to note that, from
the range of collision energy 50 $\sim$ 200 GeV to that one 500 $\sim$ 900
GeV, the inelasticity shows a difference about 60 percent in its values for
$b$ $\lesssim 0.5$ _fm_. We can also see that, at fixed $b$, the inelasticity
$K$ decreases as $\sqrt{s}$ increases in the interval 52.6 - 900 GeV. The Eq.
(22) depends on the eikonal and $\beta$ parameter. With the
Figure 7: Impact parameter dependence of the inelasticity by using the formula
obtained in this work, Eq. (22).
eikonal as determined phenomenologically in [6] as input, where high energy
cross sections grow with energy as a consequence of the increasing number of
soft partons populating the colliding particles $(pp/p\overline{p})$, it seems
quite natural to expect that multiparton interactions leads to larger
multiplicities/inelasticities as consequence to the full development of the
gluonic structure. However, looking the same impact parameter dependence of
inelasticity functions at 546 and 900 GeV, Fig. 7, we would be tempted to
conclude that we are observing saturation effects due gluon recombination in
the inelasticity, but there is no evidence of saturation in this range of
energy (52.6 - 900 GeV). We have also applied our approach at energies
$\sqrt{s}$=2.36 and 7 TeV (LHC), as shown in Fig. 7, and the results suggest
that the inelasticity is an increasing function of energy for the interval
2.36 - 7 TeV. As mentioned before, our main purpose is study features of
multiplicity distributions deriving information on inelasticity and our
analysis is based on the model in which is assumed that in parton-parton
collision there is formation of a string. Thus, despite some simplifications
made in the _One String Model_ , the results seems to be consistent with the
multiplicity distributions data in a wide interval of energy (52.6 - 900 GeV).
Hence, the computed inelasticities, in this range of energy, are reliable
results. In counterpart, the results at $\sqrt{s}$=2.36 and 7 TeV are
inconclusive in the context our analysis, because the _One String Model_
probably underestimates the high multiplicities events due to the lack of the
multicomponent structure in its formulation. In fact, recent results reported
by CMS Collaboration pointed out the importance of a multicomponent structure
in hadron-hadron inelastic interactions, in agreement with previous
experimental results (for details see [15]). Inelasticity also has been
studied recently in Ref. [19], where, in the context of both Wdowczyk and
Wolfendale model and UHECR data analysis, it was found that the inelasticity
decreases in very high energy interactions, and, in the same work and by using
the modified Feynman scaling formula, the inelasticity is an increasing
function of the energy. It reflects the subtlety of the theme. We note that at
ISR Energies (30-60 GeV), where the leading particle spectrum could be
measured, the inelasticity is defined to be about 0.5. This value can be
identified with $pp$ collision taking place at $b\sim$ 0.6 _fm_ , Fig. 7. The
_One String Model_ has been used to study the influence on $\Phi$ considering
possible values of $A$ parameter at $\sqrt{s}$=546 GeV, Fig. 6. In addition,
we have also computed $\Phi$ at energies 52.6, 200 and 900 GeV using the
different values of $A$ (0.258, 0.198 and 0.11) and the results, in each
energy, are similar with that obtained in Fig. 6. Finally, we emphasize that
the curves in Figs. 1-4 has not been fit to data, except for the values of $A$
and $k$ (fixed) no experimental information about multiplicity distribution
has gone into the calculation. Hence, the energy evolution of the multiplicity
distributions, from ISR to Collider ($30\sim 900\,$GeV), is correctly
reproduced by changing only the Overlap Function, Eq.(2), without changing the
underlying elementary interaction, in agreement with what could be expected
from QCD.
## 4 Concluding Remarks
Being the impact parameter $b$ an essential variable in a geometrical
description of hadronic collisions, we have investigated the $b$ dependence of
inelasticity and also inferred some information on its energetic behavior. By
using a geometrical model we have derived an expression for $K$ based on the
hypothesis of connection between multiplicity distribution and inelasticity.
We have adopted the Block _et al._ model in our analysis, where the eikonal
functions $\chi_{qq}{(s,b)}$ and $\chi_{qg}{(s,b)}$ are needed to describe the
lower energy forward data, while $\chi_{gg}{(s,b)}$ contribution dominates at
high energy and determines the asymptotic behavior of cross sections. We
believe that the same impact parameter dependence of the inelasticity at
energies 546 and 900 GeV can have important implications for the underlying
gluon-gluon dynamics. In fact, we are testing our formalism by using the QCD
Eikonal Model in which the gluon may develop a dynamical mass [20], [21]. At
energies $\sqrt{s}$=2.36 and 7 TeV the _One String Model_ parametrization can
not be tested, hence the results do not allow for any conclusion. Finally, the
results suggest that there are relationships between the inelasticity and the
eikonal function.
I am grateful to E.G.S. Luna for several instructive discussions.
## References
* [1] G. Musulmanbekov, Physics of Atomic Nuclei 67 no. 1, (2004) 90.
* [2] F.O. Duraes, F.S. Navarra and G. Wilk, Braz. J. Phys, 35 no. 01, (2005) 3.
* [3] Y. Hama and S. Paiva, Phys. Rev. Lett. 78, no. 16 (1997) 3070.
* [4] P.C. Beggio, M.J. Menon and P. Valin, Phys. Rev. D61, (2000) 034015.
* [5] P.C. Beggio and Y. Hama, Braz. J. Phys, 37 no. 3B, (2007) 1164.
* [6] M.M. Block, E.M. Gregores, F. Halzen and G. Pancheri , Phys. Rev. D60, (1999) 054024-1.
* [7] I.M. Dremin, JETP Letters 84 no. 5, (2006) 235.
* [8] W.D. Walker, Phys. Rev. D69, (2004) 034007.
* [9] I.M. Dremin and J.W. Gary, Phys.Rep. 349, (2001) 301.
* [10] S. Barshay , Phys. Rev. Lett. 49, (1982) 380.
* [11] Eugene Levin and Amir H. Rezaeian, hep-ph/1102.2385v1 11/Feb/2011.
* [12] ABCDWH Collaboration; A. Breakstone _et al._ , Phys. Rev. D30, (1984) 528.
* [13] UA5 Collaboration, R.E. Ansorge et al., Z. Phys. C - Particles and Fields 43, (1989) 357.
* [14] UA5 Collaboration, G.J. Alner et al., Phys. Rep. 154 n s 5 and 6, (1987) 247.
* [15] CMS Collaboration, hep-ex/1011.5531v1 24/Nov/2010; JHEP 01 (2011) 079.
* [16] K. Kadija and M. Martinis, Phys. Rev. D48, no. 5 (1993) 2027.
* [17] M. Basile et al., Phys. Lett. 92B, (1980) 367; B.B. Back et al., nucl-ex/0301017 v1 (2003).
* [18] Jan Fiete Grosse-Oetringhaus and Klaus Reygers, hep-ex/0912.0023v1 03/Nov/2009; J. Phys. G: Nucl. Part. Phys. 37 (2010) 083001\.
* [19] Tadeusz Wibig, hep-ph/1102.1385v1 07/Feb/2011.
* [20] E.G.S. Luna et al., Phys. Rev. D72, (2005) 034019.
* [21] E.G.S. Luna, Phys. Lett. B641, (2006) 171.
|
arxiv-papers
| 2011-06-13T18:58:39 |
2024-09-04T02:49:19.604951
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. C. Beggio",
"submitter": "Paulo Beggio Cesar",
"url": "https://arxiv.org/abs/1106.2518"
}
|
1106.2601
|
# Knowledge Dispersion Index for Measuring Intellectual Capital
Vikram Dhillon Electronic address: dhillonv10@gmail.com
###### Abstract
In this paper we propose a novel index to quantify and measure the flow of
information on macro and micro scales. We discuss the implications of this
index for knowledge management fields and also as intellectual capital that
can thus be utilized by entrepreneurs. We explore different function and human
oriented metrics that can be used at micro-scales to process the flow of
information. We present a table of about 23 metrics, such as change in IT
inventory and percentage of employees with advanced degrees, that can be used
at micro scales to wholly quantify knowledge dispersion as intellectual
capital. At macro scales we split the economy in an industrial and consumer
sector where the flow of information in each determines how fast an economy is
going to grow and how overall an economy will perform given the aggregate
demand. Lastly, we propose a model for knowledge dispersion based on graph
theory and show how corrections in the flow become self-evident. Through the
principals of flow conservation and capacity constrains we also speculate how
this flow might seeks some equilibrium and exhibit self-correction codes. This
proposed model allows us to account for perturbations in form of local noise,
evolution of networks, provide robustness against local damage from lower
nodes, and help determine the underlying classification into network super-
families.
###### Index Terms:
network flow, perturbations, intellectual capital, metrics.
## I Introduction
Lord Kelvin is often quoted for saying that if one can’t measure something
they can’t improve it. This is critical in the field of knowledge management
as metric provide convincing data for entrepreneurs to make decisions and help
capitalize it for economic and social progress. The aspiration behind
developing this index arose in response to lack of a quantifying measure for
the following:
* •
The exchange of tactical knowledge concerning one particular field between
employees.
* •
Capture of that information in a knowledge-base for use by other employees.
* •
The reuse rate of frequently accessed knowledge.
This index will be divided into two levels, each further consisting of its own
metrics, the outputs from each level can be used as independent indicators and
the gross measure will complete the Knowledge Dispersion Index (KDI) for a
given location. It must be noted that this analysis involves human input
therefore some aspects can not be captured effectively, although we do include
an assumption called the Theory Y of management which asserts that management
assumes employees may be ambitious, self-motivated, exercise self-control and
if allocated adequate resources, they will perform at their very best. Given
that assumption, we establish the two levels of our index:
* •
Organizational knowledge metric: This metric will account for aggregate
dispersion at micro scales and indicate the growth of the industrial sector.
* •
General dispersion metric: This metric will account for dispersion at macro
scales and indicate how the consumer side is performing.
## II Micro Scale (Organizational Knowledge Metric)
At the micro scale industries play a prominent role, the reason for including
industries at the micro scale pertains to their role in the economy. The
distributer’s side is generally smaller in comparison to the consumer side. In
addition to that, there are some particular mechanisms that each of the firms
employ to enhance their success and if they are not fine-tuned, the results
can be disastrous. For that very reason we present a comprehensible list (see
Appendix A for the list) to account for the flow of information, the gross
result obtained from the table in the Appendix will present a reliable measure
of the information infrastructure of a company111In this paper we don’t
provide definite benchmarks which can then be used to compare, a case study
however, regarding those benchmarks is underway. . There is however another
very important role that the organizational knowledge metric plays, that of
identifying the most efficient method of dispersion and the faults that may
occur in the dispersion through the use of a theoricial construction called a
flow networks.
### II-A Flow Networks
Flow networks are the key structure in network analysis and here we show how
the traditional flow network model can be employed by a company to analyze the
flow of information in their infrastructure. First we review the traditional
flow network model [2], a flow network is a directed graph where each edge has
a capacity and each edge receives a flow. The amount of flow on an edge cannot
exceed the capacity of the edge. A flow must satisfy the restriction that the
amount of flow into a node equals the amount of flow out of it, except when it
is a source, which has more outgoing flow, or sink, which has more incoming
flow.
Formally, $G(V,\,E)$ is a directed graph where each edge
$(u$$\,v)\>\epsilon\,E$ has a non-negative, real valued capacity given by a
function $c(u\,v)$. There are however two special vertices where the capacity
constrains don’t hold: a source $s$ and a sink $t$. The rest of the nodes
$u,\,v$ follow the following rules:
* •
Capacity Limit: $f\,(u,\,v)\>\leq c\,(u,\,v).$ This relation implies that the
flow to one node can not exceed the capacity limit given to that node.
* •
Skew Symmetry: $f\,(u,\,v)\;=-f\,(u,\,v).$ This relation implies that the net
flow from one node to another or from $u$$\rightarrow v$ must be the same from
$v\rightarrow u$.
* •
Flow conservation: $\sum\,f\,(u,\,w)=k.$ This relation can be seen from the
two above, the net flow to a node is some constant $k$ which holds the same
value for each node at the same hierarchy but changes value at different
levels of management, except for the two cases of source and sink.
Now we will explore how a model following these rules can help us explore the
efficiency of the information infrastructure. First we see that the capacity
limit holds to how much information should be made avialable to employees at
each stage, this limit helps us understand better where in the graph is too
much information is present as that in itself is a loss or waste since one
employee can only utilize so much of what is given to them. Knowing the
capacity limit for the nodes can also help us by reducing the flow to one
stressed node by spreading it amongst other available nodes at the same level
of hierarchy. The skew symmetry serves as a test that can be employeed by the
management to asses whether each employee at their current level is utilizing
their resources to their maximum or not, the full employement of the resources
at each level would result in for instance completion of more patents being
filed or more solutions being completed by a company as the demand for that
particular solution increases. It must be noted here that this relation holds
dual values, the flow of information decreases as it reaches the top as can be
seen in the case of a software development company, the CEO of the company is
generally well-versed in business techniques and not in lower-level coding
therefore the higher-level overview being presented to them in the form of a
flowchart would be represented by a decrease in the flow of information
reaching them. This should not be considered a short-coming of the model
because each level of hierarchy will have a constant flow, so as information
moves up that hierarchy and reached the management positions of a company, the
flow will decrease and stabalize. The third rule indicates the communication
infrastructure present in a company occuring among various nodes (employees)
at each level. A company generally has several different departments or
smaller level committes to perform various tasks, this rule applies to how
well each of those departments interact with the other ones to finish a given
task. The other two rules provided before can also be applied here to each of
the departments to obtain more specifics. We argue that modelling a company as
a flow model and following a combination of these three rules will provide a
very good estimate of the information flow in a company. The data requried to
asses these rules will be provided by the gross result from the list in
Appendix A.
#### II-A1 Self Correction Codes
Using the flow model for information dispersion also gives us the ability to
use the inherent correction features present in the flow model. We assert that
a company modeled as a traditional flow model will have the tendency to
correct itself as needed, this is also a basic assumption in game theory, that
if each employee is seen a player in a game, each player will make the best
possible move. We generalize that assertion in the following:
###### Conjecture 1
If leadership is emplyoed in a multi-player show definitegame, the leader will
make the best possible choice for the players.
This will help in the evolution of the network, as the company prospers the
network will expand and stabalize for definite capacity limits at every level,
this is true of a successful company. For a newly developing company, the
governing dynamics will be different, each new employee is in a learning stage
but are required to follow skew symmetry and flow conservation. As a result of
that, there will be a discrepancy in the rules as the new employee will be a
pseudo-node and this discrepancy can be analyzed in terms of a pertubation to
the model. The flow model approach provides a solution for the model to be
robust against such changes, to address local pertubations, we propose the
following:
###### Conjecture 2
Given a recently established company, and a measure of reliability determined
from our Appendix, some nodes with a high measure of reliability should be
overloaded.
It should be noted here that overloading those nodes doesn’t necessarily imply
a decrease in efficiency of those employees, the idea is to vary the flow
amongst those reliable nodes and split flow conservation into two levels, one
for the newer employees and the other for the reliable ones. This allows the
learning stage of the new employees which acts as a pertubation to be absorbed
in by the overloading of those nodes. The output provided by the newer pseudo-
nodes 222The new employees only act as pseudo-nodes while they are under some
initial training, as soon as they are ready, we decrease the flow to the nodes
at which the new employees are to work, that allows a smooth introduction of
new nodes in the flow. will also go through a similar process but the
difference is that the output gets absorbed by the these overloaded nodes.
Now we will address how network super-families arise in this evolved flow,
from Conjecture 2.2 we determined a reliability measure that can help use
overload some nodes, those nodes at each hierarchical level will constitute
our network super-family. Formally, the members of the yet to be formed
superfamiliy will appear as outliers if a statistical analysis such as ANOVA
is performed because of their reliability profile established from the
Appendix metrics. Once super-families are identified in a flow, our network
becomes very strong and is robust to local and internal damage.
###### Conjecture 3
Given a flow network with identified network super-families, local damage can
be diverted to specified nodes, decreasing its impact.
Local or internal damage lacks a formal operational definition so we have to
rely on a situational definition. Any intentional efforts to disrupt a well
established flow of information constitutes local damage, if the causes of
which are identified, for instance an attack targeting some confidential
information can be diverted to the pseudo-nodes, rendering it useless since
the pseudo-nodes have lower level clearance.
## III Macro Scale (General Dispersion Metric)
We now analyze how information disperses amongst the general public or the
consumer side. It must be noted here that KDI would only be compelte as an
index once we can get results from both the metrics, implying that we need to
quantify how information flows amongst industries which form the micro scale
and the consumer side or the general public which form the macro scale. The
methods of analysis being employed before simply become inefficient here
because of the graphes modelling a large amount people would easily break down
and is quite inefficient computationally. There are however some metrics that
can help us here such as:
* •
Transport infrastructure
* •
Availability of communications facilities such as television and internet
* •
Public spending on broadcasting
* •
Access to internet
* •
Frequency of local awareness campaigns such as seminars and so on.
We will here employ another theoretical construction to analyze how
information is spread to the general population or the consumers sector. Some
charateristics of this study are that it must be general enough to account for
a vast variety of sources and at the same time be able to provide a definite
analysis for the comparison of the output from each source.
###### Definition 1
There are several sources that aid the process, such as television, newspaper,
etc. however, for simplicity of analysis, we will collectively refer to these
different sources as media.
### III-A Deep Current Sets
The most common source for information available to the public333We simply
refer it to as a source, its arguable which source constitues as the most
widely available for different regions. outputs vast amounts of interpreted
information, this output can be viewed as a surface wave or a current in an
ocean. This analogy of ocean currents is being used because the currents
themselves are very dynamic and based on very delicate balance, we imply that
in the same degree as ocean currents on the surface can influence the weather
pattern of the atmosphere above, the general croud is influenced by the larger
waves or the media source that is most influencial, imposing similar constant
weather conditions over a large portion of the atmosphere. This construction
helps analyze the behaviour and choices of the people in the given region in
the past to make deductions about the future, from that we determine how the
information flow that once only occured amongst a small group of people has
evolved now to spread to much larger networks. We present a worked example of
deep current sets in the light of a hypothetical developing situation.
###### Conjecture 4
Given an area and a developing situation, the flow of information is given as
follows:
Before the development: Those involved with an event may share or record their
ideas, theories, or plans alone in a lab or personal journal, with friends or
colleagues. The sharing of this information forms the "invisible college."
Usually, the ideas generated by the "invisible college" are simply not
available to the public and is nonexistent as a form of information today,
especially for things in the distant past. This sets the stage for the
currents to arise in the ocean.
As the development occurs: Early news releases may appear on TV or radio, in
newspapers, over newswires, and on the Web. The initial information about
these events generally accompnies the "who", "what", "where" and "when" and
most often exclude the "why"
The next day or the days after: Articles appear in newspapers/newswires;
information is disseminated on TV, radio, and the Web. Depending on the event,
the information may be prolific or sparse.
In a week or the weeks after: Articles may appear in general or subject-
focused popular magazines.
In a month or time that follows: Articles appear in scholarly journals. This
is also when scholars and researchers may start holding conferences on the
topic and eventually, conference papers will be published.
We see how through the development of this situation the ocean currents are
forming, the before sets up the currents and the next phase is when the
currents are out in the ocean and their relative size determines how the
weather is changes. From the changes in weather which are easily observable,
we can reliably make deduction about the past history of the people. The
advantage of this analysis provides is that even though a given region may
have changed now, we can still see the effects of the currents in the
future444This in some sense is similar to a pebble skipping across water,
whereever it lands, even for a brief moment of time, it leaves behind ripples
that can be the analyzed to trace its path., this analysis of the past can
also assist in the development of better dynamics modelling the behaviour of
that region. In addition to that, we can also determine what forms of media
are most prevalent in that given region. This analysis can be conducted based
on a past event and hold its validity for a long time.
As we see, the quantization of information flow here can be further assisted
through the use of metrics such as % access to cable, people with advanced
degrees, average household income and so on. This can help determine the
interests of people and classify them into sets, however in our construction,
we can construct computer models based on using restricted boltzmann machines
through a technique called deep learning. If a technology firm sees an
opportunity to start developing in a given region, if they can identify the
type of issues being discussed amongst the community (where people can be
organized into different sets based on interests), they can identify the
region555Formally, where the largest set is interested in the same objective
as the company where it is best for them to work and since people in that
community are willing to work on those issues, establishing new businesses
there to give them an opportuinity is the best way to unleash intellectual
power.
## IV Conclusion
In this paper, we presented how intellectual capital can be measured through
the use of a novel index, KDI. We also discussed how the metrics present in
the index can help increase the efficiency of an organization by modeling the
structure of a company using network flow models. The last section discussed
how the information flow among people in a given region can help organizations
unleash intellectual power and capitalize on it by providing the people the
necessary resources. We are currently unable to provide numerical analysis of
our index and therefore we are lacking milestones that can provide comparative
analysis however a case study to quantify those milestones is being
undertaken. Further research would include the aforementioned milestones and a
more sophisticated models of flow networks.
I would like to thank Christopher Norris for helpful discussions in the
writing of this paper.
## Appendix A Appendix
Here is the list of metrics [1] that we will be using at micro scales (mostly
applicable to companies/firms), we do make two assumptions here:
* •
These metrics can be easily obtained from available data and this will become
self-evident to some extent.
* •
The use of IT and various related means yields to the best dispersion of
knowledge/information within an organization.
Patents pending
---
% Investment in IT
% R&D invested in basic research
Average years of service with the company
Number of employees
Number of managers
Average duration of employment
Number of new solutions/products suggested
New patents/software/etc. filed
IT development expense/IT expense
Average age of employees
IT literacy of a staff
Company managers with advanced degrees
Revenues resulting from new business operations
IT performance/employee
IT capacity (CPU and DASD)
Changes in IT inventory
## References
* [1] Jay Liebowitz, Ching Y. Suen. Developing knowledge management metrics for measuring intellectual capital. _Journal of Intellectual Capital,_ Vol. 1 Iss: 1, pp.54 - 67, 2000.
* [2] Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows: Theory, Algorithms and Applications. Prentice Hall, 1993.
|
arxiv-papers
| 2011-06-14T03:02:58 |
2024-09-04T02:49:19.612941
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vikram Dhillon",
"submitter": "Vikram Dhillon",
"url": "https://arxiv.org/abs/1106.2601"
}
|
1106.2705
|
# Sub-barrier capture with quantum diffusion approach: actinide-based
reactions
V.V.Sargsyan1, G.G.Adamian1,2, N.V.Antonenko1, W.Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2Institute of Nuclear Physics, 702132 Tashkent, Uzbekistan
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China
###### Abstract
With the quantum diffusion approach the behavior of capture cross sections and
mean-square angular momenta of captured systems are revealed in the reactions
with deformed nuclei at sub-barrier energies. The calculated results are in a
good agreement with existing experimental data. With decreasing bombarding
energy under the barrier the external turning point of the nucleus-nucleus
potential leaves the region of short-range nuclear interaction and action of
friction. Because of this change of the regime of interaction, an unexpected
enhancement of the capture cross section is expected at bombarding energies
far below the Coulomb barrier. This effect is shown its worth in the
dependence of mean-square angular momentum of captured system on the
bombarding energy. From the comparison of calculated and experimental capture
cross sections, the importance of quasifission near the entrance channel is
shown for the actinide-based reactions leading to superheavy nuclei.
###### pacs:
25.70.Ji, 24.10.Eq, 03.65.-w
Key words: astrophysical $S$-factor; dissipative dynamics; sub-barrier capture
## I Introduction
The measurement of excitation functions down to the extreme sub-barrier energy
region is important for studying the nucleus-nucleus interaction as well as
the coupling of relative motion with other degrees of freedom, and very little
data exist on the fusion, fission and capture cross sections at extreme sub-
barrier energies ZhangOth ; ZhangOU ; Og ; ZuhuaFTh ; Nadkarni ; trotta ; Ji1
; Tr1 ; NishioOU ; Ji2 ; NishioSiU ; Vino ; Dg ; NishioSU ; HindeSTh ; ItkisSU
; akn . The experimental data obtained are of interest for solving
astrophysical problems related to nuclear synthesis. Indications for an
enhancement of the $S$-factor, $S=E_{\rm c.m.}\sigma\exp(2\pi\eta)$ Zvezda ;
Zvezda2 , where $\eta(E_{\rm c.m.})=Z_{1}Z_{2}e^{2}\sqrt{\mu/(2\hbar^{2}E_{\rm
c.m.})}$ is the Sommerfeld parameter, at energies $E_{\rm c.m.}$ below the
Coulomb barrier have been found in Refs. Ji1 ; Ji2 ; Dg . However, its origin
is still under discussion.
To clarify the behavior of capture cross sections at sub-barrier energies, a
further development of the theoretical methods is required Gomes . The
conventional coupled-channel approach with realistic set of parameters is not
able to describe the capture cross sections either below or above the Coulomb
barrier Dg . The use of a quite shallow nucleus-nucleus potential Es with an
adjusted repulsive core considerably improves the agreement between the
calculated and experimental data. Besides the coupling with collective
excitations, the dissipation, which is simulated by an imaginary potential in
Ref. Es or by damping in each channel in Ref. Hag1 , seems to be important.
The quantum diffusion approach based on the quantum master-equation for the
reduced density matrix has been suggested in Ref. EPJSub . This model takes
into consideration the fluctuation and dissipation effects in collisions of
heavy ions which model the coupling with various channels. As demonstrated in
Ref. EPJSub , this approach is successful for describing the capture cross
sections at energies near and below the Coulomb barrier for interacting
spherical nuclei. An unexpected enhancement of the capture cross section at
bombarding energies far below the Coulomb barrier has been predicted in EPJSub
. This effect is related to the switching off of the nuclear interaction at
the external turning point $r_{ex}$ (Fig. 1). If the colliding nuclei approach
the distance $R_{int}$ between their centers, the nuclear forces start to act
in addition to the Coulomb interaction. Thus, at $R<R_{int}$ the relative
motion is coupled strongly with other degrees of freedom. At $R>R_{int}$ the
relative motion is almost independent of the internal degrees of freedom.
Depending on whether the value of $r_{ex}$ is larger or smaller than the
interaction radius $R_{int}$, the impact of coupling with other degrees of
freedom upon the barrier passage seems to be different.
In the present paper we apply the approach of Ref. EPJSub to the description
of the capture process of deformed nuclei in a wide energy interval including
the extreme sub-barrier region. The used formalism is presented in Sect. II.
The results of our calculations for the reactions 16O,19F,32S,48Ca+232Th,
4He,16O,20Ne,30Si,36S,48Ca+238U, 36S,48Ca,50Ti+244Pu, 48Ca+246,248Cm, and
36S+248Cm are discussed in Sect. III. The conclusions are given in Sect. IV.
## II Model
### II.1 The nucleus-nucleus potential
Figure 1: (Upper part) The nucleus-nucleus potentials calculated at $J$ = 0
(solid curve), 30 (dashed curve), 50 (dotted curve), and 65 (dash-dotted
curve) for the 16O + 238U reaction. The interacting nuclei are assumed to be
spherical in the calculation. (Lower part) The position $R_{b}$ of the Coulomb
barrier, radius of interaction $R_{int}$, and external and internal turning
points for some values of $E_{\rm c.m.}$ are indicated at the nucleus-nucleus
potential for the same reaction at $J$=0.
The potential describing the interaction of two nuclei can be written in the
form poten
$\displaystyle V(R,Z_{i},A_{i},\theta_{i},J)$ $\displaystyle=$ $\displaystyle
V_{C}(R,Z_{i},A_{i},\theta_{i},J)$ $\displaystyle+$ $\displaystyle
V_{N}(R,Z_{i},A_{i},\theta_{i},J)+\frac{\hbar^{2}J(J+1)}{2\mu R^{2}},$
where $V_{N}$, $V_{C}$, and the last summand stand for the nuclear, the
Coulomb, and the centrifugal potentials, respectively. The nuclei are proposed
to be spherical or deformed. The potential depends on the distance $R$ between
the center of mass of two interacting nuclei, mass $A_{i}$ and charge $Z_{i}$
of nuclei ($i=1,2$), the orientation angles $\theta_{i}$ of the deformed (with
the quadrupole deformation parameters $\beta_{i}$) nuclei and the angular
momentum $J$. The static quadrupole deformation parameters are taken from Ref.
Ram for the even-even deformed nuclei. For the nuclear part of the nucleus-
nucleus potential, we use the double-folding formalism, in the form
$\displaystyle
V_{N}=\int\rho_{1}(\mathbb{r_{1}})\rho_{2}(\mathbb{R}-\mathbb{r_{2}})F(\mathbb{r_{1}}-\mathbb{r_{2}})d\mathbb{r_{1}}d\mathbb{r_{2}},$
(2)
where $F(\mathbb{r_{1}}-\mathbb{r_{2}})=C_{0}[F_{\rm
in}\frac{\rho_{0}(\mathbb{r_{1}})}{\rho_{00}}+F_{\rm
ex}(1-\frac{\rho_{0}(\mathbb{r_{1}})}{\rho_{00}})]\delta(\mathbb{r_{1}}-\mathbb{r_{2}})$
is the density-dependent effective nucleon-nucleon interaction and
$\rho_{0}(\mathbb{r})=\rho_{1}(\mathbb{r})+\rho_{2}(\mathbb{R}-\mathbb{r})$,
$F_{\rm in,ex}=f_{\rm in,ex}+f_{\rm
in,ex}^{{}^{\prime}}\frac{(N_{1}-Z_{1})(N_{2}-Z_{2})}{(N_{1}+Z_{1})(N_{2}+Z_{2})}$.
Here, $\rho_{i}(\mathbb{r_{i}})$ and $N_{i}$ are the nucleon densities and
neutron numbers of the light and the heavy nuclei of the dinuclear system,
respectively. Our calculations are performed with the following set of
parameters: $C_{0}=$ 300 MeV fm3, $f_{\rm in}=$ 0.09, $f_{\rm ex}=$ -2.59,
$f_{\rm in}^{{}^{\prime}}=$ 0.42, $f_{\rm ex}^{{}^{\prime}}=$ 0.54 and
$\rho_{00}=$ 0.17 fm-3 poten . The densities of the nuclei are taken in the
two-parameter symmetrized Woods-Saxon form with the nuclear radius parameter
$r_{0}$=1.15 fm (for the nuclei with $A_{i}\geq 16$) and the diffuseness
parameter $a$ depending on the charge and mass numbers of the nucleus poten .
We use $a$= 0.53 fm for the lighter nuclei 16O and 19F, $a$= 0.55 fm for the
intermediate nuclei (20Ne, 26Mg, 30Si, 32,34,36S, 40,48Ca, 50Ti), and $a$=
0.56 fm for the actinides. For the 4He nucleus $r_{0}$=1.02 fm and $a$=0.48
fm.
The Coulomb interaction of two deformed nuclei has the following form:
$\displaystyle V_{C}(R,Z_{i},A_{i},\theta_{i},J)=\frac{Z_{1}Z_{2}e^{2}}{R}$
(3) $\displaystyle+$
$\displaystyle\left(\frac{9}{20\pi}\right)^{1/2}\frac{Z_{1}Z_{2}e^{2}}{R^{3}}\sum_{i=1,2}R_{i}^{2}\beta_{i}\left[1+\frac{2}{7}\left(\frac{5}{\pi}\right)^{1/2}\beta_{i}\right]$
$\displaystyle\times$ $\displaystyle P_{2}(\cos\theta_{i}),$
where $P_{2}(\cos\theta_{i})$ is the Legendre polynomial.
In Fig. 1 there is shown the nucleus-nucleus potential $V$ for the 16O + 238U
reaction (for simplicity, 238U is assumed to be spherical) which has a pocket.
With increasing centrifugal part of the potential the pocket depth becomes
smaller, while the position of the pocket minimum moves towards the barrier at
the position of the Coulomb barrier $R=R_{b}\approx R_{1}+R_{2}+2$ fm, where
$R_{i}=1.15A_{i}^{1/3}$ are the radii of colliding nuclei. This pocket is
washed out at large angular momenta $J>65$. Thus, only a limited part of
angular momenta contributes to the capture process.
Figure 2: (Upper part) The nucleus-nucleus potentials calculated at $J$=0 for
the reactions 36S+238U and 16O+238U. (Lower part) The dependence of the
capture cross section for nuclei colliding with a fixed orientation on $E_{\rm
c.m.}-V_{b}^{orient}$ where $V_{b}^{orient}$ is the height of the Coulomb
barrier for certain orientations. The results of calculations for the sphere-
sphere (the interacting nuclei are spherical), sphere-pole and sphere-side
configurations are shown by solid, dashed and dotted lines, respectively. The
static quadrupole deformation parameters are: $\beta_{2}$(238U)=0.286 and
$\beta_{1}$(16O)=$\beta_{1}$(36S)=0.
For the reactions 36S + 238U and 16O + 238U (Fig. 2), the dependence of the
potential energy on the orientation of the prolate deformed nucleus 238U is
shown. The lowest Coulomb barriers are associated with collisions of the
projectile nucleus with the tips of the target nucleus, while the highest
barriers correspond to collisions with the sides of the target nucleus. The
difference of the Coulomb barriers for the sphere-pole and sphere-side
orientations is about 16 MeV (8 MeV) for the 36S + 238U (16O + 238U) system.
### II.2 Capture cross section
The capture cross section is a sum of partial capture cross sections
$\displaystyle\sigma_{cap}(E_{\rm c.m.})$ $\displaystyle=$
$\displaystyle\sum_{J}\sigma_{\rm cap}(E_{\rm c.m.},J)=$ (4) $\displaystyle=$
$\displaystyle\pi\lambdabar^{2}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{1}\sin(\theta_{1})$
$\displaystyle\times$
$\displaystyle\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})P_{\rm cap}(E_{\rm
c.m.},J,\theta_{1},\theta_{2}),$
where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie
wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$
is the nucleon mass), and the summation is over the possible values of angular
momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the
potential of the interacting nuclei for each orientation, one can obtain the
partial capture probability $P_{\rm cap}$ which is defined by the passing
probability of the potential barrier in the relative distance $R$ coordinate
at a given $J$.
Figure 3: The calculated value $\langle V_{b}\rangle$ averaged over the
orientations of the heavy deformed nucleus versus $E_{\rm c.m.}$ for the 36S +
238U reaction. The values of barriers $V^{orient}_{b}$(sphere-pole) for the
sphere-pole configuration, $V_{b}=V_{b}$(sphere-
sphere)=$V^{orient}_{b}$(sphere-sphere) for the sphere-sphere configuration
and $V^{orient}_{b}$(sphere-side) for the sphere-side configuration are
indicated by arrows. The static quadrupole deformation parameters are:
$\beta_{2}$(238U)=0.286 and $\beta_{1}$(36S)=0. Figure 4: The calculated
capture cross section (solid lines) versus $E_{\rm c.m.}$ for the reactions
16O + 232Th and 4He + 238U are compared with the available experimental data.
The experimental data in the upper part are taken from Refs. BackOTh (open
triangles), ZhangOth (closed triangles), MuakamiOTh (open squares), KailasOTh
(closed squares), ZuhuaFTh (open stars) and Nadkarni (closed stars). The
fission cross sections from Refs. trotta and ViolaOU are shown in the lower
part by open circles and solid squares, respectively. The value of the Coulomb
barrier $V_{b}$ for the spherical nuclei is indicated by arrow. The dashed
curve represents the calculation by the Wong’s formula (8). The static
quadrupole deformation parameters are: $\beta_{2}$(238U)=0.286,
$\beta_{2}$(232Th)=0.261 and $\beta_{1}$(16O)=$\beta_{1}$( 4He)=0.
The value of $P_{\rm cap}$ is obtained by integrating the propagator $G$ from
the initial state $(R_{0},P_{0})$ at time $t=0$ to the final state $(R,P)$ at
time $t$ ($P$ is a momentum):
$\displaystyle P_{\rm cap}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\int_{-\infty}^{r_{\rm
in}}dR\int_{-\infty}^{\infty}dP\ G(R,P,t|R_{0},P_{0},0)$ (5) $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{1}{2}{\rm erfc}\left[\frac{-r_{\rm
in}+\overline{R(t)}}{{\sqrt{\Sigma_{RR}(t)}}}\right].$
Figure 5: The same as in Fig. 4, but for the reactions 16O + 238U and 36S +
238U. The experimental cross sections are taken from Refs. NishioOU (open
triangles), TokeOU (closed triangles), ZuhuaFTh (open squares), ZhangOU
(closed squares), ViolaOU (open stars), ItkisSU (closed stars), and NishioSU
(rhombuses). The dashed curve represents the calculation by the Wong’s formula
(8). The static quadrupole deformation parameters are: $\beta_{2}$(238U)=0.286
and $\beta_{1}$(16O)=$\beta_{1}$(36S)=0. Figure 6: The same as in Fig. 4, but
for the reactions 32S + 232Th (solid line), 32S + 238U (dotted line) and 30Si
+ 238U. The experimental data are taken from Refs. HindeSTh (32S + 232Th,
solid squares), NishioSiU (solid circles) and Nishionew (open squares). The
static quadrupole deformation parameters are: $\beta_{2}$(238U)=0.286,
$\beta_{2}$(232Th)=0.261, $\beta_{1}$(32S)=0.312 and $\beta_{1}$(30Si)=0.315.
For the 30Si + 238U reaction, the results of calculations with
$\beta_{1}$(30Si)=0 (the predictions of the mean-field and macroscopic-
microscopic models) are presented by dashed line in the lower part of the
figure. Figure 7: The same as in Fig. 4, but for the reactions 19F + 232Th
and 20Ne + 238U. The experimental data are taken from Refs. Nadkarni (open
squares), ZuhuaFTh (closed squares), and ViolaOU (closed circles). The open
circles in the lower part are the experimental data from Ref. ViolaOU shifted
by 2 MeV to the left. The results of calculations with the static quadrupole
deformation parameters of 19F $\beta_{1}$(19F)=0.275 (as in Ref. Moel1 ),
0.41, and 0.55 are shown by the dashed, solid, and dotted lines, respectively.
The other static quadrupole deformation parameters are:
$\beta_{2}$(238U)=0.286, $\beta_{2}$(232Th)=0.261 and $\beta_{1}$(20Ne)=0.335.
Figure 8: The predicted capture cross sections for the reactions 36S +
244Pu,248Cm. The static quadrupole deformation parameters are:
$\beta_{2}$(244Pu)=0.293, $\beta_{2}$(248Cm)=0.297 and $\beta_{1}$(36S)=0.
Figure 9: The calculated mean-square angular momenta versus $E_{\rm c.m.}$
for the reactions 16O + 232Th,238U are compared with experimental data
ZuhuaFTh . The dashed curve represents the calculation by the Eq. (10). The
static quadrupole deformation parameters are: $\beta_{2}$(238U)=0.286,
$\beta_{2}$(232Th)=0.261 and $\beta_{1}$(16O)=0. The values of the Coulomb
barriers $V_{b}$ corresponding to spherical interacting nuclei are indicated
by arrows. Figure 10: The same as in Fig. 9, but for the indicated reactions
19F,48Ca + 232Th. The experimental data are taken from Ref. ZuhuaFTh . The
results of calculations with quadrupole deformation parameters
$\beta_{1}$(19F)=0.275, 0.41 and 0.55 are shown by the dashed, solid, and
dotted lines, respectively. The other static quadrupole deformation parameters
are: $\beta_{2}$(232Th)=0.261 and $\beta_{1}$(48Ca)=0. Figure 11: The
calculated values of the astrophysical $S$-factor with $\eta_{0}=\eta(E_{\rm
c.m.}=V_{b})$ for the indicated reactions 16O+232Th and 4He + 238U. The values
of the Coulomb barriers $V_{b}$ corresponding to the spherical nuclei are 78.6
and 21.2 MeV. Figure 12: The calculated values of the astrophysical
$S$-factor with $\eta_{0}=\eta(E_{\rm c.m.}=V_{b})$ (middle part), the
logarithmic derivative $L$ (upper part) and the fusion barrier distribution
$d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm c.m.}^{2}$ (lower part) for the
16O+238U reaction. The value of $L$ calculated with the assumption of
$\beta_{1}$(16O)=$\beta_{2}$(238U)=0 is shown by a dashed line. The solid and
dotted lines show the values of $d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm
c.m.}^{2}$ calculated with the increments 0.2 and 1.2 MeV, respectively. The
closed squares are the experimental data of Ref. DH . The value of the Coulomb
barrier $V_{b}$ corresponding to the spherical nuclei is 80 MeV. Figure 13:
The same as in Fig. 4, but for the 48Ca + 232Th,238U reactions. The excitation
energies $E^{*}_{CN}$ of the corresponding nuclei are indicated. The
experimental data are taken from Refs. Itkis1 (marked by squares) and Shen
(marked by circles). The static quadrupole deformation parameters are:
$\beta_{2}$(238U)=0.286, $\beta_{2}$(232Th)=0.261 and $\beta_{1}$(48Ca)=0.
Figure 14: The same as in Fig. 13, but for the reactions 48Ca + 246,248Cm. The
experimental data are from Refs. Itkis1 (squares) and Itkis2 (circles). The
static quadrupole deformation parameters are: $\beta_{2}$(246Cm)=0.298,
$\beta_{2}$(248Cm)=0.297 and $\beta_{1}$(48Ca)=0.
The second line in (5) is obtained by using the propagator
$G=\pi^{-1}|\det{\bf\Sigma}^{-1}|^{1/2}\exp(-{\bf
q}^{T}{\bf\Sigma}^{-1}{\bm{q}})$ (${\bf q}^{T}=[q_{R},q_{P}]$,
$q_{R}(t)=R-\overline{R(t)}$, $q_{P}(t)=P-\overline{P(t)}$,
$\overline{R(t=0)}=R_{0}$, $\overline{P(t=0)}=P_{0}$,
$\Sigma_{kk^{\prime}}(t)=2\overline{q_{k}(t)q_{k^{\prime}}(t)}$,
$\Sigma_{kk^{\prime}}(t=0)=0$, $k,k^{\prime}=R,P$) calculated in Ref.
DMDadonov for an inverted oscillator which approximates the nucleus-nucleus
potential $V$ in the variable $R$. The frequency $\omega$ of this oscillator
with an internal turning point $r_{\rm in}$ is defined from the condition of
equality of the classical actions of approximated and realistic potential
barriers of the same hight at given $J$. It should be noted that the passage
through the Coulomb barrier approximated by a parabola has been previously
studied in Refs. Hofman ; VAZ ; Ayik ; Hupin ; our . This approximation is
well justified for the reactions and energy range, which are here considered.
Finally, one can find the expression for the capture probability:
$\displaystyle P_{\rm cap}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
erfc}\left[\left(\frac{\pi s_{1}(\gamma-
s_{1})}{2\mu(\omega_{0}^{2}-s_{1}^{2})}\right)^{1/2}\frac{\mu\omega_{0}^{2}R_{0}/s_{1}+P_{0}}{\left[\gamma\ln(\gamma/s_{1})\right]^{1/2}}\right],$
(6)
where $\gamma$ is the internal-excitation width,
$\omega_{0}^{2}=\omega^{2}\\{1-\hbar\tilde{\lambda}\gamma/[\mu(s_{1}+\gamma)(s_{2}+\gamma)]\\}$
is the renormalized frequency in the Markovian limit, the value of
$\tilde{\lambda}$ is related to the strength of linear coupling in coordinates
between collective and internal subsystems. The $s_{i}$ are the real roots
($s_{1}\geq 0>s_{2}\geq s_{3}$) of the following equation:
$\displaystyle(s+\gamma)(s^{2}-\omega_{0}^{2})+\hbar\tilde{\lambda}\gamma
s/\mu=0.$ (7)
The details of the used formalism are presented in EPJSub . We have to mention
that most of the quantum-mechanical, dissipative effects and non-Markovian
effects accompanying the passage through the potential barrier are taken into
consideration in our formalism EPJSub ; our . For example, the non-Markovian
effects appear in the calculations through the internal-excitation width
$\gamma$.
As shown in EPJSub , the nuclear forces start to play a role at
$R_{int}=R_{b}+1.1$ fm where the nucleon density of colliding nuclei
approximately reaches 10% of the saturation density. If the value of $r_{\rm
ex}$ corresponding to the external turning point is larger than the
interaction radius $R_{int}$, we take $R_{0}=r_{\rm ex}$ and $P_{0}=0$ in Eq.
(6). For $r_{\rm ex}<R_{int}$, it is naturally to start our treatment with
$R_{0}=R_{int}$ and $P_{0}$ defined by the kinetic energy at $R=R_{0}$. In
this case the friction hinders the classical motion to proceed towards smaller
values of $R$. If $P_{0}=0$ at $R_{0}>R_{int}$, the friction almost does not
play a role in the transition through the barrier. Thus, two regimes of
interaction at sub-barrier energies differ by the action of the nuclear forces
and the role of friction at $R=r_{\rm ex}$.
## III Calculated results
Besides the parameters related to the nucleus-nucleus potential, two
parameters $\hbar\gamma$=32 MeV and the friction coefficient
$\hbar\lambda=-\hbar(s_{1}+s_{2})$=2 MeV are used for calculating the capture
probability in reactions with deformed actinides. The value of
$\tilde{\lambda}$ is set to obtain this value of $\hbar\lambda$. The most
realistic friction coefficients in the range of $\hbar\lambda\approx 1-2$ MeV
are suggested from the study of deep inelastic and fusion reactions Obzor .
These values are close to those calculated within the mean field approach Den
. All calculated results presented are obtained with the same set of
parameters and are rather insensitive to a reasonable variation of them EPJSub
; VAZ ; our .
### III.1 Effect of orientation
The influence of orientation of the deformed heavy nucleus on the capture
process in the reactions 36S + 238U and 16O + 238U is studied in Fig. 2. We
demonstrate that the capture cross section $\sigma_{cap}$ at fixed orientation
as a function of $E_{\rm c.m.}-V_{b}^{orient}$, where $V_{b}^{orient}$ is the
Coulomb barrier for this orientation, is almost independent of the orientation
angle $\theta_{2}$.
In Fig. 3 the value of the Coulomb barrier
$\displaystyle<V_{b}>$ $\displaystyle=$
$\displaystyle\frac{\pi\lambdabar^{2}}{\sigma_{\rm cap}(E_{\rm
c.m.})}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})$
$\displaystyle\times$ $\displaystyle P_{\rm cap}(E_{\rm
c.m.},J,\theta_{1},\theta_{2})V(R_{b},Z_{i},A_{i},\theta_{i},J)$
averaged over all possible orientations of the heavy nucleus versus $E_{\rm
c.m.}$ is shown for the 36S + 238U reaction. With increasing (decreasing)
$E_{\rm c.m.}$ the value of $<V_{b}>$ approaches the value of the Coulomb
barrier for the sphere-sphere configuration (for the sphere-pole
configuration). The influence of deformation on the capture cross section is
very weak already at bombarding energies about 15 MeV above the Coulomb
barrier corresponding to spherical nuclei.
### III.2 Comparison with experimental data and predictions
In Figs. 4–6 the calculated capture cross sections for the reactions
16O,19F,32S+232Th and 4He,16O,30Si,32,36S+238U are in a rather good agreement
with the available experimental data NishioOU ; TokeOU ; ZuhuaFTh ; ZhangOU ;
ViolaOU ; ItkisSU ; NishioSU ; BackOTh ; ZhangOth ; MuakamiOTh ; KailasOTh ;
Nadkarni ; trotta ; HindeSTh ; NishioSiU ; Nishionew . Because of the
uncertainties in the definition of the deformation of the light nucleus and in
the experimental data NishioSiU ; Nishionew in Fig. 6, we show the calculated
results for the 30Si+238U reaction with $\beta_{1}$(30Si) from Ref. Ram as
well as with $\beta_{1}$(30Si)=0 (lower part of Fig. 6). Note that
$\beta_{1}$(30Si)=0 for the ground state were predicted within the mean-field
and macroscopic-microscopic models.
In Fig. 7 (upper part) we are not able to describe well the data of Ref.
Nadkarni for the 19F+232Th reaction at $E_{\rm c.m.}<74$ MeV, even by varying
the static quadrupole deformation parameters $\beta_{1}$ of 19F. However, the
deviations of the solid curve in the upper part of Fig. 7 from the
experimental data are within the uncertainty of these data. Note that the
value of $\beta_{1}$ mainly influences the slope of curve at $E_{\rm
c.m.}<V_{b}$ and one can extract the ground state deformation of nucleus from
the experimental capture cross section data. For the 20Ne + 238U reaction, the
calculated capture cross sections in Fig. 7 are consistent with the
experimental data ViolaOU if the latter ones are shifted by 2 MeV to lower
energies. For the 20Ne nucleus, the experimental quadrupole deformation
parameter $\beta_{1}$=0.727 related in Ref. Ram to the first 2+ state seems to
be unrealistically large and we take $\beta_{1}$=0.335 as predicted in Ref.
Moel1 . The capture cross sections for the reactions 32S + 238U and 36S +
244Pu,248Cm are shown in Figs. 6 and 8, respectively.
One can see in Figs. 4–8 that there is a sharp fall-off of the cross sections
just under the Coulomb barrier corresponding to undeformed nuclei. With
decreasing $E_{\rm c.m.}$ up to about 8–10 MeV (when the projectile is
spherical) and 15–20 MeV (when both projectile and target are deformed nuclei)
below the Coulomb barrier the regime of interaction is changed because at the
external turning point the colliding nuclei do not reach the region of nuclear
interaction where the friction plays a role. As result, at smaller $E_{\rm
c.m.}$ the cross sections fall with a smaller rate. With larger values of
$R_{int}$ the change of fall rate occurs at smaller $E_{\rm c.m.}$. However,
the uncertainty in the definition of $R_{int}$ is rather small. Therefore, an
effect of the change of fall rate of sub-barrier capture cross section should
be in the data if we assume that the friction starts to act only when the
colliding nuclei approach the barrier. Note that at energies of 10–20 MeV
below the barrier the experimental data have still large uncertainties to make
a firm experimental conclusion about this effect. The effect seems to be more
pronounced in collisions of spherical nuclei, where the regime of interaction
is changed at $E_{\rm c.m.}$ up to about 3–5 MeV below the Coulomb barrier
EPJSub . The collisions of deformed nuclei occur at various mutual
orientations affecting the value of $R_{int}$.
The well-known Wong formula for the capture cross section is
$\displaystyle\sigma(E_{\rm c.m.})$ $\displaystyle=$
$\displaystyle\frac{R_{b}^{2}\hbar\omega}{2E_{\rm
c.m.}}\int_{0}^{\pi/2}d\theta_{1}\sin\theta_{1}\int_{0}^{\pi/2}d\theta_{2}\sin\theta_{2}$
(8) $\displaystyle\times$ $\displaystyle\ln(1+\exp[2\pi(E_{\rm
c.m.}-E_{b}(\theta_{1},\theta_{2}))/\hbar\omega]),$
where $E_{b}(\theta_{1},\theta_{2})$ is value of the Coulomb barrier which
depends on the orientations of the deformed nuclei Wong . As seen from Figs. 4
and 5 (dashed lines) the Wong formula (8) does not reproduce the capture cross
section at $E_{\rm c.m.}<V_{b}$ even taking into consideration the static
quadrupole deformation of target-nucleus.
The calculated mean-square angular momenta
$\displaystyle\langle J^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\pi\lambdabar^{2}\sum_{J}J(J+1)(2J+1)}{\sigma_{cap}(E_{\rm
c.m.})}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\pi/2}d\theta_{1}\sin(\theta_{1})\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})P_{\rm
cap}(E_{\rm c.m.},J,\theta_{1},\theta_{2})$
of captured systems versus $E_{\rm c.m.}$ are presented in Figs. 9–10 for the
reactions mentioned above. At energies below the barrier the value of $\langle
J^{2}\rangle$ has a minimum. This minimum depends on the deformations of
nuclei and on the factor $Z_{1}\times Z_{2}$. For the reactions 16O + 232Th,
16O + 238U, 19F + 232Th and 48Ca + 232Th, these minima are about 7, 8, 12 and
15 MeV below the corresponding Coulomb barriers, respectively. The
experimental data Vand indicate the presence of the minimum as well. On the
left-hand side of this minimum the dependence of $\langle J^{2}\rangle$ on
$E_{\rm c.m.}$ is rather weak. A similar weak dependence has been found in
Refs. Bala in the extreme sub-barrier region. Note that the found behavior of
$\langle J^{2}\rangle$, which is related to the change of the regime of
interaction between the colliding nuclei, would affect the angular anisotropy
of the products of fission-like fragments following capture. Indeed, the
values of $\langle J^{2}\rangle$ are extracted from data on angular
distribution of fission-like fragments akn .
In the Wong model Wong the value of the mean-square angular momentum is
determined as
$\displaystyle\langle J^{2}\rangle$ $\displaystyle=$ $\displaystyle\frac{\mu
R_{b}^{2}\hbar\omega}{\pi\hbar^{2}}\int_{0}^{\pi/2}d\theta_{1}\sin\theta_{1}\int_{0}^{\pi/2}d\theta_{2}\sin\theta_{2}$
(10) $\displaystyle\times$ $\displaystyle\frac{-Li_{2}(-\exp[2\pi(E_{\rm
c.m.}-E_{b}(\theta_{1},\theta_{2}))/\hbar\omega])}{\ln(1+\exp[2\pi(E_{\rm
c.m.}-E_{b}(\theta_{1},\theta_{2}))/\hbar\omega])}.$
Here, the $Li_{2}(z)$ is the polylogarithm function. At $\exp[2\pi(E_{\rm
c.m.}-E_{b})/\hbar\omega])\ll$1 (much below the Coulomb barrier),
$\frac{-Li_{2}(-\exp[2\pi(E_{\rm
c.m.}-E_{b})/\hbar\omega])}{\ln(1+\exp[2\pi(E_{\rm
c.m.}-E_{b})/\hbar\omega])}\approx 1$ and one can obtain the saturation value
of the mean-square angular momentum Gomes :
$\displaystyle\langle J^{2}\rangle=\frac{\mu
R_{b}^{2}\hbar\omega}{\pi\hbar^{2}}.$ (11)
The agreement between $\langle J^{2}\rangle$ calculated with Eq. (10) and
experimental $\langle J^{2}\rangle$ is not good. At energies below the barrier
$\langle J^{2}\rangle$ has no a minimum (see Fig. 9). However, for the
considered reactions the saturation values of $\langle J^{2}\rangle$ are close
to those obtained with our formalism.
### III.3 Astrophysical factor, L-factor and barrier distribution
In Figs. 11 and 12 the calculated astrophysical $S$–factors versus $E_{\rm
c.m.}$ are shown for the reactions 4He,16O+238U and 16O+232Th. The $S$-factor
has a maximum for which there are experimental indications in Refs. Ji1 ; Ji2
; Es . After this maximum $S$-factor slightly decreases with decreasing
$E_{\rm c.m.}$ and then starts to increase. This effect seems to be more
pronounced in collisions of spherical nuclei EPJSub . The same behavior has
been revealed in Refs. LANG by extracting the $S$-factor from the experimental
data.
In Fig. 12, the so-called logarithmic derivative, $L(E_{\rm
c.m.})=d(\ln(E_{\rm c.m.}\sigma_{cap}))/dE_{\rm c.m.},$ and the barrier
distribution $d^{2}(E_{\rm c.m.}\sigma_{cap})/dE_{\rm c.m.}^{2}$ are presented
for the 16O+238U reaction. The logarithmic derivative strongly increases below
the barrier and then has a maximum at $E_{\rm c.m.}\approx
V_{b}^{orient}$(sphere-pole)-3 MeV (at $E_{\rm c.m.}\approx V_{b}$-3 MeV for
the case of spherical nuclei). The maximum of $L$ corresponds to the minimum
of the $S$-factor.
The barrier distributions calculated with an energy increment 0.2 MeV have
only one maximum at $E_{\rm c.m.}\approx V_{b}^{orient}$(sphere-
sphere)$=V_{b}$ as in the experiment DH . With increasing increment the
barrier distribution is shifted to lower energies. Assuming a spherical target
nucleus in the calculations, we obtain a more narrow barrier distribution (see
Fig. 12).
### III.4 Capture cross sections in reactions with large fraction of
quasifission
In the case of large values of $Z_{1}\times Z_{2}$ the quasifission process
competes with complete fusion at energies near barrier and can lead to a large
hindrance for fusion, thus ruling the probability for producing superheavy
elements in the actinide-based reactions trota ; nasha . Since the sum of the
fusion cross section $\sigma_{fus}$, and the quasifission cross section
$\sigma_{qf}$ gives the capture cross section,
$\sigma_{cap}=\sigma_{fus}+\sigma_{qf},$
and $\sigma_{fus}\ll\sigma_{qf}$ in the actinide-based reactions 48Ca +
232Th,238U,244Pu,246,248Cm and 50Ti + 244Pu nasha , we have
$\sigma_{cap}\approx\sigma_{qf}.$
In a wide mass-range near the entrance channel, the quasifission events
overlap with the products of deep-inelastic collisions and can not be firmly
distinguished. Because of this the mass region near the entrance channel is
taken out in the experimental analyses of Refs. Itkis1 ; Itkis2 . Thus, by
comparing the calculated and experimental capture cross sections one can study
the importance of quasifission near the entrance channel for the actinide-
based reactions leading to superheavy nuclei.
The capture cross sections for the quasifission reactions Shen ; Itkis1 ;
Itkis2 are shown in Figs. 13-15. One can observe a large deviations of the
experimental data of Refs. Itkis1 ; Itkis2 from the the calculated results.
The possible reason is an underestimation of the quasifission yields measured
in these reactions. Thus, the quasifission yields near the entrance channel
are important. Note that there are the experimental uncertainties in the
bombarding energies.
Figure 15: The same as in Fig. 13, but for the indicated 48Ca,50Ti + 244Pu
reactions. The experimental data are from Refs. Itkis2 (squares) and Itkis1
(circles). The static quadrupole deformation parameters are:
$\beta_{2}$(244Pu)=0.293, and $\beta_{1}$(48Ca)=$\beta_{1}$(50Ti)=0. Figure
16: The ratio of theoretical and experimental capture cross sections versus
the excitation energy $E_{\rm c.m.}$ of the compound nucleus for the reactions
48Ca+238U (closed stars), 48Ca+244Pu (closed triangles), 48Ca+246Cm (closed
squares), 48Ca+248Cm (closed circles), and 50Ti+244Pu (closed rhombuses).
One can see in Fig. 16 that the experimental and the theoretical cross
sections become closer with increasing bombarding energy. This means that with
increasing bombarding energy the quasifission yields near the entrance channel
mass-region decrease with respect to the quasifission yields in other mass-
regions. The quasifission yields near the entrance channel increase with
$Z_{1}\times Z_{2}$.
## IV Summary
The quantum diffusion approach is applied to study the capture process in the
reactions with deformed nuclei at sub-barrier energies. The available
experimental data at energies above and below the Coulomb barrier are well
described, showing that the static quadrupole deformations of the interacting
nuclei are the main reasons for the capture cross section enhancement at sub-
barrier energies. Since the deformations of the interacting nuclei mainly
influence the slope of curve at $E_{\rm c.m.}<V_{b}$ and one can extract the
ground state deformation of projectile or target from the experimental capture
cross section data.
Due to a change of the regime of interaction (the turning-off of the nuclear
forces and friction) at sub-barrier energies, the curve related to the capture
cross section as a function of bombarding energy has smaller slope $E_{\rm
c.m.}-V_{b}<$ – 5 MeV. This change is also reflected in the functions $\langle
J^{2}\rangle$, $L(E_{\rm c.m.})$, and $S(E_{\rm c.m.})$. The mean-square
angular momentum of captured system versus $E_{\rm c.m.}$ has a minimum and
then saturates at sub-barrier energies. This behavior of $\langle
J^{2}\rangle$ would increase the expected anisotropy of the angular
distribution of the products of fission and quasifission following capture.
The astrophysical factor has a maximum and a minimum at energies below the
barrier. The maximum of $L$-factor corresponds to the minimum of the
$S$-factor. One can suggest the experiments to check these predictions.
The importance of quasifission near the entrance channel is shown for the
actinide-based reactions leading to superheavy nuclei.
## V acknowledgements
This work was supported by DFG, NSFC, and RFBR. The IN2P3-JINR, MTA-JINR and
Polish-JINR Cooperation programs are gratefully acknowledged.
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|
arxiv-papers
| 2011-06-14T13:12:24 |
2024-09-04T02:49:19.624586
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang",
"submitter": "Vazgen Sargsyan Dr.",
"url": "https://arxiv.org/abs/1106.2705"
}
|
1106.2714
|
RBC and UKQCD Collaborations
# $K$ to $\pi\pi$ Decay amplitudes from Lattice QCD
T. Blum Physics Department, University of Connecticut, Storrs, CT 06269-3046,
USA P.A. Boyle SUPA, School of Physics, The University of Edinburgh,
Edinburgh EH9 3JZ, UK N.H. Christ Physics Department, Columbia University,
New York, NY 10027, USA N. Garron SUPA, School of Physics, The University of
Edinburgh, Edinburgh EH9 3JZ, UK E. Goode School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, UK T. Izubuchi Brookhaven
National Laboratory, Upton, NY 11973, USA RIKEN-BNL Research Center,
Brookhaven National Laboratory, Upton, NY 11973, USA C. Lehner RIKEN-BNL
Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Q. Liu
Physics Department, Columbia University, New York, NY 10027, USA R.D.
Mawhinney Physics Department, Columbia University, New York, NY 10027, USA
C.T. Sachrajda School of Physics and Astronomy, University of Southampton,
Southampton SO17 1BJ, UK A. Soni Brookhaven National Laboratory, Upton, NY
11973, USA C. Sturm Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany H. Yin Physics Department, Columbia University, New York,
NY 10027, USA R. Zhou Physics Department, University of Connecticut, Storrs,
CT 06269-3046, USA Department of Physics, Indiana University, Bloomington, IN
47405, USA
(June 09, 2011)
###### Abstract
We report a direct lattice calculation of the $K$ to $\pi\pi$ decay matrix
elements for both the $\Delta I=1/2$ and $3/2$ amplitudes $A_{0}$ and $A_{2}$
on 2+1 flavor, domain wall fermion, $16^{3}\times 32\times 16$ lattices. This
is a complete calculation in which all contractions for the required ten,
four-quark operators are evaluated, including the disconnected graphs in which
no quark line connects the initial kaon and final two-pion states. These
lattice operators are non-perturbatively renormalized using the Rome-
Southampton method and the quadratic divergences are studied and removed. This
is an important but notoriously difficult calculation, requiring high
statistics on a large volume. In this paper we take a major step towards the
computation of the physical $K\to\pi\pi$ amplitudes by performing a complete
calculation at unphysical kinematics with pions of mass 422 MeV at rest in the
kaon rest frame. With this simplification we are able to resolve Re$(A_{0})$
from zero for the first time, with a 25% statistical error and can develop and
evaluate methods for computing the complete, complex amplitude $A_{0}$, a
calculation central to understanding the $\Delta=1/2$ rule and testing the
standard model of CP violation in the kaon system.
###### pacs:
11.15.Ha, 12.38.Gc 14.40.Be 13.25.Es
††preprint: CU-TP-1197, MPP-2011-51
## I Introduction
The Cabibbo-Kobayashi-Maskawa (CKM) theory for the weak interactions of the
quarks when combined with QCD provides a framework describing in complete
detail all the properties and interactions of the six quarks. This framework
incorporates the most general assignment of masses and couplings and appears
able to explain all observed phenomena in which these quarks participate.
However, to date, the non-perturbative character of low energy QCD has
obscured many of the consequences of the CKM theory. In particular, both the
direct CP violation seen in K meson decay and the factor of 22.5 enhancement
of the $I=0$, $K\rightarrow\pi\pi$ decay amplitude $A_{0}$ relative to the
$I=2$ amplitude $A_{2}$ (the $\Delta I=1/2$ rule) lack a quantitative
explanation.
Wilson coefficients evaluated at a QCD scale of about $2$ GeV represent the
short distance physics and can be evaluated from the CKM theory using QCD and
electro-weak perturbation theory. However, these factors explain only a factor
of two enhancement of the $I=0$ amplitude Gaillard:1974nj ; Altarelli:1974exa
. The remaining enhancement must arise from the hadronic matrix elements which
require non-perturbative treatment.
Direct CP violation in kaon decays provides a critical test of the standard
model’s CKM mechanism of CP violation. While forty years of experimental
effort have produced the measured result
Re$(\epsilon^{\prime}/\epsilon)=1.65(26)\times 10^{-3}$ Nakamura:2010zzi ,
with only a 16% error, there is no reliable theoretical calculation of this
quantity based on the standard model. A previous lattice QCD calculation using
2+1 dynamical domain wall fermions failed to give a conclusive result because
of the large systematic errors associated with the use of chiral perturbation
theory at the scale of the kaon mass Li:2008kc . (However, there are on-going
efforts using chiral perturbation theory Laiho:2010ir .) Earlier quenched
results Blum:2001xb ; Noaki:2001un are subject to this same difficulty
together with uncontrolled uncertainties associated with quenching
Golterman:2001qj ; Golterman:2002us ; Aubin:2006vt .
A direct lattice calculation of $K\rightarrow\pi\pi$ decay is extremely
important to provide an explanation for the $\Delta I=1/2$ rule and to test
the standard model of CP violation from first principles. This is an unusually
difficult calculation because of the presence of disconnected graphs. However,
with the continuing increase of available computing power and the development
of improved algorithms, calculations with disconnected graphs are now no
longer out of reach. In fact, our recent successful calculation of the masses
and mixing of the $\eta^{\prime}$ and $\eta$ mesons Christ:2010dd was carried
out in part to develop and test the methods needed for the calculation
presented here. In this paper, we present a first direct calculation of the
complete $K^{0}\rightarrow\pi\pi$ decay amplitude. At this stage, we work with
the simplified kinematics of a threshold decay in which the kaon is at rest
and decays into two pions each with zero momentum and with mass one-half that
of the kaon. The calculation with this choice of kinematics still contains the
main difficulties we need to overcome in order to be able to compute the
physical $K\to\pi\pi$ decay amplitudes; i.e. the presence of disconnected
diagrams coupled with the need to subtract ultraviolet power divergences.
However, as explained below, with the pions at rest we are able to generate
sufficient statistics to explore how to handle these difficulties. We stress
that at this simplified choice of kinematics, we compute the $K\to\pi\pi$
amplitudes directly and completely.
In order to calculate the decay amplitudes, we perform a direct, brute force
calculation of the required weak matrix elements. The isospin zero $\pi-\pi$
final state implies the presence of disconnected graphs in correlation
functions and makes the calculation very difficult. For these graphs, the
noise does not decrease with increasing time separation between the source and
sink, while the signal does. Therefore, substantial statistics are needed to
get a clear signal. This difficulty is compounded by the presence of diagrams
which diverge as $1/a^{2}$ as the continuum limit is approached ($a$ is the
lattice spacing). While these divergent amplitudes must vanish for a physical,
on-shell decay they substantially degrade the signal to noise ratio even for
an energy-conserving calculation such as this one. Studying the properties of
the $1/a^{2}$ terms and learning how to successfully subtract them is one of
the important objectives of this calculation. The chiral symmetry needed to
control operator mixing is provided by our use of domain wall fermions.
Recognizing the difficulty of this problem, we choose to perform this first
calculation on a lattice which is relatively small compared to those used in
other recent work and to use a somewhat heavy pion mass ($m_{\pi}\approx$ 421
MeV) so we can more easily collect large statistics. We concentrate on
exploring and reducing the statistical uncertainty since the primary goal of
this work is to extract a clear signal for these amplitudes. Therefore, the
quoted errors on our results are statistical only.
The main objective of this paper is to calculate the $\Delta I=1/2$ decay
amplitude $A_{0}$. A calculation of the $\Delta I=3/2$ part is included here
for comparison and completeness. A much more physical calculation of this
$\Delta I=3/2$ amplitude alone can be found in Goode:2011kb . In the case of
the $I=2$ final state no disconnected diagrams appear, there are no divergent
eye diagrams and isospin conservation requires that four valence quark
propagators must join the kaon and weak operator with the operators creating
the two final-state pions. This allows physical kinematics with non-zero final
momenta to be achieved by imposing anti-periodic boundary conditions on one
species of valence quark Kim:2003xt ; Sachrajda:2004mi . As a result, the
preliminary calculation of $A_{2}$ reported in Ref. Goode:2011kb is performed
at almost physical kinematics on a lattice of spatial size 4.5 fm and
determines complex $A_{2}$ with controlled errors of $O(10\%)$. The present
work is intended as the first step toward an equally physical but much more
challenging calculation of $A_{0}$.
While we do not employ physical kinematics, the final results for the complex
amplitudes $A_{0}$ and $A_{2}$ presented in this paper are otherwise physical.
In particular, we use Rome-Southampton methods Martinelli:1995ty to change
the normalization of our bare lattice four-quark operators to that of the
RI/MOM scheme. A second conversion to the $\overline{\mbox{MS}}$ scheme is
then performed using the recent results of Ref. Lehner:2011fz . Finally these
$\overline{\mbox{MS}}$-normalized matrix elements are combined with the
appropriate Wilson coefficients Buchalla:1995vs , determined in this same
scheme, to obtain our results for $A_{0}$ and $A_{2}$. Because of our
unphysical, threshold kinematics and focus on controlling the statistical
errors associated with the disconnected diagrams, we do not estimate the size
of possible systematic errors. Similarly we do not include the systematic or
statistical errors associated with the Rome-Southampton renormalization
factors, both of which could be made substantially than our statistical errors
when required.
This paper is organized as follows. We first summarize our computational
setup, including our strategy to collect large statistics. Next we discuss our
results for $\pi-\pi$ scattering which are a by-product of the necessary
characterization of the operator creating the $\pi-\pi$ final state and are
also needed to evaluate the Lellouch-Lüscher, finite-volume correction
Lellouch:2000pv . After a section giving the details of the
$K^{0}\rightarrow\pi\pi$ contractions, we provide our numerical results for
the $K^{0}\rightarrow\pi\pi$ decay amplitudes for both the $\Delta I=3/2$ and
$1/2$ channels. The details of the operator renormalization required by the
Wilson coefficients which we use are presented in Appendix A. Finally we
present our conclusions and discuss future prospects.
## II Computational Details
Our calculation uses the Iwasaki gauge action with $\beta=2.13$ and 2+1
flavors of domain wall fermions (DWF). While the computational costs of DWF
are much greater than those of Wilson or staggered fermions, as has been shown
in earlier papers Bernard:1988zj ; Dawson:1997ic ; Blum:2001xb ; Noaki:2001un
, accurate chiral symmetry at short distances is critical to avoid extensive
operator mixing, which would make the lattice treatment of $\Delta S=1$
processes much more difficult.
We use a single lattice ensemble with space-time volume $16^{3}\times 32$, a
fifth-dimensional extent of $L_{s}=16$ and light and strange quark masses of
$m_{l}=0.01$, $m_{s}=0.032$, respectively. This ensemble is similar to the
$m_{l}=0.01$ ensemble reported in Ref. Allton:2007hx except we use the
improved RHMC-II algorithm of Ref. Allton:2008pn and a more physical value
for the strange quark mass. The inverse lattice spacing for these input
parameters was determined to be 1.73(3)GeV and the residual mass is $m_{\rm
res}=0.00308(4)$ Allton:2008pn . The total number of configurations we used is
800, each separated by 10 time units. We initially generated an ensemble one-
half of this size. When our analysis showed a non-zero result for Re$A_{0}$,
we then doubled the size of the ensemble to assure ourselves that the result
was trustworthy and to reduce the resulting error. We have performed the
analysis described below both by treating the results from each configuration
as independent and by grouping them into blocks. The resulting statistical
errors are independent of block size suggesting that the individual
configurations are essentially uncorrelated for our observables.
We use anti-periodic boundary conditions in the time direction, and periodic
boundary conditions in the space directions for the Dirac operator. The
propagators (inverses of the Dirac operator) are calculated using a Coulomb
gauge fixed wall source (used for meson propagators) and a random wall source
(used to calculate the loops in the $type3$ and $type4$ graphs shown in Figs.
5 and 6 below) for each of the 32 time slices in our lattice volume. For each
time slice and source type, twelve inversions are required corresponding to
the possible 3 color and 4 spin choices for the source. Thus, all together we
carry out 768 inversions for each quark mass on a given configuration. As will
be shown below, this large number of inversions, performed on 800
configurations, provides the substantial statistics needed to resolve the real
part of the $I=0$ amplitude $A_{0}$ with $25\%$ accuracy.
The situation described above in which 768 Dirac propagators must be computed
on a single gauge background is an excellent candidate for the use of
deflation techniques. The overhead associated with determining a set of low
eigenmodes of this single Dirac operator can be effectively amortized over the
many inversions in which those low modes can be used. Our $m_{l}=0.01$, light
quark inversions are accelerated by a factor of 2-3 by using exact, low-mode
deflation Giusti:2002sm in which we compute the Dirac eigenvectors with the
smallest 35 eigenvalues and limit the conjugate gradient inversion to the
remaining orthogonal subspace.
Table 1: Masses of pion and kaons and energies of the two-pion states. Here the subscript $I=0$ or 2 on the $\pi-\pi$ energy, $E_{I}^{\pi\pi}$, labels the isospin of the state and $E_{0}^{\pi\pi\prime}$ represents the isospin zero, two-pion energy obtained when the disconnected graph V is ignored. The superscript (0), (1) or (2) on the kaon mass distinguishes our three choices of valence strange quark mass, $m_{s}=0.066$, 0.099 and 0.165 respectively. $m_{\pi}$ | $E_{0}^{\pi\pi}$ | $E_{0}^{\pi\pi\prime}$ | $E_{2}^{\pi\pi}$ | $m_{K}^{(0)}$ | $m_{K}^{(1)}$ | $m_{K}^{(2)}$
---|---|---|---|---|---|---
0.24373(47) | 0.443(13) | 0.4393(41) | 0.5066(11) | 0.42599(42) | 0.50729(44) | 0.64540(49)
In order to obtain energy-conserving $K^{0}\rightarrow\pi\pi$ decay
amplitudes, the mass of the valence strange quark in the kaon is assigned a
value different from that appearing in the fermion determinant used to
generate the ensembles, i.e. the strange quark is partially quenched. Since
the mass of the dynamical strange quark is expected to have a small effect on
amplitudes of the sort considered here Allton:2008pn ; Lightman:2009ka , this
use of partial quenching is appropriate for the purposes of this paper.
Valence strange quark masses are chosen to be $m_{s}=0.066$, 0.099 and 0.165,
which are labeled 0, 1 and 2 respectively. The resulting kaon masses are shown
in Tab. 1. In the following section we will see that by using these values for
$m_{s}$ we can interpolate to energy-conserving decay kinematics for both the
$I=2$ and $I=0$ channels.
## III Two-pion Scattering
The $\pi-\pi$ scattering calculation requires 4 contractions which we have
labeled direct (D), cross (C), rectangle (R), and vacuum (V) as in Ref.
Liu:2009uw and which are shown in Fig. 1. For convenience, the minus sign
arising from the number of fermion loops is not included in the definition of
these contractions. The vacuum contraction should be accompanied by a vacuum
subtraction. These contractions can be calculated in terms of the light quark
propagator $L(t_{\mathrm{snk}},t_{\mathrm{src}})$ for a Coulomb gauge fixed
wall source located at the time $t_{\mathrm{src}}$ and a similar wall sink
located at $t_{\mathrm{snk}}$. The resulting complete vacuum amplitude,
including the vacuum subtraction, is given by
$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\frac{1}{32}\sum_{t^{\prime}=0}^{31}\Biggl{\\{}\Bigl{\langle}\mbox{tr}[L(t^{\prime},t^{\prime})L(t^{\prime},t^{\prime})^{\dagger}]\mbox{tr}[L(t+t^{\prime},t+t^{\prime})L(t+t^{\prime},t+t^{\prime})^{\dagger}]\Bigr{\rangle}$
$\displaystyle\hskip
21.68121pt-\Bigl{\langle}\mbox{tr}[L(t^{\prime},t^{\prime})L(t^{\prime},t^{\prime})^{\dagger}]\Bigr{\rangle}\Bigl{\langle}\mbox{tr}[L(t+t^{\prime},t+t^{\prime})L(t+t^{\prime},t+t^{\prime})^{\dagger}]\Bigr{\rangle}\Biggr{\\}},$
where the indicated traces are taken over spin and color, the hermiticity
properties of the domain wall propagator have been used to eliminate factors
of $\gamma^{5}$ and we are explicitly combining the results from each of the
32 time slices.
Our results for each of these four types of contractions are shown in the left
panel of Fig. 2. Notice that the disconnected (vacuum) graph has an almost
constant error with increasing time separation between the source and sink, so
it appears to have an increasing error bar in the log plot, while the signal
decreases exponentially.
Figure 1: The four diagrams which contribute to $\pi-\pi$ scattering: direct
(D), cross (C), rectangle (R), and vacuum (V), arranged from the left top to
right bottom.
These four types of correlators can be combined to construct physical
correlation functions for two-pion states with definite isospin:
$\displaystyle\left\langle
O_{2}^{\pi\pi}(t+t^{\prime})^{\dagger}O_{2}^{\pi\pi}(t^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle 2\bigl{(}D(t)-C(t)\bigr{)}$ (2)
$\displaystyle\left\langle
O_{0}^{\pi\pi}(t+t^{\prime})^{\dagger}O_{0}^{\pi\pi}(t^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle 2D(t)+C(t)-6R(t)+3V(t).$ (3)
Here the operator $O_{I}^{\pi\pi}(t)$ creates a two-pion state with total
isospin $I$ and $z$-component of isospin $I_{z}=0$ using two quark and two
anti-quark wall-sources located at the time-slice $t$. As in Eq. III we will
average over all 32 possible values of common time displacement $t^{\prime}$
to improve statistics.
The two-pion correlation functions for isospin $I$ and $I_{z}=0$ are fit with
a functional form
Corr${}_{I}(t)=N_{I}^{2}\\{\exp(-E_{I}^{\pi\pi}t)+\exp(-E_{I}^{\pi\pi}(T-t))+C_{I}\\}$,
where the constant $C_{I}$ comes from the case in which the two pions
propagate in opposite time directions. The fitted energies are summarized in
Tab. 1. In order to see clearly the effect of the disconnected graph, we also
perform the calculation for the $I=0$ channel without the disconnected graphs.
This result is given in Tab. 1 with a label with an additional prime
($\prime$) symbol. The resulting effective mass plots for each case are shown
in the right panel of Fig. 2. For comparison, a plot of twice the pion
effective mass is also shown. This figure clearly demonstrates that the two-
pion interaction is attractive in the $I=0$ channel with the finite volume,
$I=0$ $\pi-\pi$ energy $E_{0}^{\pi\pi}$ lower than $2m_{\pi}$. In contrast,
the $I=2$ channel is repulsive with $E_{2}^{\pi\pi}$ larger than $2m_{\pi}$.
The fitted parameters $N_{I}^{\pi\pi}$ and $E_{I}^{\pi\pi}$ will be used to
extract weak matrix elements from the $K^{0}\rightarrow\pi\pi$ correlation
functions discussed below in which these same operators $O_{I}^{\pi\pi}(t)$
are used to construct the two-pion states.
|
---|---
Figure 2: Left: Results for the four types of contractions, direct (D), cross
(C), rectangle (R), and vacuum(V) represented by the graphs in Fig. 1. Right:
Effective mass plots for correlation functions for states with isospin two
($I_{2}$), isospin zero ($I_{0}$), isospin zero without the disconnected graph
($I_{0}^{\prime}$) and twice the pion effective mass ($2m_{\pi}$).
## IV Contractions for $K^{0}\rightarrow\pi\pi$ Decays
The effective weak Hamiltonian describing $K^{0}\rightarrow\pi\pi$ decay
including the $u$, $d$, and $s$ flavors as dynamical variables is
$H_{w}=\frac{G_{F}}{\sqrt{2}}V_{ud}^{*}V_{us}\sum_{i=1}^{10}[(z_{i}(\mu)+\tau
y_{i}(\mu))]Q_{i}.$ (4)
Throughout this paper we follow the conventions and notation of Ref.
Blum:2001xb . In Eq. 4 the $Q_{i}$ are the ten conventional four-quark
operators, $z_{i}$ and $y_{i}$ are the Wilson coefficients, and $\tau$
represents a combination of CKM matrix elements:
$\tau=-V_{ts}^{*}V_{td}/V_{ud}V_{us}^{*}$. To calculate the decay amplitudes
$A_{2}$ and $A_{0}$, we need to calculate the matrix elements
$\langle\pi\pi|Q_{i}|K^{0}\rangle$ on the lattice.
|
---|---
{\scriptsize1}⃝/{\scriptsize3}⃝ | {\scriptsize2}⃝/{\scriptsize4}⃝
|
{\scriptsize5}⃝/{\scriptsize7}⃝ | {\scriptsize6}⃝/{\scriptsize8}⃝
Figure 3: Diagrams representing the eight $K^{0}\rightarrow\pi\pi$ contractions of $type1$, where $\Gamma_{V\pm A}=\gamma_{\mu}(1\pm\gamma_{5})$. The black dot indicates a $\gamma_{5}$ matrix, which is present in each operator creating or destroying a pseudoscalar meson. |
---|---
{\scriptsize9}⃝/{\scriptsize11}⃝ | {\scriptsize10}⃝/{\scriptsize12}⃝
|
{\scriptsize13}⃝/{\scriptsize15}⃝ | {\scriptsize14}⃝/{\scriptsize16}⃝
Figure 4: Diagrams for the eight $type2$ $K^{0}\rightarrow\pi\pi$ contractions. |
---|---
{\scriptsize17}⃝/{\scriptsize19}⃝ | {\scriptsize18}⃝/{\scriptsize20}⃝
|
{\scriptsize21}⃝/{\scriptsize23}⃝ | {\scriptsize22}⃝/{\scriptsize24}⃝
|
{\scriptsize25}⃝/{\scriptsize27}⃝ | {\scriptsize26}⃝/{\scriptsize28}⃝
|
{\scriptsize29}⃝/{\scriptsize31}⃝ | {\scriptsize30}⃝/{\scriptsize32}⃝
Figure 5: Diagrams for the 16 $type3$ $K^{0}\rightarrow\pi\pi$ contractions. |
---|---
{\scriptsize33}⃝/{\scriptsize35}⃝ | {\scriptsize34}⃝/{\scriptsize36}⃝
|
{\scriptsize37}⃝/{\scriptsize39}⃝ | {\scriptsize38}⃝/{\scriptsize40}⃝
|
{\scriptsize41}⃝/{\scriptsize43}⃝ | {\scriptsize42}⃝/{\scriptsize44}⃝
|
{\scriptsize45}⃝/{\scriptsize47}⃝ | {\scriptsize46}⃝/{\scriptsize48}⃝
Figure 6: Diagrams for the sixteen $type4$ $K^{0}\rightarrow\pi\pi$
contractions.
We list all of the possible contractions contributing to the matrix elements
$\left<\pi\pi|Q_{i}|K^{0}\right>$ in Figs. 3-6. There are 48 different
contractions which are labeled by circled numbers ranging from 1 to 48, and
grouped into four categories labeled as $type1$, $type2$, $type3$, and $type4$
according to their topology. Once we have calculated all of these
contractions, the correlation functions
$\left<O^{\pi\pi}_{I}(t_{\pi})Q_{i}(t_{\rm op})K^{0}(t_{K})\right>$ are then
obtained as combinations of these contractions. In order to simplify the
following formulae, we use the amplitude $A_{I,i}(t_{\pi},t,t_{K})$ to
represent three point function $\langle O^{\pi\pi}_{I}(t_{\pi})Q_{i}(t_{\rm
op})K(t_{K})\rangle$. Using this notation, the $I=2$ amplitudes can be
written,
$\displaystyle A_{2,1}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{2}{3}}\\{\mbox{{\scriptsize1}⃝}-\mbox{{\scriptsize5}⃝}\\}$ (5a)
$\displaystyle A_{2,2}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{2}{3}}\\{\mbox{{\scriptsize2}⃝}-\mbox{{\scriptsize6}⃝}\\}$ (5b)
$\displaystyle A_{2,3}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle 0$ (5c) $\displaystyle A_{2,4}(t_{\pi},t_{\rm op},t_{K})$
$\displaystyle=$ $\displaystyle 0$ (5d) $\displaystyle A_{2,5}(t_{\pi},t_{\rm
op},t_{K})$ $\displaystyle=$ $\displaystyle 0$ (5e) $\displaystyle
A_{2,6}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$ $\displaystyle 0$ (5f)
$\displaystyle A_{2,7}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{3}{2}}\\{\mbox{{\scriptsize3}⃝}-\mbox{{\scriptsize7}⃝}\\}$ (5g)
$\displaystyle A_{2,8}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{3}{2}}\\{\mbox{{\scriptsize4}⃝}-\mbox{{\scriptsize8}⃝}\\}$ (5h)
$\displaystyle A_{2,9}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{3}{2}}\\{\mbox{{\scriptsize1}⃝}-\mbox{{\scriptsize5}⃝}\\}$ (5i)
$\displaystyle A_{2,10}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{\frac{3}{2}}\\{\mbox{{\scriptsize2}⃝}-\mbox{{\scriptsize6}⃝}\\}$ (5j)
and in the I=0 case,
$\displaystyle A_{0,1}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{1}{\sqrt{3}}\\{-\mbox{{\scriptsize1}⃝}-2\cdot\mbox{{\scriptsize5}⃝}+3\cdot\mbox{{\scriptsize9}⃝}+3\cdot\mbox{{\scriptsize17}⃝}-3\cdot\mbox{{\scriptsize33}⃝}\\}$
(6a) $\displaystyle A_{0,2}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{1}{\sqrt{3}}\\{-\mbox{{\scriptsize2}⃝}-2\cdot\mbox{{\scriptsize6}⃝}+3\cdot\mbox{{\scriptsize10}⃝}+3\cdot\mbox{{\scriptsize18}⃝}-3\cdot\mbox{{\scriptsize34}⃝}\\}$
(6b) $\displaystyle A_{0,3}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{3}\\{-\mbox{{\scriptsize5}⃝}+2\cdot\mbox{{\scriptsize9}⃝}-\mbox{{\scriptsize13}⃝}+2\cdot\mbox{{\scriptsize17}⃝}+\mbox{{\scriptsize21}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize25}⃝}-\mbox{{\scriptsize29}⃝}-2\cdot\mbox{{\scriptsize33}⃝}-\mbox{{\scriptsize37}⃝}+\mbox{{\scriptsize41}⃝}+\mbox{{\scriptsize45}⃝}\\}$
$\displaystyle A_{0,4}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{3}\\{-\mbox{{\scriptsize6}⃝}+2\cdot\mbox{{\scriptsize10}⃝}-\mbox{{\scriptsize14}⃝}+2\cdot\mbox{{\scriptsize18}⃝}+\mbox{{\scriptsize22}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize26}⃝}-\mbox{{\scriptsize30}⃝}-2\cdot\mbox{{\scriptsize34}⃝}-\mbox{{\scriptsize38}⃝}+\mbox{{\scriptsize42}⃝}+\mbox{{\scriptsize46}⃝}\\}$
$\displaystyle A_{0,5}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{3}\\{-\mbox{{\scriptsize7}⃝}+2\cdot\mbox{{\scriptsize11}⃝}-\mbox{{\scriptsize15}⃝}+2\cdot\mbox{{\scriptsize19}⃝}+\mbox{{\scriptsize23}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize27}⃝}-\mbox{{\scriptsize31}⃝}-2\cdot\mbox{{\scriptsize35}⃝}-\mbox{{\scriptsize39}⃝}+\mbox{{\scriptsize43}⃝}+\mbox{{\scriptsize47}⃝}\\}$
$\displaystyle A_{0,6}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\sqrt{3}\\{-\mbox{{\scriptsize8}⃝}+2\cdot\mbox{{\scriptsize12}⃝}-\mbox{{\scriptsize16}⃝}+2\cdot\mbox{{\scriptsize20}⃝}+\mbox{{\scriptsize24}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize28}⃝}-\mbox{{\scriptsize32}⃝}-2\cdot\mbox{{\scriptsize36}⃝}-\mbox{{\scriptsize40}⃝}+\mbox{{\scriptsize44}⃝}+\mbox{{\scriptsize48}⃝}\\}$
$\displaystyle A_{0,7}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{\sqrt{3}}{2}\\{-\mbox{{\scriptsize3}⃝}-\mbox{{\scriptsize7}⃝}+\mbox{{\scriptsize11}⃝}+\mbox{{\scriptsize15}⃝}+\mbox{{\scriptsize19}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize23}⃝}+\mbox{{\scriptsize27}⃝}+\mbox{{\scriptsize31}⃝}-\mbox{{\scriptsize35}⃝}+\mbox{{\scriptsize39}⃝}-\mbox{{\scriptsize43}⃝}-\mbox{{\scriptsize47}⃝}\\}$
$\displaystyle A_{0,8}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{\sqrt{3}}{2}\\{-\mbox{{\scriptsize4}⃝}-\mbox{{\scriptsize8}⃝}+\mbox{{\scriptsize12}⃝}+\mbox{{\scriptsize16}⃝}+\mbox{{\scriptsize20}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize24}⃝}+\mbox{{\scriptsize28}⃝}+\mbox{{\scriptsize32}⃝}-\mbox{{\scriptsize36}⃝}+\mbox{{\scriptsize40}⃝}-\mbox{{\scriptsize44}⃝}-\mbox{{\scriptsize48}⃝}\\}$
$\displaystyle A_{0,9}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{\sqrt{3}}{2}\\{-\mbox{{\scriptsize1}⃝}-\mbox{{\scriptsize5}⃝}+\mbox{{\scriptsize9}⃝}+\mbox{{\scriptsize13}⃝}+\mbox{{\scriptsize17}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize21}⃝}+\mbox{{\scriptsize25}⃝}+\mbox{{\scriptsize29}⃝}-\mbox{{\scriptsize33}⃝}+\mbox{{\scriptsize37}⃝}-\mbox{{\scriptsize41}⃝}-\mbox{{\scriptsize45}⃝}\\}$
$\displaystyle A_{0,10}(t_{\pi},t_{\rm op},t_{K})$ $\displaystyle=$
$\displaystyle
i\frac{\sqrt{3}}{2}\\{-\mbox{{\scriptsize2}⃝}-\mbox{{\scriptsize6}⃝}+\mbox{{\scriptsize10}⃝}+\mbox{{\scriptsize14}⃝}+\mbox{{\scriptsize18}⃝}$
$\displaystyle\hskip
36.135pt-\mbox{{\scriptsize22}⃝}+\mbox{{\scriptsize26}⃝}+\mbox{{\scriptsize30}⃝}-\mbox{{\scriptsize34}⃝}+\mbox{{\scriptsize38}⃝}-\mbox{{\scriptsize42}⃝}-\mbox{{\scriptsize46}⃝}\\},$
where the factor $i$ comes from our definition of the interpolation operator
for the mesons, e.g. $K^{0}=i(\overline{d}\gamma_{5}s)$.
A few notes about the contractions shown in the Figs. 3 \- 6 may be useful:
1. 1.
The contractions identified by circled numbers do not carry the minus sign
required when there is an odd number of fermion loops. Instead, the signs are
included explicitly in Eqs. 5 and 6.
2. 2.
The routing of the solid line indicates spin contraction while that of the
dashed line indicates the contraction of color indices. If there is no dashed
line, then solid line indicates connections implied by the trace over both
color and spin indices. (This will be explained in more detail below.)
3. 3.
A line represents a light quark propagator if it is not explicitly labeled
with ’s’. Up and down quarks and particular flavors of pion are not
distinguished in Figs. 3 \- 6. Instead these specific contractions of strange
and light quark propagators are combined in Eqs. 5 and 6 to give the $I=2$ and
$I=0$ amplitudes directly.
4. 4.
Using Fierz symmetry, it can be shown that there are 12 identities among these
contractions:
$\displaystyle\mbox{{\scriptsize6}⃝}=-\mbox{{\scriptsize1}⃝},\quad\mbox{{\scriptsize5}⃝}=-\mbox{{\scriptsize2}⃝},\quad\mbox{{\scriptsize14}⃝}=-\mbox{{\scriptsize9}⃝},\quad\mbox{{\scriptsize13}⃝}=-\mbox{{\scriptsize10}⃝},$
(7a)
$\displaystyle\mbox{{\scriptsize26}⃝}=-\mbox{{\scriptsize17}⃝},\quad\mbox{{\scriptsize25}⃝}=-\mbox{{\scriptsize18}⃝},\quad\mbox{{\scriptsize29}⃝}=-\mbox{{\scriptsize22}⃝},\quad\mbox{{\scriptsize30}⃝}=-\mbox{{\scriptsize21}⃝},$
(7b)
$\displaystyle\mbox{{\scriptsize42}⃝}=-\mbox{{\scriptsize33}⃝},\quad\mbox{{\scriptsize41}⃝}=-\mbox{{\scriptsize34}⃝},\quad\mbox{{\scriptsize45}⃝}=-\mbox{{\scriptsize38}⃝},\quad\mbox{{\scriptsize46}⃝}=-\mbox{{\scriptsize37}⃝}.$
(7c)
A consequence of these identities is that Eq. 6 is consistent with only seven
of the ten operators $Q_{i}$ being linearly independent and with the three
usual relations:
$\displaystyle Q_{10}-Q_{9}$ $\displaystyle=$ $\displaystyle Q_{4}-Q_{3}$ (8a)
$\displaystyle Q_{4}-Q_{3}$ $\displaystyle=$ $\displaystyle Q_{2}-Q_{1}$ (8b)
$\displaystyle 2Q_{9}$ $\displaystyle=$ $\displaystyle 3Q_{1}-Q_{3}.$ (8c)
5. 5.
Based on charge conjugation symmetry and $\gamma^{5}$ hermiticity, the gauge
field average of each of these contractions is real.
6. 6.
The loop contractions of $type3$ and $type4$ are calculated using the
Gaussian, stochastic wall sources described in Sec. II.
In order to make our approach more explicit, we will discuss some examples.
First consider the two contractions of $type1$ identified as {\scriptsize1}⃝
and {\scriptsize2}⃝ and shown in the top half of Fig. 3:
{\scriptsize1}⃝ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Bigl{\\{}\gamma_{\mu}(1-\gamma_{5})L(x_{op},t_{\pi})L(x_{op},t_{\pi})^{\dagger}\Bigr{\\}}$
$\displaystyle\hskip
36.135pt\cdot\mathrm{Tr}\Bigl{\\{}\gamma^{\mu}(1-\gamma_{5})L(x_{op},t_{\pi})\gamma^{5}\left[\sum_{\vec{x}_{\pi}}L((\vec{x}_{\pi},t_{\pi}),t_{K})\right]S(x_{op},t_{K})^{\dagger}\Bigr{\\}}$
{\scriptsize2}⃝ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{c}\Biggl{\\{}\mathrm{Tr}_{s}\Bigl{\\{}\gamma_{\mu}(1-\gamma_{5})L(x_{op},t_{\pi})L(x_{op},t_{\pi})^{\dagger}\Bigr{\\}}$
$\displaystyle\hskip
36.135pt\cdot\mathrm{Tr}_{s}\Bigl{\\{}\gamma^{\mu}(1-\gamma_{5})L(x_{op},t_{\pi})\gamma_{5}\left[\sum_{\vec{x}_{\pi}}L((\vec{x}_{\pi},t_{\pi}),t_{K})\right]S(x_{op},t_{K})^{\dagger}\Bigr{\\}}\Biggr{\\}},$
where $t_{K}$ is the time of the kaon wall source, $t_{\pi}$ the time at which
the two pions are absorbed and $x_{op}=(\vec{x}_{op},t_{op})$ the location of
the weak operator. The function $L(x_{\rm sink},t_{\rm src})$ is the light
quark propagator, a $12\times 12$ spin-color matrix, while $S(x_{\rm
sink},t_{\rm src})$ is the strange quark propagator. The hermitian conjugation
operation, $\dagger$, operates on these $12\times 12$ matrices. We use Trc to
indicate a color trace, Trs a spin trace, and Tr, with no subscript, stands
for both a spin and color trace. We have also used the $\gamma^{5}$
hermiticity of the quark propagators to realize the combination of quark
propagators given in Eqs. IV and IV, allowing both contractions to be
constructed from light and strange propagators computed using Coulomb gauge
fixed wall sources located only at the times $t_{\pi}$ and $t_{K}$. Note the
sum over the spatial components of the sink $\vec{x}_{\pi}$ creates a
symmetrical wall sink provided that the appropriate Coulomb gauge
transformation matrix has been applied to the sink color index of this
propagator to duplicate the Coulomb gauge transformation that was used to
create the Coulomb gauge fixed wall source. We will sum over the spatial
location, $\vec{x}_{op}$, of the weak operator, to project onto zero spatial
momentum and improve statistics. Below we will show results as a function of
the separations between $t_{\pi}$, $t_{\rm op}$ and $t_{K}$.
As a third example, which illustrates the use of random wall sources, consider
contraction {\scriptsize19}⃝ shown in Fig. 5. Using the notation introduced
above, this contraction is given by
{\scriptsize19}⃝ $\displaystyle=$
$\displaystyle\mathrm{Tr}\Bigl{\\{}\gamma_{\mu}(1+\gamma_{5})L^{R}(x_{\rm
op},t_{\rm op})\Bigr{\\}}\eta(x_{\rm op})^{*}$ $\displaystyle\hskip
7.22743pt\cdot\mathrm{Tr}\Bigl{\\{}\gamma^{\mu}(1-\gamma_{5})L(x_{\rm
op},t_{\pi})\Biggl{[}\sum_{\vec{x}_{\pi}^{\prime}}L\Bigl{(}(\vec{x}_{\pi}^{\prime},t_{\pi}),t_{\pi}\Bigr{)}^{\dagger}\Biggr{]}\Biggl{[}\sum_{\vec{x}_{\pi}}L\Bigl{(}(\vec{x}_{\pi},t_{\pi}),t_{K}\Bigr{)}\Biggr{]}S(x_{\rm
op},t_{K})^{\dagger}\Bigr{\\}}.$
Here $\eta(x)$ is the value of the complex, Gaussian random wall source at the
space-time position $x$, while $L^{R}(x_{\rm sink},t_{\rm src})$ is the
propagator whose source is $\eta(x)\delta(x_{0}-t_{\rm src})$. The Dirac delta
function $\delta(x_{0}-t_{\rm src})$ restricts the source to the time plane
$t=t_{\rm src}$. In the usual way, the average over the random source
$\eta(\vec{x})$ which accompanies the configuration average, will set to zero
all terms in which the source and sink positions for the propagator
$L^{R}(x_{\rm op},t_{\rm op})$ in Eq. IV differ, giving us the contraction
implied by the closed loop in the top left panel of Fig. 5. By using 32
separate propagators each with a random source non-zero on only one of our 32
time slices we obtain more statistically accurate results than would result
from a single random source spread over all times.
An important objective of this calculation is to learn how to accurately
evaluate the quark loop integration that is present in $type3$ and $type4$
graphs and which contains a $1/a^{2}$, quadratically divergent component. As
can be recognized from the structure of the diagrams, these divergent terms
can be interpreted as arising from the mixing between the dimension-six
operators $Q_{i}$ (for all $i$ but 7 and 8) and a dimension-3 “mass” operator
of the form $\overline{s}\gamma_{5}d$. Such divergent terms are expected and
do not represent a breakdown of the standard effective Hamiltonian written in
Eq. 4. In fact, given the good chiral symmetry of domain wall fermions all
other operators with dimension less than six which might potentially mix with
those in Eq. 4 will vanish if the equations of motion are imposed. Therefore
these operators cannot contribute to the Green’s functions evaluated in Eqs. 5
and 6 where the operators in $H_{W}$ are separated in space-time from those
operators creating the $K$ meson and destroying the $\pi$ mesons, a
circumstance in which the equations of motion can be applied.
The problematic operator $\overline{s}\gamma_{5}d$ is not explictly removed
from the effective Hamiltonian because, again using the equations of motion,
$\overline{s}\gamma_{5}d$ can be written as the divergence of an axial current
and hence will vanish in the physical case where the weak operator $H_{W}$
carries no four-momentum and is evaluated between on-shell states. While we
can explicitly sum the effective Hamiltonian density ${\cal H}_{W}$ over space
to ensure $H_{W}$ carries no spatial momentum, to ensure that no energy is
transferred we must arrange that the kaon mass and two-pion energy are equal.
We may achieve this condition, at least approximately, but there will be
contributions from heavier states, which are normally exponentially
suppressed, but which will violate energy conservation and hence will be
enhanced by this divergent $\overline{s}\gamma_{5}d$ term.
Since $\overline{s}\gamma_{5}d$ will not contribute to the physical, energy-
conserving $K\rightarrow\pi\pi$ amplitude, there is no theoretical requirement
that it be removed. The coefficient of this $\overline{s}\gamma_{5}d$ piece is
both regulator dependent and irrelevant. The contribution of these terms in a
lattice calculation of $K\rightarrow\pi\pi$ decay amplitudes will ultimately
vanish as the equality of the initial and final energies is made more precise
and as increased time separations are achieved. However, the unphysical
effects of this $\overline{s}\gamma_{5}d$ mixing are much more easily
suppressed by reducing the size of this irrelevant term than by dramatically
increasing the lattice size and collecting the substantially increased
statistics required to work at large time separations.
A direct way to remove this $1/a^{2}$ enhancement is to explicitly subtract an
$\alpha_{i}\overline{s}\gamma_{5}d$ term from each of the relevant operators
$Q_{i}$ where the coefficient $\alpha_{i}$ can be fixed by imposing the
condition:
$\left<0|Q_{i}-\alpha_{i}\overline{s}\gamma_{5}d|K\right>=0,$ (12)
a condition that is typically required in the chiral perturbation theory for
$K\rightarrow\pi\pi$ Blum:2001xb . Of course, this arbitrary condition will
leave a finite, regulator-dependent $\overline{s}\gamma_{5}d$ piece behind in
the subtracted operator $Q_{i}-\alpha_{i}\overline{s}\gamma_{5}d$. However,
this unphysical piece will not contribute to the energy-conserving amplitude
being evaluated. Since it is no longer $1/a^{2}$-enhanced its effects on our
calculation will be similar to those of the many other energy non-conserving
terms which we must suppress by choosing equal energy $K$ and $\pi\pi$ states
and using sufficient large time separation to suppress the contributions of
excited states.
Following Eq. 12 we will choose the coefficient $\alpha_{i}$ from the ratio
$\alpha_{i}=\frac{\left<0|Q_{i}|K^{0}\right>}{\left<0|\overline{s}\gamma_{5}d|K^{0}\right>}.$
(13)
(Note, with this definition the coefficient $\alpha_{i}$ is proportional to
the difference of the strange and light quark masses.) Thus, we will improve
the accuracy when calculating graphs of $type3$ and $type4$ by including an
explicit subtraction term for those operators $Q_{i}$ where mixing with
$\overline{s}\gamma_{5}d$ is permitted by the symmetries (all but $Q_{7}$ and
$Q_{8}$):
$\left<O^{\pi\pi}_{0}(t_{\pi})Q_{i}(t_{\rm
op})K^{0}(t_{K})\right>_{sub}=\left<O^{\pi\pi}_{0}(t_{\pi})Q_{i}(t_{\rm
op})K^{0}(t_{K})\right>-\alpha_{i}\left<O^{\pi\pi}_{0}(t_{\pi})\overline{s}\gamma_{5}d(t_{\rm
op})K^{0}(t_{K})\right>.$ (14)
We should recognize that there is a second, divergent, parity-even operator
$\overline{s}d$ which mixes with our operators $Q_{i}$. However, we choose to
neglect this effect because parity symmetry prevents it from contributing to
either the $K\rightarrow\pi\pi$ or $K\rightarrow|0\rangle$ correlation
functions being evaluated here.
The amplitude $\left<O^{\pi\pi}_{0}(t_{\pi})\overline{s}\gamma_{5}d(t_{\rm
op})K^{0}(t_{K})\right>$ includes two contractions, one connected and one
disconnected as shown in Fig. 7. These terms, which arise from the mixing of
the operators $Q_{i}$ with $\overline{s}\gamma_{5}d$, are labeled $mix3$ and
$mix4$. To better visualize the contributions from different types of
contractions, we can write the right hand side of Eq. 14 symbolically as
$\displaystyle type1+type2+type3+type4-\alpha\cdot(mix3+mix4)$ (15)
$\displaystyle=$ $\displaystyle type1+type2+sub3+sub4,$
where $sub3=type3-\alpha\cdot mix3$ and $sub4=type4-\alpha\cdot mix4$. Note,
here and in later discussions we refer to the term being subtracted as “mix”
and the final difference as the subtracted amplitude “sub”.
|
---|---
$mix3$ | $mix4$
Figure 7: Diagrams showing the contractions needed to evaluate the subtraction
terms. These are labeled $mix3$ and $mix4$ and constructed from the $type3$
and $type4$ contractions by replacing the operator $Q_{i}$ and fermion loop
with the vertex $\overline{s}\gamma_{5}d$.
## V $K^{0}\rightarrow\pi\pi$ $\Delta I=3/2$ amplitude
As Eqs. 5 and 7a show, the $\Delta I=3/2$ $K^{0}\rightarrow 2\pi$ decay
amplitude includes only $type1$ contractions and four of the correlation
functions are related
$A_{2,10}=A_{2,9}=\frac{3}{2}A_{2,1}=\frac{3}{2}A_{2,2}.$ (16)
Therefore, we need only to calculate $A_{2,1}$, $A_{2,7}$ and $A_{2,8}$. The
corresponding three correlation functions, $C_{2,i}(\Delta,t)$ for $i=1$, 7
and 8, with the choice of $m_{K}^{(1)}$ for the kaon mass, are shown in Fig.
8. Here we exploit our propagator calculation for sources on each of the 32
time slices to compute $C_{2,i}(\Delta,t)$ from an average over all 32 source
positions:
$C_{2,i}(\Delta,t)=\frac{1}{32}\sum_{t^{\prime}=0}^{31}A_{2,i}(t_{\pi}=t^{\prime}+\Delta,t_{\rm
op}=t+t^{\prime},t_{K}=t^{\prime}).$ (17)
In Fig. 8 we plot $C_{2,i}(\Delta,t)$ for $0<t<\Delta$ at fixed $\Delta=12$ or
16. Table 1 shows that $m_{K}^{(1)}$ is almost equal to the energy of $I=2$,
$\pi-\pi$ state, so the 3-point correlation function $C_{2,i}(\Delta,t)$
should be approximately independent of $t$ in the central region where the
time coordinate of the operator is far from both the kaon and the two-pion
sources, $0\ll t\ll\Delta$.
|
---|---
$\Delta=12$ | $\Delta=16$
Figure 8: Plots of the $\Delta I=3/2$ $K^{0}\rightarrow\pi-\pi$ correlation
functions for kaon source and $\pi-\pi$ sink separations of $\Delta=12$ (left
panel) and $16$ (right panel). The $x$-axis gives the time $t$ specifying the
time slice over which the operator, $Q_{i}(\vec{x},t)$, $i=1,$ 7, 8, is
averaged. The results for the operator $Q_{7}$ are divided by 12, and those
for $Q_{8}$ by 48 to allow the results to be shown in the same graph. The
correlators $C_{2,i}(\Delta,t)$ are fit using the $\Delta=12$ data with a
fitting range $5\leq t\leq 7$. The resulting constants are shown as horizontal
lines in both the $\Delta=12$ and 16 graphs. We can see that the $\Delta=16$
data are consistent with those from $\Delta=12$, but receive large
contributions from the around-the-world paths.
We fit the correlators $C_{2,i}(\Delta,t)$ using a single free parameter
$M_{i}^{3/2,{\rm lat}}$:
$\displaystyle C_{2,i}(\Delta,t)$ $\displaystyle=$ $\displaystyle
M_{i}^{3/2,{\rm
lat}}N_{\pi\pi}N_{K}e^{-E_{\pi\pi}\Delta}e^{-(m_{K}-E_{\pi\pi})t},$ (18)
where $N_{K}$, $m_{K}$ and $N_{\pi\pi}$, $E_{\pi\pi}$ are determined by
fitting the kaon and two-pion correlators respectively:
$\displaystyle\frac{1}{32}\sum_{t^{\prime}=0}^{31}\left<K(t+t^{\prime})K(t^{\prime})\right>$
$\displaystyle=$ $\displaystyle
N_{K}^{2}\left(e^{-m_{K}t}+e^{-m_{K}(T-t)}\right)$ (19)
$\displaystyle\frac{1}{32}\sum_{t^{\prime}=0}^{31}\left<O^{\pi\pi}_{2}(t+t^{\prime})O^{\pi\pi}_{2}(t^{\prime})\right>$
$\displaystyle=$ $\displaystyle
N_{\pi\pi}^{2}\left(e^{-E_{\pi\pi}t}+e^{-E_{\pi\pi}(T-t)}+C\right).$ (20)
The constant $C$ arises when the two pions join the source at $t^{\prime}$ and
sink at $t+t^{\prime}$ by traveling in opposite time directions as discussed
below. The fitted results for the matrix elements $M_{i}^{3/2,{\rm lat}}$ from
$\Delta=12$ are listed in Tab. 2 in lattice units.
Table 2: Results for the lattice $\Delta I=3/2$, $K\rightarrow\pi\pi$ transition amplitudes obtained from fitting the 3-point correlation functions to the functional form given in Eq. 18 for the six operators with $\Delta I=3/2$ components. The second column gives the lattice matrix elements $M_{i}^{3/2,{\rm lat}}(\times 10^{-2})$ while the third and fourth column give their contributions to the real and imaginary parts of $A_{2}$. i | $M_{i}^{3/2,{\rm lat}}(\times 10^{-2})$ | Re$(A_{2})$(GeV) | Im$(A_{2})$(GeV)
---|---|---|---
1 | 0.4892(16) | -1.737(11)e-08 | 0
2 | $=M_{1}$ | 6.665(42)e-08 | 0
7 | 6.080(18) | 2.422(16)e-11 | 4.070(26)e-14
8 | 21.26(6) | -1.979(13)e-10 | -9.646(61)e-12
9 | =1.5$M_{1}$ | -7.917(50)e-15 | 5.185(24)e-13
10 | =1.5$M_{1}$ | 6.103(38)e-12 | -1.448(9)e-13
Total | - | 4.911(31)e-08 | -5.502(40)e-13
Figure 8 shows that for the operators $Q_{7}$ and $Q_{8}$ the larger
separation, $\Delta=16$, between the kaon source and $\pi-\pi$ sink gives a
much shorter plateau region than the case $\Delta=12$. This behavior is
inconsistent with the usual expectation that it is the contributions from
excited states of the kaon and pion, contributions which should be suppressed
for larger $\Delta$, that cause the poor plateau. An alternative, consistent
explanation attributes the shortened plateau region seen for $\Delta=16$ to
the ‘around-the-world’ effect. This is the contribution to the correlation
function in which the two-pion interpolating operator at the sink annihilates
one pion and creates another (instead of annihilating two pions as in the
$K\to\pi\pi$ contribution we are seeking) and the process at the weak operator
is $K\pi\to\pi$ (instead of $K\to\pi\pi$). While one pion travels from the
weak operator to the $\pi-\pi$ sink the second is created at the sink and
travels forward in time, passing through the periodic boundary to reach the
weak operator together with the kaon. The corresponding dominant path is shown
in Fig. 9. The time dependence of this behavior can be estimated as
$\sim M_{i}^{3/2,{\rm
lat}}N_{\pi}^{2}N_{K}e^{-m_{\pi}T}e^{-(E_{K\pi}-m_{\pi})t}$ (21)
which is $\Delta$ independent but suppressed by the factor $\exp(-m_{\pi}T)$,
where $N_{\pi}$ is the analogue of $N_{K}$ for the case of single pion
production and $T=32$ is the temporal extent of the lattice. In contrast, the
physical contribution in Eq. 18 is suppressed by $\exp(-E_{\pi\pi}\Delta)$.
Thus, the second, standard term falls with increasing $\Delta$ and the two
factors are of similar size when $\Delta=T/2$. Therefore, we should expect to
see a large contamination from such around-the-world effects in the
$\Delta=16$ case, consistent with Fig. 8. In both panels of that figure, we
plot as three horizontal lines the fitted result from $\Delta=12$ for the
three amplitudes $M^{3/2,{\rm lat}}_{i}N_{\pi\pi}N_{K}\exp{-\Delta
E_{\pi\pi}}$ for $i=1$, 7 and 8. The agreement between these lines and the
short plateaus seen in the right-hand, $\Delta=16$ panel indicates consistency
between these two values of $\Delta$.
---
Figure 9: Diagrams showing the dominant around-the-world paths contributing to
graphs of $type1$. The space-time region between the kaon wall source at
$t_{K}$ and its periodic recurrence at $t_{K}+T$ is shown, where $T=32$ is the
extent of the periodic lattice in the time direction. For this around-the-
world path, one pion travels directly from the pion wall source at $t_{\pi}$
to the weak operator, represented by the grey dot at $t_{\rm op}$. However,
the second pion propagates in the other direction in time, passes through the
periodic boundary and combines with the kaon before reaching the weak operator
at $t_{\rm op}$.
Additional evidence supporting this explanation for the short plateau in the
case of $\Delta=16$ can be obtained by examining the explicit dependence on
$t$ given by Eq. 21 for the around-the-world contribution. Examining the
exponential decay with $t$ in the $\Delta=16$ correlators plotted in the right
panel of Fig. 8, for operators $Q_{7}$ and $Q_{8}$ we find a value for
$E_{K\pi}-m_{\pi}$ varying between 0.4 and 0.5 depending on the choice of fit
range. A more accurate value of 0.498(2) can be obtained by fitting the
corresponding correlator for $\Delta=20$ and a fit range of 5 to 11. The
strangeness-carrying state whose mass we have labeled $E_{K\pi}$ can be formed
from two quarks and must be parity even. Direct calculation of $E_{K\pi}$ from
a scalar $\overline{s}d$ correlator yields $E_{K\pi}=0.752(12)$ which is
consistent with the sum of the result above, $E_{K\pi}-m_{\pi}=0.498(2)$, and
the pion mass $m_{\pi}=0.2437(5)$. (This energy difference is also close to
the kaon mass $m_{K}^{(1)}=0.50729$ given in Tab. 1.) Thus, the time
dependence expected from the around-the-world path is quite consistent with
that seen in Fig. 8.
We conclude that it is important to increase the lattice extent in the time
direction both to suppress this around-the-world effect and to permit the use
of a larger source-sink separation giving a longer plateau. We will return to
discussion of the around-the-world effect below for the $\Delta I=1/2$ kaon
decay where it creates even greater difficulties. However, here we can begin
to appreciate the severity of this effect in the $K^{0}\rightarrow\pi\pi$
system for our temporal lattice extent of 32, given our values of the lattice
spacing and meson masses.
The Wilson coefficients and operators which appear in Eq. 4 are typically
expressed in the $\overline{\mbox{MS}}$ scheme. Thus, we must change the
normalization of our lattice operators $Q_{i}$ to that of the
$\overline{\mbox{MS}}$ scheme. We begin by converting our bare lattice
operators into the regularization invariant momentum (RI/MOM) scheme of Ref.
Martinelli:1995ty . Here we use the earlier results of Ref. Li:2008zz which
were obtained for the present lattice action using the methods of Ref.
Blum:2001xb . In this previous work off-shell, Landau-gauge-fixed Green’s
functions containing the lattice operators $Q_{i}$ are evaluated at specific
external momenta characterized by an energy scale $\mu$. These results
determine a renormalization matrix $Z_{ij}^{\mathrm{RI}}(\mu,a)$ which can be
used to convert the lattice normalization into that of the RI scheme:
$Q^{\mathrm{RI}}(\mu)_{i}=\sum_{j=1}^{7}Z_{ij}^{{\rm lat}\to{\rm
RI}}(\mu,a)Q_{j}^{\prime}.$ (22)
As explained in Appendix A, these equalities hold only when the operators
appear in physical matrix elements. The indices $i$ and $j$ take on seven
values corresponding to the seven independent operators in what will be called
the chiral basis. (The primes in this equation indicate lattice operators
defined in that basis.) This is referred to as nonperturbative renormalization
(NPR) because the matrix $Z_{ij}^{{\rm lat}\to{\rm RI}}(\mu,a)$ is computed
using a lattice evaluation of off-shell Green’s functions and perturbation
theory is not used.
Next these $Q^{\mathrm{RI}}(\mu)_{i}$ operators are converted to the
$\overline{\mbox{MS}}$ scheme in which the Wilson coefficients are evaluated
by applying a conversion matrix
$R^{\mathrm{RI}\rightarrow\overline{\mathrm{MS}}}_{ij}$ discussed in detail in
Ref. Lehner:2011fz . Finally the matrix elements of these
$\overline{\mbox{MS}}$ operators are combined with the Wilson coefficients
obtained in the $\overline{\mbox{MS}}$ scheme Buchalla:1995vs using the scale
$\mu=2.15$ GeV to determine the results given later in this section for the
$\Delta I=3/2$ amplitude $A_{2}$ and in the following section for the $\Delta
I=1/2$ $A_{0}$. These procedures are described in greater detail in Appendix
A.
A good approximation to the infinite volume decay amplitude can be obtained by
including the Lellouch-Lüscher factor ($F$) Lellouch:2000pv which relates the
$K\rightarrow\pi\pi$ matrix element $M$ of the effective weak Hamiltonian of
Eq. 4 calculated using finite volume states normalized to unity to the
infinite volume amplitude $A$: $|A|^{2}=F^{2}M^{2}$ where
$F^{2}=4\pi\left(\frac{E_{\pi\pi}^{2}m_{K}}{p^{3}}\right)\left\\{p\frac{\partial\delta_{2}(p)}{\partial
p}+q\frac{\partial\phi(q)}{\partial q}\right\\}.$ (23)
Here $p$ is defined through $E_{\pi\pi}=2\sqrt{m_{\pi}^{2}+p^{2}}$,
$q=Lp/2\pi$ and $\delta_{2}(p)$ is the $s$-wave, $I=2$, $\pi-\pi$ scattering
phase shift for pion relative momentum $p$. The function $\phi(q)$ is known
analytically and given, for example, in Ref. Lellouch:2000pv . The $I=2$ phase
shift $\delta_{2}(p)$ is determined from the measured two-pion energy
$E_{\pi\pi}=0.443(13)$ given in Tab. 1 and the finite volume quantization
condition Luscher:1990ux
$\phi(q)+\delta_{2}(p)=n\pi.$ (24)
For our threshold case we set the integer $n$ to zero and obtain
$\delta_{2}(p)=-0.0849(43)$. Because of the small value of $p$ we assume that
$\delta_{2}(p)$ is a linear homogenous function of $p$ and write
$\delta_{2}(p)=p\partial\delta_{2}(p)/\partial p$, the quantity required in
Eq. 23 and given in Tab. 3. (Equation 23 differs by a factor of two from the
expression given in the Lellouch-Lüscher paper because of our different
conventions for the decay amplitude $A$. With our conventions the experimental
value of Re$(A_{2})=1.48\times 10^{-8}$ GeV.)
In the limit of non-interacting pions, the factor $F$ becomes
$F_{\mathrm{free}}^{2}=2(2m_{\pi})^{2}m_{K}L^{3}$, which reflects the
different normalization of states in a box and plane wave states in infinite
volume. Results for $F$ in this $I=2$ case and the quantities used to
determine it are given in Tab. 3. We should note that applying the finite
volume correction of Eq. 23 gives us a finite-volume corrected amplitude for a
$\Delta I=3/2$, $K\rightarrow\pi\pi$ decay that is slightly above threshold by
the amount $E_{2}^{\pi\pi}-2m_{\pi}=33(1)$ MeV.
Table 3: The calculated quantities which appear in the Lellouch-Lüscher factor $F$ for $I=2$. The corresponding factor for the case of non-interacting particles is $F_{\rm free}=31.42$. The difference reflects the final two-pion scattering in a box. $p$ | $q\frac{\partial\phi(q)}{\partial q}$ | $p\frac{\partial\delta(p)}{\partial p}$ | $F$
---|---|---|---
0.0690(13) | 0.221(10) | -0.0849(43) | 26.01(18)
We can now combine everything and calculate the $K^{0}\rightarrow\pi\pi$ decay
amplitudes,
$A_{2/0}=F\frac{G_{F}}{\sqrt{2}}V_{ud}V_{us}\sum_{i=1}^{10}\sum_{j=1}^{7}\left[\Bigl{(}z_{i}(\mu)+\tau
y_{i}(\mu)\Bigr{)}Z_{ij}^{{\rm lat}\to{\overline{\rm
MS}}}M_{j}^{\frac{3}{2}/\frac{1}{2},{\rm lat}}\right],$ (25)
where the construction of the $10\times 7$ renormalization matrix
$Z_{ij}^{{\rm lat}\to{\overline{\rm MS}}}$ is explained in Appendix A. For
later use we have written Eq. 25 in a way which is applicable for $\Delta
I=1/2$ decays as well as for the $\Delta I=3/2$ transitions considered in this
section. The results for the complex $\Delta I=3/2$ decay amplitude $A_{2}$
are summarized in Tab. 4, including those for the other two, energy-non-
conserving choices of kaon mass. Since $m_{K}^{(1)}$ differs from the
isospin-2 $\pi-\pi$ energy by only 0.2 percent, we quote this case as our
energy-conserving kaon decay amplitude. Therefore, in physical units, we
obtain the energy-conserving $\Delta I=3/2$, $K^{0}\rightarrow\pi\pi$ complex,
threshold decay amplitude for $m_{K}=877$ MeV and $m_{\pi}=422$ MeV:
$\displaystyle\mathrm{Re}(A_{2})$ $\displaystyle=$ $\displaystyle
4.911(31)\times 10^{-8}\mathrm{GeV}$ (26) $\displaystyle\mathrm{Im}(A_{2})$
$\displaystyle=$ $\displaystyle-0.5502(40)\times 10^{-12}\mathrm{GeV}.$ (27)
This result for Re$(A_{2})$ can be compared with the experimental value of
$1.48\times 10^{-8}$ GeV given above. The larger result found in our
calculation is likely explained by our unphysically heavy kaon and pions.
Table 4: The complex, $K^{0}\rightarrow\pi\pi$, $\Delta I=3/2$ decay amplitudes in units of GeV. $m_{K}$ | Re$(A_{2})(\times 10^{-8})$ | Im$(A_{2})(\times 10^{-12})$
---|---|---
$m_{K}^{(0)}$ | 4.308(28) | -0.5596(40)
$m_{K}^{(1)}$ | 4.911(31) | -0.5502(40)
$m_{K}^{(2)}$ | 5.916(38) | -0.5316(39)
## VI $K^{0}\rightarrow\pi\pi$ $\Delta I=1/2$ amplitude
Following the prescription given by Eq. 6 we have calculated all of the
$\Delta I=1/2$ kaon decay correlation functions,
$C_{0,i}(\Delta,t)=\frac{1}{32}\sum_{t^{\prime}=0}^{31}A_{0,i}(t_{\pi}=t^{\prime}+\Delta,t_{\rm
op}=t+t^{\prime},t_{K}=t^{\prime}),$ (28)
for each of the ten effective weak operators. In the calculation we treat each
of these ten operators as independent and then verify that the identities
shown in Eq. 8 are automatically satisfied. Figures 10 and 11 show two
examples of the resulting correlation functions for the operators $Q_{2}$ and
$Q_{6}$, in the case of the lightest kaon $m_{K}^{(0)}$. Table 1 shows that
the mass of this kaon is very close to the energy of the I=0 two-pion state.
Therefore, we expect to get a reasonably flat plateau when the operator is far
from both the source and sink.
|
---|---
(a) | (b)
|
(c) | (d)
Figure 10: Plots showing the $t$ dependence of the various contractions which contribute to the $\Delta I=1/2$ correlation function $C_{0,2}(\Delta=16,t)$ for the operator $Q_{2}$. (a) Contractions of $type3$, the divergent mixing term $mix3$ that will be subtracted and the result after subtraction, $sub3$. (b) Contractions of $type4$, the divergent mixing term $mix4$ that will be subtracted and the result after subtraction, $sub4$. (c) Results for each of the four types of contraction after the needed subtractions have been performed. (d): Results for the complete $Q_{2}$ correlation function $C_{0,2}(\Delta=16,t)$ obtained by combining these four types of contractions. The solid points labeled $Q_{2}$ are the physical result while the open points labeled $Q_{2}^{\prime}$ are obtained by omitting all the vacuum graphs, $sub4$. The solid and dotted horizontal lines indicate the corresponding fitting results and the time interval, $5\leq t\leq 11$ over which the fits are performed. |
---|---
(a) | (b)
|
(c) | (d)
Figure 11: The result for each type of contraction contributing to the 3-point
correlation function $C_{0,6}(\Delta=16,t)$ for the operator $Q_{6}$ following
the same conventions as in Fig. 10.
Given this good agreement between the energies of the $K$ and $\pi-\pi$
states, we might expect that the unphysical, dimension three operator,
$\overline{s}\gamma^{5}d$ which mixes with the $(8,1)$ operators in Eq. 4 and
is itself a total divergence, will also give a negligible contribution to such
an energy and momentum conserving matrix element. However, as can be seen from
Figs. 10(a) and 11(a), the matrix element of this term is large and the
explicit subtraction described in Sec. IV is necessary.
This difficulty is created by the combination of two phenomena. First the
mixing coefficient which multiplies the $\overline{s}\gamma^{5}d$ operator
when it appears in our weak $(8,1)$ operators is large, of order
$(m_{s}-m_{l})/a^{2}$. Second, in our lattice calculation the necessary energy
conserving kinematics (needed to insure that this total divergence does not
contribute) is only approximately valid. The required equality of the spatial
momenta of the kaon and $\pi-\pi$ states is assured by our summing the
location of the weak vertex over a complete temporal hyperplane. On the other
hand, the equality of the energies of the initial and final states results
only if we have adjusted the kaon mass to approximately that of the two-pion
state and chosen the time extents sufficiently large that other states with
different energies have been suppressed. However, as can be seen in Figs.
10(a) and 11(a) the subtraction terms $mix$3 and $mix$4 show strong dependence
on the time at which they are evaluated. This implies that there are important
contributions coming from initial and final states which have significantly
different energies. One or both of these states is then not the intended $K$
or $\pi-\pi$ state but instead an unwanted contribution which has been
insufficiently suppressed by the time separations between source, weak
operator and sink.
Thus, instead of relying on large time extents and energy conserving
kinematics to suppress this unphysical, $O(1/a^{2})$ term we must explicitly
remove it. As explained in Sec. IV this can be done by including an explicit
subtraction which we fix by the requirement that the kaon to vacuum matrix
element of the complete subtracted operator vanishes as in Eq. 12. Thus, we
determine the divergent coefficient of this mixing term from the ratio
$\alpha_{i}=\langle 0|Q_{i}|K\rangle/\langle
0|\overline{s}\gamma^{5}d|K\rangle$ and then perform the explicit subtraction
of the resulting terms, labeled $\alpha_{i}\cdot mix3$ and $\alpha_{i}\cdot
mix4$ in Figs. 10 and 11.
Of course, the finite part of such a subtraction is not determined from first
principles and our choice, specified by Eq. 12 is arbitrary. Thus, we must
rely on our identification of a plateau and the approximate energy
conservation of our kinematics to make the arbitrary part of this subtraction
small, along with the other errors associated with evaluating the decay matrix
element of interest between initial and final states with slightly different
energies.
We now examine the very visible time dependence in Figs. 10(a) and 11(a) for
both the original matrix elements and the subtraction terms in greater detail.
As discussed above one might expect these divergent subtraction terms to
contribute to excited state matrix elements in which the energies of the
initial and final states are very different. Typical terms should be
exponentially suppressed as the separation between the weak operator and the
source or sink is increased, with the time behavior
$\exp\\{-(m_{K}^{*}-m_{K})t\\}$ or
$\exp\\{-(E_{\pi\pi}^{*}-E_{\pi\pi})(\Delta-t)\\}$, which ever is larger. (The
$\ast$ denotes an excited state.) However, by carefully examining the time
behavior of the $mix3$ amplitude, we find that the time dependence, at least
in the vicinity of the central region, is less rapid than might be expected
from such excited states suggesting that it is probably not due primarily to
contamination from excited states.
---
Figure 12: The dominant around-the-world paths contributing to graphs of
$type3$. As in Fig. 9 we show the space-time region between the kaon source at
$t=t_{K}$ and its periodic recurrence at $t=t_{K}+T$. The gray circle
represents the four quark operator $Q_{i}$. For the first two graphs, one of
the two pions created at the $t=t_{\pi}$ source travels directly to the
operator $Q_{i}$ while the second pion travels in the other direction in time
and reaches the kaon and weak operator by passing through the periodic lattice
boundary. In the third diagram it is the kaon which travels in the opposite to
the expected time direction.
We believe that the dominant, energy-nonconserving matrix elements which cause
the significant time dependence in Figs. 10 and 11 arise from the around-the-
world effects identified and discussed in the previous $\Delta I=3/2$ section.
In fact, for the reasons just discussed associated with divergent operator
mixing, such around-the-world effects are a more serious problem in the
$\Delta I=1/2$ case. The dominant around-the-world graphs are shown in Fig.
12. An estimate of the time dependence of these graphs gives,
$\displaystyle<K^{0}\pi|Q_{i}|\pi>N_{\pi}N_{K}N_{\pi}e^{-m_{\pi}T}e^{-(E_{K\pi}-m_{\pi})t}$
$\displaystyle+<0|Q_{i}|K^{0}\pi\pi>N_{\pi}N_{K}N_{\pi}e^{-m_{K}((T-\Delta)+(\Delta-t))}\,,$
(29)
where the first term comes from the first two graphs of Fig. 12, while the
second term comes from the third graph. (Recall that $t=t_{\mathrm{op}}-t_{K}$
and $\Delta=t_{\pi}-t_{K}$). Notice that these two terms involve amplitudes
which are far from energy conserving and therefore contain large divergent
contributions from mixing with the operator $\overline{s}\gamma_{5}d$ which
will be removed only when combined with the corresponding around-the-world
paths occuring in the $mix3$ contraction.
We conclude that it is these around-the-world matrix elements which are the
reason for the observed large divergent subtraction in the $type3$ graph. The
largest divergent contribution is thus not the subtraction for the matrix
element we are trying to evaluate, $<\pi\pi|Q_{i}|K^{0}>$; rather, it is the
divergent subtraction for the matrix elements $<K^{0}\pi|Q_{i}|\pi>$ and
$<0|Q_{i}|K^{0}\pi\pi>$ which arise from the around-the-world paths which are
not sufficiently suppressed by our lattice size. Two important lessons can be
learned from this analysis. First, it is important to perform an explicit
subtraction of the divergent mixing with the operator
$\overline{s}\gamma_{5}d$. While this term will not contribute to the energy
conserving matrix element of interest, in a Euclidean space lattice
calculation there are in general, other, unwanted, energy non-conserving terms
which may be uncomfortably large if this subtraction is not performed. Second
it would be wise to work on a lattice with a much larger size $T$ in time
direction in order to suppress further the around-the-world terms which give
such a large contribution in the present calculation. Using the average of
propagators computed with periodic plus anti-periodic boundary conditions to
effectively double the length in the time direction would be a good solution.
We should emphasize that these divergent, around-the-world contributions do
not pose a fundamental difficulty. The largest part of these amplitudes are
removed by the corresponding subtraction terms constructed from the operator
$\overline{s}\gamma_{5}d$. The remaining finite contributions from this and
other around-the-world terms are suppressed by the factor $\exp(-m_{\pi}T)$ or
$\exp(-m_{k}(T-\Delta))$. Fortunately, the large divergent subtraction also
reduces the statistical errors substantially, especially for the $type4$
vacuum graphs, which indicates the expected strong correlation between the
divergent part of the weak operator and the corresponding
$\overline{s}\gamma_{5}d$ subtraction. Our results suggest that the separation
of $\Delta=16$ gives a relatively longer plateau region, so we use that
$K-\pi\pi$ time separation in the analysis below.
The lattice matrix elements are determined by fitting the $I=1/2$ correlators
$C_{0}^{i}(\Delta,t)$ given in Eq. 28 using the fitting form:
$\displaystyle C_{0,i}(\Delta,t)$ $\displaystyle=$ $\displaystyle
M_{i}^{1/2,{\rm
lat}}N_{\pi\pi}N_{K}e^{-E_{\pi\pi}\Delta}e^{-(m_{K}-E_{\pi\pi})t}.$ (30)
The fitted results for the weak, $\Delta I=1/2$ matrix elements of all ten
operators are summarized in Tab. 5. To see the effects of the disconnected
graph clearly, a second fit is performed to the amplitude from which the
disconnected, $type4$ graphs have been omitted and the calculated results are
shown with an additional $\prime$ label, as in the earlier two-pion scattering
section.
Table 5: Fitted results for the weak, $\Delta I=1/2$ kaon decay matrix elements using the kaon mass $m_{K}^{(0)}$. The column $M_{i}^{\rm lat}$ shows the complete result from each operator. The column $M_{i}^{\prime\,{\rm lat}}$ shows the result when the disconnected graphs are omitted while the 4th and 5th columns show the contributions of each operator the real and imaginary parts of the physical decay amplitude $A_{0}$. These results are obtained using a source-sink separation $\Delta=16$, and a fit range $5\leq t\leq 11$. i | $M_{i}^{1/2,{\rm lat}}(\times 10^{-2})$ | $M_{i}^{\prime 1/2,{\rm lat}}(\times 10^{-2})$ | Re$(A_{0})$(GeV) | Im$(A_{0})$(GeV)
---|---|---|---|---
1 | -1.6(16) | -1.10(37) | 7.6(64)e-08 | 0
2 | 1.52(61) | 1.92(15) | 2.86(97)e-07 | 0
3 | -0.3(41) | 0.3(10) | 2.1(136)e-10 | 1.1(76)e-12
4 | 2.7(33) | 3.32(78) | 4.2(44)e-09 | 1.4(14)e-11
5 | -3.3(38) | -6.81(86) | 3.1(53)e-10 | 1.6(28)e-12
6 | -7.8(48) | -19.6(9) | -5.6(33)e-09 | -3.3(20)e-11
7 | 10.9(14) | 15.20(42) | 5.2(12)e-11 | 8.8(20)e-14
8 | 35.7(28) | 47.2(10) | -3.66(28)e-10 | -1.79(14)e-12
9 | -2.2(12) | -1.79(29) | 3.1(15)e-14 | -2.01(96)e-12
10 | 0.9(12) | 1.24(29) | 1.2(11)e-11 | -2.7(27)e-13
Total | - | - | 3.46(78)e-07 | -2.4(23)e-11
The calculation of the $\Delta I=1/2$ decay amplitude $A_{0}$ from the lattice
matrix elements $M_{i}^{1/2,{\rm lat}}$ given in Tab. 5 is very similar to the
$\Delta I=3/2$ case: the values of $M_{i}^{1/2,{\rm lat}}$ are simply
substituted in Eq. 25. However, the attractive character of the $I=0$,
$\pi-\pi$ interaction and resulting negative value of $p^{2}$ makes the
Lellouch-Lüscher treatment of finite volume corrections inapplicable. For the
repulsive $I=2$ case, we could apply this treatment to obtain the decay
amplitude for a two-pion final state which was slightly above threshold
corresponding to the actual finite volume kinematics. In the present case
there is no corresponding infinite-volume decay into two pions below threshold
and an unphysical increase of $m_{\pi}$ to compensate for the finite volume
$\pi-\pi$ attraction will introduce an $O(1/L^{3})$ error in the decay
amplitude of the same size as that which the Lellouch-Lüscher treatment
corrects. Thus, for this $\Delta I=1/2$ we do not include finite volume
corrections and simply use the free-field value for the factor $F$ in Eq. 25.
While we believe that we cannot consistently apply the Lellouch-Lüscher finite
volume correction factor to improve our result for the $I=0$,
$K\rightarrow\pi\pi$ decay amplitude, we might still be able to use the
quantization condition of Eq. 24 to determine the $I=0$ $\pi-\pi$ scattering
phase shift $\delta_{0}(p)$. Even though Eq. 24 can be analytically continued
to imaginary values of the momentum $p$, its application for large negative
$p^{2}$ is uncertain since the function $\phi(q)$ becomes ill defined. In
fact, our value of $p^{2}$ sits very close to a singular point of $\phi(q)$.
We believe this happens because the condition on the interaction range $R\ll
L/2$ used to derive the quantization condition in Eq. 24 is not well satisfied
for our small volume. This impediment to determining $\delta_{0}(p)$ will
naturally disappear once we work with lighter pions in a larger volume.
The results for Re($A_{0}$) and Im($A_{0}$) are summarized in Tab. 6 and the
individual contribution from each of the operators is detailed in the last two
columns of Tab. 5. Within a large uncertainty Tab. 5 shows that the largest
contribution to Re($A_{0}$) comes from operator $Q_{2}$, and that to
Im($A_{0}$) from $Q_{6}$ as found, for example, in Refs. Blum:2001xb ;
Noaki:2001un .
Since the choice $m_{K}^{(0)}$ for the kaon mass is not precisely equal to the
energy of the $I=0$ $\pi\pi$ state, we carried out a simple linear
interpolation between $m_{K}^{(0)}$ and $m_{K}^{(1)}$ to obtain an energy
conserving matrix element, which is shown in the last row of Tab 6. In terms
of physical units, therefore, our full calculation gives the energy
conserving, $K^{0}\rightarrow\pi\pi$, $\Delta I=1/2$, complex decay amplitude
$A_{0}$ for $m_{K}=766$ MeV and $m_{\pi}=422$ MeV:
$\displaystyle\mathrm{Re}(A_{0})$ $\displaystyle=$ $\displaystyle
3.80(82)\times 10^{-7}\mathrm{GeV}$ (31) $\displaystyle\mathrm{Im}(A_{0})$
$\displaystyle=$ $\displaystyle-2.5(2.2)\times 10^{-11}\mathrm{GeV}.$ (32)
These complete results can be compared with those obtained when the
disconnected graphs are neglected given in Tab. 6 and the experimental value
for Re$(A_{0})=3.3\times 10^{-7}$ GeV. As in the case of Re$(A_{2})$, our
larger value is likely the result of our unphysically heavy kaon and pion.
Table 6: Amplitudes for $\Delta I=1/2$ $K^{0}\rightarrow\pi\pi$ decay in units of GeV. The energy conserving amplitudes are obtained by a simple linear interpolation between $m_{K}^{(0)}$=0.42599 and $m_{K}^{(1)}$=0.50729 to the energy of two-pion state. As in the previous tables, the $\prime$ indicates results from which the disconnected graphs have been omitted. $m_{K}$ | Re$(A_{0})(\times 10^{-8})$ | Re$(A^{\prime}_{0})(\times 10^{-8})$ | Im$(A_{0})(\times 10^{-12})$ | Im$(A^{\prime}_{0})(\times 10^{-12})$
---|---|---|---|---
$m_{K}(0)$ | 36.1(78) | 42.3(20) | -21(21) | -66.1(43)
$m_{K}(1)$ | 45(10) | 48.8(24) | -41(26) | -74.6(47)
$m_{K}(2)$ | 65(15) | 58.6(32) | -69(39) | -89.6(63)
Energy conserving | 38.0(82) | 43.4(21) | -25(22) | -67.5(44)
## VII Discussion and Conclusions
Comparing the results of Re($A_{2}$) in Tab. 4 and Re($A_{0}$) in Tab. 6, we
find the $\Delta I=1/2$ enhancement ratio Re($A_{0}$)/Re($A_{2}$) to be
roughly 7-9. This comparison is degraded by our threshold kinematics which,
since the $I=0$ and $I=2$ two-pion states have different energies in a finite
volume, causes us to use a different kaon mass in the calculations of
$(A_{2})$ and $(A_{0})$ in order to have energy conserving decays in each
case. These two energy conserving amplitudes have a ratio of $38.0/4.911=7.7$,
while if we ignore energy conservation and use the same $m_{K}^{(1)}$ value
for kaon mass, the ratio becomes $45.0/4.911=9.2$. Of course, both estimates
are far from the experimental ratio of 22.5 suggesting that our 422 MeV pion
mass and small lattice volume are far from physical.
For completeness, we also calculate the measure of direct CP violation,
$\mathrm{Re}\left(\frac{\epsilon^{\prime}}{\epsilon}\right)=\frac{\omega}{\sqrt{2}|\epsilon|}\left[\frac{\mathrm{Im}(A_{2})}{\mathrm{Re}(A_{2})}-\frac{\mathrm{Im}(A_{0})}{\mathrm{Re}(A_{0})}\right],$
(33)
where $\omega=\mathrm{Re}(A_{2})/\mathrm{Re}(A_{0})$ is the inverse of the
$\Delta I=1/2$ enhancement factor. Using our kinematics, the kaon mass
$m_{K}^{(1)}$ and substituting the experimental value for $\epsilon$, we get
Re$(\epsilon^{\prime}/\epsilon)=(2.7\pm 2.6)\times 10^{-3}$. If we instead use
the experimental value for $\omega$, we get
Re$(\epsilon^{\prime}/\epsilon)=(1.11\pm 0.91)\times 10^{-3}$.
Our calculation is sufficiently far from physical kinematics, that it is not
appropriate to compare these results with experiment.111A further unphysical
aspect of our kinematics is the inequality of the strange quark mass used in
the fermion determinant and the self contractions appearing in the eye graphs
($m_{s}=0.032$) and strange quark masses used in the valence propagator of the
K meson ($m_{s}=0.066$, 0.99 and 0.165). Instead, our objective is to show how
well our method performs. We have been able to calculate Re($A_{0}$), the key
element needed to explain the $\Delta I=1/2$ rule, with a 25% statistical
error. Comparing our results for Re($A_{0}$) obtained on sub-samples of N=100,
400 and all 800 configurations we find that the statistical errors on the
quantities we measure do indeed scale as $1/\sqrt{N}$. Therefore, we believe
that our non-zero signal for Re($A_{0}$) is real and that we could reduce this
statistical error to 10 percent by quadrupling the size of our sample to 3200
configurations. It is interesting to note the results for primed (disconnected
graphs omitted) and unprimed (all graphs included) quantities contributing to
Re$(A_{0})$ have similar values suggesting that the disconnected graphs, while
contributing significantly to the statistical error, have an effect on the
final result for Re$(A_{0})$ at or below 25%.
In contrast, the result for Im($A_{0}$) has an 80% error. Thus, it is not
clear whether the size of the result will survive a quadrupling of the sample
with its statistical error reducing to a 40% error or whether the result
itself will shrink, remaining statistically consistent with zero. Considering
the substantial systematic errors associated with our small volume and the
fact that our kinematics are far from the physical, we present this trial
calculation as a guideline for future work and a proof of method rather than
giving accurate numbers to compare with experiment.
From our observation of the around-the-world effect, we conclude that it is
important to use the average of quark propagators obeying periodic and anti-
periodic boundary conditions to extend the lattice size in the time direction.
In addition, explicit subtraction of the divergent mixing term
$\overline{s}\gamma^{5}d$ is necessary even for kinematics which are literally
energy conserving because the around-the-world path and possibly other excited
state matrix elements are far off shell and can be substantially enhanced by
such a divergent contribution. Finally, future work should be done using a
much larger lattice which can contain two pions without any worry about finite
size effects.
The focus of this paper is on developing techniques capable of yielding
statistically meaningful results from the challenging lattice correlation
functions involved in the amplitude $A_{0}$. However, there are other
important problems that will also require careful attention if physically
meaningful results are to be obtained for this amplitude with an accuracy of
better than 20%. Two important issues are associated with operator mixing. As
discussed in Appendix A, a proper treatment of the non-perturbative
renormalization of the four independent $(8,1)$ four-quark operators requires
that additional operators containing gluonic variables (some of which are not
gauge invariant) be included. While including such operators is in principle
possible and the subject of active research, controlling such mixing using
RI/MOM methods offers significant challenges.
A second problem is operator mixing induced by the residual chiral symmetry
breaking of the DWF formulation. The mixing of such wrong-chirality operators
should be suppressed by a factor of order $m_{\rm res}$. However, the
$K\rightarrow\pi\pi$ matrix elements of the important $(8,1)$ four-quark
operators are themselves suppressed by at least one power of $m_{K}^{2}$, a
suppression that is absent from similar matrix elements of the induced, wrong-
chirality operators. Therefore, such mixing has been ignored in this paper
because its effect on the matrix elements of interest are expected to be of
order $m_{\rm res}/m_{s}\approx 0.08$, suggesting that these effects will be
smaller than our 25% statistical errors. To perform a more accurate
calculation in the future, these mixing effects may be further suppressed by
adopting a gauge action with smaller residual chiral symmetry breaking. For
example, this ratio reduces to 0.04 for the DSDR gauge action now being used
in RBC/UKQCD simulations Renfrew:2009wu and to 0.023 for those ensembles with
the smallest lattice spacing created to date using the Iwasaki gauge action
Aoki:2010dy . When greater accuracy is required either an improved fermion
action, larger $L_{s}$ or explicit subtraction of wrong-chirality mixing must
be employed.
As we move closer to the physical pion mass we must overcome a further
important difficulty: giving physical relative momentum to the two pions. This
can be accomplished while keeping the two-pion state in which we are
interested as the ground state, if the kaon is given non-zero spatial momentum
relative to the lattice. In this case the lowest energy final state can be
arranged to have one pion at rest while the other pion carries the kaon
momentum, as in the $\Delta I=3/2$ calculation of Ref. Yamazaki:2008hg .
However, this requires the momentum carried by the initial kaon and final pion
to be 739 MeV, which is 5.4 times larger than the physical pion mass. Such a
large spatial momentum will likely make the calculation extremely noisy. For
the $\Delta I=3/2$ calculation, it is possible to use anti-periodic boundary
conditions in one or more spatial directions for one of the light quarks so
that each pion necessarily carries the physical, 206 MeV momentum present in
the actual decay while the kaon can be at rest Kim:2003xt ; Goode:2011kb .
However, this approach cannot be used in the case of the $I=0$ final state
being studied here. Instead, the use of G-parity boundary conditions
Kim:2002np may be the solution to this problem.
###### Acknowledgements.
We thank Dirk Brömmel and our other colleagues in the RBC and UKQCD
collaborations for discussions, suggestions, and assistance. We acknowledge
RIKEN BNL Research Center, the Brookhaven National Laboratory and the U.S.
Department of Energy (DOE) for providing the facilities on which this work was
performed. NC, QL, RM were supported in part by U.S. DOE grant DE-
FG02-92ER40699, TB and RZ by U.S. DOE grant DE-FG02-92ER40716 and AS and TI by
DOE contract DE-AC02-98CH10886(BNL). EG was supported by an STFC studentship
and CTS was partially supported by UK STFC Grant PP/D000211/1 and by EU
contract MRTN-CT-2006-035482 (Flavianet). Finally, QL would like to thank the
U.S. DOE for support as a DOE Fellow in High Energy Theory and CL acknowledges
support of the RIKEN FPR program.
## Appendix A Operator normalization
In order to combine our lattice matrix elements with the Wilson coefficients
describing the short-distance weak interaction physics responsible for
$K\rightarrow\pi\pi$ decay we must convert our lattice operators into those
normalized according to that $\overline{\mathrm{MS}}$ scheme in which the
Wilson coefficients are evaluated. We will discuss the details of this
procedure in this appendix.
The first step is converting the lattice operators into those normalized
according to the RI/MOM scheme Martinelli:1995ty . We follow the procedure of
Ref. Blum:2001xb and make use of the fact that the ten operators which enter
the conventional expression given in Eq. 4 are linearly dependent and can be
reduced to a set of seven independent operators, $Q_{1}^{\prime}$,
$Q_{2}^{\prime}$, $Q_{3}^{\prime}$, $Q_{5}^{\prime}$, $Q_{6}^{\prime}$,
$Q_{7}^{\prime}$ and $Q_{8}^{\prime}$ defined in Eq. 172-175 Ref. Blum:2001xb
. These have been defined so that the resulting operators belong to specific
irreducible representations of $SU_{L}(3)\times SU_{R}(3)$. The operator
$Q_{1}^{\prime}$ transforms as a $(27,1)$. The four operators
$Q_{2}^{\prime}$, $Q_{3}^{\prime}$, $Q_{5}^{\prime}$ and $Q_{6}^{\prime}$ all
belong to the $(8,1)$ representation, while $Q_{7}^{\prime}$ and
$Q_{8}^{\prime}$ each transform as an $(8,8)$. Here $(m,n)$ denotes the
product of an $m$-dimensional irreducible representation of $SU_{L}(3)$ with
an $n$-dimensional irreducible representation of $SU_{R}(3)$. We refer to the
basis of these seven independent operators as the chiral basis. Because
$SU_{L}(3)\times SU_{R}(3)$ is an exact symmetry of the large momentum,
massless limit which our NPR calculation is intended to approximate, the
mixing matrix $Z^{{\rm lat}\to{\rm RI}}$ given in Eq. 22 which relates the
lattice and RI-normalized operators will be block diagonal, only connecting
operators which belong to the same irreducible representation of
$SU_{L}(3)\times SU_{R}(3)$.
The RI/MOM conditions which define the operators $O^{\mathrm{RI}}_{i}$ and
determine the $7\times 7$ matrix $Z^{{\rm lat}\to{\rm RI}}$ are imposed on the
Green’s functions:222While this equation agrees with Eqs. 143 and 152 of Ref.
Blum:2001xb , a different choice of momenta was actually used in that earlier
reference. These two equations accurately describe the earlier kinematics only
after one pair of the momenta $p_{1}$ and $p_{2}$ are exchanged:
$p_{1}\leftrightarrow p_{2}$.
$G_{i}(p_{1},p_{2})_{\alpha\beta\gamma\delta}^{f}=\prod_{i=1}^{4}\left\\{\int
d^{4}x_{i}\right\\}\left\langle
s(x_{1})_{\alpha}f(x_{2})_{\beta}Q^{\mathrm{RI}}_{i}(0)\overline{d}_{\gamma}(x_{3})\overline{f}_{\delta}(x_{4})\right\rangle
e^{-ip_{2}(x_{1}+x_{2})}e^{ip_{1}(x_{3}+x_{4})}$ (34)
evaluated for $p_{1}^{2}=p_{2}^{2}=(p_{1}-p_{2})^{2}=\mu^{2}$. Here $\alpha$,
$\beta$, $\gamma$ and $\delta$ are spin and color indices. The fields
$\overline{d}$ and $\overline{f}$ create a down quark and a quark of flavor
$f=u$ or $d$ while $s$ and $f$ destroy a strange quark and a quark of flavor
$f$. The RI/MOM conditions are imposed by removing the four external quark
propagators from the amplitudes in Eq. 34, and then contracting each of the
resulting seven amputated Green’s functions obtained from Eq. 34 with seven
projectors $\\{\Gamma^{ij;f}_{\alpha\beta\gamma\delta}\\}_{1\leq j\leq 7}$.
The matrix $Z^{{\rm lat}\to{\rm RI}}$ is then determined by requiring that the
resulting 49 quantities take their free field values, as is described in
detail in Refs. Blum:2001xb and Lehner:2011fz .
The choice of external momenta specified by Eq. 34 is non-exceptional since no
partial sum of these momenta vanish (if their signs are chosen so that all
four momenta are incoming) and is the choice used in Refs. Li:2008zz and
Lehner:2011fz . Such a choice of kinematics is expected to result in
normalization conditions which are less sensitive to non-zero quark masses and
QCD vacuum chiral symmetry breaking than would be the case if an exceptional
set of momenta had been used Aoki:2007xm . The resulting matrix $Z^{{\rm
lat}\to{\rm RI}}(\mu,a)/Z_{q}^{2}$ obtained for $\mu=2.15$ GeV in Ref.
Li:2008zz is given in Tab. 7.
Table 7: The renormalization matrix $Z^{{\rm lat}\to{\rm RI}}/Z_{q}^{2}$ in the seven operator chiral basis at the energy scale $\mu=2.15$ GeV. These values were obtained from Ref. Li:2008zz by performing an error weighted average of the values given in Tabs. 40, 41 and 42 (corresponding to bare quark masses of 0.01, 0.02 and 0.03) and inverting the resulting matrix with an uncorrelated propagation of the errors. Since the results given in these three tables are equal within errors, we chose to combine them to reduce their statistical errors rather than to perform a chiral extrapolation. | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---
1 | 0.825(7) | 0. | 0. | 0. | 0. | 0. | 0.
2 | 0. | 0.882(38) | -0.111(41) | -0.009(12) | 0.010(10) | 0. | 0.
3 | 0. | -0.029(69) | 0.962(92) | 0.013(22) | -0.011(25) | 0. | 0.
4 | 0. | -0.04(12) | -0.01(13) | 0.924(42) | -0.149(35) | 0. | 0.
5 | 0. | 0.17(18) | 0.08(23) | -0.042(55) | 0.649(63) | 0. | 0.
6 | 0. | 0. | 0. | 0. | 0. | 0.943(8) | -0.154(9)
7 | 0. | 0. | 0. | 0. | 0. | -0.0636(53) | 0.680(11)
Since these RI/MOM renormalization conditions are being imposed for off-shell,
gauge-fixed external quark lines, we must in principle include a larger number
of operators than the minimal set of seven independent operators which can
represent all gauge invariant matrix elements between physical states of
$H_{W}$. Therefore, we must also employ a correspondingly larger set of
conditions to distinguish among this larger set of operators. This larger set
of operators is required if we are to reproduce with these RI operators all
the gauge-fixed, off-shell Green’s functions that can be constructed using the
original, chiral basis of lattice operators $Q^{\prime}_{i}$. Thus, as stated
in Sec. V, the relations given in Eq. 22 between the seven lattice and the
seven RI operators are valid only when those operators appear in physical
matrix elements between on-shell states. For this equation to be valid when
the operators appear in the off-shell, gauge-fixed Green’s that define the RI
scheme, additional RI/MOM-normalized operators must be added.
However, our ultimate goal is to evaluate on-shell, physical matrix elements
of these operators. For such matrix elements there are only seven independent
operators and we can collapse the expanded set of operators referred to above
back to the seven, four-quark, chiral basis operators $Q_{i}^{\rm RI}$. This
is the meaning of the $7\times 7$ matrix $Z^{{\rm lat}\to{\rm RI}}$ matrix
given in Tab. 7: gauge symmetry and the equations of motion must be imposed to
reduce to seven the RI-normalized operators to which the seven lattice
operators are equated. In the calculation of $Z^{{\rm lat}\to{\rm RI}}$
presented in Ref. Li:2008zz such extra operators are neglected. For all but
one, this might be justified because these operators enter only at two loops
or beyond and the perturbative coefficients that we are using in later steps
are computed at only one loop. A single operator, given in Eq. 146 of Ref.
Blum:2001xb and Eq. 12 of Ref. Lehner:2011fz does appear at one loop but has
also been neglected because it is expected to give a smaller contribution than
other two-quark operators with quadratically divergent coefficients whose
effects are indeed small. A final imperfection in the results presented in
Tab. 7 is that the subtraction of a third dimension-four, two-quark operator
which contains a total derivative was not performed. However, the effect of
subtracting this third operator is expected to be similar to those of the two
operators which were subtracted, effects which were not visible outside of the
statistical errors (see e.g. Tabs. XIV and XVIII in Ref. Blum:2001xb ).
In the second step we convert the seven $RI$ operators obtained above into the
$\overline{\mathrm{MS}}$ scheme:
${Q_{i}^{\prime}}^{\overline{\mathrm{MS}}}=\sum_{j}\left(1+\Delta
r^{\mathrm{RI}\to\overline{\rm MS}}\right)_{ij}Q^{\mathrm{RI}}_{j}.$ (35)
Here the indices $i$ and $j$ run over the set $\\{1,2,3,5,6,7,8\\}$
corresponding to the chiral basis of the operators $Q_{j}$ defined above and a
set of operators ${Q_{j}^{\prime}}^{\overline{\mathrm{MS}}}$, with identical
chiral properties, which are defined in Ref. Lehner:2011fz . We use the
computational framework described in Ref. Lehner:2011fz and the resulting
$7\times 7$ matrix $\Delta r^{\mathrm{RI}\to\overline{\rm MS}}$ is given in
Tab. VIII of that reference. As in the case of Eq. 22, the two sets of seven
RI and $\overline{\rm MS}$ operators are related by this $7\times 7$ matrix
only when appearing in physical matrix elements. Since the values in this
table were obtained for the case that the wave function renormalization
constant for the quark field is the quantity $Z^{\not}{q}_{q}$ it is that
factor which we use to extract $Z^{{\rm lat}\to\mathrm{RI}}$ from the matrix
$Z^{{\rm lat}\to\mathrm{RI}}/Z_{q}^{2}$ given in Tab. 7. For our $\beta=2.13$,
Iwasaki gauge ensembles $Z^{\not}{q}_{q}=0.8016(3)$. (Note, $Z^{\not}{q}_{q}$
is the same as the quantity $Z^{\prime}_{q}$ introduced in earlier,
exceptional momentum schemes Sturm:2009kb .)
A third and final step is needed before we can combine the Wilson coefficients
with the matrix elements determined in our calculation to obtain the physical
amplitudes $A_{0}$ and $A_{2}$. The $7\times 7$ matrix given in Tab. VIII of
Ref. Lehner:2011fz gives us $\overline{\rm MS}$ operators defined in the
chiral basis. However, the Wilson coefficients which are available in Ref.
Buchalla:1995vs are defined for the ten operator basis referred to as basis I
in Ref. Lehner:2011fz . The conversion between the linearly independent, seven
operator basis and the conventional set of ten linearly dependent operators is
correctly given by the application of simple Fierz identities for the case of
the lattice and RI/MOM operators. As is explained, for example, in Ref.
Lehner:2011fz , this procedure is more complex for operators defined using
$\overline{\rm MS}$ normalization. Here subtleties of defining $\gamma^{5}$ in
dimensions different from four, result in ten $\overline{\rm MS}$-normalized
operators, $Q_{i}^{\overline{\rm MS}}$, which are not related by the usual
Fierz identities, with Fierz violating terms appearing at order $\alpha_{s}$.
Thus, the conventional ten $\overline{\rm MS}$-normalized operators
$Q_{i}^{\overline{\rm MS}}$ which appear in Eq. 4 must be constructed, again
through one-loop perturbation theory, from the seven operators
${Q_{i}^{\prime}}^{\overline{\rm MS}}$:
$Q_{i}^{\overline{\rm MS}}=\sum_{j}\left(T+\Delta T^{\overline{\rm
MS}}_{I}\right)_{ij}{Q^{\prime}}^{\overline{\rm MS}}_{j},$ (36)
in the notation of Ref. Lehner:2011fz . The $10\times 7$ matrices, $T$ and
$\Delta T^{\overline{\rm MS}}_{I}$ are given in Eqs. 59 and 65 of that
reference. (The subscript $I$ on the matrix $\Delta T^{\overline{\rm MS}}_{I}$
identifies the particular ten-operator, $\overline{\rm MS}$ basis required by
the Wilson coefficients of Ref. Buchalla:1995vs .)
This entire set of non-perturbative and perturbative transformations can be
summarized by the following equation which expresses the ten $\overline{\rm
MS}$-normalized operators $Q_{i}^{\overline{\rm MS}}$ in terms of the seven,
chiral basis, lattice operators whose matrix elements we actually compute:
$\displaystyle Q_{i}^{\overline{\rm MS}}$ $\displaystyle=$
$\displaystyle\sum_{j}\left[\left(T+\Delta T^{\overline{\rm
MS}}_{I}\right)_{10\times 7}\left(1+\Delta r^{{\rm RI}\to\overline{\rm
MS}}\right)_{7\times 7}\left(Z^{{\rm lat}\to\mathrm{RI}}\right)_{7\times
7}\right]_{ij}Q_{j}^{\rm lat}$ (37) $\displaystyle=$
$\displaystyle\sum_{j}\left[\left(Z^{{\rm lat}\to\overline{\rm
MS}}\right)_{10\times 7}\right]_{ij}Q_{j}^{\rm lat},$ (38)
where the subscripts indicate the dimensions of the matrices being multiplied
and the matrix $Z^{{\rm lat}\to\overline{\rm MS}}_{ij}$ is used in Eq. 25.
The physical matrix elements listed in Tabs. 2 and 5 are obtained by using Eq.
38 to determine the matrix elements of the ten conventional operators
$Q_{i}^{\overline{\rm MS}}$ in term of the matrix elements of the seven
lattice operators $Q_{j}$. These ten matrix elements are then combined with
the twenty Wilson coefficients computed for the renormalization scale
$\mu=2.15$ GeV using the formulae in Ref. Buchalla:1995vs . The values
obtained for these Wilson coefficients are listed in Tab. 8.
Table 8: Wilson Coefficients in the $\overline{MS}$ scheme, at energy scale $\mu=2.15$GeV. $i$ | $y_{i}^{\overline{MS}}(\mu)$ | $z_{i}^{\overline{MS}}(\mu)$
---|---|---
1 | 0 | -0.29829
2 | 0 | 1.14439
3 | 0.024141 | -0.00243827
4 | -0.058121 | 0.00995157
5 | 0.0102484 | -0.00110544
6 | -0.069971 | 0.00657457
7 | -0.000211182 | 0.0000701587
8 | 0.000779244 | -0.0000899541
9 | -0.0106787 | 0.0000150176
10 | 0.0029815 | 0.0000656482
Note, there are many important details of the RI/MOM renormalization
procedure, such as the subtraction of dimension three and four operators,
which are not repeated here because they are already discussed with some care
in Refs. Blum:2001xb and Lehner:2011fz .
## References
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* (17) G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996), [eprint hep-ph/9512380].
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* (30) T. Yamazaki (RBC), Phys. Rev. D79, 094506 (2009), [eprint 0807.3130].
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|
arxiv-papers
| 2011-06-14T13:54:58 |
2024-09-04T02:49:19.632837
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. Blum, P.A. Boyle, N.H. Christ, N. Garron, E. Goode, T. Izubuchi, C.\n Lehner, Q. Liu, R.D. Mawhinney, C.T. Sachrajda, A. Soni, C. Sturm, H. Yin, R.\n Zhou",
"submitter": "Qi Liu",
"url": "https://arxiv.org/abs/1106.2714"
}
|
1106.2751
|
# Mathematical Principles of Dynamic Systems and the Foundations of Quantum
Physics
Eric Tesse
###### Abstract
Everybody agrees that quantum physics is strange, and that the world view it
implies is elusive. However, it is rarely considered that the theory might be
opaque because the mathematical language it employs is inarticulate. Perhaps,
if a mathematical language were constructed specifically to handle the
theory’s subject matter, the theory itself would be clarified. This article
explores that possibility. It presents a simple but rigorous language for the
description of dynamics, experiments, and experimental probabilities. This
language is then used to answer a compelling question: What is the set of
allowed experiments? If an experiment is allowed, then the sum of the
probabilities of its outcomes must equal 1. If probabilities are non-additive,
there are necessarily sets of outcomes whose total probability is not equal to
1. Such experiments are therefore not allowed. That being the case, in quantum
physics, which experiments are allowed, and why are the rest disallowed? What
prevents scientists from performing the disallowed experiments? By phrasing
these questions within our mathematical language, we will uncover answers that
are complete, conceptually simple, and clearly correct. This entails no magic
or sleight of hand. To write a rigorous mathematical language, all unnecessary
assumptions must be shed. In this way, the thicket of ad hoc assumptions that
surrounds quantum physics will be cleared. Further, in developing the theory,
the logical consequences of the necessary assumptions will be laid bare.
Therefore, when a question can be phrased in such a language, one can
reasonably expect a clear, simple answer. In this way we will dispel much of
the mystery surrounding quantum measurements, and begin to understand why
quantum probabilities have their peculiar representation as products of
Hilbert space projection operators.
###### Contents
1. I Introduction
2. II Dynamics
1. II.1 Parameters
2. II.2 Parametrized Functions and Dynamic Sets
3. II.3 Dynamic Spaces
4. II.4 Special Sets
5. II.5 Limits and Closed Sets
3. III Experiments
1. III.1 Shells
2. III.2 E-Automata
1. III.2.1 Environmental Shells
2. III.2.2 The Automata Condition
3. III.2.3 Unbiased Conditions
4. III.2.4 E-Automata
3. III.3 Ideal E-Automata
1. III.3.1 Boolean E-Automata
2. III.3.2 All-Reet E-Automata
3. III.3.3 Ideal E-Automata
4. III.4 Ideal Partitions
5. III.5 Companionable Sets & Compatible Sets
4. IV Probabilities
1. IV.1 Dynamic Probability Spaces
2. IV.2 T-Algebras and GPS’s
1. IV.2.1 $\neg t$ and $[t]$
2. IV.2.2 $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$
3. IV.2.3 Convergence on a GPS
4. IV.2.4 $T_{S}$, $T_{S\mathbb{N}}$, and $T_{\mathbb{N}}^{S}$
3. IV.3 Nearly Compatible Sets
4. IV.4 Deterministic & Herodotistic Spaces
1. IV.4.1 DPS’s on Deterministic & Herodotistic Spaces
2. IV.4.2 DPS’s in Deterministic Universes
3. IV.4.3 DPS’s in Herodotistic Universes
5. V Application to Quantum Measurement
1. V.1 Preliminary Matters
1. V.1.1 A Note On Paths
2. V.1.2 Discretely Determined Partitions
3. V.1.3 Interconnected Dynamic Sets
2. V.2 Partitions of Unity for Quantum Systems
1. V.2.1 The Conditional Case
2. V.2.2 The Non-Conditional Case
3. V.3 Maximal Quantum Systems
4. V.4 Terminus & Exordium
6. A $\Delta$-Additivity
7. B Conditional Probabilities & Probability Dynamics
8. C Invariance On Dynamic Sets and DPS’s
1. C.1 Invariance On Dynamic Sets
2. C.2 Invariance on DPS’s
9. D Parameter Theory
## I Introduction
Our fascination with quantum physics has as much do to with its strangeness as
its success. This strangeness can conjure contradictory responses: on the one
hand, the sense that science has dug so deep as to touch upon profound
metaphysical questions, and on the other, the sense that something is amiss,
as science should strive to uncover simple explanations for seemingly strange
phenomena. Fans of the first response will find little of interest in the
paper, for it explores the second.
Let’s start by noting that the mathematical language employed by quantum
mechanics was not developed to investigate the types of problems that are of
interest in that field. Hilbert spaces, for example, were developed to
investigate analogies between certain function spaces & Euclidean spaces; they
were only later adopted by physicists to describe quantum systems. This is in
sharp contrast with classical mechanics; the development of differential
calculus was, to a large extent, driven by the desire to describe the observed
motion of bodies - the very question with which classical mechanics is
concerned. In consequence, it is no exaggeration to say that if a question is
well defined within classical mechanics, it can be described using calculus.
In quantum mechanics, the mathematical language is far less articulate. For
example, it leaves unclear what empirical properties a system must posses in
order for the quantum description to apply. It’s also unclear how, and even
whether, the language quantum mechanics can be used to describe the
experiments employed to test the theory. It is similarly unclear whether
and/or how quantum mechanics can be used to describe the world of our direct
experience.
Such considerations lead to a simple question: To what extent are the
difficulties of quantum theory due to limitations in our ability to phrase
relevant questions in the theory’s mathematical language? If such limitations
do play a role, it would not represent a unique state of affairs. As most
everyone knows, Zeno’s paradoxes seemed to challenge some of our most basic
notions of time and motion, until calculus resolved the paradoxes by creating
a clear understanding of the continuum. Somewhat more remotely, the drawing up
of annual calendars (and other activities founded on cyclic heavenly activity)
was once imbued with a mystery well beyond our current awe of quantum physics.
With the slow advance of the theory of numbers the mystery waned, until now
such activity requires nothing more mysterious than straightforward
arithmetic.
In this article, the question of whether a similar situation exists for
quantum mechanics will be investigated. Three simple mathematical theories
will be created, each addressing a basic aspect of dynamic systems. The
resulting mathematical language will then be used to analyze quantum systems.
If the analysis yields core characteristics of quantum theory, some portion of
the theory’s underpinnings will necessarily be revealed, and a measure of
insight into previously mentioned questions ought to be gained.
The mathematics will be constructed to speak to two of the most basic
differences between experimental results in quantum mechanics and those in
classical mechanics: in quantum mechanics experimental outcomes are non-
deterministic & the experimental probabilities are not additive. Non-
additivity in turn puts limitations on the kinds of experiments that can be
performed. To see this, note that if an experiment has outcomes of $\\{X,\neg
X\\}$, and another has outcomes of $\\{X_{1},X_{2},\neg X\\}$ then
$P(X)+P(\neg X)=1=P(X_{1})+P(X_{2})+P(\neg X)$, and so
$P(X)=P(X_{1})+P(X_{2})$; if $P(X)\neq P(X_{1})+P(X_{2})$ then one of these
experiments can not be performed (the only other possibility is that the
probability of a given outcome depends on the make-up of the experiment as a
whole, but this is not the case in quantum physics). Turning this around, if
there is no set $Y$, s.t. $\\{X\\}\bigcup Y$ and $\\{X_{1},X_{2}\\}\bigcup Y$
are both sets of experimental outcomes, one may wonder if we can expect $P(X)$
to equal $P(X_{1})+P(X_{2})$.
This line of reasoning provides some of the central questions to be answered
in this article: What sorts of experiments can be performed? What limits
scientists to only being able to perform these experiments? What does the set
of allowed experiments imply about the nature of the experimental
probabilities? How do these experimental probabilities correspond to quantum
probabilities? We will seek simple, readily understandable answers to these
questions.
Of the three mathematical theories to be constructed, the first will be a
simple theory of dynamic systems that encompasses both deterministic and non-
deterministic dynamics. Determinism refers to the particular case in which a
complete knowledge of the present grants complete knowledge of the future; all
other cases represent types of non-determinism. The theory of dynamic systems
will then be utilized to construct a theory of experiments; dynamic systems
will be used to describe both experiments as a whole, and the (sub)systems
whose natures the experiments probe. The analysis will be somewhat analogous
to that found in automata theory: when a (sub)system path is “read into” an
experimental set-up, the set-up determines which outcome the path belongs to.
By understanding how such processes can be constructed, we can obtain an
understanding of what types of experiments are performable, reproducible, and
have well defined outcomes.111It should be pointed that, while the analogy
between experiments and automata is useful for creating a quick sketch of the
theory, it is no more than an analogy. In the automata studied by computer
scientists, time is assumed to be discrete and the number of automata states
is assumed to be finite. These assumptions must be made in order to assert
that the automata are performing “calculations”, and they have significant
impact in deriving classes of calculable functions, but such assumptions would
be out of place when discussing experiments.
Finally, building on this understanding, a probability theory will be given
for collections of experiments by assuming that the usual rules of probability
and statistics hold on the set of outcomes for any individual experiment &
that if any two experiments in the collection share an outcome, then they
agree on that outcome’s probability.
These constructs will then be applied to quantum measurements, which are
commonly described in terms of projection operators in a Hilbert space. It
will be found that the nature and structure of these measurements can indeed
be reduced to a clear, simple, rational understanding. At no point will there
be any need to invoke anything the least bit strange, spooky, or beyond the
realm of human understanding, nor any need to rely on any procedures that are
utilized because they work even though we don’t understand why.
Because this paper only considers a fraction of all quantum phenomenon, no
claim can be made that all (or even most) quantum phenomenon can yield to some
simple, rational understanding. What we seek to establish is more modest -
that at least some of our bafflement in the face of quantum physics is due to
the manner with which we address the phenomenon, rather than the nature of the
phenomena themselves.
### On the Question of Interpretations
Though quantum interpretations is a large topic, in this article it will play
a small role. However, because a theory can not be fully understood without
some notion of its possible interpretations, a word or two are in order before
proceeding.
The concept of an "interpretation" will be taken here as being more or less
equivalent to the mathematical concept of a "model".222The mathematical notion
of a “model” is a basic concept from the field of mathematical logic. Since
mathematical logic is not generally a part of the scientific curriculum,
here’s a brief description of model theory. A mathematical theory is a set of
formal statements, generally taken to be closed under logical implication. A
model for the theory is a “world” in which all the statements are true. For
example, group theory starts with three statements involving a binary
function, $\cdot$, and a constant, $I$. They are: For all $x,y,z$, $(x\cdot
y)\cdot z=x\cdot(y\cdot z)$; For all $x$, $x\cdot I=x$; For all $x$ there
exists a $y$ s.t. $x\cdot y=I$. One model for this theory is the set of
integers, with “$\cdot$” meaning $+$ and “$I$” meaning $0$. Another is the set
of non-zero real numbers, with “$\cdot$” meaning $\times$ and “$I$” meaning
$1$. It is generally the case that a theory will have a more than one model. A
model may be thought of as a reality that underlies the theory (in which case
the fact that a theory has many different models simply means that it holds
under many different circumstances). It is this sense of models referring to a
theory’s underlying reality that leads to the correspondence between
scientific interpretations and mathematical models. The theories in this
article will have many models, and there will be no attempt to single out any
one as being preferred. In this section, informal sample models will be given
for the two properties of central interest: non-deterministic dynamics and
non-additive probabilities. These should help provide a background
understanding for the theories.
#### Two Models for Non-Determinism
As noted previously, non-determinism simply means that a complete knowledge of
a system’s current state does not imply a complete knowledge of all the
system’s future states. In the simplest model of non-determinism, at any given
time the system is in a single state, but that state doesn’t contain enough
information to be able to deduce what all the future states will be. This will
be referred to as _type-i_ non-determinism (the “i” standing for “individual”,
because the system is always in an individual state). Turning towards the past
rather than the future, this is akin to the situations found in archeology and
paleontology, in which greater knowledge of the present state does yield
greater knowledge of the past, but you would not expect any amount of
knowledge of the present to yield a complete knowledge of all history.
In _type-m_ non-determinism, the system takes multiple paths simultaneously -
every path it can take, it will take (the “m” in “type-m” stands for
“multiple”). Quantum mechanics is often interpreted as displaying type-m non-
determinism; for example, in a double slit experiment, the particle is viewed
as traveling through both slits.
There are also mixed models. In non-deterministic automata, the input shows
type-i non-determinism (in that a single character in the input string will
not determine what the rest of the input must be), while the automata has
type-m non-determinism (as individual characters are read in, the automata
“non-deterministically samples” all possible transitions). Some quantum
mechanical interpretations invert that view - the system being experimented on
is seen as having type-m non-determinism while the experimental set-up is seen
as showing type-i non-determinism (it’s often further assumed that if the
system were to be deterministic, then the experimental set-up would also be
deterministic, and more specifically, classical). In a similar manner,
decoherence often entails an unspoken assumption that a system displays type-m
non-determinism if the non-diagonal elements of the density matrix do not
vanish, and displays type-i non-determinism otherwise; in this sense, the non-
determinism is considered to be type-i in so far the probabilities are
additive, and type-m in so far as the probabilities are non-additive.
It’s important to stress that type-i, type-m, and mixed models are not the
only possible models for non-determinism; many-worlds interpretations, for
example, provide yet another type of model.
#### Two Models for Non-Additivity
Two models will now be given for systems that display both non-determinism &
non-additive probabilities.
Systems with type-i non-determinism can have non-additive probabilities if
interactions with the measuring devices can not be made arbitrarily small
(e.g., if the fields mediating the interactions are quantized). As an example,
imagine experimental outcome $X$ and outcomes $\\{X_{1},X_{2}\\}$ s.t
$X=X_{1}\bigcup X_{2}$ and $X_{1}\bigcap X_{2}=\emptyset$. The minimal
interactions required to determine $X$ will in general be different from the
minimal interactions to determine $X_{1}$ or $X_{2}$. These differing
interactions will cause different deflections to the system paths, which can
result in differences between the statistical likelihood of the outcome being
$X$ and the statistical likelihood of the outcome being $X_{1}$ or $X_{2}$.
Such a state of affairs can be referred to as the “intuitive model”, because
it allows our physical intuition to be applied.
For type-m non-determinism, one manner in which non-additive probabilities can
appear is if the various paths that the system takes interfere with one other.
In this case, for outcome $X$, all paths corresponding to $X$ may interfere,
whereas for $\\{X_{1},X_{2}\\}$ paths can only interfere if they correspond to
the same $X_{i}$; the differences in interference then lead to different
probabilities for the outcomes. This can be referred to as the “orthodox
model”, because it is shared by many of the most widely accepted
interpretations of quantum mechanics.
Once again, these are just two “sample” models for the non-additivity; there
are many others.
## II Dynamics
To begin, we require a theory of dynamic systems. Existing theories tend to
make assumptions that are violated by quantum systems, so here a simple,
lightweight theory will be presented; a theory in which all excess assumptions
will be stripped away.
### II.1 Parameters
Dynamic systems are systems that can change over time; they are parametrized
by time. In a sense, the one feature that all dynamic systems have in common
is that the are parametrized. We therefore begin by quickly reviewing the
concept of a parameter.
Parameters come in variety of forms. For example, some systems have discrete
parameters, while others have continuous parameters. None the less, all
parameters share several basic features. Specifically, a parameter must be
totally ordered, and must support addition. Thus, parameters are structures
with signature $(\Lambda,<,+,0)$, where $<$ is a total ordering, $+$ is the
usual addition function, and $0$ is the additive identity. It is shown in
Appendix D that this, together with the requirement that there be an element
greater than $0$, yields the general concept of a parameter. The set of Real
numbers are parameters in this sense, as are the Integers , the Rationals, and
all infinite ordinals.
This is, however, a little too general. First, in this article we will only be
interested in parameters whose values are finite. Second, for reasons of
analysis, we will only be interested in parameters that are Cauchy convergent.
These two requirements are equivalent to adding a completeness axiom; this
axiom states that all subsets of $\Lambda$ that are bounded from above have a
least upper bound. This limits the models to only four, classified by whether
the parameter are discrete or continuous, and whether or not they are bounded
from below by $0$. These four types of parameters are closely related to the
canonical number systems, and can be readily constructed from them by
introducing the parameter value $\mathbf{1}$, and multiplication by a number.
If the parameter is discrete, assign “$\mathbf{1}$” to be the successor to
$0$; if it’s continuous, choose “$\mathbf{1}$” to be any parameter value
greater than $0$. The choice of $\mathbf{1}$ sets the scale (e.g., 1 second);
it is because parameters have a scale that multiplication is not defined on
parameters.
If $\Lambda$ is a parameter and $\lambda\in\Lambda$, $\lambda$ added to itself
$n$ times will be denoted $n\lambda$ (for example,
$3\lambda\equiv\lambda+\lambda+\lambda$). $0\lambda\equiv 0$.
As shown in Appendix D, constructing the four types of parameters from numbers
is now straightforward. For any discrete, bounded from below parameter,
$(\Lambda,<,+,0)$, the following will hold:
$\Lambda=\\{n\mathbf{1}:n\in\mathbb{N}\\}$,
$n\mathbf{1}+m\mathbf{1}=(n+m)\mathbf{1}$, and $n\mathbf{1}>m\mathbf{1}$ iff
$n>m$. Discrete, unbounded parameters are similar, but with the Integers
replacing the natural numbers. For continuous, unbounded parameters, take
$\mathbf{1}$ to be any positive value, and replace the natural numbers with
the Reals (multiplication by a real number is defined in Appendix D).
Similarly, for continuous, bounded parameters, replace the natural numbers
with the non-negative Reals. This close correspondence between parameters and
numbers is why the two concepts are often treated interchangeably.
We conclude this section by reviewing some standard notation. First, the
notation for parameter intervals:
###### Definition 1.
If $\Lambda$ is a parameter and $\lambda_{1},\lambda_{2}\in\Lambda$
$[\lambda_{1},\lambda_{2}]=\\{\lambda\in\Lambda\,:\,\lambda_{1}\leq\lambda\leq\lambda_{2}\\}$
$(\lambda_{1},\lambda_{2})=\\{\lambda\in\Lambda\,:\,\lambda_{1}<\lambda<\lambda_{2}\\}$
and similarly for $[\lambda_{1},\lambda_{2})$ and $(\lambda,\lambda_{2}]$.
$[\lambda_{1},\infty]=\\{\lambda\in\Lambda\,:\,\lambda_{1}\leq\lambda\\}$
$[-\infty,\lambda_{1}]=\\{\lambda\in\Lambda\,:\,\lambda\leq\lambda_{1}\\}$
and similarly for $(\lambda,\infty)$, etc. (note that
$[\lambda_{1},\infty)=[\lambda_{1},\infty]$).
Next, functions for least upper bound and greatest lower bound:
###### Definition 2.
If $\Lambda$ is a parameter and $\chi\subset\Lambda$ then if $\chi$ is bounded
from above, $lub(\chi)$ is $\chi$’s least upper bound, and if $\chi$ is
bounded from below, $glb(\chi)$ is $\chi$’s greatest lower bound.
$Min(\chi)$ is equal to $glb(\chi)$ if $\chi$ is bounded from below, and
$-\infty$ otherwise. Similarly, $Max(\chi)$ is equal to $lub(\chi)$ if $\chi$
is bounded from above, and $\infty$ otherwise.
And lastly, the definition of subtraction:
###### Definition 3.
If $\Lambda$ is a parameter and $\lambda,\lambda^{\prime}\in\Lambda$:
$\lambda-\lambda^{\prime}\equiv\begin{cases}0&if\,Min(\Lambda)=0\,and\,\lambda<\lambda^{\prime}\\\
\Delta\lambda&where\,\Delta\lambda+\lambda=\lambda^{\prime},\>otherwise\end{cases}$
It follows from the numeric constructions outlined above that
$\lambda-\lambda^{\prime}$ always yields a unique element of $\Lambda$.
### II.2 Parametrized Functions and Dynamic Sets
Parametrized functions and dynamic sets are the rudimentary concepts on which
all else will be built. This section introduces them, along with their
notational language. We start with parametrized functions.
###### Definition 4.
A _parametrized function_ is a function whose domain is a parameter.
If $f$ is a parametrized function, and $[x_{1},x_{2}]$ is an interval of
$Dom(f)$, then $f[x_{1},x_{2}]$ is $f$ restricted to domain $[x_{1},\,x_{2}]$
(values of $x_{1}=-\infty$ and/or $x_{2}=\infty$ are allowed.)
We define one operation on parametrized functions, concatenation.
###### Definition 5.
If $f$ and $g$ are parametrized functions, $Dom(f)=Dom(g)$, and
$f(\lambda)=g(\lambda)$, then $f[x_{1},\lambda]\circ g[\lambda,x_{2}]$ is the
function on domain $[x_{1},x_{2}]$ s.t.
$f[x_{1},\lambda]\circ
g[\lambda,x_{2}](\lambda^{\prime})=\begin{cases}f(\lambda^{\prime})&if\,\lambda^{\prime}\in[x_{1},\lambda]\\\
g(\lambda^{\prime})&if\,\lambda^{\prime}\in[\lambda,x_{2}]\end{cases}$
That is all that’s needed for parametrized functions. They will now be used to
define dynamic sets.
###### Definition 6.
A _dynamic set_ , $S$, is any non-empty set of parametrized functions s.t. all
elements share the same parameter
If $f\in S$ then $f$ can be referred to as a _path_ , and will be generally be
written $\bar{p}$
With $\bar{p}\in S$, $\Lambda_{S}\equiv Dom(\bar{p})$ (that is, $\Lambda_{S}$
is the parameter that all elements of $S$ share)
$\mathcal{P}_{S}\equiv\bigcup_{\bar{p}\in S}Ran(\bar{p})$; the elements of
$\mathcal{P}_{S}$ are _states_
$S[x_{1},x_{2}]\equiv\\{\bar{p}[x_{1},x_{2}]:\bar{p}\in S\\}$
For $\lambda\in\Lambda_{S}$,
$S(\lambda)\equiv\\{p\in\mathcal{P}_{S}:for\>some\>\bar{p}\in
S,\>\bar{p}(\lambda)=p\\}$ (That is, $S(\lambda)$ is the set of possible
states at time $\lambda$)
$Uni(S)\equiv\\{(\lambda,p)\in\Lambda_{S}\otimes\mathcal{P}_{S}:p\in
S(\lambda)\\}$ (“$Uni(S)$” is the “universe” of $S$ \- the set of all possible
time-state pairs)
In the above definition, “time” was occasionally mentioned. In what follows,
we will refer only to “the parameter”, and not “time”. This is because “time”
has grown into an over-loaded concept. For example, in relativity the time
measured on a clock is a function of the path the clock takes. Time measured
by a clock is generally referred to as the “proper time”. On the other hand,
in relativity theory a particle state is generally taken to be a four
dimensional vector, $\overrightarrow{x}$, and coordinates are often chosen so
that the “$0^{th}$” element parametrizes the dynamics. This coordinate is
generally referred to as the “time coordinate”, and this notion of time
referred to as “coordinate time”. For such single particle systems, the
elements of $Uni(S)$ will be of the form $(\lambda,\overrightarrow{x})$, and
for coordinates that have a “time coordinate”, the $0^{th}$ element of
$\overrightarrow{x}$ will always equal $\lambda$. Since coordinates may be
used to describe all paths, and “proper time” is path dependent, it follows
that in such this case $\lambda$ may not equal the proper time.
In this paper, all such complications will be shrugged off. First, we will
make no attempt to map states onto coordinates. Moreover, there will be a
preference to reserve the term “time” for the quantity that is measured by
clocks. Since this may or may not equal the quantity used to parametrize the
system dynamics (depending on the path the clock takes), we retire the term
“time” and speak only of “parameters”.
Back to dynamic sets.
The concatenation operation can be extended to sets of path-segments.
###### Definition 7.
If $S$ is a dynamic set and $A$ and $B$ are sets of partial paths of $S$ then
$A\circ
B\equiv\\{\bar{p}_{1}[x_{1},\lambda]\circ\bar{p}_{2}[\lambda,x_{2}]:\bar{p}_{1}[x_{1},\lambda]\in
A,\>\bar{p}_{2}[\lambda,x_{2}]\in
B\>and\>\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)\\}$
The following notation for various sets of path-segments will prove quite
useful.
###### Definition 8.
If $S$ is a dynamic set,
$(\lambda,p),(\lambda_{1},p_{1}),(\lambda_{2},p_{2})\in Uni(S)$, and
$\lambda_{1}\leq\lambda_{2}$
$S_{\rightarrow(\lambda,p)}\equiv\\{\bar{p}[-\infty,\lambda]:\bar{p}\in
S\>and\>\bar{p}(\lambda)=p\\}$
$S_{(\lambda,p)\rightarrow}\equiv\\{\bar{p}[\lambda,\infty]:\bar{p}\in
S\>and\>\bar{p}(\lambda)=p\\}$
$S_{(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2})}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}\in
S,\>\bar{p}(\lambda_{1})=p_{1},\>and\>\bar{p}(\lambda_{2})=p_{2}\\}$
If $S$ is a dynamic set and $p,p_{1},p_{2}\in\mathcal{P}_{S}$
$S_{\rightarrow p}\equiv\\{\bar{p}[-\infty,\lambda]:\bar{p}\in
S,\>\lambda\in\Lambda_{S},\>and\>\bar{p}(\lambda)=p\\}$
$S_{p\rightarrow}\equiv\\{\bar{p}[\lambda,\infty]:\bar{p}\in
S,\>\lambda\in\Lambda_{S},\>and\>\bar{p}(\lambda)=p\\}$
$S_{p_{1}\rightarrow
p_{2}}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}\in
S,\>\lambda_{1},\lambda_{2}\in\Lambda_{S},\>\bar{p}(\lambda_{1})=p_{1},\>and\>\bar{p}(\lambda_{2})=p_{2}\\}$
If $S$ is a dynamic set, $Y,Z\subset\mathcal{P}_{S}$
$S_{Y\rightarrow Z}\equiv\bigcup_{p\in Y,p^{\prime}\in Z}S_{p\rightarrow
p^{\prime}}$
…and similarly for $S_{\rightarrow Z}$, $S_{Y\rightarrow}$,
$S_{(\lambda,Y)\rightarrow(\lambda,Z)}$, etc.
This notation may be extended as needed. For example, $S_{\rightarrow
p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}\rightarrow}$ is the set
of all paths in $S$ that pass through $p_{1}$, then through
$(\lambda_{2},p_{2})$, then through $p_{3}$.
One of the most basic properties of a dynamic systems is whether or not its
dynamics can change over time. A dynamic set is homogeneous if its dynamics
are the same regardless of when a state occurs. More formally:
###### Definition 9.
A dynamic set, $S$, is _homogeneous_ if for all
$(\lambda_{1},p),(\lambda_{2},p)\in Uni(S)$, $\lambda_{1}\leq\lambda_{2}$,
$\bar{p}\in S_{\rightarrow(\lambda_{1},p)\rightarrow}$ iff there exists a
$\bar{p}^{\prime}\in S_{\rightarrow(\lambda_{2},p)\rightarrow}$ s.t. for all
$\lambda\in\Lambda_{S}$,
$\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda+\lambda_{2}-\lambda_{1})$.
Note that if $\Lambda_{D}$ is bounded from below, $S$ can still be
homogeneous; it would simply mean that the paths running through
$(\lambda_{1},p)$ and those running through $(\lambda_{2},p)$ only differ by a
shift & by the fact the initial part of the paths running though
$(\lambda_{1},p)$ are cut off.
A related property is whether or not a given state, or set of states, can
occur at any time.
###### Definition 10.
If $S$ is a dynamic set, $p\in\mathcal{P}_{S}$ is _homogeneously realized_ if
for every $\lambda\in\Lambda_{S}$, $p\in S(\lambda)$.
$A\subset\mathcal{P}_{S}$ is _homogeneously realized_ if every $p\in A$ is.
### II.3 Dynamic Spaces
Dynamic sets embrace quite a broad concept of dynamics. This can make them
difficult to work with, and so it is often helpful to make further assumptions
about the system dynamics. A common assumption for closed systems is that the
system’s possible future paths are determined entirely by its current state.
This notion is captured by the following type of dynamic set:
###### Definition 11.
A _dynamic space_ , $D$, is a dynamic set s.t. if $\bar{p},\bar{p}^{\prime}\in
D$, $\lambda\in\Lambda_{D}$, and $\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda)$,
then $\bar{p}[-\infty,\lambda]\circ\bar{p}^{\prime}[\lambda,\infty]\in D$.
Thus, dynamic spaces are closed under concatenation. This will prove to be an
enormous simplification. Closed systems are generally assumed to have this
property, and so in what follows, closed systems will always be assumed to be
dynamic spaces. (Note that a system being experimented on interacts with
experimental equipment, and so is not closed. As a result, such a system might
not be a dynamic space.)
###### Definition 12.
If $D$ is a dynamic space, $D_{\square_{1}\rightarrow\square_{2}}$ may be
written $\square_{1}\rightarrow\square_{2}$. (For example,
$D_{p_{1}\rightarrow p_{2}}$ may by written $p_{1}\rightarrow p_{2}$).
Similarly, $D_{\square_{1}\rightarrow\square_{2}}\circ
D_{\square_{2}\rightarrow\square_{3}}$ may be written
$\square_{1}\rightarrow\square_{2}\rightarrow\square_{3}$,
$D_{\rightarrow\square_{1}}\circ D_{\square_{1}\rightarrow\square_{2}}\circ
D_{\square_{2}\rightarrow}$ may be written
$\rightarrow\square_{1}\rightarrow\square_{2}\rightarrow$, etc.
If $D$ is a dynamic space then, for example,
$D_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow
p_{3}}=D_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ
D_{(\lambda_{2},p_{2})\rightarrow p_{3}}$; if, on the other hand, $S$ is
simply a dynamic set, we could only assert that
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}}\subset
S_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ S_{(\lambda_{2},p_{2})\rightarrow
p_{3}}$. This is why, for dynamic spaces, we may relax the notation and simply
refer to $p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}$, while for
dynamic sets we need to be clear about whether we mean
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})}\circ
S_{(\lambda_{2},p_{2})\rightarrow p_{3}}$ or
$S_{p_{1}\rightarrow(\lambda_{2},p_{2})\rightarrow p_{3}}$.
For dynamic spaces the use of outer arrows, such as
$\rightarrow(\lambda_{1},p_{1})\rightarrow p_{2}\rightarrow$, may be extended
to arbitrary sets of partial paths:
###### Definition 13.
If $D$ is a dynamic space, and $A$ is a set of partial paths s.t. if
$\bar{p}[x_{1},x_{2}]\in A$ then $\bar{p}[x_{1},x_{2}]\in D[x_{1},x_{2}]$,
$\bar{p}\in\,\rightarrow A\rightarrow$ if $\bar{p}\in D$ and for some
$\bar{p}^{\prime}[x_{1},x_{2}]\in A$,
$\bar{p}[x_{1},x_{2}]=\bar{p}^{\prime}[x_{1},x_{2}]$.
$\bar{p}[x,\infty]\in\,A\rightarrow$ if $\bar{p}[x,\infty]\in D[x,\infty]$ and
for some $\bar{p}^{\prime}[x,x_{1}]\in A$,
$\bar{p}[x,x_{1}]=\bar{p}^{\prime}[x,x_{1}]$.
$\bar{p}[-\infty,x]\in\,\rightarrow A$ if $\bar{p}[-\infty,x]\in D[-\infty,x]$
and for some $\bar{p}^{\prime}[x_{0},x]\in A$,
$\bar{p}[x_{0},x]=\bar{p}^{\prime}[x_{0},x]$.
### II.4 Special Sets
Although the notation introduced in Secs II.2 and II.3 will be sufficient for
nearly all circumstances, two straightforward additions will prove useful in
the discussion of experiments.
First, it will be useful to isolate the subset of $X\rightarrow Y$ consisting
of the path-segments that don’t re-enter $X$ after their start; it will
similarly be useful to isolate the path-segments that don’t enter $Y$ until
their end.
###### Definition 14.
If $S$ is a dynamic set and $X,Y,Z\subset\mathcal{P}_{S}$:
$S_{X\rightarrow Y\upharpoonright
Z}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\in S_{X\rightarrow
Y}:for\>\lambda\in[\lambda_{1},\lambda_{2}),\>\bar{p}(\lambda)\notin Z\\}$
$S_{Z\upharpoonleft X\rightarrow
Y}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\in S_{X\rightarrow
Y}:for\>\lambda\in(\lambda,\lambda_{2}],\>\bar{p}(\lambda)\notin Z\\}$
$S_{X\rightarrow Y\upharpoonright Y}$ is then the set of path segments from
$S$ that start at $X$ and end at $Y$, but do not enter $Y$ before the end of
the segment. Similarly for $S_{X\upharpoonleft X\rightarrow Y}$.
$S_{X/Y}$, to be introduces momentarily, bears a resemblance to
$S_{X\rightarrow Y\upharpoonright Y}$, but overcomes a difficulty that occurs
in the continuum:
###### Definition 15.
$S_{X/Y}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]:\bar{p}[\lambda_{1},\infty]\in
S_{X\rightarrow},\>Y\bigcap
Ran(\bar{p}[\lambda_{1},\infty])\neq\emptyset,\>and\>\lambda_{2}=glb(\bar{p}^{-1}[\lambda_{1},\infty][Y])\\}$
In the above definition, $\lambda_{2}$ is either the first time $Y$ occurs in
$\bar{p}[\lambda_{1},\infty]$ or, if the “first time” can’t be obtained in the
continuum, the moment before $Y$ first appears.
###### Definition 16.
If $S$ is a dynamic set and $X,Y\subset\mathcal{P}_{S}$,
$(X/Y)\equiv\\{p\in\mathcal{P}_{S}\,:\,for\>some\>\bar{p}[\lambda_{1},\lambda_{2}]\in
S_{X/Y},\>p=\bar{p}(\lambda_{2})\\}$
###### Theorem 17.
If $D$ is a homogeneous dynamic space then
$D_{X/Y}=X\rightarrow(X/Y)\upharpoonright Y$
###### Proof.
If $\bar{p}[\lambda_{1},\lambda_{2}]\in D_{X/Y}$ then $\bar{p}(\lambda_{1})\in
X$, $\bar{p}(\lambda_{2})\in(X/Y)$, and for all
$\lambda\in[\lambda_{1},\lambda_{2})$, $\bar{p}(\lambda)\notin Y$, so
$D_{X/Y}\subset X\rightarrow(X/Y)\upharpoonright Y$. (Note that this doesn’t
require $D$ to be homogeneous, or a dynamic space.)
It remains to show that $X\rightarrow(X/Y)\upharpoonright Y\subset D_{X/Y}$.
Take $\bar{p}[\lambda_{1},\lambda_{2}]\in X\rightarrow(X/Y)\upharpoonright Y$,
$p\equiv\bar{p}(\lambda_{2})\in(X/Y)$. By the definition of $(X/Y)$, there’s a
$\bar{p}^{\prime}[\lambda_{3},\infty]\in p\rightarrow$ s.t.
$\lambda_{3}=glb(\bar{p}^{\prime}[\lambda_{3},\infty]^{-1}[Y])$. By
homogeneity, there’s a $\bar{p}^{\prime\prime}$ s.t. for all
$\lambda\in\Lambda_{D}$,
$\bar{p}^{\prime\prime}(\lambda)=\bar{p}^{\prime}(\lambda+(\lambda_{3}-\lambda_{2}))$
(assuming $\lambda_{3}\geq\lambda_{2}$ or $\Lambda_{D}$ is unbounded from
below; otherwise
$\bar{p}^{\prime}(\lambda)=\bar{p}^{\prime\prime}(\lambda+(\lambda_{2}-\lambda_{3}))$
may be asserted for all $\lambda\in\Lambda_{D}$). By considering
$\bar{p}[\lambda_{1},\lambda_{2}]\circ\bar{p}^{\prime\prime}[\lambda_{2},\infty]$,
it follows that $\bar{p}[\lambda_{1},\lambda_{2}]\in D_{X/Y}$. ∎
### II.5 Limits and Closed Sets
n this final section, system dynamics will be used to define limits. Four ways
in which a point $p\in\mathcal{P}_{S}$ can be a limit point of set
$X\subset\mathcal{P}_{S}$ at $\lambda$ will be defined. They correspond to: if
the system is in state $p$ at $\lambda$ then $X$ must be about to occur, if
the system is in state $p$ at $\lambda$ then $X$ might be about to occur, if
the system is in state $p$ at $\lambda$ then $X$ might have just happened, and
if the system is in state $p$ at $\lambda$ then $X$ must have just happened.
For formally:
###### Definition 18.
If $S$ is a dynamic set, $p\in\mathcal{P}_{S}$ , $\lambda\in\Lambda_{S},$ and
$X$ is a non-empty subset of $\mathcal{P}_{S}$ then
$p\in lim_{\lambda}^{\vartriangleright}X$ if for every
$\bar{p}[\lambda,\infty]\in p\rightarrow$, every $\lambda^{\prime}>\lambda$,
there’s a $\lambda^{\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$
$p\in lim_{\lambda}^{>}X$ if for every $\lambda^{\prime}>\lambda$, there’s a
$\bar{p}[\lambda,\infty]\in p\rightarrow$ and a
$\lambda^{\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$
$p\in lim_{\lambda}^{<}X$ if $\lambda>Min(\Lambda_{D})$ and for every
$\lambda^{\prime}<\lambda$, there’s $\bar{p}[-\infty,\lambda]\in\,\rightarrow
p$ and a $\lambda^{\prime\prime}\in(\lambda^{\prime},\lambda)$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$
$p\in lim_{\lambda}^{\vartriangleleft}X$ if $\lambda>Min(\Lambda_{D})$ and for
every $\bar{p}[-\infty,\lambda]\in\,\rightarrow p$ s.t. , every
$\lambda^{\prime}<\lambda$, there’s a
$\lambda^{\prime\prime}\in(\lambda^{\prime},\lambda)$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in X$
If $S$ is homogeneous, then the $\lambda$ subscript may be dropped, and we may
simply write $p\in lim^{\vartriangleright}X$, etc.
###### Definition 19.
If $S$ is a dynamic set, $X\subset\mathcal{P}_{S}$, and
$\lambda\in\Lambda_{S}$ then $X_{\lambda}^{\vartriangleright}\equiv X\bigcup
lim^{\vartriangleright}X$, $X_{\lambda}^{>}\equiv X\bigcup lim^{>}X$,
$X_{\lambda}^{<}\equiv X\bigcup lim^{<}X$, and
$X_{\lambda}^{\vartriangleleft}\equiv X\bigcup lim^{\vartriangleleft}X$.
Once again, if $S$ is homogeneous, then the $\lambda$ subscript may be
dropped, and one may simply write $X^{\vartriangleright}$, etc.
A set is closed if it contains all its limit points. The following theorem
lays out conditions under which $X^{\vartriangleright}$, etc., are closed.
###### Theorem 20.
1) If $S$ is a homogeneous dynamic set, and $X\subset\mathcal{P}_{S}$, then
$X^{\vartriangleright\vartriangleright}=X^{\vartriangleright}$ and
$X^{\vartriangleleft\vartriangleleft}=X^{\vartriangleleft}$.
1) If $D$ is a homogeneous dynamic space, and $X\subset\mathcal{P}_{D}$, then
$X^{>>}=X^{>}$ and $X^{<<}=X^{<}$.
###### Proof.
If $\Lambda_{D}$ is discrete, this holds because if $\Lambda_{D}$ is discrete
then for all $X\subset D$, $lim^{\square}X=\emptyset$.
Take $\Lambda_{D}$ to be continuous.
1) Assume $p\in lim^{\vartriangleright}X^{\vartriangleright}$ and take
$\bar{p}[\lambda,\infty]\in S_{p\rightarrow}$ & $\lambda^{\prime}>\lambda$.
There’s a
$\lambda^{\prime\prime}\in(\lambda,\lambda+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime})\in X^{\vartriangleright}$. Because
$\bar{p}(\lambda^{\prime\prime})\in X^{\vartriangleright}$, either
$\bar{p}(\lambda^{\prime\prime})\in X$ or $\bar{p}(\lambda^{\prime\prime})\in
lim^{\vartriangleright}X$, so there’s a
$\lambda^{\prime\prime\prime}\in[\lambda^{\prime\prime},\lambda^{\prime\prime}+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime\prime})\in X$. Therefore there’s a
$\lambda^{\prime\prime\prime}\in(\lambda,\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime\prime})\in X$, and so $p\in
lim^{\vartriangleright}X$, and so $p\in X^{\vartriangleright}$.
$\vartriangleleft$ is similar
2) Assume $p\in lim^{>}X^{>}$. For any $\lambda$, $\lambda^{\prime}>\lambda$
there’s a $\bar{p}[\lambda,\infty]\in p\rightarrow$ and a
$\lambda^{\prime\prime}\in(\lambda,\lambda+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}(\lambda^{\prime\prime})\in X^{>}$. Because
$\bar{p}(\lambda^{\prime\prime})\in X^{>}$, either
$\bar{p}(\lambda^{\prime\prime})\in X$ or $\bar{p}(\lambda^{\prime\prime})\in
lim^{>}X$, so there’s a $\bar{p}^{\prime}[\lambda^{\prime\prime},\infty]\in
p\rightarrow$,
$\lambda^{\prime\prime\prime}\in[\lambda^{\prime\prime},\lambda^{\prime\prime}+\frac{1}{2}(\lambda^{\prime}-\lambda))$
s.t. $\bar{p}^{\prime}(\lambda^{\prime\prime\prime})\in X$. Take
$\bar{p}^{\prime\prime}[\lambda,\infty]=\bar{p}[\lambda,\lambda^{\prime\prime}]\circ\bar{p}^{\prime}[\lambda^{\prime\prime},\infty]$.
$\bar{p}^{\prime\prime}\in p\rightarrow$,
$\lambda^{\prime\prime\prime}\in(\lambda,\lambda^{\prime})$, and
$\bar{p}^{\prime\prime}(\lambda^{\prime\prime\prime})\in X$, so $p\in
lim^{\vartriangleright}X$, and so $p\in X^{\vartriangleright}$.
$<$ is similar. ∎
Under the conditions of Theorem 20, the collection of all closed sets of type
“$>$” are closed under intersections and finite unions. They therefore form a
topology on $\mathcal{P}_{D}$; the lower-limit topology. The closed sets of
type “$<$” also form a topology; the upper-limit topology.
The closed sets of type “$\vartriangleright$”, and those of type
“$\vartriangleleft$”, are closed under intersection, but not union. They
therefore do not form topologies. That is because these limits are fairly
strict, and so can not always determine whether or not some point $p$ is local
to some set $X$.
###### Theorem 21.
If $S$ is a homogeneous dynamic set, and $X,Y\subset\mathcal{P}_{S}$, then
$(X/Y)\subset Y^{>}$
###### Proof.
By the definition of $(X/Y)$, $p\in(X/Y)$ iff for some $\bar{p}\in S$,
$\lambda_{1},\lambda_{2}\in\Lambda_{D}$ ($\lambda_{1}\leq\lambda_{2}$),
$\bar{p}(\lambda_{2})=p$, $\bar{p}(\lambda_{1})\in X$,
$[\lambda_{1},\lambda_{2})\bigcap\bar{p}^{-1}[Y]=\emptyset$, and for all
$\lambda^{\prime}>\lambda_{2}$
$[\lambda_{2},\lambda^{\prime})\bigcap\bar{p}^{-1}[Y]\neq\emptyset$. In this
case, $\bar{p}[\lambda_{2},\infty]\in p\rightarrow$ and for all
$\lambda^{\prime}>\lambda_{2}$ there’s a
$\lambda^{\prime\prime}\in[\lambda_{2},\lambda^{\prime})$ s.t.
$\bar{p}(\lambda^{\prime\prime})\in Y$, so either $p\in Y$ or $p\in lim^{>}Y$.
∎
## III Experiments
### III.1 Shells
In this part, experimentation on dynamic systems will be formalized. As
mentioned in the Introduction, this will be analogous to creating a formal
theory of computation via automata. In computation theory, sequences of
characters are read into an automata, and the automata determines whether they
are sentences in a given language. One of the goals of the theory is to
determine what languages automata are capable of deciding. Experiments are
similar; system paths are “read into” an experimental set-up, which then
determines which outcomes these paths belong to. Our goal will be to determine
what sets of outcomes experiments are capable of deciding.
In this first section, we will not divide the experiment into the system being
experimented on, and the environment containing the experimental apparatus;
for now, we consider the closed system that encompasses both. The dynamic
space of this closed system will be used to formalize some of the external
properties of experiments; namely, that experiments are re-runnable, they have
a clearly defined start, once started they must complete, and once complete
they “remember” that the experiment took place.
###### Definition 22.
A _shell_ is a triple, $(D,I,F)$, where $D$ is a dynamic space and
$I,F\subset\mathcal{P}_{D}$ are the sets of initial and final states. These
must satisfy:
1) $D$ is homogeneous
2) $I$ is homogeneously realized
3) $I\rightarrow\,=(I\upharpoonleft I\rightarrow F)\rightarrow$ (That is,
$I\rightarrow\,=D_{I\upharpoonleft I\rightarrow F}\rightarrow$)
4) $\rightarrow F=\,\rightarrow(I\upharpoonleft I\rightarrow F)$
In the above definition, $I$ represents the set of initial states, or start
states; when the system enters into one of these states, the experiment
starts. $F$ represents the set of final states; when the system enters into
one of these states, the experiment has ended.
Axioms (1) and (2) ensure that the experiment is reproducible. Axiom (3)
ensures that once the experiment begins, it must end. Axiom (4) ensures that
the system can only enter a final state via the experiment.
A note of explanation may be in order for this final axiom. $D$ is assumed to
be a closed system encompassing everything that has bearing on the experiment,
including the person performing the experiment. As a result, all recording
equipment is considered part of $D$, including the experimenter’s memory. So
if the final axiom were violated, when the final state is reached, you
wouldn’t be able to remember whether or not the experiment had taken place.
In (3) and (4), $I\upharpoonleft I\rightarrow F$ is used rather than
$I\rightarrow F$ in order to ensure that there’s a clearly defined space in
which the experiment takes place. For example, $\rightarrow F=\,\rightarrow
I\rightarrow F$ would allow for a path, $\bar{p}[-\infty,0]\in\,\rightarrow
F$, s.t. for all $n\in\mathbb{N}^{+}$, $\bar{p}(\frac{-1}{2n})\in I$ and
$\bar{p}(\frac{-1}{2n+1})\in F$; in this case, the $F$ state at $\lambda=0$
can not be paired with any particular experimental run. Similarly,
$I\rightarrow\,=I\rightarrow F\rightarrow$ would allow a path to cross $I$
several times before crossing $F$; in this case it would not be clear which
crossing of $I$ represented the start of the experiment.
An experiment is considered to be in progress while the space is in a path
segment that runs from $I$ to $F$. The set of states in those path segments
constitute the shell interior, or more formally:
###### Definition 23.
If $Z=(D,I,F)$ is a shell, $Int(Z)=\\{p\in\mathcal{P}_{D}:I\rightarrow
p\upharpoonright F\neq\emptyset\\}$ is the _shell interior_.
###### Theorem 24.
If $Z=(D,I,F)$ is a shell then
$Int(Z)=Ran(D_{I/F})=Ran(I\rightarrow(I/F)\upharpoonright F)$. (The
abbreviation “$Ran(A)$” stands for the set of states
$\bigcup_{\bar{p}[x_{1},x_{2}]\in A}Ran(\bar{p}[x_{1},x_{2}])$.)
###### Proof.
By Thm 17 and shell axiom 1, $D_{I/F}=I\rightarrow(I/F)\upharpoonright F$.
If $p\in Ran(I\rightarrow(I/F)\upharpoonright F)$ then for some
$\bar{p}[\lambda_{1},\lambda_{2}]\in I\rightarrow(I/F)\upharpoonright F$, some
$\lambda_{3}\in[\lambda_{1},\lambda_{2}]$, $\bar{p}(\lambda_{3})=p$, so
$\bar{p}[\lambda_{1},\lambda_{3}]\in I\rightarrow p\upharpoonright F$, and so
$p\in Int(Z)$.
If $p\in Int(Z)$ then there’s a $\bar{p}[\lambda_{1},\lambda_{2}]\in
I\rightarrow p\upharpoonright F$. Since $I\rightarrow\,=(I\upharpoonleft
I\rightarrow F)\rightarrow$ there must be a
$\bar{p}^{\prime}[\lambda_{1},\lambda_{3}]\in D_{I/F}$ s.t.
$\bar{p}^{\prime}[\lambda_{1},\lambda_{2}]=\bar{p}[\lambda_{1},\lambda_{2}]$,
so $p\in Ran(D_{I/F})$. ∎
The following items will prove quite useful:.
###### Definition 25.
If $Z=(D,I,F)$ is a shell, $Dom(Z)\equiv F\bigcup Int(Z)$ is the _shell
domain_
If $A\subset Dom(Z)$, $\omega_{A}\equiv\\{\bar{p}[0,\lambda]\in
I\upharpoonleft I\rightarrow A\\}=I\upharpoonleft(0,I)\rightarrow A$
If $A\subset F$, $\Theta_{A}\equiv\\{\bar{p}[0,\lambda]\in
D_{I/F}:for\>some\>\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{A},\>\bar{p}^{\prime}[0,\lambda]=\bar{p}[0,\lambda]\\}$
Given that our current state is in $A$, $\omega_{A}$ tells us what has
happened in the experiment; because shell dynamics are homogeneous, and $I$ is
homogeneously realized, it is sufficient only consider paths that start at
$\lambda_{0}=0$.
Given that the final state is in $A$, $\Theta_{A}$ captures what occurred
during the experiment.
The shell axioms validate these interpretations of $\omega_{A}$ and
$\Theta_{A}$.
### III.2 E-Automata
Further structure will now be added to shells, starting with dividing the
shell domain into a system and its environment.
#### III.2.1 Environmental Shells
In the following definition: “$\bigotimes$” will refer to the Cartesian
product, and for n-ary relation, $R$, on $A_{1}\bigotimes...\bigotimes A_{n}$,
$\textrm{P}_{i}(R)$ refers the set of $a\in A_{i}$ s.t. for some $r\in R$,
$a=r_{i}$.
###### Definition 26.
An _environmental shell_ , $Z=(D,I,F)$, is a shell together with three sets,
$\mathcal{S}_{Z}$, $\mathcal{E}_{Int(Z)}$, and $\mathcal{E}_{F}$ satisfying
1)
$(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{Int(Z)})\bigcap\mathcal{P}_{D}=Int(Z)$
2) $(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{F})\bigcap\mathcal{P}_{D}=F$
3) $\mathcal{S}_{Z}=\textrm{P}_{1}(Dom(Z))$,
$\mathcal{E}_{Int(Z)}=\textrm{P}_{2}(Int(Z))$, and
$\mathcal{E}_{F}=\textrm{P}_{2}(F)$
$\mathcal{S}_{Z}$ is the _system_ and
$\mathcal{E}_{Z}\equiv\mathcal{E}_{Int(Z)}\bigcup\mathcal{E}_{F}$ is the
_environment._
Informally speaking, it’s assumed that the system ($\mathcal{S}_{Z}$) is being
observed, measured, recorded, etc., and that any observers, measuring devices,
recording equipment, etc., are in the environment. Therefore, it is components
in the environment which decide whether the experiment is complete; this is
why a subset of the environmental states, $\mathcal{E}_{F}$, determine whether
the shell is on $F$.
If $\bar{p}[\lambda_{1},\lambda_{2}]$ is a path segment from $I\upharpoonleft
I\rightarrow F$, $\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}$ is the
system component of the path segment, and
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{E}$ is the environmental
portion. More formally:
###### Definition 27.
For environmental shell $Z$:
If
$\bar{s}[\lambda_{1},\lambda_{2}]:[\lambda_{1},\lambda_{2}]\rightarrow\mathcal{S}_{Z}$,
$\bar{e}[\lambda_{1},\lambda_{2}]:[\lambda_{1},\lambda_{2}]\rightarrow\mathcal{E}_{Z}$,
and
$\bar{p}[\lambda_{1},\lambda_{2}]=(\bar{s}[\lambda_{1},\lambda_{2}],\bar{e}[\lambda_{1},\lambda_{2}])$
then
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}\equiv\bar{s}[\lambda_{1},\lambda_{2}]$
and
$\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{E}\equiv\bar{e}[\lambda_{1},\lambda_{2}]$.
If $A$ is any set of shell domain path segments,
$A\cdot\mathcal{S}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}:\bar{p}[\lambda_{1},\lambda_{2}]\in
A\\}$ and
$A\cdot\mathcal{E}\equiv\\{\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{\mathcal{E}}:\bar{p}[\lambda_{1},\lambda_{2}]\in
A\\}$.
This can be applied to $\omega_{A}$ and $\Theta_{A}$ to extract the system
information:
###### Definition 28.
If $Z$ is an environmental shell:
If $X\subset\mathcal{E}_{F}$,
$\mathcal{O}_{X}\equiv\Theta_{\mathcal{S}_{Z}\bigotimes X}\cdot\mathcal{S}$
If $A\subset Int(Z)$:
$\mathcal{\mbox{$\Sigma$}}_{A}\equiv\\{\bar{s}[0,\lambda_{1}]:for\>some\>\bar{p}[0,\lambda_{2}]\in\Theta_{F},\>\bar{p}(\lambda_{1})\in
A\>\&\>\bar{p}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}[0,\lambda_{1}]\\}$
$\Sigma_{A\rightarrow}\equiv\\{\bar{s}[\lambda_{1},\lambda_{2}]:for\>some\>\bar{p}[0,\lambda_{2}]\in\Theta_{F},\>\bar{p}(\lambda_{1})\in
A\>\&\>\bar{p}[\lambda_{1},\lambda_{2}]\cdot\mathcal{S}=\bar{s}[\lambda_{1},\lambda_{2}]\\}$.
If an experiment completes with the environment in $X$, $\mathcal{O}_{X}$
tells us what happened in the system during the experimental run (the
“$\mathcal{O}$” stands for “outcome”). $\mathcal{O}_{\mathcal{E}_{F}}$ may be
abbreviated $\mathcal{O}_{F}$.
$\Sigma_{A}$ and $\Sigma_{A\rightarrow}$ give insight into what’s happening to
the system in the shell interior: $\Sigma_{A}$ gives the system paths from $I$
to $A$ and $\Sigma_{A\rightarrow}$ gives the system paths from $A$ to $(I/F)$.
For convenience, $\Sigma_{\sigma\bigotimes\mathcal{E}_{Int(Z)}}$ may be
written $\Sigma_{\sigma}$.
###### Theorem 29.
$\mathcal{O}_{F}=\Sigma_{(I/F)}$
###### Proof.
It’s clear that $\mathcal{O}_{F}\subset\Sigma_{(I/F)}$.
Assume $\bar{s}[0,\lambda]\in\Sigma_{(I/F)}$. For some
$\bar{p}[0,\lambda_{2}]\in\Theta_{F}$, $\bar{p}(\lambda_{1})\equiv p\in(I/F)$
and $\bar{p}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}[0,\lambda_{1}]$. Since
$p\in(I/F)$ and $D$ is homogeneous there must be a
$\bar{p}^{\prime}[\lambda_{1},\infty]\in p\rightarrow$ s.t. for all
$\lambda_{2}>\lambda_{1}$ there’s a
$\lambda^{\prime}\in[\lambda_{1},\lambda_{2})$ s.t.
$\bar{p}^{\prime}(\lambda^{\prime})\in F$. Therefore
$\bar{p}[0,\lambda_{1}]\circ\bar{p}^{\prime}[\lambda_{1},\infty]\in
I\rightarrow$ and $\bar{p}[0,\lambda_{1}]\in D_{I/F}$; this means that
$\bar{p}[0,\lambda_{1}]\in\Theta_{F}$, and so
$\bar{s}[0,\lambda_{1}]\in\mathcal{O}_{F}$. ∎
#### III.2.2 The Automata Condition
The automata condition demands that an experiment will terminate based solely
on what has transpired in the system. In particular, it specifies that if
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\omega_{F}$,
$\bar{p}_{1}[0,\lambda]\in\Theta_{F}$, and
$\bar{p}_{1}[0,\lambda]\cdot\mathcal{S}=\bar{p}_{2}[0,\lambda]\cdot\mathcal{S}$
then $\bar{p}_{2}[0,\lambda]\in\Theta_{F}$. Given shell axiom (3), this may be
rephrased as follows:
###### Definition 30.
An environmental shell satisfies _the automata condition_ if for every
$\bar{s}_{1}[0,\lambda_{1}],\,\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$
s.t. $\lambda_{2}>\lambda_{1}$,
$\bar{s}_{1}[0,\lambda_{1}]\neq\bar{s}_{2}[0,\lambda_{1}]$
The constraints this places on the closed system’s dynamics are summarized by
the following theorem.
###### Theorem 31.
The following assertions on environmental shell $Z$ are equivalent:
1) $Z$ satisfies the automata condition
2) For $p\in Int(Z)$, if $\Sigma_{p}\bigcap\mathcal{O}_{F}\neq\emptyset$ then
$p\in F^{\vartriangleright}$
3) $(I/F)\subset F^{\vartriangleright}$ and
$\Sigma_{Int(Z)-(I/F)}\bigcap\Sigma_{(I/F)}=\emptyset$
###### Proof.
$1\Rightarrow 2$: In order for the automata condition to hold, If $p\in
Int(Z)$ and $\Sigma_{p}\bigcap\mathcal{O}_{F}\neq\emptyset$ then either $p\in
F\bigcap(I/F)$, or all paths leaving $p$ must immediately enter $F$; either
way, $p\in F^{\vartriangleright}$.
$2\Rightarrow 3$: Since $\mathcal{O}_{F}=\Sigma_{(I/F)}$,
$\Sigma_{(I/F)}\bigcap\mathcal{O}_{F}\neq\emptyset$, and so $(I/F)\subset
F^{\vartriangleright}$.
If $p\in Int(Z)-(I/F)$ then the shell is not in $F$ and is not about to
transition into $F$, so $p\notin F^{\vartriangleright}$, and so
$\Sigma_{p}\bigcap\mathcal{O}_{F}=\emptyset$.
$3\Rightarrow 1$: Take
$\bar{p}_{1}[0,\lambda_{1}],\,\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ and
$\lambda_{2}\geq\lambda_{1}$. If
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{p}_{2}[0,\lambda_{1}]\cdot\mathcal{S}$
then
$\Sigma_{\bar{p}_{1}(\lambda_{1})}\bigcap\Sigma_{\bar{p}_{2}(\lambda_{1})}\neq\emptyset$
and so $\bar{p}_{2}(\lambda_{1})\in(I/F)$ (since
$\bar{p}_{1}(\lambda_{1})\in(I/F)$ and
$\Sigma_{Int(Z)-(I/F)}\bigcap\Sigma_{(I/F)}=\emptyset$). Since
$\bar{p}_{2}(\lambda_{1})\in(I/F)$ , $\bar{p}_{2}(\lambda_{1})\in
F^{\vartriangleright}$, and so $\bar{p}_{2}[0,\lambda_{1}]\in\Theta_{F}$, and
for all $\lambda_{3}>\lambda_{1}$
$\bar{p}_{2}[0,\lambda_{3}]\notin\Theta_{F}$. Therefore
$\lambda_{2}=\lambda_{1}$. So if
$\bar{p}_{1}[0,\lambda_{1}],\,\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ and
$\lambda_{2}>\lambda_{1}$ then
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}\neq\bar{p}_{2}[0,\lambda_{1}]\cdot\mathcal{S}$
∎
#### III.2.3 Unbiased Conditions
An environmental shell is unbiased if the environment does not influence the
outcome. In the strongest sense this demands that, while the environment may
record the system’s past, it has no effect on the system’s future. Naively,
this may be written: For every $(s,e_{1}),(s.e_{2})\in Int(Z)$,
$\Sigma_{(s,e_{1})\rightarrow}=\Sigma_{(s,e_{2})\rightarrow}$. However,
because in general $\Sigma_{(s,e_{1})}\neq\Sigma_{(s,e_{2})}$, the elements of
$\Sigma_{(s,e_{1})\rightarrow}$ and $\Sigma_{(s,e_{2})\rightarrow}$ may
terminate at different times; this leads to a definition whose wording is more
convoluted, but whose meaning is essentially the same.
###### Definition 32.
An environmental shell, $(D,I,F)$ is _strongly unbiased_ if for every
$\lambda\in Dom(\Theta_{F})$, every
$(s,e_{1}),(s.e_{2})\in\Theta_{F}(\lambda)$,
$\bar{s}_{1}[\lambda,\lambda_{1}]\in\Sigma_{(s,e_{1})\rightarrow}$ iff there
exists a $\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$
s.t., with $\lambda^{\prime}\equiv Min(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$.
This condition means that any effect the environment may have on the system
can be incorporated into the system dynamics, allowing the system to be
comprehensible without having to reference its environment. The following
theorem shows that if an environmental shell is strongly unbiased,
$\mathcal{O}_{F}$ behaves like a dynamic space.
###### Theorem 33.
If an environmental shell is strongly unbiased then for every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$,
s.t. for some $\lambda\in[0,Min(\lambda_{1},\lambda_{2})]$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)$, there exists an
$\bar{s}_{3}[0,\lambda_{3}]\in\mathcal{O}_{F}$ s.t., with
$\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$,
$\bar{s}_{3}[0,\lambda^{\prime}]=\bar{s}_{1}[0,\lambda]\circ\bar{s}_{2}[\lambda,\lambda^{\prime}]$
###### Proof.
First note that, regardless of whether the environmental shell is unbiased,
for any $p\in Int(Z)$, if $\bar{s}[0,\lambda]\in\Sigma_{p}$ and
$\bar{s}^{\prime}[\lambda,\lambda^{\prime}]\in\Sigma_{p\rightarrow}$ then
$\bar{s}[0,\lambda]\circ\bar{s}^{\prime}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}$.
Taking $s=\bar{s}_{1}(\lambda)$, for some
$e_{1},e_{2}\in\mathcal{E}_{Int(Z)}$,
$\bar{s}_{1}[0,\lambda]\in\Sigma_{(s,e_{1})}$ and
$\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$. Since the
shell is unbiased, there is a
$\bar{s}_{3}[\lambda,\lambda_{3}]\in\Sigma_{(s,e_{1})\rightarrow}$ s.t.
$\bar{s}_{3}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$
($\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$). By the considerations
of the prior paragraph,
$\bar{s}_{1}[0,\lambda]\circ\bar{s}_{3}[\lambda,\lambda_{3}]\in\mathcal{O}_{F}$.
∎
There are experiments for which the strong unbiased condition can fail, but
the experiment still be accepted as valid. Taking an example from quantum
mechanics, consider the case in which a particle’s position is measured
$\lambda_{1}$, and if the particle is in region $A$, the particle’s spin will
be measured along the y-axis at $\lambda_{2}$, and if particle is not in
region $A$, the spin will be measured along the z-axis at $\lambda_{2}$. Now
consider two paths, $\bar{s}_{1}$ and $\bar{s}_{2}$, s.t. $\bar{s}_{1}$ is in
region $A$ at $\lambda_{1}$, $\bar{s}_{2}$ is not in region $A$ at
$\lambda_{1}$, and for some $\lambda\in(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)\equiv s$.
$\bar{s}_{1}[-\infty,\lambda]\circ\bar{s}_{2}[\lambda,\infty]$ is not a
possible path because a particle can’t be in region $A$ at $\lambda_{1}$ and
have its spin polarized along the z-axis at $\lambda_{2}$. Therefore the set
of particle paths is not a dynamic space (though it is still a dynamic set).
Since the strong unbiased condition insures that the system can be described
by a dynamic space, the unbiased condition must have failed.
To see this, assume that the e-automata is in state $(s,e_{1})$ at $\lambda$
in the case where the system takes path $\bar{s}_{1}$, and state $(s,e_{2})$
in the case where the system takes path $\bar{s}_{2}$. $e_{1}$ and $e_{2}$
determine different futures for the particle paths, $e_{1}$ insures that the
spin will be measured along the y-axis at $\lambda_{2}$ while $e_{2}$ insures
that the spin will be measured along the z-axis at $\lambda_{2}$. This
violates the strong unbiased condition. However, it doesn’t necessarily create
a problem for the experiment because $e_{1}$ and $e_{2}$ know enough about the
system history to know $s_{1}[-\infty,\lambda]$ and $s_{2}[-\infty,\lambda]$
must reside in separate outcomes (one belongs to the set of “$A$” outcomes,
the other to the set of “not $A$” outcomes); once paths have been
differentiated into separate outcomes, they may be treated differently by the
environment. This motivates a weak version of the unbiased condition, one that
holds only when $(s,e_{1})$ and $(s,e_{2})$ have not yet been differentiated
into separate outcomes.
For an environmental shell, $Z=(D,I,F)$, $p,p^{\prime}\in Int(Z)$ have not
been differentiated if for some $e\in\mathcal{E}_{F}$, $\lambda\in\Lambda$,
$\Sigma_{p}\bigcap\mathcal{O}_{e}[0,\lambda]\neq\emptyset$ and
$\Sigma_{p^{\prime}}\bigcap\mathcal{O}_{e}[0,\lambda]\neq\emptyset$. If
$(s,e)$ and $(s,e^{\prime})$ have not been differentiated then the unbiased
condition should hold for them. Similarly, if $(s,e_{1})$ and $(s,e_{2})$ have
not been differentiated, and neither have $(s,e_{2})$ and $(s,e_{3})$, then
the unbiased condition should hold for $(s,e_{1})$ and $(s,e_{3})$. This
motivates the following definition
###### Definition 34.
For e-automata $Z$, $e\in\mathcal{E}_{F}$, $\lambda\geq 0$
$|\mathcal{O}_{e}[0,\lambda]|_{0}\equiv\mathcal{O}_{e}[0,\lambda]$
$|\mathcal{O}_{e}[0,\lambda]|_{n+1}\equiv\\{\bar{s}[0,\lambda]:for\>some\>e^{\prime}\in\mathcal{E}_{F},\>\mathcal{O}_{e^{\prime}}[0,\lambda]\bigcap|\mathcal{O}_{e}[0,\lambda]|_{n}\neq\emptyset,\>\&\>\bar{s}[0,\lambda]\in\mathcal{O}_{e^{\prime}}[0,\lambda]\\}$
$|\mathcal{O}_{e}[0,\lambda]|\equiv\bigcup_{n\in\mathbb{N}}|\mathcal{O}_{e}[0,\lambda]|_{n}$
Note that if $\bar{s}[0,\lambda_{0}]\in\mathcal{O}_{e}$ then
$\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$ iff
$\lambda_{0}\geq\lambda$.
###### Definition 35.
For environmental shell, $Z=(D,I,F)$, $p,p^{\prime}\in Int(Z)$ are
_undifferentiated_ if for some $e\in\mathcal{E}_{F}$, $\lambda\in\Lambda$,
$\Sigma_{p}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$ and
$\Sigma_{p^{\prime}}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$
Now define the weak unbiased condition in the same way as the strong
condition, except that it only applies to undifferentiated states.
###### Definition 36.
An environmental shell, $(D,I,F)$ is _weakly unbiased_ if for every
$\lambda\in Dom(\Theta_{F})$, every
$(s,e_{1}),(s.e_{2})\in\Theta_{F}(\lambda)$ s.t. $(s,e_{1})$ and $(s.e_{2})$
are undifferentiated,
$\bar{s}_{1}[\lambda,\lambda_{1}]\in\Sigma_{(s,e_{1})\rightarrow}$ iff there
exists a $\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$
s.t. with $\lambda^{\prime}\equiv Min(\lambda_{1},\lambda_{2})$,
$\bar{s}_{1}[\lambda,\lambda^{\prime}]=\bar{s}_{2}[\lambda,\lambda^{\prime}]$.
###### Theorem 37.
If an environmental shell is weakly unbiased then for every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$,
s.t. for some $\lambda\in[0,Min(\lambda_{1},\lambda_{2})]$,
$\bar{s}_{1}(\lambda)=\bar{s}_{2}(\lambda)=s$ and for some
$e\in\mathcal{E}_{F}$,
$\bar{s}_{1}[0,\lambda],\bar{s}_{2}[0,\lambda]\in|\mathcal{O}_{e}[0,\lambda]|$,
there exists an $\bar{s}_{3}[0,\lambda_{3}]\in\mathcal{O}_{F}$ s.t., with
$\lambda^{\prime}\equiv Min(\lambda_{2},\lambda_{3})$,
$\bar{s}_{3}[0,\lambda^{\prime}]=\bar{s}_{1}[0,\lambda]\circ\bar{s}_{2}[\lambda,\lambda^{\prime}]$
###### Proof.
Choose $e_{1},e_{2}\in\mathcal{E}_{Int(Z)}$ s.t.
$\bar{s}_{1}[0,\lambda]\in\Sigma_{(s,e_{1})}$,
$\bar{s}_{2}[\lambda,\lambda_{2}]\in\Sigma_{(s,e_{2})\rightarrow}$,
$\Sigma_{(s,e_{1})}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$, and
$\Sigma_{(s,e_{2})}\bigcap|\mathcal{O}_{e}[0,\lambda]|\neq\emptyset$. Such
$e_{1}$and $e_{2}$ must exist because
$\bar{s}_{1}[0,\lambda],\bar{s}_{2}[0,\lambda]\in|\mathcal{O}_{e}[0,\lambda]|$.
$(s,e_{1})$ and $(s,e_{2})$ are undifferentiated, and so the proof now
proceeds similarly to that for Thm 33. ∎
#### III.2.4 E-Automata
###### Definition 38.
An environmental shell is an _e-automata_ if it is weakly unbiased and
satisfies the automata condition.
###### Theorem 39.
If $Z$ is an e-automata, $p_{1},p_{2}\in Int(Z)$, and
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}\neq\emptyset$ then
$\Sigma_{p_{1}\rightarrow}=\Sigma_{p_{2}\rightarrow}$
###### Proof.
Since $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}\neq\emptyset$, $p$ and $p^{\prime}$
are undifferentiated, so the result follows from $Z$ being weakly unbiased &
satisfying the automata condition ∎
As mentioned earlier, an important goal will be to determine what sets of
outcomes can be decided by various types of e-automata. Towards that end, the
meaning of an e-automata “deciding a set of outcomes” will now be defined. For
the moment we’ll concentrate measurements on dynamic spaces; the more general
case will be taken up in a later section.
###### Definition 40.
If $X$ is a set, $C$ is a _covering_ of $X$ if it is a set of subsets of $X$
and $\bigcup C=X$. (Note: it is more common to demand that $X\subset\bigcup
C$; the restriction to $\bigcup C=X$ will allow for some mild simplification.)
$C$ is a _partition_ of $X$ if it is a pairwise disjoint covering
…
###### Definition 41.
If $Z=(D,I,F)$ is an e-automata, $\lambda\in\Lambda_{D}$, $A\subset F$, and
$X\subset\mathcal{E}_{F}$:
$\Theta_{A}^{\lambda}\equiv\\{\bar{p}[\lambda,\lambda^{\prime}]\in
D_{I/F}:for\>some\>\bar{p}^{\prime}[\lambda,\lambda^{\prime\prime}]\in
I\upharpoonleft I\rightarrow
A,\>\bar{p}^{\prime}[\lambda,\lambda^{\prime}]=\bar{p}[\lambda,\lambda^{\prime}]\\}$
$\mathcal{O}_{X}^{\lambda}\equiv\Theta_{\mathcal{S}_{Z}\bigotimes
X}^{\lambda}\cdot\mathcal{S}$
Because $D$ is homogeneous and $I$ is homogeneously realized,
$\Theta_{A}^{\lambda}$ and $\mathcal{O}_{X}^{\lambda}$ are simply $\Theta_{A}$
and $\mathcal{O}_{X}$ shifted by $\lambda$, and theorems for $\Theta_{A}$ and
$\mathcal{O}_{X}$ can be readily translated to theorems for
$\Theta_{A}^{\lambda}$ and $\mathcal{O}_{X}^{\lambda}$ .
When an e-automata decides a set of outcomes on a dynamic space, it is fairly
straightforward to define the relationship between the dynamic space and the
e-automata’s outcomes.
###### Definition 42.
A covering, $K$, of dynamic space $D$ is _decided_ by e-automata
$Z=(D_{Z},I,F)$ if $\Lambda_{D}=\Lambda_{D_{Z}}$ and, for some
$\lambda\in\Lambda_{D}$,
$K\equiv\\{\rightarrow\mathcal{O}_{e}^{\lambda}\rightarrow:e\in\mathcal{E}_{F}\\}$.
Note that $Z$ can not decide $K$ if starts after some $\lambda$. This may seem
to violate experimental reproducibility. In a later section it will be seen
that any such a measurement can be consistent with experimental
reproducibility. It’s not difficult to see how. If we were interested, for
example, in the likelihood that a system is in state $y$ at time $1\>sec$
given that is was in state $x$ at time $0$, we do not mean that we are only
interested in those probabilities at a particular moment in the history of the
universe; it is assumed that the system can be assembled at any time, and the
time at which it is assembled is simply assigned the value of $0$.
If $D$ is a dynamic space, and $A\subset D$, an e-automata can determine
whether or not $A$ occurs if it decides a covering on $D$, $K$, s.t. and for
all $\alpha\in K$, either $\alpha\subset A$ or $\alpha\bigcap A\,=\emptyset$.
While this concept of measurement is adequate for many purposes, it’s a little
too broad for quantum physics; in the next section a subclass of e-automata
will be introduced that will eliminate the problematic measurements.
### III.3 Ideal E-Automata
As they stand, e-automata draw no clear distinction between two different
types of uncertainty: extrinsic uncertainty - uncertainty about the state of
the environment, particularly with regard to knowing precisely which
$e\in\mathcal{E}_{F}$ occurred, and intrinsic uncertainty - uncertainty about
the system given complete knowledge of the environment.
It is often necessary to treat with these two type of uncertainty separately.
Such is the case in quantum physics: if $\bar{s}_{1}$ and $\bar{s}_{2}$ are
system paths, there’s a difference between “either $\bar{s}_{1}$ or
$\bar{s}_{2}$ was measured, but we don’t know which” (extrinsic uncertainty)
and “$\\{\bar{s}_{1},\bar{s}_{2}\\}$ was measured” (intrinsic uncertainty),
because in general $P(A\bigcup B)\neq P(A)+P(B)$.
In this section two properties will be introduced that together will remove
the ambiguity between these types of uncertainty. These are the only
refinement to e-automata that will be required.
#### III.3.1 Boolean E-Automata
###### Definition 43.
An e-automata is _boolean_ if for every $e,e^{\prime}\in\mathcal{E}_{F}$
either $\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}$ or
$\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}=\emptyset$.
###### Theorem 44.
If $K$ is decided by a boolean e-automata then it is a partition
###### Proof.
Follows from the boolean & automata conditions. ∎
If an e-automata is not boolean, it’s possible to have
$e,e^{\prime}\in\mathcal{E}_{F}$ s.t.
$\mathcal{O}_{e}\neq\mathcal{O}_{e^{\prime}}$ and
$\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}\neq\emptyset$. If it’s known
that $e$ occurred, and so $e^{\prime}$ did not occur, it’s not clear whether
the elements of $\mathcal{O}_{e}\bigcap\mathcal{O}_{e^{\prime}}$ could have
occurred. Boolean e-automata eliminate this sort of ambiguity; if $e$ occurred
and $\mathcal{O}_{e}\neq\mathcal{O}_{e^{\prime}}$ then none of the paths in
$\mathcal{O}_{e^{\prime}}$ could have been taken. This means that for boolean
e-automata, one can always separate uncertainty about which path(s)
corresponding to $\mathcal{O}_{e}$ occurred from uncertainty about which
$e\in\mathcal{E}_{F}$ occurred.
###### Definition 45.
For a boolean e-automata, $e\in\mathcal{E}_{F}$,
$[e]\equiv\\{e^{\prime}\in\mathcal{E}_{F}:\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}\\}$
#### III.3.2 All-Reet E-Automata
It is possible for the environment to record a piece of information about the
system at one point, and then later forget it, so that it is not ultimately
reflected in the experimental outcome. This leads ambiguity of the type
discussed above; should the outcome be understood to be as $A\bigcup B$ or as
$A\>or\>B$ (or as something distinct from either)?
An all-reet e-automata is all retaining, once it records a bit of information
about the system, it never forgets it. In order to define this type of
e-automate, it will be helpful to first define the subsets of
$\omega_{Dom(Z)}$, $\Sigma_{Int(Z)}$, $\Theta_{F}$, and $\mathcal{O}_{F}$
containing just those path-segments that are consistent with a given
environmental path-segment.
###### Definition 46.
For $\bar{e}:\Lambda\rightarrow\mathcal{E}_{Z}$,
$\omega_{\bar{e}[0,\lambda]}\equiv\\{\bar{p}[0,\lambda]\in\omega_{Dom(Z)}\,:\,\bar{p}[0,\lambda]\cdot\mathcal{E}=\bar{e}[0,\lambda]\\}$
$\Sigma_{\bar{e}[0,\lambda]}\equiv\\{\bar{s}[0,\lambda]:(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\in\omega_{Int(Z)}\\}$
If $\bar{e}(\lambda)\in\mathcal{E}_{F}$:
$\Theta_{\bar{e}[0,\lambda]}\equiv\\{\bar{p}[0,\lambda^{\prime}]\in\Theta_{F}:for\>some\>\bar{p}^{\prime}[0,\lambda]\in\omega_{\bar{e}[0,\lambda]},\>\bar{p}^{\prime}[0,\lambda^{\prime}]=\bar{p}[0,\lambda^{\prime}]\\}$
$\mathcal{O}_{\bar{e}[0,\lambda]}\equiv\Theta_{\bar{e}[0,\lambda]}\cdot\mathcal{S}$
###### Theorem 47.
If $Z$ is an e-automata and $e\in\mathcal{E}_{F}$ then
$\mathcal{O}_{e}=\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$
###### Proof.
Since $F=(\mathcal{S}_{Z}\bigotimes\mathcal{E}_{F})\bigcap\mathcal{P}_{D}$,
for any
$(\bar{s}_{1}[0,\lambda],\bar{e}[0,\lambda]),(\bar{s}_{2}[0,\lambda],\bar{e}[0,\lambda])\in\omega_{F}$,
any $\lambda^{\prime}\leq\lambda$,
$\bar{s}_{1}[0,\lambda^{\prime}]\in\mathcal{O}_{\bar{e}(\lambda)}$ iff
$\bar{s}_{2}[0,\lambda^{\prime}]\in\mathcal{O}_{\bar{e}(\lambda)}$; therefore
$\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}\subset\mathcal{O}_{e}$.
If $\bar{s}[0,\lambda^{\prime}]\in\mathcal{O}_{e}$ then there must exist a
$\bar{p}[0,\lambda]\in\omega_{e}$ s.t.consistent
$\bar{s}[0,\lambda^{\prime}]=\bar{p}[0,\lambda^{\prime}]\cdot\mathcal{S}$, in
which case
$\bar{s}[0,\lambda^{\prime}]=\mathcal{O}_{\bar{p}[0,\lambda^{\prime}]\cdot\mathcal{E}}$.
Therefore
$\mathcal{O}_{e}\subset\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$.
∎
Let’s assume that as an experiment unfolds, the environment sequentially
writes everything it discovers about the system to incorruptible memory. The
state of the environment would include the state of this memory, so for every
$\bar{e}_{1}[0,\lambda_{1}],\bar{e}_{2}[0,\lambda_{2}]\in\omega_{F}\cdot\mathcal{E}$,
if
$\mathcal{O}_{\bar{e}_{1}[0,\lambda_{1}]}\neq\mathcal{O}_{\bar{e}_{2}[0,\lambda_{2}]}$
then $\bar{e}_{1}(\lambda_{1})\neq\bar{e}_{2}(\lambda_{2})$. Since
$\mathcal{O}_{e}=\bigcup_{\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}}\mathcal{O}_{\bar{e}[0,\lambda]}$,
we are lead to the following definition.
###### Definition 48.
For e-automata $Z=(D,I,F)$, $e\in\mathcal{E}_{F}$ is _reet_ if for all
$\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$,
$\mathcal{O}_{\bar{e}[0,\lambda]}=\mathcal{O}_{e}$
$Z$ is _all-reet_ if every $e\in\mathcal{E}_{F}$ is reet.
###### Theorem 49.
If $Z$ is an all-reet e-automata, $e\in\mathcal{E}_{F}$,
$\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$, and
$\bar{e}[0,\lambda](\lambda^{\prime})=e^{\prime}\in\mathcal{E}_{F}$ then
$\mathcal{O}_{e}=\mathcal{O}_{e^{\prime}}$
###### Proof.
Take $\lambda_{1}=glb(\bar{e}^{-1}[0,\lambda][\mathcal{E}_{F}])$; since
$\bar{e}[0,\lambda](\lambda^{\prime})\in\mathcal{E}_{F}$,
$\lambda_{1}\leq\lambda^{\prime}$.
$\Theta_{\bar{e}[0,\lambda]}=\Theta_{\bar{e}[0,\lambda^{\prime}]}=\omega_{\bar{e}[0,\lambda_{1}]}$,
and so
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}=\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\mathcal{O}_{e^{\prime}}$.∎
###### Theorem 50.
If $Z$ is an all-reet e-automata then for every $e\in\mathcal{E}_{F}$, every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{e}$,
$\lambda_{1}=\lambda_{2}$
###### Proof.
Take any $\bar{e}[0,\lambda]\in\omega_{e}\cdot\mathcal{E}$;
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}$. Assume that
$\lambda_{1}<\lambda_{2}$; since $(I/F)\subset F^{\vartriangleright}$ there’s
a $\lambda^{\prime}\in[\lambda_{1},\lambda_{2})$ s.t.
$\bar{e}(\lambda^{\prime})\in\mathcal{E}_{F}$. However, in that case
$\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{e}$ and
$\bar{s}_{2}[0,\lambda_{2}]\notin\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\mathcal{O}_{\bar{e}(\lambda^{\prime})}$,
contrary to Thm 49.∎
###### Definition 51.
If $Z$ is an all-reet e-automata, $e\in\mathcal{E}_{F}$ and
$\bar{s}[0,\lambda]\in\mathcal{O}_{e}$ then
$\Lambda(\mathcal{O}_{e})\equiv\lambda$.
Thm 50 implies a similar statement for all of $Int(Z)$.
###### Theorem 52.
If $Z=(D,I,F)$ is an all-reet e-automata then for every $p\in Int(Z)$, every
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\Sigma_{p}$,
$\lambda_{1}=\lambda_{2}$
###### Proof.
Assume $\lambda_{1}\leq\lambda_{2}$. Take any
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\omega_{p}$ and any
$\bar{p}[\lambda_{1},\lambda_{3}]\in I\upharpoonleft p\rightarrow F$. Because
$D$ is homogeneous there must be a
$\bar{p}^{\prime}[\lambda_{2},\lambda_{3}+\lambda_{2}-\lambda_{1}]\in
I\upharpoonleft p\rightarrow F$ s.t. for all
$\lambda\in[\lambda_{1},\lambda_{3}]$,
$\bar{p}(\lambda)=\bar{p}^{\prime}(\lambda+\lambda_{2}-\lambda_{1})$. Define
$\bar{p}_{3}[0,\lambda_{3}]\equiv\bar{p}_{1}[0,\lambda_{1}]\circ\bar{p}[\lambda_{1},\lambda_{3}]$
and
$\bar{p}_{4}[0,\lambda_{3}+\lambda_{2}-\lambda_{1}]\equiv\bar{p}_{2}[0,\lambda_{2}]\circ\bar{p}^{\prime}[\lambda_{2},\lambda_{3}+\lambda_{2}-\lambda_{1}]$.
If $\bar{p}_{3}[0,\lambda_{x}]\in\Theta_{F}$ then
$\bar{p}_{4}[0,\lambda_{x}+\lambda_{2}-\lambda_{1}]\in\Theta_{F}$, so by Thm
50, $\lambda_{1}=\lambda_{2}$.consistent ∎
The next two theorems will prove to be quite useful.
###### Theorem 53.
If $Z$ is an all-reet e-automata then for any $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$\mathcal{O}_{e}=\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$.
###### Proof.
Clearly
$\mathcal{O}_{e}\subset\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$.
Take any
$\bar{s}[0,\Lambda(\mathcal{O}_{e})],\bar{s}^{\prime}[0,\Lambda(\mathcal{O}_{e})]\in\mathcal{O}_{e}$
s.t. $\bar{s}(\lambda)=\bar{s}^{\prime}(\lambda)$. For
$\bar{e}[0,\lambda^{\prime}]\in\omega_{e}\cdot\mathcal{E}$,
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}=\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}\cdot\mathcal{S}$.
$(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\circ(\bar{s}_{1}[\lambda,\Lambda(\mathcal{O}_{e})],\bar{e}[\lambda,\Lambda(\mathcal{O}_{e})])\in\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}$,
so since
$\mathcal{O}_{e}=\omega_{\bar{e}[0,\Lambda(\mathcal{O}_{e})]}\cdot\mathcal{S}$,
$\bar{s}[0,\lambda]\circ\bar{s}[\lambda,\Lambda(\mathcal{O}_{e})]\in\mathcal{O}_{e}$.∎
###### Theorem 54.
If $Z$ is an all-reet e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{e}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$
###### Proof.
First note that there exists a
$\bar{e}[0,\lambda^{\prime}]\in\omega_{e}\cdot\mathcal{E}$ s.t.
$\bar{e}(\lambda)=e_{\lambda}$.
A:
$\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}\subset\mathcal{O}_{e}[0,\lambda]$
\- Take any $s_{\lambda}\in\mathcal{O}_{e}(\lambda)$. Since $Z$ is all-reet,
$s_{\lambda}\in\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}(\lambda)$, so there
must be a
$\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{\bar{e}[0,\lambda^{\prime}]}$
s.t. $\bar{p}^{\prime}(\lambda)=(s_{\lambda},e_{\lambda})$. Since
$\bar{p}^{\prime}[0,\lambda^{\prime}]\in\omega_{\bar{e}[0,\lambda^{\prime}]}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in
I\upharpoonleft(s_{\lambda},e_{\lambda})\rightarrow e$. Now take any
$\bar{s}[0,\lambda]\in\Sigma_{(s_{\lambda},e_{\lambda})}$; there must exist a
$(\bar{s}[0,\lambda],\bar{e}^{\prime}[0,\lambda])\in\omega_{(s_{\lambda},e_{\lambda})}$.
$(\bar{s}[0,\lambda],\bar{e}^{\prime}[0,\lambda])\circ\bar{p}[\lambda,\Lambda(\mathcal{O}_{e})]\in\Theta_{e}$
so $\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$. -
B:
$\mathcal{O}_{e}[0,\lambda]\subset\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$
\- If $\bar{s}[0,\lambda]\in\mathcal{O}_{e}[0,\lambda]$ then
$\bar{s}[0,\lambda]\in\mathcal{O}_{\bar{e}[0,\lambda^{\prime}]}[0,\lambda]$,
so with $s=\bar{s}(\lambda)\in\mathcal{O}_{e}(\lambda)$,
$\bar{s}[0,\lambda]\in\Sigma_{(s,e_{\lambda})}$. \- ∎
The following are some of the consequences of Thm 54.
###### Theorem 55.
If $Z$ is an all-reet e-automata, $p,p_{1},p_{2}\in Int(Z)$,
$e\in\mathcal{E}_{F}$, and $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$
1) If $(s_{1},e_{1}),(s_{2},e_{2})\in\Theta_{e}(\lambda)$ then
$(s_{1},e_{2})\in\Theta_{e}(\lambda)$
2) If $(s,e_{1}),(s,e_{2})\in\Theta_{e}(\lambda)$ then
$\Sigma_{(s,e_{1})}=\Sigma_{(s,e_{2})}$
3) If $Z$ is boolean and all-reet and
$(s,e_{1}),(s,e_{2})\in\Theta_{[e]}(\lambda)$ then
$\Sigma_{(s,e_{1})}=\Sigma_{(s,e_{2})}$
###### Proof.
1 and 2 are immediate from Thm 54. 3 follows from Thm 54 and the definition of
$[e]$. ∎
#### III.3.3 Ideal E-Automata
###### Definition 56.
An e-automata is _ideal_ if it is boolean and all-reet.
Ideal e-automata lack the ambiguities mentioned at the beginning of the
section. The remainder of this section will seek to establish a central
property ideal e-automata, to be given in Thm 59.
###### Theorem 57.
For ideal e-automata $Z$, for any $p_{1},p_{2}\in Int(Z)$
1) If $p_{1}\cdot\mathcal{S}=p_{2}\cdot\mathcal{S}$, and for some
$e\in\mathcal{E}_{F}$, $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$p_{1},p_{2}\in\Theta_{[e]}(\lambda)$ then $\Sigma_{p_{1}}=\Sigma_{p_{2}}$, in
all other cases $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$.
2) If $\Sigma_{p_{1}}=\Sigma_{p_{2}}$ then for all $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, $p_{1}\in\Theta_{[e]}(\lambda)$ iff
$p_{2}\in\Theta_{[e]}(\lambda)$
###### Proof.
1) If the conditions hold then $\Sigma_{p_{1}}=\Sigma_{p_{2}}$by Thm 55.3.
If $p_{1}\cdot\mathcal{S}\neq p_{2}\cdot\mathcal{S}$ then clearly
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$.
If there’s doesn’t exist a $e\in\mathcal{E}_{F}$ s.t. $p_{1},p_{2}\in
Ran(\Theta_{[e]})$ then, since $Z$ is boolean,
$\Sigma_{p_{1}}\circ\Sigma_{p_{1}\rightarrow}\bigcap\Sigma_{p_{2}}\circ\Sigma_{p_{2}\rightarrow}=\emptyset$.
By Thm 39, $\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$
If there exits an $e\in\mathcal{E}_{F}$ s.t. $p_{1},p_{2}\in
Ran(\Theta_{[e]})$, but no $\lambda\in[0,\Lambda(\mathcal{O}_{e})]$ s.t.
$p_{1},p_{2}\in\Theta_{[e]}(\lambda)$ then
$\Sigma_{p_{1}}\bigcap\Sigma_{p_{2}}=\emptyset$ by Thm 52.
2) By Thm 39
$\Sigma_{p_{1}}\circ\Sigma_{p_{1}\rightarrow}=\Sigma_{p_{2}}\circ\Sigma_{p_{2}\rightarrow}$;
since $Z$ is boolean $p_{1}\in Ran(\Theta_{[e]})$ iff $p_{2}\in
Ran(\Theta_{[e]})$. From Thm 52 it then follows that
$p_{1}\in\Theta_{[e]}(\lambda)$ iff $p_{2}\in\Theta_{[e]}(\lambda)$. ∎
For ideal e-automata, the sets $|\mathcal{O}_{e}[0,\lambda]|$ (see Section
III.2.3) are particularly useful; they tell you what has been measured as of
$\lambda$.
###### Theorem 58.
If $Z$ is an ideal e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{[e]}\cdot\mathcal{E})(\lambda)$,
$|\mathcal{O}_{e}[0,\lambda]|=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|(\lambda)}\Sigma_{(s,e_{\lambda})}$
###### Proof.
A: If $Z$ is an ideal e-automata then for every $e\in\mathcal{E}_{F}$, every
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$, every
$e_{\lambda}\in(\Theta_{[e]}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})}$
\- Immediate from Thm 54 and Thm 55.3 -
It is sufficient to show that for every $n\in\mathbb{N},$
$|\mathcal{O}_{e}[0,\lambda]|_{n}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{n}(\lambda)}\Sigma_{(s,e_{\lambda})}$.
By (A) this holds for $n=0$. Assume it holds for $n=i$ and consider $n=i+1$.
For any $\mathcal{O}_{e^{\prime}}$ s.t.
$\mathcal{O}_{e^{\prime}}[0,\lambda]\bigcap|\mathcal{O}_{e}[0,\lambda]|_{i}\neq\emptyset$
for every $e_{2}\in(\Theta_{[e^{\prime}]}\cdot\mathcal{E})(\lambda)$,
$\mathcal{O}_{e^{\prime}}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e^{\prime}}(\lambda)}\Sigma_{(s,e_{2})}$.
By assumption
$|\mathcal{O}_{e}[0,\lambda]|_{i}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{i}(\lambda)}\Sigma_{(s,e_{\lambda})}$,
so by Thm 57.1 there’s an
$s\in|\mathcal{O}_{e}[0,\lambda]|_{i}(\lambda)\bigcap\mathcal{O}_{e^{\prime}}(\lambda)$
s.t. $\Sigma_{(s,e_{2})}=\Sigma_{(s,e_{\lambda})}$; by Thm 57.2
$(s,e_{\lambda})\in\Theta_{[e^{\prime}]}(\lambda)$, so by (A)
$\mathcal{O}_{e^{\prime}}[0,\lambda]=\bigcup_{s\in\mathcal{O}_{e^{\prime}}(\lambda)}\Sigma_{(s,e_{\lambda})}$
. Since $|\mathcal{O}_{e}[0,\lambda]|_{i+1}$is the union over all such
$\mathcal{O}_{e^{\prime}}[0,\lambda]$ it follows immediately that
$|\mathcal{O}_{e}[0,\lambda]|_{i+1}=\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|_{i+1}(\lambda)}\Sigma_{(s,e_{\lambda})}$.∎
###### Theorem 59.
If $Z$ is an ideal e-automata then for any $e\in\mathcal{E}_{F}$,
$\lambda\in[0,\Lambda(\mathcal{O}_{e})]$,
$\mathcal{O}_{e}=|\mathcal{O}_{e}[0,\lambda]|\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
###### Proof.
With $e_{\lambda}\in(\Theta_{e}\cdot\mathcal{E})(\lambda)$:
$\mathcal{O}_{e}=\mathcal{O}_{e}[0,\lambda]\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 53)
$=(\bigcup_{s\in\mathcal{O}_{e}(\lambda)}\Sigma_{(s,e_{\lambda})})\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 54)
$=(\bigcup_{s\in|\mathcal{O}_{e}[0,\lambda]|(\lambda)}\Sigma_{(s,e_{\lambda})})\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
$=|\mathcal{O}_{e}[0,\lambda]|\circ\mathcal{O}_{e}[\lambda,\Lambda(\mathcal{O}_{e})]$
(Thm 58) ∎
### III.4 Ideal Partitions
This section will be concerned with the necessary and sufficient conditions
for a partition of a dynamic set to be the set of outcomes of an ideal
e-automata. As a warm up, we’ll start by picking up where we left off in
Section III.2.4 and consider the case where the system is a dynamic space; the
more general case, where the system is a dynamic set, will be handled
immediately thereafter. In order to attack these problems, several basic
definitions first have to be given.
###### Definition 60.
If $S$ is a dynamic set, $\alpha\subset S$ is _bounded from above (by_
$\lambda$) if $\alpha=\alpha[-\infty,\lambda]\circ S[\lambda,\infty]$,
$\alpha$ is _bounded from below (by_ $\lambda$) if
$\alpha=S[-\infty,\lambda]\circ\alpha[\lambda,\infty]$, and $\alpha$ is
_bounded_ if it is both bounded from above and bounded from below.
If $A$ is a set of subsets of $S$, $A$ is _bounded from above/below (by_
$\lambda$) every element of $A$ is bounded from above/below by $\lambda$; it’s
_bounded_ if it’s bounded from both above and below.
“Bounded from below by $\lambda$” may be abbreviated “$bb\lambda$” and
“Bounded from above by $\lambda$” may be abbreviated “$ba\lambda$”.
Next to transfer the concept of $|\mathcal{O}_{e}[0,\lambda]|$ to partitions.
###### Definition 61.
If $K$ is a covering of dynamic set $S$, $\alpha\in\gamma$,
$\lambda\in\Lambda_{D}$, and $x\leq\lambda$ is either $-\infty$ or an element
of $\Lambda_{S}$
$|\alpha[x,\lambda]|_{0}\equiv\alpha[x,\lambda]$
$|\alpha[x,\lambda]|_{n+1}\equiv\\{\bar{s}[x,\lambda]:for\>some\>\beta\in
K\>s.t.\>\beta[x,\lambda]\bigcap|\alpha[x,\lambda]|_{n}\neq\emptyset,\>\bar{s}[x,\lambda]\in\beta[x,\lambda]\\}$
$|\alpha[x,\lambda]|\equiv\bigcup_{n\in\mathbb{N}}|\alpha[x,\lambda]|_{n}$
By far the most important case is when $x=-\infty$; this is a because of its
use in defining ideal partitions. Ideal partitions will initially be defined
just for dynamic spaces.
###### Definition 62.
If $\gamma$ a partition of dynamic space $D$, it is an _ideal partition_
(_ip_) if it is bounded from below, all $\alpha\in\gamma$ are bounded from
above, and for all $\alpha\in\gamma$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.
The result to be attained in this section is that a covering is decidable by
an ideal e-automata if and only if the covering is an ideal partition. The
first step towards that result is the following theorem.
###### Theorem 63.
If a covering, $K$, of dynamic space $D$ is decided by an ideal e-automata,
$Z=(D_{Z},I,F)$, then $K$ is an ideal partition
###### Proof.
Since $K$ is decided by $Z$, for some $\lambda_{0}\in\Lambda_{D}$,
$K\equiv\\{\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow:e\in\mathcal{E}_{F}\\}$.
For each $\alpha=\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow\,\in K$
define $\lambda_{\alpha}\equiv\lambda_{0}+\Lambda(\mathcal{O}_{e})$.
A: $K$ is $bb\lambda_{0}$ :
\- Immediate from the definition of
$\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow$ -
B: Every $\alpha\in K$ is bounded from above
\- Immediate from Thm 50 and
$K=\\{\rightarrow\mathcal{O}_{e}^{\lambda_{0}}\rightarrow:e\in\mathcal{E}_{F}\\}$
-
C: $K$ is a partition
\- Immediate from Thm 44 -
D: For all $\lambda>\lambda_{0}$, $\alpha\in K$,
$|\alpha[-\infty,\lambda]|=\,\rightarrow|\alpha[\lambda_{0},\lambda]|$
\- It is sufficient to show that for all $n\in\mathbb{N}$,
$|\alpha[-\infty,\lambda]|_{n}=\rightarrow|\alpha[\lambda_{0},\lambda]|_{n}$.
By (A), this holds for $n=0$. Assume it holds for $n=i$. By (A) and assumption
on $i$, for all $\beta\in K$,
$\beta[-\infty,\lambda]\bigcap|\alpha[-\infty,\lambda]|_{i}\neq\emptyset$ iff
$\beta[\lambda_{0},\lambda]\bigcap|\alpha[\lambda_{0},\lambda]|_{i}\neq\emptyset$.
Since $|\alpha[x,\lambda]|_{i+1}$ is equal to the union over the $\beta$’s
that intersect $|\alpha[x,\lambda]|_{i}$, $|\alpha[-\infty,\lambda]|_{i}$ &
$|\alpha[\lambda_{0},\lambda]|_{i}$ are intersected by the same set of
$\beta\in K$, and because all such $\beta$ are $bb\lambda$,
$|\alpha[-\infty,\lambda]|_{i+1}=\rightarrow|\alpha[\lambda_{0},\lambda]|_{i+1}$
-
It remains to show that for all $\alpha\in K$, $\lambda\in\Lambda_{D}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$. Clearly
$\alpha\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$, so it
only needs to be shown that
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]\subset\alpha$.
For $\lambda\leq\lambda_{0}$: Immediate from (A).
For $\lambda\in(\lambda_{0},\lambda_{\alpha})$: By Thm 59,
$\alpha[\lambda_{0},\lambda_{\alpha}]=|\alpha[\lambda_{0},\lambda]|\circ\alpha[\lambda,\lambda_{\alpha}]$.
By (B) and (D)
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=(\rightarrow|\alpha[\lambda_{0},\lambda]|)\circ(\alpha[\lambda,\lambda_{\alpha}]\rightarrow)=\,\rightarrow(|\alpha[\lambda_{0},\lambda]|\circ\alpha[\lambda,\lambda_{\alpha}])\rightarrow\,=\,\rightarrow\alpha[\lambda_{0},\lambda_{\alpha}]\rightarrow\,=\alpha$.
For $\lambda\geq\lambda_{\alpha}$: By (B) and (C),
$|\alpha[-\infty,\lambda]|=\alpha[-\infty,\lambda]$, so
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset\alpha[-\infty,\lambda]\rightarrow\,=\alpha$.
∎
Because ideal e-automata need only be weakly unbiased, it is insufficient to
only consider measurements on dynamic spaces, so let’s now to consider the
more general case of a system that’s a dynamic set.
The first thing to do is to extend the “outer $\rightarrow$” notation to
dynamic sets.
###### Definition 64.
If $S$ is a dynamic set and $A$ is a set of partial paths then $\rightarrow
A\rightarrow$, $\rightarrow A$, and $A\rightarrow$ are defined as before:
$\bar{p}\in\rightarrow A\rightarrow$ if there exists a
$\bar{p}^{\prime}[x_{1},x_{2}]\in A$ s.t. $\bar{p}\in
S[-\infty,x_{1}]\circ\bar{p}^{\prime}[x_{1},x_{2}]\circ S[x_{2},\infty]$, etc.
$\rightharpoondown A\rightharpoondown$ is $\rightarrow A\rightarrow$
relativized to $S$: $\rightharpoondown A\rightharpoondown\equiv(\rightarrow
A\rightarrow)\bigcap S$;
similarly for $\rightharpoondown A$ and $A\rightharpoondown$.
As a convenient shorthand, $+A\equiv\rightharpoondown A\rightharpoondown$ .
If $S$ is a dynamic space then $\rightharpoondown A=\,\rightarrow A$ and
$A\rightharpoondown\,=A\rightarrow$ .
It will also be useful to also extend the notion of boundedness to cover
various situations that can arise with dynamic sets. (Note that the definition
for bounded from above/below was given for dynamic sets, and so still holds.)
###### Definition 65.
If $S$ is a dynamic set and $\alpha\subset S$:
If $\alpha=\alpha[-\infty,\lambda]\rightharpoondown$ then $\alpha$ is _weakly
bounded from above (by $\lambda$)_; if
$\alpha=\,\rightharpoondown\alpha[\lambda,\infty]$ then $\alpha$ is _weakly
bounded from below (by $\lambda$)_.
If for all $\lambda^{\prime}\leq\lambda$,
$\alpha=\,\rightarrow\alpha[-\infty,\lambda^{\prime}],$ $\alpha$ is _strongly
bounded from below (by $\lambda$);_ similarly for _strongly bounded from above
(by $\lambda$)._
If $A$ is a set of subsets of $S$, $A$ is _strongly/weakly bounded from
above/below (by_ $\lambda$) every element of $A$ is strongly/weakly bounded
from above/below (by $\lambda$).
“Weakly bounded from above by $\lambda$” may be abbreviated “$wba\lambda$”,
“strongly bounded from above by $\lambda$” may be abbreviated “$sba\lambda$”,
etc.
If $S$ is a dynamic space there is no difference between being bounded from
above/below, weakly bounded from above/below, and strongly bounded from
above/below.
###### Theorem 66.
If $\alpha$ is $wba\lambda$ and $\lambda^{\prime}>\lambda$ then $\alpha$ is
$wba\lambda^{\prime}$
###### Proof.
Assume
$\bar{p}_{1}[-\infty,\lambda^{\prime}]\in\alpha[-\infty,\lambda^{\prime}]$,
$\bar{p}_{2}[\lambda^{\prime},\infty]\in S[\lambda^{\prime},\infty]$, and
$\bar{p}=\bar{p}_{1}[-\infty,\lambda^{\prime}]\circ\bar{p}_{2}[\lambda^{\prime},\infty]\in
S$. Since $\alpha$ is $wba\lambda$ and
$\bar{p}[-\infty,\lambda]=\bar{p}_{1}[-\infty,\lambda]\in\alpha[-\infty,\lambda]$,
$\bar{p}\in\alpha$. ∎
The most inclusive notion of decidability on a dynamic set would be:
e-automata $(D_{Z},I,F)$ decides covering $K$ on dynamic set $S$ if for some
$\lambda\in\Lambda_{S}$
1) For all $\bar{p}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}^{\lambda}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in S[\lambda,\lambda^{\prime}]$
2) $K=\\{+\mathcal{O}_{e}^{\lambda}:e\in\mathcal{E}_{F}\\}$.
However, this would mean that the interactions between the system and the
environment are only regulated while the experiment is taking place; before
and after the experiment, any kind of interactions would be allowed, even
those that would undermine the autonomy of the system. It is more sensible to
assume that these interactions are always unbiased. If the system and
environment don’t interact outside of the experiment, which would be a natural
assumption, then it’s certainly the case that the interactions are always
unbiased. Because nothing is measured prior to $\lambda$, prior to $\lambda$
there is no distinction between being weakly & strongly unbiased. From
$\lambda$ onward, Thm 37 ought to hold. This leads to the following with
regard to the make up of a system:
###### Definition 67.
If $S$ is a dynamic set, $K$ is a covering of $S$, and
$\lambda\in\Lambda_{S}$, _$S$ is unbiased with respect to $(\lambda,K)$_ if
1) $S$ is $sbb\lambda$
2) For all $a\in K$, all $\lambda^{\prime}>\lambda$, all
$\bar{p}_{1},\bar{p}_{2}\in+|a[-\infty,\lambda^{\prime}]|$, if
$\bar{p}_{1}(\lambda^{\prime})=\bar{p}_{2}(\lambda^{\prime})$ then
$\bar{p}_{1}[-\infty,\lambda^{\prime}]\circ\bar{p}_{2}[\lambda^{\prime},\infty]\in
S$.
It may be assumed that $K$ is $sbb\lambda$, but it is not demanded.
We’re now in a position to define decidability in general.
###### Definition 68.
A covering, $K$, of dynamic set $S$ is _decided_ by e-automata $Z=(D_{Z},I,F)$
if $\Lambda_{S}=\Lambda_{D_{Z}}$ and for some $\lambda\in\Lambda_{D}$:
1) For all $\bar{p}[\lambda,\lambda^{\prime}]\in\mathcal{O}_{F}^{\lambda}$,
$\bar{p}[\lambda,\lambda^{\prime}]\in S[\lambda,\lambda^{\prime}]$.
2) $S$ is unbiased with respect to $(\lambda,K)$.
3) $K\equiv\\{+\mathcal{O}_{e}^{\lambda}:e\in\mathcal{E}_{F}\\}$.
Ideal partitions now also need to be generalized for dynamic sets.
###### Definition 69.
If $\gamma$ a partition of dynamic set $S$, it is an _ideal partition_ (_ip_)
if it is strongly bounded from below, all $\alpha\in\gamma$ are weakly bounded
from above, and for all $\alpha\in\gamma$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.
In the case where $S$ is a dynamic set, this reduces to the prior definition.
Now to generalize Thm 63.
###### Theorem 70.
If a covering, $K$, of dynamic set $S$ is decided by an ideal e-automata,
$Z=(D_{Z},I,F)$, then $K$ is an ideal partition
###### Proof.
Since $K$ is decided by $Z$, for some$\lambda_{0}\in\Lambda_{S}$,
$K\equiv\\{+\mathcal{O}_{e}^{\lambda_{0}}:e\in\mathcal{E}_{F}\\}$. For each
$\alpha=+\mathcal{O}_{e}^{\lambda_{0}}\in K$ define
$\lambda_{\alpha}\equiv\lambda_{0}+\Lambda(\mathcal{O}_{e})$.
A: $K$ is $sbb\lambda_{0}$
\- Take any $\alpha\in K$, $\lambda\leq\lambda_{0}$; it is necessary to show
that $\alpha=\,\rightarrow\alpha[\lambda,\infty]$. For any
$\bar{p}_{1}[-\infty,\lambda]\in S[-\infty,\lambda]$,
$\bar{p}_{2}[\lambda,\infty]\in\alpha[\lambda,\infty]$ s.t.
$\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)$,
$\bar{p}=\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in S$
because $S$ is unbiased with respect to $(\lambda,K)$. Since
$\lambda\leq\lambda_{0}$,
$\bar{p}_{2}[\lambda_{0},\infty]\in\alpha[\lambda_{0},\infty]$, so since
$\alpha=+\alpha[\lambda_{0},\lambda_{\alpha}]$, $\bar{p}\in\alpha$.
B: Every $\alpha\in K$ is weakly bounded from above
-Immediate from Thm 50 and $K=\\{\rightharpoondown\mathcal{O}_{e}^{\lambda_{0}}\rightharpoondown:e\in\mathcal{E}_{F}\\}$ -
C: $K$ is a partition
\- Follows from the boolean & automata conditions -
D: For all $\lambda\in\Lambda_{S}$, $\alpha\in K$,
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset S$
\- Follows from $S$ being unbiased with respect to $(\lambda,K)$. -
E: For all $\lambda>\lambda_{0}$, $\alpha\in K$,
$|\alpha[-\infty,\lambda]|=\rightarrow|\alpha[\lambda_{0},\lambda]|$
\- Identical to (D) in Thm 63 -
It remains to show that for all $\alpha\in K$, $\lambda\in\Lambda_{S}$,
$\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$. As before,
$\alpha\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$, so it
only needs to be shown that
$|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]\subset\alpha$.
For $\lambda\leq\lambda_{0}$: Immediate from (A).
For $\lambda\in(\lambda_{0},\lambda_{\alpha})$: Assume
$\bar{p}_{1}[-\infty,\lambda]\in|\alpha[-\infty,\lambda]|$ and
$\bar{p}_{2}[\lambda,\infty]\in\alpha[\lambda,\infty]$. It follows from (E)
that $\bar{p}_{1}[\lambda_{0},\lambda]\in|\alpha[\lambda_{0},\lambda]|$, and
so from Thm 59 that
$\bar{p}_{1}[\lambda_{0},\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha[\lambda_{0},\lambda_{\alpha}]$.
It then follows from (A) that
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha[-\infty,\lambda_{\alpha}]$.
Since $S$ is unbiased with respect to $(\lambda,K)$,
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in S$.
Therefore, since $\alpha=\alpha[-\infty,\lambda_{\alpha}]\rightharpoondown$,
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\lambda_{\alpha}]\in\alpha$.
For $\lambda\geq\lambda_{\alpha}$: By (B) and (C),
$|\alpha[-\infty,\lambda]|=\alpha[-\infty,\lambda]$, and by (D)
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset\alpha[-\infty,\lambda]\rightharpoondown$.
By (B) $\alpha=\mathbf{\alpha[-\infty,\lambda]\rightharpoondown}$. ∎
And now establish the inverse, that all ip’s are ideally decidable.
###### Theorem 71.
All ideal partitions are ideally decidable
###### Proof.
Assume $\gamma$ is an ideal partition of dynamic set $S$. An ideal e-automata,
$Z=(D,I,F)$, that decides $\gamma$ will be constructed.
1) Start by constructing a new dynamic space $D_{0}=S\otimes E_{0}$ as
follows: For every $\lambda\in\Lambda_{D}$ create a set $E(\lambda),$ and a
bijection $b_{\lambda}:|\gamma[-\infty,\lambda]|\rightarrow E(\lambda)$ s.t.
for all $\lambda_{1}\neq\lambda_{2}$, $E(\lambda_{1})\bigcap
E(\lambda_{2})=\emptyset$; $(\bar{s},\bar{e})\in D_{0}$ iff $\bar{s}\in S$ and
for all $\lambda\in\Lambda$, if $\bar{s}\in\alpha\in\gamma$ then
$\bar{e}(\lambda)=b_{\lambda}(|\alpha[-\infty,\lambda]|)$
2) From $D_{0}$ construct $Z$’s dynamic space, $D$:
If $\Lambda_{S}$ is unbounded from below, for every $\lambda\in\Lambda_{D}$
define $D_{\lambda}\equiv\\{\bar{p}:for\>some\>\bar{p}^{\prime}\in
D_{0},\>for\>all\>\lambda^{\prime}\in\Lambda_{S},\>\bar{p}^{\prime}(\lambda^{\prime}+\lambda)=\bar{p}(\lambda^{\prime})\\}$.
$D_{Z}\equiv\bigcup_{\lambda\in\Lambda}D_{\lambda}$.
If $\Lambda_{D}$ is bounded from below, take $D_{0}^{*}$ to be any dynamic
space s.t. $\Lambda_{D_{0}^{*}}$, is unbounded from below,
$D_{0}^{*}[0,\infty]=D_{0}$, and for all $\lambda<\lambda^{\prime}<0$,
$D_{0}^{*}(\lambda)\bigcap D_{0}^{*}(\lambda^{\prime})=\emptyset$ (this
already holds for all $\lambda>\lambda^{\prime}\geq 0$). Construct $D^{*}$ as
above and take $D=D^{*}[0,\infty]$.
3) Construct $I$: Select any $\lambda_{0}\in\Lambda_{D}$ s.t. $\gamma$ is
$bb\lambda_{0}$, $I\equiv D_{0}(\lambda_{0})$
4) Construct $F$: For every $\alpha\in\gamma$ select a
$\lambda_{\alpha}\in\Lambda_{D}$ s.t.
$\alpha=+\alpha[-\infty,\lambda_{\alpha}]$;
$F\equiv\bigcup_{\alpha\in\gamma}\alpha(\lambda_{\alpha})\otimes\\{b_{\lambda_{a}}(|\alpha[-\infty,\lambda_{\alpha}]|)\\}$.
Now to show that $(D,I,Z)$ is an ideal e-automata that decides $\gamma$.
A: $D_{0}$ is a dynamic space
\- Take any $(\bar{s}_{1},\bar{e}_{1}),(\bar{s}_{2},\bar{e}_{2})\in D_{0}$
s.t. for some $\lambda\in\Lambda_{S}$,
$(\bar{s}_{1}(\lambda),\bar{e}_{1}(\lambda))=(\bar{s}_{2}(\lambda),\bar{e}_{2}(\lambda))$.
Take $\bar{s}_{1}\in\alpha_{1}\in\gamma$ and
$\bar{s}_{2}\in\alpha_{2}\in\gamma$; note that since
$\bar{e}_{1}(\lambda)=\bar{e}_{2}(\lambda)$,
$|\alpha_{1}[-\infty,\lambda]|=|\alpha_{2}[-\infty,\lambda]|$.
Take
$(\bar{s},\bar{e})\equiv(\bar{s}_{1}[-\infty,\lambda]\circ\bar{s}_{2}[\lambda,\infty],\bar{e}_{1}[-\infty,\lambda]\circ,\bar{e}_{2}[\lambda,\infty])$.
$\bar{s}\in|\alpha_{1}[-\infty,\lambda]|\circ\alpha_{2}[\lambda,\infty]=|\alpha_{2}[-\infty,\lambda]|\circ\alpha_{2}[\lambda,\infty]=\alpha_{2}$,
so $(\bar{s},\bar{e})\in D_{0}$ iff for all $\lambda^{\prime}\in\Lambda_{D}$,
$\bar{e}(\lambda^{\prime})=b_{\lambda^{\prime}}(|\alpha_{2}[-\infty,\lambda^{\prime}]|)$,
which follows from the fact that $\bar{s}\in\alpha_{2}$. -
B: $D$ is a dynamic space
\- For any $p\in\mathcal{P}_{D_{0}}=\mathcal{P}_{D}$, $p$ is only realized at
a single $\lambda\in\Lambda_{S}$ in $D_{0}$. Because $D$ is a union over
“shifted” copies of $D_{0}$, paths in $D$ can only intersect if the belong to
the same copy; it follows that $D$ is a dynamic space if $D_{0}$ is. -
C: $D$ is homogeneous; $I$ is homogeneously realized
\- Because for any $p\in\mathcal{P}_{D_{0}}$, $p$ is only realized at a single
$\lambda\in\Lambda_{S}$ in $D_{0}$, $D_{0}$ must be homogeneous. From the
nature of the construction of $D$ from $D_{0}$ it’s clear that $D$ is also
homogeneous, and that all of $\mathcal{P}_{D_{0}}$ is homogeneously realized
in $D$. -
D: $I\rightarrow\>=I\upharpoonleft(I\rightarrow F)\rightarrow$
$\rightarrow F=\>\rightarrow I\upharpoonleft(I\rightarrow F)$
\- $D_{0}=\>\rightarrow I\upharpoonleft I\rightarrow F\rightarrow$ and so for
$D_{0}$, $I\rightarrow\>=I\upharpoonleft(I\rightarrow F)\rightarrow$ and
$\rightarrow F=\>\rightarrow I\upharpoonleft(I\rightarrow F)$. If this holds
for $D_{0}$, it must also hold for $D$ -
E: $Z$ is an environmental shell
\- All elements of $Dom(Z)$ can be decomposed into system & environmental
states, and $F$ has its own set of environmental states -
F: $Z$ is weakly unbiased
\- Recall that $I=D_{0}(\lambda_{0})$. Choose any $e_{1}\in\mathcal{E}_{F}$,
and take $\alpha\in\gamma$, $\lambda_{1}+\lambda_{0}\in\Lambda_{S}$ s.t.
$b_{\lambda_{0}+\lambda_{1}}(|\alpha[-\infty,\lambda_{0}+\lambda_{1}]|)=e_{1}$
(since $e_{1}\in\mathcal{E}_{F}$ there can only be one such $\alpha$). Take
any $e\in\mathcal{E}_{Int(Z)}$, $\lambda\in\Lambda_{D}$ s.t.
$\Sigma_{e}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$. It follows
that $e=b_{\lambda_{0}+\lambda}(|\alpha[-\infty,\lambda_{0}+\lambda]|)$.
Therefore, if $e,e^{\prime}\in\mathcal{E}_{Int(Z)}$ and for some
$\lambda\in\Lambda$,
$\Sigma_{e}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$ and
$\Sigma_{e^{\prime}}\bigcap|\mathcal{O}_{e_{1}}[0,\lambda]|\neq\emptyset$ then
$e=e^{\prime}$. $Z$ must then be weakly unbiased -
G: $Z$ is an e-automata
\- Take any
$\bar{s}_{1}[0,\lambda_{1}],\bar{s}_{2}[0,\lambda_{2}]\in\mathcal{O}_{F}$ s.t.
$\lambda_{2}\geq\lambda_{1}$ and
$\bar{s}_{1}[0,\lambda_{1}]=\bar{s}_{2}[0,\lambda_{1}]$. For
$\bar{p}_{1}[0,\lambda_{1}],\bar{p}_{2}[0,\lambda_{2}]\in\Theta_{F}$ s.t.
$\bar{p}_{1}[0,\lambda_{1}]\cdot\mathcal{S}=\bar{s}_{1}[0,\lambda_{1}]$ and
$\bar{p}_{2}[0,\lambda_{2}]\cdot\mathcal{S}=\bar{s}_{2}[0,\lambda_{2}]$,
$\bar{p}_{1}(\lambda_{1})=\bar{p}_{2}(\lambda_{1})\in F$, so
$\lambda_{2}=\lambda_{1}$. -
H: $Z$ is all-reet
\- For any $e\in\mathcal{E}_{F}$ there’s only a single
$\bar{e}[0,\lambda]\in\omega_{F}\cdot\mathcal{E}$, so naturally
$\mathcal{O}_{e}=\mathcal{O}_{\bar{e}[0,\lambda]}$ -
I: $Z$ is boolean
\- For any $\bar{s}[0,\lambda]\in\mathcal{O}_{F}$ there’s only one
$(\bar{s}[0,\lambda],\bar{e}[0,\lambda])\in\Theta_{F}$. Since
$\bar{e}(\lambda)\in\mathcal{E}_{F}$ there’s only one $e\in\mathcal{E}_{F}$
s.t. $\bar{s}[0,\lambda]\in\mathcal{O}_{e}$ -
J: $Z$ decides $\gamma$
\- Given any $\alpha\in\gamma$, take
$e=b_{\lambda_{\alpha}}(|\alpha[-\infty,\lambda_{\alpha}]|)$;
$e\in\mathcal{E}_{F}$ and $\alpha=+\mathcal{O}_{e}^{\lambda_{0}}$ \- ∎
Note that $S$ need not be homogeneous, nor does any part of $\mathcal{P}_{S}$
need to be homogeneously realized.333Had any part of $\mathcal{P}_{S}$ needed
to be homogeneously realized it could have presented a problem with viewing
$S$ as playing out on the stage of space-time, because every point in space-
time can only be realized at a single $\lambda$.
###### Theorem 72.
If $\gamma$ is an ip and $\bigcup\gamma$ is a dynamic space, then $\gamma$ is
decided by a strongly unbiased ideal e-automate.
###### Proof.
Use the same construction as the prior theorem. For every $(s,e)\in D_{0}$,
$e=b_{\lambda}(|\alpha[-\infty,\lambda]|)$,
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}=(+|\alpha[-\infty,\lambda]|)_{(\lambda,s)\rightarrow}$.
If $\bigcup\gamma$ is a dynamic space, this means that
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}=(\bigcup\gamma)_{(\lambda,s)\rightarrow}$.
Since $\Sigma_{(s,e)\rightarrow}$ is simply
$(D_{0})_{(s,e)\rightarrow}\cdot\mathcal{S}$ shifted by $\lambda_{0}$ and
truncated at the $\lambda_{\alpha}$’s (for definitions of $\lambda_{0}$ and
$\lambda_{\alpha}$, see (3) and (4) in the proof of the prior theorem), it
follows immediately that the e-automata is strongly unbiased. ∎
### III.5 Companionable Sets & Compatible Sets
In the final section of this part, the make-up of ip’s will be investigated.
###### Definition 73.
If $S$ is a dynamic set:
$A\subset S$ is a _subspace_ (of $S$) if it is non-empty and for every
$\lambda\in\Lambda_{S}$, $A=A[-\infty,\lambda]\circ A[\lambda,\infty]$.
$A\subset S$ is _companionable_ if it is a subspace that’s weakly bounded from
above and strongly bounded from below.
###### Theorem 74.
If $\gamma$ is an ip and $\alpha\in\gamma$ then $\alpha$ is companionable
###### Proof.
For any $\lambda\in\Lambda$,
$\alpha\subset\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$.
Since $\alpha=|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]$,
$\alpha=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]$.
By the definition of an ip, $\alpha$ must be strongly bounded from below and
weakly bounded from above.∎
###### Definition 75.
If $S$ is a dynamic set, $\Gamma$ is a non-empty set of subsets of $S$, and
$\alpha\in\Gamma$:
$|\alpha[-\infty,\lambda]|_{\Gamma}$ is defined identically to how it’s
defined for covering in definition 61
$(\alpha)_{\lambda}^{\Gamma}\equiv\\{\beta\in\Gamma:\beta[-\infty,\lambda]\subset|\alpha[-\infty,\lambda]|_{\Gamma}\\}$.
$\Gamma$ is _compatible_ if it is a pairwise disjoint set of companionable
sets, strongly bounded from below, and for all $\alpha,\beta\in\Gamma$, all
$\lambda\in\Lambda_{S}$, if $\beta\in(\alpha)_{\lambda}^{\Gamma}$ and
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$ then
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.
If the set $\Gamma$ is understood, $|\alpha[-\infty,\lambda]|_{\Gamma}$ may be
written $|\alpha[-\infty,\lambda]|$ and $(\alpha)_{\lambda}^{\Gamma}$ may be
written $(\alpha)_{\lambda}$.
###### Theorem 76.
If $S$ is a dynamic set, $\Gamma$ is a non-empty set of subsets of $S$, and
$\alpha,\beta\in\Gamma$, then
1) $\beta\in(\alpha)_{\lambda}^{\Gamma}$ iff there’s a finite sequence of
elements of $\Gamma$, $(a_{i})_{i\leq n}$, s.t. $a_{1}=\alpha$, $a_{n}=\beta$,
and for all $1\leq i<n$, $a_{i}[-\infty,\lambda]\bigcap
a_{i+1}[-\infty,\lambda]\neq\emptyset$.
2) $\beta\in(\alpha)_{\lambda}^{\Gamma}$ iff
$|\beta[-\infty,\lambda]|_{\Gamma}\bigcap|\alpha[-\infty,\lambda]|_{\Gamma}\neq\emptyset$
iff $|\beta[-\infty,\lambda]|_{\Gamma}=|\alpha[-\infty,\lambda]|_{\Gamma}$.
###### Proof.
Immediate from the definition of $|\alpha[-\infty,\lambda]|_{\Gamma}$.∎
###### Theorem 77.
$\Gamma$ is compatible iff it’s pairwise disjoint, strongly bounded from
below, all $\alpha\in\Gamma$ are weakly bounded from above, and for all
$\lambda\in\Lambda$,
$\alpha=|\alpha[-\infty,\lambda]|_{\Gamma}\circ\alpha[\lambda,\infty]$.
###### Proof.
$\Rightarrow$
$|\alpha[-\infty,\lambda]|_{\Gamma}=\bigcup_{\beta\in(\alpha)_{\lambda}^{\Gamma}}\beta[-\infty,\lambda]$.
Since $\Gamma$ is compatible, for all $\beta\in(\alpha)_{\lambda}^{\Gamma}$,
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$,
$\beta_{\rightarrow(\lambda,p)}=\alpha_{\rightarrow(\lambda,p)}$. Therefore
$\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]=|\alpha[-\infty,\lambda]|_{\Gamma}\circ\alpha[\lambda,\infty]$.
Since $\alpha$ is companionable,
$\alpha=\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]$.
$\Leftarrow$ For $\alpha,\beta\in\Gamma$, if
$\beta\in(\alpha)_{\lambda}^{\Gamma}$ then
$|\alpha[0,\lambda]|=|\beta[0,\lambda]|$, so
$\beta=|\alpha[-\infty,\lambda]|\circ\beta[\lambda,\infty]$, and so for any
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$,
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.
$\alpha\subset\alpha[-\infty,\lambda]\circ\alpha[\lambda,\infty]\subset|\alpha[-\infty,\lambda]|\circ\alpha[\lambda,\infty]=\alpha$,
so $\alpha$ is companionable.∎
###### Theorem 78.
A covering of a dynamic set is an ip iff it’s compatible
###### Proof.
Immediate from Thm 77. ∎
This establishes that if $\alpha$ is the element of some ip then it’s
companionable, and if $t$ is the subset of some ip then it’s compatible. For
dynamic spaces, the converse also holds. For any compatible set, $t$, an “all-
reet not $t$” can be constructed by taking the paths not in $t$ and grouping
them by when they broke off from $t$, and which $|\alpha[-\infty,\lambda]|$
they broke off from. $t$ together with the “all-reet not $t$” form an ip. When
the parameter is not discrete one mild complication is that, in dealing with
paths that broke off from $t$ at $\lambda$, we’ll need to distinguish between
paths that were in $t$ at $\lambda$, but aren’t in $t$ at any
$\lambda^{\prime}>\lambda$ from those that are not in $t$ at $\lambda$, but
were in $t$ at all $\lambda^{\prime}<\lambda$.
###### Definition 79.
If $t$ is compatible, $S$ is a dynamic set s.t. $\bigcup t\subset S$, and
$\alpha\in t$ then
$\sim t_{\lambda}\equiv\\{\bar{p}\in S:\bar{p}[-\infty,\lambda]\notin\bigcup
t[-\infty,\lambda]\\}$
$\sim|\alpha|_{\lambda}^{+}\equiv\\{\bar{p}\in
S:\bar{p}[-\infty,\lambda]\in|\alpha[-\infty,\lambda]|\>and\>for\>all\>\lambda^{\prime}>\lambda,\>\bar{p}[-\infty,\lambda^{\prime}]\notin\bigcup
t[-\infty,\lambda]|\\}$
$\sim|\alpha|_{\lambda}^{-}\equiv\\{\bar{p}\in
S:\bar{p}[-\infty,\lambda]\notin\bigcup
t[-\infty,\lambda]|\>and\>for\>all\>\lambda^{\prime}<\lambda,\>\bar{p}[-\infty,\lambda^{\prime}]\in|\alpha[-\infty,\lambda^{\prime}]|\\}$
These sets can be used to construct an ip containing $t$.
###### Theorem 80.
If $D$ is a dynamic space, $\alpha$ is a subset of $D$, and $t$ is a set of
subsets of $D$ then
1) If $\alpha$ is companionable then there exists an ip, $\gamma$, s.t.
$\alpha\in\gamma$
2) If $t$ is compatible then there exists an ip, $\gamma$, s.t.
$t\subset\gamma$
###### Proof.
(1) Follows immediately from (2). For (2), assume $t$ is $bb\lambda$ and take
$\gamma$ to be the set s.t. $t\subset\gamma$, if $\sim
t_{\lambda}\neq\emptyset$ then $\sim t_{\lambda}\in\gamma$, and for all
$\alpha\in t$, $\lambda^{\prime}>\lambda$, if
$\sim|\alpha|_{\lambda^{\prime}}^{+/-}\neq\emptyset$ then
$\sim|\alpha|_{\lambda^{\prime}}^{+/-}\in\gamma$. $\gamma$ is then partition
of $D$. (From here on, the qualifiers “if $\sim t_{\lambda}\neq\emptyset$” and
“if $\sim|\alpha|_{\lambda^{\prime}}^{+/-}\neq\emptyset$” will be understood.)
The following is key to establishing that $\gamma$ is an ip:
A: For all $\lambda_{1}<\lambda_{2}$,
$p\in\,\sim|\alpha|_{\lambda_{2}}^{-}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$
\- By the definition of $\sim|\alpha|_{\lambda}^{-}$ it’s clearly the case
that
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}\subset|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$.
Take any $\bar{p}_{1}\in+|\alpha[-\infty,\lambda_{2}]|$ and any
$\bar{p}_{2}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$ s.t.
$\bar{p}_{1}(\lambda_{1})=\bar{p}_{2}(\lambda_{1})=p$. Take
$\bar{p}\equiv\bar{p}_{1}[-\infty,\lambda_{1}]\circ\bar{p}_{2}[\lambda_{1},\infty]\in
D$ and note that for all $\lambda^{\prime}<\lambda_{2}$
$\bar{p}[-\infty,\lambda^{\prime}]\in|\alpha[-\infty,\lambda^{\prime}]|$.
Assume that for some $\beta\in t$,
$\bar{p}[-\infty,\lambda_{2}]\in\beta[-\infty,\lambda_{2}]$. That would mean
$|\beta[-\infty,\lambda_{1}]|=|\alpha[-\infty,\lambda_{1}]|$ and so
$\bar{p}_{2}[-\infty,\lambda_{1}]\circ\bar{p}[\lambda_{1},\lambda_{2}]=\bar{p}_{2}[-\infty,\lambda_{2}]$$\in\beta[-\infty,\lambda_{2}]$.
This contradicts $\bar{p}_{2}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$, so there
can be no such $\beta\in t$. Therefore
$\bar{p}\in\,\sim|\alpha|_{\lambda_{2}}^{-}$, and so for all
$\lambda_{1}<\lambda_{2}$,
$(\sim|\alpha|_{\lambda_{2}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$.
-
Similarly, for all $\lambda_{1}\leq\lambda_{2}$,
$p\in\,\sim|\alpha|_{\lambda_{2}}^{+}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda_{2}}^{+})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda_{2}]|_{\rightarrow(\lambda_{1},p)}$
For any $\lambda^{\prime}>\lambda$ the following properties therefore hold for
$\sim|\alpha|_{\lambda^{\prime}}^{-}$ and
$\sim|\alpha|_{\lambda^{\prime}}^{+}$:
$\sim|\alpha|_{\lambda^{\prime}}^{-}$ is $ba\lambda^{\prime}$ and for all all
$\lambda_{1}<\lambda^{\prime}$,
$p\in\,\sim|\alpha|_{\lambda^{\prime}}^{-}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda^{\prime}}^{-})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda^{\prime}]|_{\rightarrow(\lambda_{1},p)}$.
Similarly, for all $\lambda^{\prime\prime}>\lambda^{\prime}$,
$\sim|\alpha|_{\lambda^{\prime}}^{+}$ is $ba\lambda^{\prime\prime}$ and for
all $\lambda_{1}\leq\lambda^{\prime}$,
$p\in\,\sim|\alpha|_{\lambda^{\prime}}^{+}(\lambda_{1})$,
$(\sim|\alpha|_{\lambda^{\prime}}^{+})_{\rightarrow(\lambda_{1},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda_{1},p)}$
.
As an aside, it follows that the $\sim|\alpha|_{\lambda^{\prime}}^{+/-}$ are
$bb\lambda$.
Finally, since $t$ is $bb\lambda$, $\sim
t_{\lambda}=\,\rightarrow(\lambda,D(\lambda)-\bigcup t(\lambda))\rightarrow$.
These properties are sufficient to establish that $\gamma$ is an ip. ∎
This result may be generalized for systems that are not dynamic spaces.
###### Theorem 81.
If $S$ is a dynamic set and $\gamma$ is an ip of $S$ then
1) If $\alpha\subset\beta\in\gamma$ and $\alpha$ is companionable then there
exists an ip of $S$, $\gamma^{\prime}$, s.t. $\alpha\in\gamma^{\prime}$
2) If $t$ is a compatible set of subsets of $S$ s.t. for each $\alpha\in t$
there’s a $\alpha\subset\beta\in\gamma$ then there exists an ip of $S$,
$\gamma^{\prime}$, s.t. $t\subset\gamma^{\prime}$
###### Proof.
Once again, (1) follows from (2).
Assume $t$ is $bb\lambda$ and this time take $\gamma^{\prime}$ to be the set
s.t. $t\subset\gamma^{\prime}$ and for all $\beta\in\gamma$ if
$\beta\bigcap(\sim t_{\lambda})\neq\emptyset$ then $\beta\bigcap(\sim
t_{\lambda})\in\gamma^{\prime}$, and for all $\alpha\in t$,
$\lambda^{\prime}>\lambda$, if
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+/-})\neq\emptyset$ then
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+/-})\in\gamma^{\prime}$.
The nature of these sets are similar to those of the prior proof, except that
they’re relativized to each $\beta\in\gamma$. The following properties are
sufficient to establish that $\gamma^{\prime}$ is an ip:
For all $\lambda^{\prime\prime}<\lambda^{\prime}$,
$p\in(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))(\lambda^{\prime\prime})$,
$(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))_{\rightarrow(\lambda^{\prime\prime},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$
(Assume $\alpha\subset\eta\in\gamma$; since
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-})\neq\emptyset$, and
$\lambda^{\prime\prime}<\lambda^{\prime}$,
$\beta[-\infty,\lambda^{\prime\prime}]\subset|\eta[-\infty,\lambda^{\prime\prime}]|$,
so
$\beta_{\rightarrow(\lambda^{\prime\prime},p)}=|\eta[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$,
and so
$|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}\subset\beta_{\rightarrow(\lambda^{\prime\prime},p)}$.)
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-})=(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{-}))[-\infty,\lambda^{\prime}]\circ\beta[\lambda^{\prime},\infty]$.
For all $\lambda^{\prime\prime}\leq\lambda^{\prime}$,
$p\in(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})(\lambda^{\prime\prime})$,
$(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})_{\rightarrow(\lambda^{\prime\prime},p)}=|\alpha[-\infty,\lambda]|_{\rightarrow(\lambda^{\prime\prime},p)}$
For all $\lambda^{\prime\prime}>\lambda^{\prime}$,
$\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+})=(\beta\bigcap(\sim|\alpha|_{\lambda^{\prime}}^{+}))[-\infty,\lambda^{\prime\prime}]\circ\beta[\lambda^{\prime\prime},\infty]$.
With $X\equiv[\beta\bigcap(\sim t_{\lambda})](\lambda)$, $\beta\bigcap(\sim
t_{\lambda})=\beta_{\rightarrow(\lambda,X)}\circ\beta_{(\lambda,X)\rightarrow}$.
∎
The first theorem follows from the second, since if $D$ is a dynamic space
then $\\{D\\}$ is an ip.
## IV Probabilities
### IV.1 Dynamic Probability Spaces
For a single ip, probabilities are no different than in classic probability
and statistics444In the Introduction two models were for non-determinism were
given. One of them, type-m non-determinism, encounters a well known difficulty
at this point: in order for the statistical view of probabilities to be
applicable, every individual run of an experiment must result in an individual
outcome being obtained. However, if an e-automata displays pure type-m non-
determinism, it will simultaneously take all paths for all possible outcomes,
not just paths which cross some particular $[e]$, and this will result in
multiple outcomes. In this controversy, experimental results have decided in
favor of individual outcomes; experimental apparatus always end up in a single
state, and as outcomes have been defined, each individual final environmental
state corresponds to an individual outcome.:
###### Definition 82.
An _ip probability space_ is a triple, $(\gamma,\Sigma,P)$, where $\gamma$ is
an ip, $\Sigma$ is a set of subsets of $\gamma$, and
$P:\Sigma\rightarrow[0,1]$ s.t.:
1) $\gamma\in\Sigma$
2) If $\sigma\in\Sigma$ then $\gamma-\sigma\in\Sigma$
3) If $\psi\subset\Sigma$ is finite then $\bigcup\psi\in\Sigma$
4) $P(\gamma)=1$
5) If $\psi\subset\Sigma$ is finite and pairwise disjoint then
$P(\bigcup\psi)=\Sigma_{\sigma\in\psi}P(\sigma)$
Very commonly, “finite” in (3) and (5) is replaced with “countable”. Countable
additivity is invaluable when dealing with questions of convergence, however
convergence will become a more multifaceted issue in the structures to be
introduced, so the countable condition has been relaxed & questions of
convergence delayed.
###### Theorem 83.
In the definition of an ip probability space, (2) and (3) can be replaced
with: If $\sigma_{1},\sigma_{2}\in\Sigma$ then
$\sigma_{1}-\sigma_{2}\in\Sigma$
###### Proof.
If (2) holds then (3) is equivalent to “If $\sigma_{1},\sigma_{2}\in\Sigma$
then $\sigma_{1}\bigcap\sigma_{2}\in\Sigma$”. The equivalence now follows from
$\sigma_{1}-(\sigma_{1}-\sigma_{2})=\sigma_{1}\bigcap\sigma_{2}$ and
$\sigma_{1}-\sigma_{2}=\sigma_{1}\bigcap(\gamma-\sigma_{2})$.∎
###### Definition 84.
If $S_{1}$ and $S_{2}$ are dynamic spaces and $\gamma_{1}$ and $\gamma_{2}$
are ip’s of $S_{1}$ and $S_{2}$ respectively, then ip probability spaces
$(\gamma_{1},\Sigma_{1},P_{1})$ and $(\gamma_{2},\Sigma_{2},P_{2})$ are
_consistent_ if
1) $\gamma_{1}\bigcap\gamma_{2}\in\Sigma_{1}$ and
$\gamma_{1}\bigcap\gamma_{2}\in\Sigma_{2}$
2) If $\sigma\subset\gamma_{1}\bigcap\gamma_{2}$ then $\sigma\in\Sigma_{1}$
iff $\sigma\in\Sigma_{2}$
3) For any $t\in\Sigma_{1}\bigcap\Sigma_{2}$, $P_{1}(t)=P_{2}(t)$
If $Y$ is a set of ip probability spaces, $Y$ is _consistent_ if, for any
$x,y\in Y$, $x$ and $y$ are consistent.
A dynamic probability space is, essentially, a consistent collection of ip
probability spaces.
###### Definition 85.
A _dynamic probability space_ (dps) is a triple $(X,T,P)$ where $X$ is a set
of dynamic sets, $T$ is a set of compatible sets, and $P:T\rightarrow[0,1]$
s.t.:
1) For every $S\in X$ there’s a $\gamma\in T$ s.t. $\gamma$ is an ip of $S$
2) For every $t\in T$ there’s a $\gamma\in T$ s.t. $\gamma$ is an ip of some
$S\in X$ and $t\subset\gamma$
3) If $t_{1},t_{2}\in T$ then $t_{1}-t_{2}\in T$
4) If $\gamma\in T$ is an ip of some $S\in X$ then $P(\gamma)=1$
5) If $t_{1},t_{2}\in T$ are disjoint, and $t_{1}\bigcup t_{2}\in T$ then
$P(t_{1}\bigcup t_{2})=P(t_{1})+P(t_{2})$
For dynamic probability space $(X,T,P)$
$G_{T}\equiv\\{\gamma\in T:\gamma\>is\>a\>partition\>of\>some\>S\in X\\}$
If $S$ is a dynamic space and $\alpha,\beta\subset S$, it’s important to
stress that axiom 5 means that
$P(\\{\alpha,\beta\\})=P(\\{\alpha\\})+P(\\{\beta\\})$ (assuming
$\\{\alpha,\beta\\},\\{\alpha\\},\\{\beta\\}\in T$); it does not mean that
$P(\\{\alpha,\beta\\})=P(\\{\alpha\bigcup\beta\\})$ (even if
$\\{\alpha,\beta\\},\\{\alpha\bigcup\beta\\}\in T$), and so in general
$P(\\{\alpha\bigcup\beta\\})\neq P(\\{\alpha\\})+P(\\{\beta\\})$.
Note that Thm 83 does not hold for dps’s. This is essentially because there is
no $Z\in T$ s.t. for all $t\in T$, $t\subset Z$; as a result, “not $t$” is not
uniquely defined, and arbitrary finite unions can not be expected to be
elements of $T$ (though arbitrary finite intersections are elements of $T$).
To formally describe the connection between dynamic probability spaces and
classic probability theory, the following definitions will be useful.
###### Definition 86.
If $(X,T,P)$ is a dps and $\gamma\in G_{T}$ then $T_{\gamma}\equiv\\{t\in
T\,:\,t\subset\gamma\\}$ and $P_{\gamma}\equiv P\mid_{T_{\gamma}}$ (that is,
$Dom(P_{\gamma})=T_{\gamma}$ and for all $t\in T_{\gamma}$,
$P_{\gamma}(t)=P(t)$)
If $\pi=(X,T,P)$ is a dps and $A\subset G_{T}$,
$A_{\pi}\equiv\\{(\gamma,T_{\gamma},P_{\gamma}):\gamma\in A\\}$
If $Y$ is a consistent set of ip probability spaces,
$X_{Y}\equiv\\{\bigcup\gamma:(\gamma,\Sigma,P)\in Y\\}$,
$T_{Y}\equiv\bigcup_{(\gamma,\Sigma,P)\in Y}\Sigma$ and
$P_{Y}:T_{Y}\rightarrow[0,1]$ s.t. if $(\gamma,\Sigma,P)\in Y$ and
$t\in\Sigma$ then $P_{Y}(t)=P(t)$.
The following theorem states that a dynamic probability space is simply a
consistent set of ip probability spaces.
###### Theorem 87.
1) If $\pi=(X,T,P)$ is a dps and $A\subset G_{T}$ then $A_{\pi}$ is a
consistent set of ip probability spaces
2) If $Y$ is a consistent set of ip probability spaces then
$(X_{Y},T_{Y},P_{Y})$ is dps
3) If $(X,T,P)$ is a dps and
$Y\equiv\\{(\gamma,T_{\gamma},P_{\gamma})\,:\,\gamma\in G_{T}\\}$ then
$X=X_{Y}$, $T=T_{Y}$ and $P=P_{Y}$
4) If $Y$ is a consistent set of ip probability spaces then
$Y=\\{(\gamma,(T_{Y})_{\gamma},(P_{Y})_{\gamma}):\gamma\in G_{(T_{Y})}\\}$
###### Proof.
(1) says that a dps can be rewritten as a consistent set of ip probability
spaces, (2) says that a consistent set of ip probability spaces can be
rewritten as a dps, and (3) & (4) say that in moving between dps’s &
consistent sets of ip probability spaces, no information is lost.
A: For dps $(X,T,P)$, if $t_{1},t_{2}\in T$ and $\gamma$ is any element of
$G_{T}$ s.t. $t_{1}\subset\gamma$ then $t_{1}-t_{2}\in T_{\gamma}$
\- Follows from $t_{1}-t_{2}=t_{1}-t_{1}\bigcap t_{2}=t_{1}-\gamma\bigcap
t_{2}$ (note that $\gamma\bigcap t_{2}\in T_{\gamma}$) -
(1) That the elements of $A_{\pi}$ are ip probability spaces follows from Thm
83. That they’re consistent follows from
$\gamma_{1}\bigcap\gamma_{1}=\gamma_{1}-(\gamma_{1}-\gamma_{2})$
(2) Axioms 1, 2, and 4 clearly hold. Axiom 3 follows from Thm 83 and (A).
Axiom 5 follows from axiom 2.
(3) & (4) That no information is lost in going from a dps to a set of ip
probability spaces follows from (A). It’s clear that no information is lost
when going from a set of ip probability spaces to a dps. ∎
### IV.2 T-Algebras and GPS’s
Very little in the definition of a dps depends on $X$ being a set of dynamic
sets or $G_{T}$ being a set of ip’s. As things often get simpler as they get
more abstract, it will be useful to take a step back & generalize the
probability theory.
###### Definition 88.
A _t-algebra_ is a double, $(X,T)$, where $X$ is a set of sets and
1) For every $x\in X$ there’s a $\gamma\in T$ s.t. $\gamma$ is a partition of
$x$
2) For every $t\in T$ there’s a $\gamma\in T$ s.t. $\gamma$ is a partition of
some $x\in X$ and $t\subset\gamma$
3) If $t_{1},t_{2}\in T$ then $t_{1}-t_{2}\in T$
A t-algebra, $(X,T)$, may be referred to simply by $T$. As before,
$G_{T}\equiv\\{\gamma\in T:\gamma\>is\>a\>partition\>of\>some\>x\in X\\}$.
###### Definition 89.
A _generalized probability space_ (gps) is a triple, $(X,T,P)$, where $(X,T)$
is a t-algebra and $P:T\rightarrow[0,1]$ s.t.
1) If $t\in T$ is a partition of some $x\in X$ then $P(t)=1$
2) If $t_{1},t_{2}\in T$ are disjoint and $t_{1}\bigcup t_{2}\in T$ then
$P(t_{1}\bigcup t_{2})=P(t_{1})+P(t_{2})$
#### IV.2.1 $\neg t$ and $[t]$
###### Definition 90.
For t-algebra $(X,T)$, $t\in T$, $\neg t\equiv\\{t^{\prime}\in
T:t^{\prime}\bigcap t=\emptyset\>and\>t^{\prime}\bigcup t\in G_{T}\\}$.
For $A\subset T$, $\neg A\equiv\bigcup_{t\in A}\neg t$.
$\neg^{(1)}t\equiv\neg t$, $\neg^{(2)}t\equiv\neg\neg t$,
$\neg^{(3)}t\equiv\neg\neg\neg t$, etc.
###### Theorem 91.
For gps $(X,T,P)$, $t\in T$, $t^{\prime}\in\neg^{(n)}t$
1) If $n$ is odd, $P(t)+P(t^{\prime})=1$
2) If $n$ is even, $P(t)=P(t^{\prime})$
###### Proof.
Simple induction over $\mathbb{N}^{+}$.∎
###### Theorem 92.
For gps $(X,T,P)$, $t,t^{\prime}\in T$, $n,m>0$
1) $t^{\prime}\in\neg^{(n+m)}t$ iff for some $t^{\prime\prime}\in T$,
$t^{\prime\prime}\in\neg^{(n)}t$ and $t^{\prime}\in\neg^{(m)}t^{\prime\prime}$
2) $t^{\prime}\in\neg^{(n)}t$ iff $t\in\neg^{(n)}t^{\prime}$
3) $t\in\neg^{(2n)}t$
4) If $m<n$ then $\neg^{(2m)}t\subset\neg^{(2n)}t$ and
$\neg^{(2m+1)}t\subset\neg^{(2n+1)}t$
###### Proof.
(1) follows from induction over $m$. (2) follows from induction over $n$. (3)
follows from induction over $n$ and (1). (4) follows from (1) and (3).∎
###### Definition 93.
If $(X,T)$ is a t-algebra and $t\in T$,
$[t]\equiv\bigcup_{n\in\mathbb{N}^{+}}\neg^{(2n)}t$.
###### Theorem 94.
$[t]$ is an equivalence class
###### Proof.
By Thm 92.3, $t\in[t]$
By Thm 92.2, if $t_{1}\in[t_{2}]$ then $t_{2}\in[t_{1}]$
By Thm 92.1, if $t_{2}\in[t_{1}]$ and $t_{3}\in[t_{2}]$ then $t_{3}\in[t_{1}]$
∎
###### Theorem 95.
$\neg[t]=\bigcup_{n\in\mathbb{N}^{+}}\neg^{(2n+1)}t$
###### Proof.
$\neg[t]\equiv\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$.
$t^{\prime\prime}\in\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$ iff for some
$t^{\prime}\in[t]$, $t^{\prime\prime}\in\neg t^{\prime}$. $t^{\prime}\in[t]$
iff for some $n\in\mathbb{N}^{+}$, $t^{\prime}\in\neg^{(2n)}t$. So
$t^{\prime\prime}\in\bigcup_{t^{\prime}\in[t]}\neg t^{\prime}$ iff for some
$n\in\mathbb{N}^{+}$, $t^{\prime\prime}\in\neg^{(2n+1)}t$.∎
###### Theorem 96.
If $t^{\prime}\in\neg[t]$ then $t\in\neg[t^{\prime}]$
###### Proof.
Follows from Thm 95 and Thm 92.2∎
###### Definition 97.
A t-algebra is _simple_ if for every $t\in T$, $\neg t=\neg\neg\neg t$.
###### Theorem 98.
If $T$ is a simple t-algebra and $t\in T$ then $[t]=\neg\neg t$ and
$\neg[t]=\neg t$
###### Proof.
Clear ∎
#### IV.2.2 $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$
In general, dps’s will not have simple t-algebras. This can be seen from the
fact that, if $t_{1}\in\neg t_{2}$, $t_{2}\in\neg t_{3}$, and $t_{4}\in\neg
t_{3}$, $t_{1}\bigcup t_{4}$ will be a partition of $D$, but will generally
not be an ip.
When t-algebras are not simple, they can get fairly opaque. For example, it’s
possible to have $t_{1}\in[t_{2}]$, and $t=t_{1}\bigcap t_{2}$, but
$t_{1}-t\notin[t_{2}-t]$, even though it’s clear that for any probability
function $P$, $P(t_{1}-t)=P(t_{2}-t)$. This could never happen for a simple
t-algebra. Fortunately, starting with any t-algebra, it’s always possible to
use $\neg$ and $[t]$ to build an equivalent simple t-algebra. This will be
done iteratively, the first iteration being the t-algebra $T_{1}$:
###### Definition 99.
For t-algebra $(X,T)$,
For $t,t^{\prime}\in T$, $t\perp t^{\prime}$ if for some $t_{1},t_{2}\in T$,
$t_{1}\in\neg[t_{2}]$, $t\subset t_{1}$ and $t^{\prime}\subset t_{2}$
$T_{1}\equiv\\{t\bigcup t^{\prime}\,:\,t\perp t^{\prime}\\}$
$X_{1}\equiv\\{(\cup t)\bigcup(\cup t^{\prime})\,:\,t\in
T\>and\>t^{\prime}\in\neg[t]\\}$
###### Theorem 100.
If $(X,T)$ is a t-algebra then $(X_{1},T_{1})$ is a t-algebra
###### Proof.
Axioms (1) and (2) hold by the definitions of $\perp$ and $X_{1}$.
For axiom (3), given any $t_{1},t_{2}\in T_{1}$, there exist
$t_{1}^{\prime},t_{1}^{\prime\prime},t_{2}^{\prime},t_{2}^{\prime\prime}\in T$
s.t. $t_{i}^{\prime}\perp t_{i}^{\prime\prime}$ and
$t_{i}=t_{i}^{\prime}\bigcup t_{i}^{\prime\prime}$. Define
$t_{3}\equiv(t_{1}^{\prime}-t_{2}^{\prime})-t_{2}^{\prime\prime}$ and
$t_{4}\equiv(t_{1}^{\prime\prime}-t_{2}^{\prime})-t_{2}^{\prime\prime}$;
$t_{3},t_{4}\in T$; since $t_{3}\subset t_{1}^{\prime}$ and $t_{4}\subset
t_{1}^{\prime\prime}$, $t_{3}\perp t_{4}$, so $t_{3}\bigcup t_{4}\in T_{1}$.
$t_{1}-t_{2}=t_{3}\bigcup t_{4}$ so $t_{1}-t_{2}\in T_{1}$. ∎
We now turn to constructing a probability function for $T_{1}$.
###### Theorem 101.
If $(X,T,P)$ is a gps, $t_{1}\perp t_{2}$, $t_{3}\perp t_{4}$, and
$t_{1}\bigcup t_{2}=t_{3}\bigcup t_{4}$, then
$P(t_{1})+P(t_{2})=P(t_{3})+P(t_{4})\in[0,1]$; further, if
$t_{1}\in\neg[t_{2}]$ then $P(t_{1})+P(t_{2})=1$
###### Proof.
Take $t_{1}^{\prime}=t_{1}\bigcap t_{3}$, $t_{2}^{\prime}=t_{1}\bigcap t_{4}$,
$t_{3}^{\prime}=t_{2}\bigcap t_{3}$, $t_{4}^{\prime}=t_{2}\bigcap t_{4}$. All
$t_{i}^{\prime}$ are disjoint, and each $t_{i}$ is a union of two of the
$t_{j}^{\prime}$, so
$P(t_{1})+P(t_{2})=P(t_{1}^{\prime})+P(t_{2}^{\prime})+P(t_{3}^{\prime})+P(t_{4}^{\prime})=P(t_{3})+P(t_{4})$.
If $t_{1}\in\neg[t_{2}]$ then $P(t_{1})+P(t_{2})=1$ by Thm 91.1.
More generally, if $t_{1}\perp t_{2}$, for some $t_{5},t_{6}\in T$,
$t_{6}\in\neg[t_{5}]$, $t_{1}\subset t_{5}$, and $t_{2}\subset t_{6}$.
$P(t_{5})+P(t_{6})=1$, so $P(t_{1})+P(t_{2})\in[0,1]$.∎
###### Definition 102.
If $(X,T,P)$ is a gps, $P_{1}:T_{1}\rightarrow[0,1]$ s.t. for $t_{1},t_{2}\in
T$, $t_{1}\perp t_{2}$, $P_{1}(t_{1}\bigcup t_{2})=P(t_{1})+P(t_{2})$.
###### Theorem 103.
If $(X,T,P)$ is a gps then $(X_{1},T_{1},P_{1})$ is a gps.
###### Proof.
Follows from Thm 100 and Thm 101. ∎
This process can now be repeated; starting with $(X_{1},T_{1},P_{1})$,
$(X_{2},T_{2},P_{2})$ can be constructed as
$((X_{1})_{1},(T_{1})_{1},(P_{1})_{1})$.
###### Definition 104.
If $(X,T,P)$ is a gps, $(X_{0},T_{0},P_{0})\equiv(X,T,P)$ and
$(X_{n+1},T_{n+1},P_{n+1})\equiv((X_{n})_{1},(T_{n})_{1},(P_{n})_{1})$.
###### Theorem 105.
If $(X,T,P)$ is a gps
1) $(X_{n},T_{n},P_{n})$ is a gps
2) $t\in T_{n}$ iff for some finite, pairwise disjoint $A\subset T$ with not
more than $2^{n}$ members s.t. $\bigcup A\in G_{Tn}$, some $B\subset A$,
$t=\bigcup B$
3) If $t\in T_{n}$ and $A\subset T$ is finite, pairwise disjoint, and
$t=\bigcup A$ then $P_{n}(t)=\sum_{x\in A}P(x)$
4) If $t\in T_{n}$ and $t\in T_{m}$ then $P_{n}(t)=P_{m}(t)$
###### Proof.
For 1-3, Straightforward induction. (4) follows from (2) & (3).∎
###### Definition 106.
If $(X,T)$ is a t-algebra
$X_{\mathbb{N}}\equiv\bigcup_{n\in\mathbb{N}^{+}}X_{n}$
$T_{\mathbb{N}}\equiv\bigcup_{n\in\mathbb{N}^{+}}T_{n}$
If $(X,T,P)$ is a dps, $P_{\mathbb{N}}:T_{\mathbb{N}}\rightarrow[0,1]$ s.t. if
$t\in T_{n}$ then $P_{\mathbb{N}}(t)=P_{n}(t)$
###### Theorem 107.
If $(X,T,P)$ is a gps then $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ is
a simple gps
###### Proof.
That $(X_{\mathbb{N}},T_{\mathbb{N}})$ is a t-algebra follows from the fact
that if $t\in T_{\mathbb{N}}$ then for some $m\in\mathbb{N}^{+}$, all $n>m$,
$t$ is an element of $T_{n}$, and all $T_{n}$ are t-algebras. That
$(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ is a gps follows similarly.
It remains to show that $T_{\mathbb{N}}$ is simple. Take “$\neg_{n}$” to be
“$\neg$” defined on $T_{n}$ and “$\neg_{\mathbb{N}}$” to be “$\neg$” defined
on $T_{\mathbb{N}}$. For $t\in T_{\mathbb{N}}$, if
$t^{\prime}\in\neg_{\mathbb{N}}\neg_{\mathbb{N}}\neg_{\mathbb{N}}t$ then for
some $t_{1},t_{2}\in T_{\mathbb{N}}$, $t^{\prime}\in\neg_{\mathbb{N}}t_{2}$,
$t_{2}\in\neg_{\mathbb{N}}t_{1}$, and $t_{1}\in\neg_{\mathbb{N}}t$, so for
some $m,n,p\in\mathbb{N}$, $t^{\prime}\in\neg_{m}t_{2}$,
$t_{2}\in\neg_{n}t_{1}$, and $t_{1}\in\neg_{n}t$, in which case, with
$q=lub(\\{m,n,p\\})$, $t^{\prime}\in\neg_{q+1}t$, and so
$t^{\prime}\in\neg_{\mathbb{N}}t$ ∎
The following will be of occasional use.
###### Definition 108.
$[t]_{n}$ and $\neg_{n}t$ may be use to refer to $[t]$ and $\neg t$ on
$(X_{n},T_{n})$. $[t]_{\mathbb{N}}$ and $\neg_{\mathbb{N}}t$ may be use to
refer to $[t]$ and $\neg t$ on $(X_{\mathbb{N}},T_{\mathbb{N}})$.
###### Theorem 109.
If $(X,T,P)$ is a gps
1) $t\in T_{\mathbb{N}}$ iff for some finite, pairwise disjoint $A\subset T$
s.t. $\bigcup A\in G_{T\mathbb{N}}$, some $B\subset A$, $t=\bigcup B$
2) If $A\subset T$ is finite, pairwise disjoint, and $\bigcup A\in
T_{\mathbb{N}}$ then $P_{\mathbb{N}}(\bigcup A)=\Sigma_{t\in A}P(t)$
###### Proof.
Follows from Thm 105, ∎
#### IV.2.3 Convergence on a GPS
Earlier, the question of countable convergence was deferred. Some basic
concepts will now be presented.
###### Definition 110.
For t-algebra $(X,T)$, $\mathcal{A}$ is a _c-set_ if it is a countable
partition of some $y\in X$, and there exists a sequence on $\mathcal{A}$,
$(A_{n})_{n\in\mathbb{N}}$, $(\mathcal{A}=\\{A_{n}:n\in\mathbb{N}\\})$ s.t.
for all $n\in\mathbb{N}$, $\bigcup_{i\leq n}A_{i}\in T$.
###### Theorem 111.
If $(X,T)$ is a t-algebra, $\mathcal{A}$ is a c-set iff it is a countable
partition of some $y\in X$ and for any finite $B\subset\mathcal{A}$, $\bigcup
B\in T$
###### Proof.
$\Rightarrow$ There exists a sequence $(A_{n})_{n\in\mathbb{N}}$ s.t.
$\mathcal{A}=\\{A_{n}:n\in\mathbb{N}\\}$ and for all $n\in\mathbb{N}$,
$\bigcup_{i\leq n}A_{i}\in T$.
For all $A_{i}\in\mathcal{A}$, $A_{i}=\bigcup_{j\leq i}A_{j}-\bigcup_{k\leq
i-1}A_{k}$, so $A_{i}\in T$.
For each $b\in B$ there exists a unique $i\in\mathbb{N}$ s.t. $A_{i}=b$; take
$k$ to be the largest such number. $\bigcup B\subset\bigcup_{i\leq k}A_{i}\in
T$. Since $(X,T)$ is a t-algebra, $B$ is a finite subset of $T$, and $\bigcup
B$ is a subset of an element of $T$, $\bigcup B\in T$.
$\Leftarrow$ Immediate. ∎
If $(X,T)$ is a t-algebra and $(t_{n})_{n\in\mathbb{N}}$ is a sequence of
elements of $T$ s.t. for all $i\in\mathbb{N}$, $t_{i}\subset t_{i+1}$ and
$\bigcup_{i\in\mathbb{N}}t_{i}$ is the partition of some $S\in X$, then
$\mathcal{A}\equiv\\{t_{i+1}-t_{i}:i\in\mathbb{N}\\}$ is a c-set. Further, if
$\mathcal{A}$ is a c-set and $(A_{n})_{n\in\mathbb{N}}$ is any sequence of
it’s elements then $t_{i}=\bigcup_{j\leq i}A_{j}$ form such a sequence of
elements of $T$, so these two notions are equivalent.
###### Theorem 112.
If $(X,T,P)$ is a gps and $\mathcal{A}$ is a c-set then $\sum_{x\in A}P(x)\leq
1$.
###### Proof.
If $(A_{n})_{n\in\mathbb{N}}$ is any sequence on $\mathcal{A}$, for all
$N\in\mathbb{N}$, $\sum_{n\leq N}P(A_{n})=P(\bigcup_{n\leq N}A_{n})\leq 1$.∎
###### Definition 113.
A gps, $(X,T,P)$, is _convergent_ if for every c-set, $\mathcal{A}$,
$\sum_{x\in A}P(x)=1$.
###### Theorem 114.
If $(X,T,P)$ is a convergent dps, $\mathcal{A}$ and $\mathcal{B}$ are c-sets,
$Z\subset\mathcal{A}$, $Y\subset\mathcal{B}$, and $\bigcup Z=\bigcup Y$ then
$\sum_{x\in Z}P(x)=\sum_{x\in Y}P(x)$.
###### Proof.
Take $C\equiv\\{\alpha:for\>some\>a\in Y,\>b\in Z,\>\alpha=a\bigcap
b\>and\>\alpha\neq\emptyset\\}$ and $V\equiv(\mathcal{B}-Y)\bigcup C$. $V$ has
the following properties: every element of $V$ is an element of $T$; for every
finite $\alpha\subset V$ there exits a finite $b\subset B$ s.t.
$\bigcup\alpha\subset\bigcup b$, and so $\bigcup\alpha\in T$; and finally $V$
is pairwise disjoint & $\bigcup V=\bigcup B$. Therefore $V$ is a c-set.
$\sum_{x\in B-Y}P(x)+\sum_{x\in C}P(x)=\sum_{x\in V}P(x)=1=\sum_{x\in
B}P(x)=\sum_{x\in B-Y}P(x)+\sum_{x\in Y}P(x)$, so $\sum_{x\in
C}P(x)=\sum_{x\in Y}P(x)$. Proceeding similarly with $(A-Z)\bigcup C$ yields
$\sum_{x\in C}P(x)=\sum_{x\in Z}P(x)$. Therefore $\sum_{x\in Z}P(x)=\sum_{x\in
Y}P(x)$.∎
###### Definition 115.
If $(X,T)$ is a t-algebra,
$T_{c}\equiv\\{x:For\>some\>c\textrm{-}set,\>A,\>some\>Z\subset A,\>x=\bigcup
Z\\}$.
If $(X,T,P)$ is a convergent dps, $P_{c}:T_{c}\rightarrow[0,1]$ s.t. if $A$ is
a c-set and $B\subset A$ then $P_{c}(\bigcup B)=\sum_{x\in B}P(x)$.
###### Theorem 116.
If $(X,T,P)$ is convergent $A\subset T_{c}$ is countable, pairwise disjoint,
and $\bigcup A\in T_{c}$ then $\sum_{x\in A}P_{c}(x)=P_{c}(\bigcup A)$.
###### Proof.
For every $x\in A$ take $Y_{x}$ to a be a c-set s.t. for $B_{x}\subset Y_{x}$,
$\bigcup B_{x}=x$. Take $Z$ to be a c-set s.t. for $D\subset Z$, $\bigcup
D=\bigcup A$. Finally, for each $x\in A$, take
$C_{x}\equiv\\{\alpha:for\>some\>a\in B_{x},\>b\in D,\>\alpha=a\bigcap
b\>\&\>\alpha\neq\emptyset\\}$. Note that $\bigcup C_{x}=x$.
Each $(Y_{x}-B_{x})\bigcup C_{x}$ is a c-set, so $\sum_{t\in
C_{x}}P_{c}(t)=P_{c}(x)$. With $C\equiv\bigcup_{x\in A}C_{x}$, $(Z-D)\bigcup
C$ is also a c-set, so $P_{c}(\bigcup A)=\sum_{x\in A}\sum_{t\in
C_{x}}P_{c}(t)=\sum_{x\in A}P_{c}(x)$. ∎
If $(X,T)$ is a t-algebra, there’s no guarantee that $(X,T_{c})$ will be a
t-algebra; in particular, if $A$ and $B$ are c-sets, $(\bigcup
A)\bigcap(\bigcup B)$ might not be an element of $T_{c}$.555$(X,T_{c},P_{c})$
does, however, meet all the other requirements for being a gps; that is, if
for all pairs of c-sets, $A$ and $B$, $(\bigcup A)\bigcap(\bigcup B)\in T_{c}$
then $(X,T_{c},P_{c})$ is a gps Also, if $(X,T,P)$ is convergent,
$(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ may not be (though if
$(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ is convergent then $(X,T,P)$
is). Because convergence on $(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$
is a useful quality for a gps to posses, the following simplified notation
will be used.
###### Definition 117.
If $(X,T)$ is a t-algebra, $T_{\Upsilon}\equiv(T_{\mathbb{N}})_{c}$. If
$(X,T,P)$ is a dps, it is $\Upsilon$-convergent if
$(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ is convergent. If $(X,T,P)$
is an $\Upsilon$-convergent dps then $P_{\Upsilon}\equiv(P_{\mathbb{N}})_{c}$.
The easiest way to remember this is that $\Upsilon$ has absolutely nothing to
do with these properties.
#### IV.2.4 $T_{S}$, $T_{S\mathbb{N}}$, and $T_{\mathbb{N}}^{S}$
###### Definition 118.
If $(X,T)$ is a t-algebra and $S\in X$ then
$T_{S}\equiv\\{t\in T:for\>some\>\gamma\in
G_{T},\>\bigcup\gamma=S\>and\>t\subset\gamma\\}$
$T_{S\mathbb{N}}\equiv(T_{S})_{\mathbb{N}}$ and
$T_{\mathbb{N}}^{S}\equiv(T_{\mathbb{N}})_{S}$.
It follows readily that $(\\{S\\},T_{S})$ is a t-algebra, both
$(\\{S\\},T_{S\mathbb{N}})$ and $(\\{S\\},T_{\mathbb{N}}^{S})$ are simple
t-algebras, and $T_{S\mathbb{N}}\subset T_{\mathbb{N}}^{S}$. One question that
arises is, under what conditions will $T_{S\mathbb{N}}=T_{\mathbb{N}}^{S}$;
that is, when does the rest of $T$ yield no further information about the
probabilities on $S$? There are a number of ways in which this can occur; the
most obvious is: for all $S^{\prime}\in X$, $S^{\prime}\neq S$, $S\bigcap
S^{\prime}=\emptyset$ (or more generally, for all $\gamma\in G_{TS}$,
$\gamma^{\prime}\in G_{TS^{\prime}}$, if $S\neq S^{\prime}$ then
$\gamma\bigcap\gamma^{\prime}=\emptyset$). In such cases, $T_{S}$ is
completely independent of the rest of $T$. Another way in which
$T_{S\mathbb{N}}=T_{\mathbb{N}}^{S}$ is if the structure of $T$ as a whole can
be mapped onto $T_{S}$; this possibility will now be sketched.
(In the following definition, $\mathbf{P}(S)$ is the power set of $S$)
###### Definition 119.
If $(X,T)$ and $(\\{S\\},T^{\prime})$ are t-algebras, $f:\bigcup
T\rightarrow\mathbf{P}(\bigcup T^{\prime})$ _reduces $T$ to $T^{\prime}$_ if
1) If $\gamma\in G_{T}$ then $\bigcup f[\gamma]\in G_{T^{\prime}}$
2) If $\gamma\in G_{T}$, $\alpha,\beta\in\gamma$, and $\alpha\neq\beta$ then
$f(\alpha)\bigcap f(\beta)=\emptyset$
3) If $\alpha\in\bigcup T$ and $\alpha\subset S$ then $f(\alpha)=\\{\alpha\\}$
###### Theorem 120.
If $f$ reduces $T$ to $T^{\prime}$ then it reduces $T_{1}$ to $T_{1}^{\prime}$
###### Proof.
For $t_{1},t_{2}\in T$, assume $t_{1}\in\neg t_{2}$; by (2), $f[t_{1}]\bigcap
f[t_{2}]=\emptyset$ and by (1) $\bigcup f[t_{1}\bigcup t_{2}]\in
G_{T^{\prime}}$, so $\bigcup f[t_{1}]\in\neg\bigcup f[t_{2}]$. It follows from
the definition of $\neg^{(n)}$ that if $t_{1}\in\neg^{(n)}t_{2}$ then $\bigcup
f[t_{1}]\in\neg^{(n)}\bigcup f[t_{2}]$. (1) & (2) in the definition of
reduction follow immediately from this. (3) continues to hold because $\bigcup
T_{1}=\bigcup T$.∎
###### Theorem 121.
If $f$ reduces $T$ to $T^{\prime}$ then it reduces $T_{\mathbb{N}}$ to
$T_{\mathbb{N}}^{\prime}$
###### Proof.
By Thm 120, $f$ reduces $T_{n}$ to $T_{n}^{\prime}$.
If $\gamma\in G_{T\mathbb{N}}$ then for some $n$, $\gamma\in G_{Tn}$, and so
(1) and (2) hold. (3) hold because$\bigcup T_{\mathbb{N}}=\bigcup T$.∎
###### Theorem 122.
If $(X,T)$ is a t-algebra, $S\in X$, and there exists an $f$ that reduces $T$
to $T_{S}$ then $T_{\mathbb{N}}^{S}=T_{S\mathbb{N}}$.
###### Proof.
In all cases, $T_{S\mathbb{N}}\subset T_{\mathbb{N}}^{S}$
Take any $t\in T_{\mathbb{N}}^{S}$ and any $\gamma\in G_{T\mathbb{N}}$ s.t.
$t\subset\gamma$. By Thm 121 $f$ reduces $T_{\mathbb{N}}$ to
$T_{S\mathbb{N}}$, so by (1) and (3) in the definition of a reduction,
$t\subset\bigcup f[\gamma]\in G_{T_{S\mathbb{N}}}$; therefore $t\in
T_{S\mathbb{N}}$ and so $T_{\mathbb{N}}^{S}\subset T_{S\mathbb{N}}$. ∎
### IV.3 Nearly Compatible Sets
Now to return to dps’s and, in particular, the
$(X_{\mathbb{N}},T_{\mathbb{N}},P_{\mathbb{N}})$ construction for dps’s.
Because the only additional requirement imposed on dps’s is that all
$\gamma\in G_{T}$ must be ip’s, and because this only places an indirect
restriction on the makeup of “$X$” in dps $(X,T,P)$, dps’s of the form
$(\\{S\\},T,P)$ will be of greatest interest. Naturally, any dps may be
decomposed into a set such dps’s, one for each $S\in X$, and the results of
this section may be applied to those component parts.
###### Definition 123.
A set of companionable sets, $A$, is _nearly compatible_ if it is pairwise
disjoint and for every $\alpha,\beta\in A$, every $(\lambda,p)\in Uni(\bigcup
A)$ either $\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$ or
$\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}=\emptyset$.
All compatible sets are nearly compatible, but the converse does not hold. The
basic result of this section is that for a dps, $(\\{S\\},T,P)$, while
elements of $G_{T\mathbb{N}}$ need not be compatible, they must be nearly
compatible.
###### Definition 124.
$\mathcal{A}$ is a _nearly compatible collection_(_ncc_) if it is pairwise
disjoint and $\bigcup\mathcal{A}$ is nearly compatible. $\mathcal{A}$ is an
_$S$ -nccp_ if it is an ncc and $\bigcup\mathcal{A}$ is a partition of $S$.
###### Theorem 125.
If $S$ is a dynamic set and $\\{t_{1},t_{2}\\}$, $\\{t_{2},t_{3}\\}$, and
$\\{t_{3},t_{4}\\}$ are $S$-nccp’s, then so is $\\{t_{1},t_{4}\\}$.
###### Proof.
Take $\alpha\in t_{1}$, $\beta\in t_{4}$ and assume
$\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}\neq\emptyset$.
With
$\bar{s}[-\infty,\lambda]\in\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}$,
there must be a $\eta\in t_{2}$ and a $\nu\in t_{3}$ s.t.
$\bar{s}[-\infty,\lambda]\in\eta_{\rightarrow(\lambda,p)}$ and
$\bar{s}[-\infty,\lambda]\in\nu_{\rightarrow(\lambda,p)}$. Since $t_{1}\bigcup
t_{2}$, $t_{2}\bigcup t_{3}$, and $t_{3}\bigcup t_{4}$ are nearly compatible,
$\alpha_{\rightarrow(\lambda,p)}=\eta_{\rightarrow(\lambda,p)}=\nu_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.
∎
Thm 125 is the key property of nearly compatible sets. Essentially, if $Y$ is
an attribute s.t. all ip’s are $Y$, and $Y$ is transitive in the sense of Thm
125, then all $\gamma\in G_{T\mathbb{N}}$ are $Y$. Near compatibility is a
close approximation of compatibility that possesses this property.
###### Theorem 126.
If $(\\{S\\},T,P)$ is a dps, $t_{1}\in T$, and $t_{2}\in\neg[t_{1}]$ then
$\\{t_{1},t_{2}\\}$ is an $S$-nccp.
###### Proof.
If $t_{1}\in\neg t_{2}$ then $t_{1}$ and $t_{2}$ are disjoint and
$t_{1}\bigcup t_{2}$ is a compatible (and so nearly compatible) partition, so
$\\{t_{1},t_{2}\\}$ is an $T$-nccp. The result is now immediate from Thm 125.
∎
Note that if $t_{1}\in\neg[t_{2}]$ there are no grounds to conclude that
$t_{1}\bigcup t_{2}$ is compatible, and in general it won’t be.
###### Theorem 127.
If $(\\{S\\},T,P)$ is a dps, $x_{1}\in T_{n}$, and
$x_{2}\in\neg_{n}[x_{1}]_{n}$ then $\\{x_{1},x_{2}\\}$ is a $S$-nccp.
###### Proof.
Holds for $n=0$ by Thm 126. Assume it holds for $n=m$. For $n=m+1$, if
$x_{2}\in\neg_{m+1}x_{1}$ then $x_{1}\bigcup x_{2}\in G_{T(m+1)}$; therefore
there must be a $y_{1},y_{2}\in T_{m}$ s.t. $y_{2}\in\neg_{m}[y_{1}]_{m}$ and
$y_{1}\bigcup y_{2}=x_{1}\bigcup x_{2}$. By assumption on $n=m$, $x_{1}\bigcup
x_{2}=y_{1}\bigcup y_{2}$ is nearly compatible, so $\\{x_{1},x_{2}\\}$ is a
$S$-nccp. From Thm 125 it then follows that if
$x_{2}\in\neg_{m+1}[x_{1}]_{m+1}$ then $\\{x_{1},x_{2}\\}$ is an $S$-nccp.∎
###### Theorem 128.
If $(\\{S\\},T,P)$ is a dps then
1) If $\gamma\in G_{T\mathbb{N}}$ then for some finite $S$-nccp $A\subset T$,
$\gamma=\bigcup A$
2) If $t\in T_{\mathbb{N}}$ then for some finite ncc, $A\subset T$, $t=\bigcup
A$
###### Proof.
Immediate from Thm 127. ∎
The richer a dps’s t-algebra is, the more information the dps carries. A
t-algebra contains minimal information if $G_{T\mathbb{N}}$ contains only the
original ip’s; in that case, no information can be derived from the dps as a
whole that isn’t known by considering the individual
$(\gamma,T_{\gamma},P_{\gamma})$ in isolation. As the t-algebra grows richer,
more information can be derived from it. Thm 128 indicates the outer limit on
the information that can be derived from a dps on a single dynamic set.
###### Definition 129.
If $(\\{S\\},T,P)$ is a dps, it is _maximal_ if $\gamma\in G_{T\mathbb{N}}$
iff there’s a finite $S$-nccp, $A\subset T$, s.t. $\bigcup A=\gamma$.
If $(\\{S\\},T,P)$ and $(\\{S\\},T^{\prime},P^{\prime})$ are dps’s, $T\subset
T^{\prime}$, and $P=P^{\prime}|_{T}$, then if $T$ is maximal,
$(\\{S\\},T^{\prime},P^{\prime})$ will give no further information about
$(\\{S\\},T,P)$. However, if $(\\{S\\},T,P)$ is not maximal, but
$(\\{S\\},T^{\prime},P^{\prime})$ is, this immediately yields a wealth of
information about $(\\{S\\},T,P)$, indeed more information than
$(\\{S\\},T_{\mathbb{N}},P_{\mathbb{N}})$ yields; it means that the following
construct forms a simple gps:
$G_{Te}\equiv\\{\gamma:for\>some\>finite\>S\textrm{-}nccp,\>A\subset
T,\>\gamma\equiv\bigcup A\\}$
$T_{e}\equiv\\{t:for\>some\>finite\>ncc,\>A\subset T,\>some\>\gamma\in
G_{Te},\>t=\bigcup A\subset\gamma\\}$
$P_{e}:T_{e}\rightarrow[0,1]$ s.t. if $A\subset T$ is a finite ncc and
$\bigcup A\equiv t\in T_{e}$ then
$P_{e}(t)=P_{\mathbb{N}}^{\prime}(t)=\sum_{t^{\prime}\in A}P(t^{\prime})$.
Note that $T_{e}=T_{\mathbb{N}}$ only if $(\\{S\\},T,P)$ is maximal. In all
other cases, $(\\{S\\},T_{e},P_{e})$ contains more information than
$(\\{S\\},T_{\mathbb{N}},P_{\mathbb{N}})$.
For $\Upsilon$-convergent dps’s, this can be pushed further.
###### Definition 130.
If $(\\{S\\},T,P)$ is a dps, it is _$\omega$ -maximal_ if for every countable
$S$-nccp, $A\subset T$, every finite $B\subset A$, $\bigcup B\in
T_{\mathbb{N}}$.
$(\\{S\\},T,P)$ is _$\Upsilon$ -maximal_ if it is $\omega$-maximal and
$\Upsilon$-convergent.
For an $\omega$-maximal dps, every countable $S$-nccp composed of elements of
$T$ is a c-set in $T_{\mathbb{N}}$. This means that for a $\Upsilon$-maximal
dps, $\gamma\in G{}_{T\Upsilon}$ iff there’s a countable $S$-nccp, $A\subset
T$, s.t. $\bigcup A=\gamma$. It was mentioned previously that, in general,
$(\\{S\\},T_{\Upsilon},P_{\Upsilon})$ is not a gps; however it can be seen
that for $\Upsilon$-maximal dps’s, $(\\{S\\},T_{\Upsilon},P_{\Upsilon})$ is
not only a gps, it’s a simple gps.
Returning to the case where $(\\{S\\},T,P)$ and
$(\\{S\\},T^{\prime},P^{\prime})$ are dps’s, $T\subset T^{\prime}$, and
$P=P^{\prime}|_{T}$, if $(\\{S\\},T^{\prime},P^{\prime})$ is
$\Upsilon$-maximal then the following construct forms a simple gps:
$G_{T\varepsilon}\equiv\\{\gamma:for\>some\>countable\>S\textrm{-}nccp,\>A\subset
T,\>\gamma\equiv\bigcup A\\}$
$T_{\varepsilon}\equiv\\{t:for\>some\>countable\>ncc,\>A\subset
T,\>some\>\gamma\in G_{Te},\>t=\bigcup A\subset\gamma\\}$
$P_{\varepsilon}:T_{\varepsilon}\rightarrow[0,1]$ s.t. if $A$ is a countable
ncc and $\bigcup A\equiv t\in T_{\varepsilon}$ then
$P_{\varepsilon}(t)=P_{\Upsilon}^{\prime}(t)=\sum_{t^{\prime}\in
A}P(t^{\prime})$.
When it’s applicable, this is, of course, a very useful construct.
### IV.4 Deterministic & Herodotistic Spaces
A system is deterministic if a complete knowledge of the present yields a
complete knowledge of the future; it is herodotistic if complete knowledge of
the present yields a complete knowledge of the past. More formally:
###### Definition 131.
A dynamic set, $S$, is _deterministic_ if for every $(\lambda,p)\in Uni(S)$,
$S_{(\lambda,p)\rightarrow}$ is a singleton (that is,
$S_{(\lambda,p)\rightarrow}$ has only one element).
$S$ is _herodotistic_ if for every $(\lambda,p)\in Uni(S)$,
$S_{\rightarrow(\lambda,p)}$ is a singleton.
In this section deterministic and herodotistic dps’s will be investigated, as
will deterministic/herodotistic “universes”, where the e-automata as whole is
deterministic/herodotistic though the system being measured might not be. It
will follow from the various results that for the probabilities seen in
quantum physics to hold, neither the systems being investigated nor the
universe as a whole can be either deterministic or herodotistic.
No other material in this article will depend on the material in this section.
#### IV.4.1 DPS’s on Deterministic & Herodotistic Spaces
###### Theorem 132.
If a dynamic set, $S$, is deterministic or herodotistic then $S$ is a dynamic
space all its subsets are subspaces.
###### Proof.
For $A\subset S$, $\bar{p}_{1},\bar{p}_{2}\in A$,
$\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)$, if $S$ is herodotistic then
$\bar{p}_{1}[-\infty,\lambda]=\bar{p}_{2}[-\infty,\lambda]$ so
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]=\bar{p}_{2}\in
A$. Similarly, if $S$ is deterministic then
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]=\bar{p}_{1}\in
A$∎
###### Theorem 133.
1) If $D$ is deterministic then for any $X\subset D$, $\lambda\in\Lambda_{D}$,
$X=+X[-\infty,\lambda]$
2) If $D$ is herodotistic then for any $X\subset D$, $\lambda\in\Lambda_{D}$,
$X=+X[\lambda,\infty]$
###### Proof.
1) For any $\bar{p}[-\infty,\lambda]\in X[-\infty,\lambda]$ there exists only
one $\bar{p}^{\prime}\in D$ s.t.
$\bar{p}^{\prime}[-\infty,\lambda]=\bar{p}[-\infty,\lambda]$.
2) Similar∎
###### Theorem 134.
1) If $D$ is deterministic then $A,B\subset D$ are compatible iff they are
disjoint and bounded from below
2) If $D$ is herodotistic then $A,B\subset D$ are compatible iff they are
disjoint and bounded from above
###### Proof.
1) If $A$ and $B$ are disjoint then $A[-\infty,\lambda]\bigcap
B[-\infty,\lambda]=\emptyset$ for all $\lambda$.
2) If $p\in A(\lambda)\bigcap B(\lambda)$ then since $\rightarrow(\lambda,p)$
has only one element,
$A_{\rightarrow(\lambda,p)}=B_{\rightarrow(\lambda,p)}$.∎
###### Definition 135.
If $(\\{S\\},T,P)$ is a dps, it is _closed under combination_ if for all
pairwise disjoint $A\subset T$ s.t. $\bigcup A$ is an ip, $\bigcup A\in
G_{T}$.
###### Theorem 136.
If $(X,T,P)$ is a dps and for all $S\in X$, $S$ is either deterministic or
herodotistic and $(\\{S\\},T_{S},P_{s})$ is closed under combination, then for
all $t_{1},t_{2}\in T$ s.t $\bigcup t_{1}=\bigcup t_{2}$
1) If $\gamma\in G_{T}$ and $t_{1}\subset\gamma$ then $(\gamma-t_{1})\bigcup
t_{2}\in G_{T}$
2) $P(t_{1})=P(t_{2})$
###### Proof.
(1) follows from Thm 134 and (2) follows from (1). ∎
Note the importance of closure under combination. One can map any dps,
$(X,T,P)$, it onto a herodotistic dps as follows. For every $S\in X$,
$\bar{p}\in S$, define $\overline{p}_{h}$ by
$\overline{p}_{h}(\lambda)=(\bar{p}(\lambda),\bar{p}[0,\lambda])$. With
$S_{h}\equiv\\{\overline{p}_{h}:\bar{p}\in S\\}$, $S_{h}$ is herodotistic. One
can then readily construct $(X_{h},T_{h},P_{h})$, which will be a dps, but in
most cases will not be closed under combination; indeed, since $\bigcup
t_{h}=\bigcup t_{h}^{\prime}$ iff $\bigcup t=\bigcup t^{\prime}$ and
$P(t)=P_{h}(t_{h})$, the conclusion of above theorem will generally not apply
to $(X_{h},T_{h},P_{h})$.
#### IV.4.2 DPS’s in Deterministic Universes
###### Theorem 137.
If partition $\gamma$ is decided by an idea e-automata, $Z=(D,I,F)$ s.t. $D$
is deterministic, then for some $\lambda_{0}\in\Lambda$, all
$\alpha\in\gamma$, $\alpha$ is $sbb\lambda_{0}$ and $wba\lambda_{0}$.
###### Proof.
A: For every $(s,e),(s,e^{\prime})\in I$,
$\Sigma_{(s,e)\rightarrow}=\Sigma_{(s,e^{\prime})\rightarrow}$
\- With $\bar{s}[0,0]$ s.t. $\bar{s}[0,0](0)=s$,
$\bar{s}[0,0]\in\Sigma_{(s,e)}\bigcap\Sigma_{(s,e^{\prime})}$ (since
$(s,e),(s,e^{\prime})\in I$), so the result follows from Thm 39. -
B: Taking $S\equiv\bigcup\gamma$, for any $s\in S(0)$,
$(\mathcal{O}_{F})_{(0,s)\rightarrow}$ is either a singleton or empty
\- Follows from definition of “deterministic” and (A) -
C: For some $\lambda_{0}\in\Lambda$, all $\alpha\in\gamma$, there’s a
$\lambda_{\alpha}\in\Lambda$, s.t. $\alpha$ is $sbb\lambda_{0}$,
$wba\lambda_{\alpha}$, and for all $s\in\alpha(\lambda_{0})$,
$\alpha_{s\rightarrow}[\lambda_{0},\lambda_{\alpha}]$ is a singleton
\- Follows from (B) and the definition of being decided by an e-automata -
D: For all $\alpha,\beta\in\gamma$, if $\alpha\neq\beta$ then
$\alpha(\lambda_{0})\bigcap\beta(\lambda_{0})=\emptyset$
\- Follows from (B) and the definition of being decided by an e-automata -
The theorem is immediate from (C) & (D).∎
###### Definition 138.
If $Y$ is a collection subsets of dynamic set $S$ then $Y$ is _cross-section
(at $\lambda$)_ if it is pairwise disjoint and for all $y\in Y$, $y$ is a
subspace, $sbb\lambda$, and $wba\lambda$.
###### Theorem 139.
1) If $Y$ is a cross-section then it’s compatible.
2) If $Y$ is cross-section at $\lambda$ then $\\{\bigcup Y\\}$ is a cross-
section at $\lambda$
###### Proof.
Clear∎
###### Definition 140.
A t-algebra is _deterministically decidable_ if for all $t\in T$, $t$ is a
cross-section.
It is _deterministically normal_ if it is deterministically decidable and
1) If $t\in T$ then $\\{\bigcup t\\}\in T$
2) If $A\subset T$ is finite and $\bigcup A$ is a partition & a cross-section
then $\bigcup A\in G_{T}$.
$(X,T,P)$ is a _deterministically decidable dps_ if $(X,T)$ is
deterministically decidable & _deterministically normal_ if $(X,T)$ is
deterministically normal.
###### Theorem 141.
If $(X,T,P)$ is deterministically normal dps
1) For any $t,t^{\prime}\in T$, if $\\{\bigcup t,\bigcup t^{\prime}\\}$ is a
cross-section and a partition of some $S\in X$ then $t^{\prime}\in\neg[t]$.
2) For all $t_{1},t_{2}\in T$ s.t. $\bigcup t_{1}=\bigcup t_{2}$,
$P(t_{1})=P(t_{2})$
###### Proof.
1) For any $\gamma\in G_{T}$ s.t. $t\subset\gamma$,
$t\bigcup\\{\bigcup(\gamma-t)\\}$ is a cross-section. Since $\bigcup
t^{\prime}=\bigcup(\gamma-t)$, $t\bigcup\\{\cup t^{\prime}\\}$ is a cross-
section. Therefore $t\bigcup\\{\cup t^{\prime}\\}$, $\\{\cup t\\}\bigcup
t^{\prime}$, and $\\{\cup t\\}\bigcup\\{\cup t^{\prime}\\}$ are all elements
of $G_{T}$.
2) Take any $\gamma\in G_{T}$ s.t. $t_{1}\subset\gamma$. It follows from (1)
that $\gamma-t_{1}\in\neg[t_{2}]$, so $t_{1}\in[t_{2}]$ ∎
If $X=\\{S\\}$ then (1) becomes - For any $t,t^{\prime}\in T$,
$t^{\prime}\in\neg[t]$ iff $\\{\bigcup t,\bigcup t^{\prime}\\}$ is a cross-
section and a partition of $S$.
#### IV.4.3 DPS’s in Herodotistic Universes
###### Definition 142.
If $t$ is a compatible set, it is _herodotistically decidable_ if for some
$\lambda_{0}\in\Lambda$, all $\alpha\in t$, there’s a
$\lambda_{\alpha}\in\Lambda$ s.t.
$\alpha=+\alpha[\lambda_{0},\lambda_{\alpha}]$, and for all
$s\in\alpha(\lambda_{\alpha})$, $\alpha_{\rightarrow
s}[\lambda_{0},\lambda_{\alpha}]$ is a singleton.
Under these conditions, $t$ is said to be _born at $\lambda_{0}$_
A t-algebra $(X.T)$ is _herodotistically decidable_ if every $t\in T$ is
herodotistically decidable (which holds iff every $\gamma\in G_{T}$ is
herodotistically decidable).
###### Theorem 143.
If ip $\gamma$ is decided by an ideal e-automata, $(D,I,F)$, and $D$ is
herodotistic, then $\gamma$ is herodotistically decidable.
###### Proof.
If $D$ is herodotistic then for any $e\in F$, any
$(s,e^{\prime})\in\mathcal{\theta}_{e}(\Lambda(\mathcal{O}_{e}))$,
$\Sigma_{(s,e^{\prime})}$ is a singleton, and by Thm 55.3 for any
$(s,e^{\prime}),(s,e^{\prime\prime})\in\mathcal{\theta}_{e}(\Lambda(\mathcal{O}_{e}))$,
$\Sigma_{(s,e^{\prime})}=\Sigma_{(s,e^{\prime\prime})}$. The theorem then
follows immediately from the definition of being decided by an ideal
e-automata.∎
###### Theorem 144.
If $(\cup t_{1})\bigcap(\cup t_{2})=\emptyset$, both $t_{1}$ and $t_{2}$ are
herodotistically decidable, and there exists a $\lambda$ s.t. both $t_{1}$ and
$t_{2}$ are born at $\lambda$, then $t_{1}$ and $t_{2}$ are nearly compatible
###### Proof.
Immediate from the definitions of herodotistically decidable and nearly
compatible. ∎
Let’s say that for herodotistically decidable t-algebra, $(X,T)$,
$t_{1},t_{2}\in T$ are _sympathetic_ if $\bigcup t_{1}=\bigcup t_{2}$ and
there exists a $t^{\prime}\in\neg t_{1}$, $\lambda\in\Lambda$ s.t. both
$t_{2}$ and $t^{\prime}$are born at $\lambda$. Thm 144 means that for a
maximal dps (or a dps that can be embedded in a maximal dps), if $t_{1}$ and
$t_{2}$ are sympathetic then $P(t_{1})=P(t_{2})$. It will soon be seen that
this is incompatible with quantum probabilities.
Note that being herodotistic has a less dramatic effect on an ideal e-automata
than being deterministic. This is because, for ideal e-automata, the
environment is assumed to remember something about the past (due being all-
reet), but know little about the future (due to being unbiased).
## V Application to Quantum Measurement
### V.1 Preliminary Matters
In this part, results from the prior sections will be applied to quantum
physics as described by the Hilbert Space formalism. In order to do this, a
few preliminary matters need to be addressed.
#### V.1.1 A Note On Paths
Quantum physics is often viewed in terms of transitions between states, rather
than in terms of system paths; an obvious exception being the path integration
formalism. Before proceeding, it may be helpful to start by describing
conditions under which state transitions can be re-represented as paths.
Start by taking $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$ to mean
that state $s_{1}$ at time $\lambda_{1}$ can transition to state $s_{2}$ at
time $\lambda_{2}$. Paths may then be defined as any parametrized function,
$\bar{s}$, s.t. for all $\lambda_{1}<\lambda_{2}$,
$(\lambda_{1},\bar{s}(\lambda_{1}))\Rightarrow(\lambda_{2},\bar{s}(\lambda_{2}))$.
To show that this set of paths is equivalent to the transition relation, it
must be shown that if $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$ then
there exists a path, $\bar{s}$, s.t. $\bar{s}(\lambda_{1})=s_{1}$ and
$\bar{s}(\lambda_{2})=s_{2}$ (the converse clearly holds).
In quantum mechanics, the $\Rightarrow$ relation has the following properties:
1) If $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$ then
$\lambda_{1}<\lambda_{2}$
2) For all $(\lambda,s)$, all $\lambda^{\prime}<\lambda$, there’s a
$s^{\prime}$ s.t. $(\lambda^{\prime},s^{\prime})\Rightarrow(\lambda,s)$
3) For all $(\lambda,s)$, all $\lambda^{\prime}>\lambda$, there’s a
$s^{\prime}$ s.t. $(\lambda,s)\Rightarrow(\lambda^{\prime},s^{\prime})$
4) For all $(\lambda_{1},s_{1})$, $(\lambda_{2},s_{2})$ s.t.
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$, all
$\lambda_{1}<\lambda^{\prime}<\lambda_{2}$, there’s a $s^{\prime}$ s.t.
$(\lambda_{1},s_{1})\Rightarrow(\lambda^{\prime},s^{\prime})$ and
$(\lambda^{\prime},s^{\prime})\Rightarrow(\lambda_{2},s_{2})$.
If the measurements are strongly unbiased, then the following holds:
5) If $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$, and
$(\lambda_{2},s_{2})\Rightarrow(\lambda_{3},s_{3})$, then
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{3},s_{3})$.666If
$<\lambda_{1},s_{1}|\lambda_{2},s_{2}>\neq 0$, then it’s clear that
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$. However, there is some
ambiguity as to whether $<\lambda_{1},s_{1}|\lambda_{2},s_{2}>=0$ necessarily
implies that the transition can not take place, or if it simply demands that
the transition occurs with probability $0$. This question grows acute if
there’s a $(\lambda,s)$ s.t. $\lambda_{1}<\lambda<\lambda_{2}$,
$<\lambda_{1},s_{1}|\lambda,s>\neq 0$, and $<\lambda,s|\lambda_{2},s_{2}>\neq
0$. In that case, in order for (5) to hold, we would have to allow the
transition $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$, but say that
it occurs with probability $0$. There’s no necessity to define the
$\Rightarrow$ relation in this manner, but doing so is in keeping both with
the path integral formalism and the orthodox interpretations of quantum
mechanics. There one would say that there exist possible paths from
$(\lambda_{1},s_{1})$ to $(\lambda_{2},s_{2})$, but they interfere with each
other in such a way as to keep the total amplitude of the transition $0$;
however, if a further measurement caused only a subset of these paths to be
taken, then the probability can become non-zero.
If measurements are unbiased, but not strongly unbiased, there still exists a
covering of $\Rightarrow$ s.t. within each element of the covering 1-5 hold.
To establish $(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$ iff there
exists a path, $\bar{s}$, s.t. $\bar{s}(\lambda_{1})=s_{1}$ and
$\bar{s}(\lambda_{2})=s_{2}$, it is sufficient to establish it within each
element of the covering. As it turns out, statements 1-5 are sufficient for
accomplishing this.
The key step is to show that if
$(\lambda_{0},s_{0})\Rightarrow(\lambda_{1},s_{1})$ then there’s a partial
path, $\bar{s}[\lambda_{0},\lambda_{1}]$ s.t.
$\bar{s}[\lambda_{0},\lambda_{1}](\lambda_{0})=s_{0}$ and
$\bar{s}[\lambda_{0},\lambda_{1}](\lambda_{1})=s_{1}$. If $\Lambda$ is
discrete this can be readily shown using statement 5 and induction. The proof
when $\Lambda$ a continuum will now be briefly sketched. First, for each
$\lambda_{0}<\lambda<\lambda_{1}$, form the set of all $s$ s.t.
$(\lambda_{0},s_{0})\Rightarrow(\lambda,s)$ and
$(\lambda,s)\Rightarrow(\lambda_{1},s_{1})$. For
$\lambda_{1/2}\equiv\frac{1}{2}(\lambda_{0}+\lambda_{1})$ choose a $s_{1/2}$
from $\lambda_{1/2}$’s set. Now for each $\lambda_{0}<\lambda<\lambda_{1/2}$
form the set of all $s$ s.t. $(\lambda_{0},s_{0})\Rightarrow(\lambda,s)$ and
$(\lambda,s)\Rightarrow(\lambda_{1/2},s_{1/2})$, and for each
$\lambda_{1/2}<\lambda<\lambda_{1}$ form the set of all $s$ s.t.
$(\lambda_{1/2},s_{1/2})\Rightarrow(\lambda,s)$ and
$(\lambda,s)\Rightarrow(\lambda_{1},s_{1})$. For
$\lambda_{1/4}\equiv\frac{1}{2}(\lambda_{0}+\lambda_{1/2})$ and
$\lambda_{3/4}\equiv\frac{1}{2}(\lambda_{1/2}+\lambda_{1})$, choose a
$s_{1/4}$ and a $s_{3/4}$ from the newly formed sets and repeat the process.
When this has been done for all $\lambda=m/2^{n}$, at each of the other
$\lambda$’s take the intersection of all the formed sets, and select one
element (the intersection must be non-empty). The set of all the selected
$(\lambda_{m/2^{n}},s_{m/2^{n}})$ and all the $(\lambda,s)$’s chosen from the
intersections is the graph of a partial path running from
$(\lambda_{0},s_{0})$ to $(\lambda_{1},s_{1})$.
Just as as one can start with the transition relation and use it to define a
set of paths, one can also start with a set of paths, and from it derive the
transition relation: If $S$ is a dynamic set,
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$ if
$S_{(\lambda_{1},s_{1})\rightarrow(\lambda_{2},s_{2})}\neq\emptyset$. With the
$\Rightarrow$ relation defined in this way, 1-4 above will always hold; if $S$
is a dynamic space, 5 will also hold. However, if the dynamic set is not a
dynamic space, the two representations may not be equivalent, paths can
contain more information. To see this, consider the case where
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{2},s_{2})$,
$(\lambda_{2},s_{2})\Rightarrow(\lambda_{3},s_{3})$, and
$(\lambda_{1},s_{1})\Rightarrow(\lambda_{3},s_{3})$; this says that a path
runs from $(\lambda_{1},s_{1})$ to $(\lambda_{2},s_{2})$, a path runs from
$(\lambda_{2},s_{2})$ to $(\lambda_{3},s_{3})$, and a path run from
$(\lambda_{1},s_{1})$ to $(\lambda_{3},s_{3})$, but it doesn’t guarantee that
any individual path runs through all three points. However, when constructing
paths from the transition relation, it might be possible to construct such a
path. Therefore, starting with a dynamic set $S$, using $S$ to construct the
$\Rightarrow$ relation, then using $\Rightarrow$ to construct the set of
paths, $S^{\prime}$, you can have $S\varsubsetneq S^{\prime}$, the elements of
$S^{\prime}-S$ being non-existent paths that can’t be ruled out based on the
transition relation alone.
Form here on out we will revert to the path formalism.
#### V.1.2 Discretely Determined Partitions
The mathematical framework employed in quantum mechanics limits the types of
measurements that the theory can talk about. These limitations are of
essentially two types. First, because the probabilities are defined directly
on the outcomes, rather than on the sets of outcomes, partitions have to be at
most countable. Second, because measurements are described by projection
operators on the state space, experimental outcomes correspond to sequences of
measurements of $S(\lambda)$ at discrete values of $\Lambda_{S}$ ($S$ being
the system’s dynamic set). The following definitions will allow us to work
within these constraints.
###### Definition 145.
For dynamic set, $S$, and $L\subset\Lambda_{S}$:
$\alpha\subset S$ is _determined on $L$_ if $\alpha=\bigcap_{\lambda\in
L}S_{\rightarrow(\lambda,\alpha(\lambda))\rightarrow}=\\{\bar{p}\in
S:for\,all\>\lambda\in L,\>\bar{p}(\lambda)\in\alpha(\lambda)\\}$
(essentially,
$\alpha=S_{\rightarrow(\lambda_{1},\alpha(\lambda_{1}))\rightarrow...\rightarrow(\lambda_{i},\alpha(\lambda_{i}))\rightarrow_{...}}$)
If $\gamma$ is a partition of $S$, $\gamma$ is determined on $L$ if all
$\alpha\in\gamma$ are determined on $L$.
$\alpha$ is _discretely determined_ if it is determined on some discrete $L$;
similarly for partitions.
Note that if $\alpha$ is determined on $L$, and $L\subset L^{\prime}$, then
$\alpha$ is determined on $L^{\prime}$.
Quantum probabilities are calculated using equations of the form
$P(A_{1},A_{2},...|S)=<S,\lambda_{0}|\mathbb{P}(A_{1};\lambda_{1})\mathbb{P}(A_{2};\lambda_{2})...\mathbb{P}(A_{n};\lambda_{n})\mathbb{P}(A_{n};\lambda_{n})...\mathbb{P}(A_{2};\lambda_{2})\mathbb{P}(A_{1};\lambda_{1})|S,\lambda_{0}>$
where $S$ is the initial system state and $\mathbb{P}(A_{i};\lambda_{i})$ is
the projection operator onto state space region $A_{i}$ at $\lambda_{i}$. This
manner of calculation necessarily limits the formalism to partitions composed
of discretely determined measurements. Before moving on, let’s briefly take a
closer look at the nature of this discreteness.
To be able to calculate (or even represent)
$|\psi>\equiv...\mathbb{P}(A_{i};\lambda_{i})...\mathbb{P}(A_{2};\lambda_{2})\mathbb{P}(A_{1};\lambda_{1})|S,\lambda_{0}>$,
$L\equiv\\{\lambda_{1},\lambda_{2},...\lambda_{i},...\\}$ must have a least
element (which should not be less than $\lambda_{0}$), and for every
$\lambda\in L$, the set of elements of $L$ that are greater than $\lambda$
must have a least element; in other words, $L$ must be well ordered. Under
these conditions, we have
$|\psi_{1}>\equiv\mathbb{P}(A_{1};\lambda_{1})|S,\lambda_{0}>$,
$|\psi_{2}>\equiv\mathbb{P}(A_{2};\lambda_{2})|\psi_{1}>$, etc. If the
sequence is finite, terminating at $n$, then $|\psi>=|\psi_{n}>$. Otherwise,
$|\psi>$ is the limit of the sequence. The required probability is then
$<\psi|\psi>$.
Similarly, to be able to calculate
$O\equiv\mathbb{P}(A_{1};\lambda_{1})\mathbb{P}(A_{2};\lambda_{2})...\mathbb{P}(A_{i};\lambda_{i})...\mathbb{P}(A_{i};\lambda_{i})...\mathbb{P}(A_{2};\lambda_{2})\mathbb{P}(A_{1};\lambda_{1})$,
$L\equiv\\{\lambda_{1},\lambda_{2},...\lambda_{i},...\\}$ must have a greatest
element, and for every $\lambda\in L$, the set of elements of $L$ that are
less than $\lambda$ must have a greatest element; in other words, $L$ must be
“upwardly well ordered”. Under these conditions, the probability is
$<S,\lambda_{0}|O|S,\lambda_{0}>$.
Because we are interested in partitions with total probability that’s
guaranteed to be $1$ based only on the structure of the measurements, and
independent of the details of the inner-products, we are limited to partitions
that are determined on parameter sets of this second type. It should be noted,
however, that this further qualification is of little consequence, because all
finitely determined partitions are allowed, and countable cases can be taken
as the limit of a sequence of finite cases.
###### Definition 146.
If $L$ is the subset of a parameter, it is _upwardly well-ordered_ if every
non-empty subset of $L$ has a greatest element.
If $\gamma$ is a partition of $S$, $\gamma\in\daleth_{S}$ if $\gamma$ is
countable and for some upwardly well-ordered, bounded from below
$L\subset\Lambda_{S}$, $\gamma$ is determined on $L$.
If $\gamma\in\daleth_{S}$, $L(\gamma)$ is the set of upwardly well-ordered,
bounded from below sets, $l\subset\Lambda$, s.t.$\gamma$ is determined on $l$.
From here on out, “discretely determined” will refer to being determined on an
$L$ that’s upwardly well-ordered and bounded from below.
Regardless of how “discrete” is interpreted, being limited to discretely
determined measurements is a surprisingly strong constraint. To see this,
consider the following simple type of measurement:
###### Definition 147.
If $S$ is a dynamic space and $\alpha\subset S$, $\alpha$ is a _moment_ if for
all $\lambda\in\Lambda_{S}$, $\alpha$ is either $bb\lambda$ or $ba\lambda$.
Assume that $\alpha$ is $bb\lambda$ and for all $\lambda^{\prime}>\lambda$,
$\alpha$ is $ba\lambda^{\prime}$. This makes $\alpha$ a moment. If $\alpha$ is
$ba\lambda$ then $\alpha$ is a measurement of the system state at $\lambda$.
Otherwise, if $\alpha$ is not $ba\lambda$, $\alpha$ may be thought of as a
measurement of the system’s state and its rate of change. Colloquially, it may
be thought of as a measurement of position and velocity777The
$\sim|\alpha|_{\lambda}^{+}$ encountered in Section III.5 are examples of
measurements of rate of change; the $\sim|\alpha|_{\lambda}^{+}$ contain paths
that are in some compatible set, $t$, as of $\lambda$, but whose “velocities”
ensure that they will exit $t$ immediately after $\lambda$. One limitation of
the Hilbert Space formulation of quantum physics is that it can only be used
to describe measurements of system state alone.
###### Theorem 148.
If $X$ is a pairwise-disjoint collection of moments of a dynamic space, then
$X$ is compatible.
###### Proof.
Moments of a dynamic space are clearly companionable. Since $X$ is pairwise-
disjoint, it only remains to show that for all $\alpha\in X$, if
$\beta\in(\alpha)_{\lambda}^{X}$ and
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$ then
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.
A: If $(\alpha)_{\lambda}^{X}\neq\\{\alpha\\}$ then $\alpha$ is $bb\lambda$.
\- Assume $\alpha[-\infty,\lambda]\bigcap\beta[-\infty,\lambda]\neq\emptyset$
and $\alpha\neq\beta$. If $\alpha$ is $ba\lambda$ then
$\alpha\bigcap\beta\neq\emptyset$ which contradicts $X$ being pairwise-
disjoint. Since $\alpha$ is a moment, and is not $ba\lambda$, it is
$bb\lambda$. -
Assume $\beta\in(\alpha)_{\lambda}^{X}$ and $\beta\neq\alpha$, then by (A),
$\alpha$ is $bb\lambda$. Similarly, because $\alpha\in(\beta)_{\lambda}^{X}$,
$\beta$ is $bb\lambda$. Therefore, for any
$p\in\alpha(\lambda)\bigcap\beta(\lambda)$,
$\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$. ∎
It follows that any partition composed of moments is an ip. We can reasonably
assume that all moment measurements can be performed; indeed, they appear to
be quite useful. However, they can not all be represented using the quantum
formalism. This represents a rather severe limitation in the mathematical
language of quantum physics.
In spite of such limitations, there is one way in which $\daleth_{S}$ may be
considered overly inclusive. Dynamic sets are a very broad concept, which can
make them unwieldy to use. To reign in their unruliness, we generally assume
that they can be decomposed into ip’s, and that these ip’s are related to the
possible measurement on the set. The subset of $\daleth_{S}$ containing
elements that support to such decompositions will prove useful.
###### Definition 149.
If $S$ is a dynamic set, $\Gamma_{S}$ is the set of $\gamma\in\daleth_{S}$
s.t. for every $\alpha\in\gamma$ there’s an ip of $S$, $\varsigma$, s.t. for
some $\nu\in\varsigma$, $\alpha\subset\nu$.
If $D$ is a dynamic space then $\Gamma_{S}=\daleth_{S}$. In the quantum
formalism, when $\Gamma_{S}\subsetneq\daleth_{S}$, it is $\Gamma_{S}$ that is
of interest, for if $\alpha$ is an outcome of a quantum measurement, then it
must be an element of an ip. To see why, start with the a time ordered product
of projection operators corresponding to the measurement,
$...\mathbb{P}(A_{i};\lambda_{i})...\mathbb{P}(A_{2};\lambda_{2})\mathbb{P}(A_{1};\lambda_{1})$.
Define $\mathbb{P}(\neg
A_{j};\lambda_{j})\equiv\mathbb{I}-\mathbb{P}(A_{j};\lambda_{j})$; with
$X_{j}$ equal to either $A_{j}$ or $\neg A_{j}$, form the set of all time
ordered products of the form
$...\mathbb{P}(X_{i};\lambda_{i})...\mathbb{P}(X_{2};\lambda_{2})\mathbb{P}(X_{1};\lambda_{1})$.
This set corresponds to an ip.
###### Theorem 150.
If $S$ is a dynamic set and $\alpha\in\gamma\in\Gamma_{S}$ then $\alpha$ is
companionable
###### Proof.
Take $\varsigma$ to be an ip and $\alpha\subset\nu\in\varsigma$. If
$\bar{p}_{1},\bar{p}_{2}\in\alpha$ and
$\bar{p}_{1}(\lambda)=\bar{p}_{2}(\lambda)$ then
$\bar{p}=\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in\nu$.
Take any $L\in L(\alpha)$. For all $\lambda\in L$,
$\bar{p}(\lambda)\in\alpha(\lambda)$, so $\bar{p}\in\alpha$, and so $\alpha$
is a subspace. With $\lambda_{0}=glb(L)$, there must be a
$\lambda\leq\lambda_{0}$ s.t. $\nu$ is $sbb\lambda$. Since $\alpha$ is
determined on $L$, $\alpha$ is then also $sbb\lambda$. Taking $\lambda_{1}$ to
be $L$’s greatest element, $\alpha$ is $wba\lambda_{1}$.∎
###### Theorem 151.
If $S$ is a dynamic set then given any $\alpha,\beta\in\bigcup\Gamma_{S}$ s.t.
$\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}\neq\emptyset$,
if $\bar{p}_{1}[-\infty,\lambda]\in\alpha_{\rightarrow(\lambda,p)}$ and
$\bar{p}_{2}[\lambda,\infty]\in\beta_{(\lambda,p)\rightarrow}$ then
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in S$.
###### Proof.
Take
$\bar{p}[-\infty,\lambda]\in\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}$,
$\bar{p}\equiv\bar{p}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]$. Since
$\beta$ is a subspace, $\bar{p}\in S$. Take $\varsigma$ to be an ip and
$\alpha\subset\nu\in\varsigma$. For some $\eta\in\gamma$, $\bar{p}\in\eta$.
$\bar{p}[-\infty,\lambda]\in\nu_{\rightarrow(\lambda,p)}\bigcap\eta_{\rightarrow(\lambda,p)}$,
so $\eta_{\rightarrow(\lambda,p)}=\nu_{\rightarrow(\lambda,p)}$, and so
$\bar{p}_{1}[-\infty,\lambda]\in\eta_{\rightarrow(\lambda,p)}$. Since $\eta$
is a subspace and
$\bar{p}[\lambda,\infty]=\bar{p}_{2}[\lambda,\infty]\in\eta_{(\lambda,p)\rightarrow}$,
$\bar{p}_{1}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in\eta\subset
S$. ∎
#### V.1.3 Interconnected Dynamic Sets
An all-reet e-automata can not forget anything its ever known about the
system. Under the right conditions, however, an e-automata can discover
something about the system’s past that is not implied by the current state.
This is most readily seen if the system parameter is discrete.
For an e-automata with a discrete parameter, assume that at $\lambda$ the
e-automata is in state $(s_{\lambda},e_{\lambda})$ and at $\lambda+1$ it’s in
state $(s_{\lambda+1},e_{\lambda+1})$. It would be reasonable (though not
necessary) to assume that the system state of $s_{\lambda}$ is not reflected
in $e_{\lambda}$, and doesn’t get reflected in the environment until
$e_{\lambda+1}$. In this case, it takes one step in time for the environment
to learn about the system. More generally, of course, the set of allowed
environmental states at $\lambda+1$ will be a function of both $s_{\lambda}$
and $s_{\lambda+1}$, as well as $e_{\lambda}$, but this still asserts that it
is possible to learn something about the system’s past that’s not reflected in
the current state of the system.
If the parameter is continuous and the measurements are at discrete times we
wouldn’t expect this to happen, though it can; the environment may gain
information at $\lambda_{2}$ about the state of the system at $\lambda_{1}$
that isn’t implied by the state of the system at $\lambda_{2}$. In order for
this to occur the system itself would have to remember something about its
state at $\lambda_{1}$ until $\lambda_{2}$, at which point the knowledge is
simultaneously passed to the environment and forgotten by the system. This is
clearly an edge case, but a pernicious one, as it allows for anomalous ip’s
with discretely determined measurements; in order to regularize this set of
ip’s, this edge case needs to be eliminated. In order to be eliminated, the
system dynamics need to be interconnected, a property that will defined first
for dynamic spaces, and then for dynamic sets.
###### Definition 152.
A dynamic space, $D$, is _interconnected_ if
for all $\lambda_{1}<\lambda_{2}$, every $p_{1},p_{1}^{\prime}\in
D(\lambda_{1})$, $p_{2},p_{2}^{\prime}\in D(\lambda_{2})$ s.t.
$(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2})\neq\emptyset$,
$(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2}^{\prime})\neq\emptyset$,
$(\lambda_{1},p_{1}^{\prime})\rightarrow(\lambda_{2},p_{2})\neq\emptyset$, and
$(\lambda_{1},p_{1}^{\prime})\rightarrow(\lambda_{2},p_{2}^{\prime})\neq\emptyset$,
there exists a $\lambda\in[\lambda_{1},\lambda_{2}]$ and a $p\in D(\lambda)$
s.t. $(\lambda_{1},p_{1})\rightarrow(\lambda,p)\neq\emptyset$,
$(\lambda_{1},p_{1}^{\prime})\rightarrow(\lambda,p)\neq\emptyset$,
$(\lambda,p)\rightarrow(\lambda_{2},p_{2})\neq\emptyset$, and
$(\lambda,p)\rightarrow(\lambda_{2},p_{2}^{\prime})\neq\emptyset$.
Being interconnected is equivalent to saying that if $p_{1}\neq
p_{1}^{\prime}$ then
$\rightarrow\\{(\lambda_{1},p_{1}),(\lambda_{1},p_{1}^{\prime})\\}\rightarrow(\lambda_{2},p_{2})\rightarrow$,
$\rightarrow(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2}^{\prime})\rightarrow$,
and
$\rightarrow(\lambda_{1},p_{1}^{\prime})\rightarrow(\lambda_{2},p_{2}^{\prime})\rightarrow$
can not be mutually compatible. Essentially, for any
$\bar{p}[-\infty,\lambda_{1}]\in\,\rightarrow(\lambda_{1},p_{1})$,
$\bar{p}^{\prime}[-\infty,\lambda_{1}]\in\,\rightarrow(\lambda_{1},p_{1}^{\prime})$,
if $\bar{p}[-\infty,\lambda_{1}]$ and $\bar{p}^{\prime}[-\infty,\lambda_{1}]$
have not been distinguished by $\lambda_{1}$, and there is no measurement
between $\lambda_{1}$ and $\lambda_{2}$, then they can not be distinguished at
$\lambda_{2}$ because the point $(\lambda,p)$ destroys the ability to
distinguish them. Nearly all actively studied dynamic systems are
interconnected, and quantum systems always are.
Now to generalize interconnectedness for dynamic sets.
###### Definition 153.
If $S$ is a dynamic set and
$\bar{p}_{11},\bar{p}_{21},\bar{p}_{12},\bar{p}_{22}\in S$,
$(\bar{p}_{11},\bar{p}_{21},\bar{p}_{12},\bar{p}_{22})$ is an _interconnect on
$[\lambda_{1},\lambda_{2}]$_ if
$\bar{p}_{11}[-\infty,\lambda_{1}]=\bar{p}_{12}[-\infty,\lambda_{1}]$
$\bar{p}_{21}[-\infty,\lambda_{1}]=\bar{p}_{22}[-\infty,\lambda_{1}]$
$\bar{p}_{11}[\lambda_{2},\infty]=\bar{p}_{21}[\lambda_{2},\infty]$
$\bar{p}_{12}[\lambda_{2},\infty]=\bar{p}_{22}[\lambda_{2},\infty]$.
The paths $\bar{p}_{11},\bar{p}_{21},\bar{p}_{12},\bar{p}_{22}$ can be thought
of as sample paths from the sets
$\rightarrow(\lambda_{1},p_{1})\rightarrow(\lambda_{2},p_{2})\rightarrow$,
etc., that were used in the dynamic space definition of interconnectedness.
Because e-automata are unbiased, if there are no measurements on these paths
between $\lambda_{1}$ and $\lambda_{2}$ then
$+\\{\bar{p}_{11}[-\infty,\lambda_{1}],\bar{p}_{21}[-\infty,\lambda_{1}]\\}$
should behave like a dynamic space in $[\lambda_{1},\lambda_{2}]$.
###### Definition 154.
If $A$ is a dynamic set, $[\lambda_{1},\lambda_{2}]$ is a _space-segment of
$A$_ if for all $\bar{p}\in A$,
$[\lambda,\lambda^{\prime}]\subset[\lambda_{1},\lambda_{2}]$,
$\bar{p}^{\prime}[\lambda,\lambda^{\prime}]\in
A_{(\lambda,\bar{p}(\lambda))\rightarrow(\lambda^{\prime},\bar{p}(\lambda^{\prime}))}$,
$\bar{p}[-\infty,\lambda]\circ\bar{p}^{\prime}[\lambda,\lambda^{\prime}]\circ\bar{p}[\lambda^{\prime},\infty]=A$.
If an experiment has not distinguished between the paths of an interconnect on
_$[\lambda_{1},\lambda_{2}]$_ by $\lambda_{1}$, and there is no measurement
between $\lambda_{1}$ and $\lambda_{2}$, then interconnectedness will insure
that they can not be distinguished at $\lambda_{2}$:
###### Definition 155.
A dynamic set, $S$, is _interconnected_ if for every
$\lambda_{1}<\lambda_{2}$, every interconnect on _$[\lambda_{1},\lambda_{2}]$
, _$(\bar{p}_{11},\bar{p}_{21},\bar{p}_{12},\bar{p}_{22})$, s.t.
$[\lambda_{1},\lambda_{2}]$ is a space-segment of
$+\\{\bar{p}_{11}[-\infty,\lambda_{1}],\bar{p}_{21}[-\infty,\lambda_{1}]\\}$
there exists a $\lambda\in[\lambda_{1},\lambda_{2}]$ and an interconnect on
$[\lambda,\lambda_{2}]$,
$(\bar{p}_{11}^{\prime},\bar{p}_{21}^{\prime},\bar{p}_{12}^{\prime},\bar{p}_{22}^{\prime})$,
s.t.
$\bar{p}_{ij}^{\prime}[-\infty,\lambda_{1}]=\bar{p}_{ij}[-\infty,\lambda_{1}]$,
$\bar{p}_{ij}^{\prime}[\lambda_{2},\infty]=\bar{p}_{ij}[\lambda_{2},\infty]$,
and all $\bar{p}_{ij}^{\prime}(\lambda)=\bar{p}_{mn}^{\prime}(\lambda)$.
$[\lambda_{1},\lambda_{2}]$ being a space-segment of
$+\\{\bar{p}_{11}[-\infty,\lambda_{1}],\bar{p}_{21}[-\infty,\lambda_{1}]\\}$
helps to insure that there exists an interconnect,
$(\bar{p}_{11}^{\prime},\bar{p}_{21}^{\prime},\bar{p}_{12}^{\prime},\bar{p}_{22}^{\prime})$,
rather than just four paths that share the same point. If $S$ is a dynamic
space, this definition is equivalent to the prior one.
Finally, to make interconnectedness more readily applicable to $\Gamma_{S}$
for the case when the system is not a dynamic space.
###### Definition 156.
If $L\subset\Lambda$ is upwardly well-ordered, and $\lambda$ is not a lower-
bound of $L$, $pred_{L}(\lambda)\equiv lub(L\bigcap[-\infty,\lambda))$.
$pred_{L}(\lambda)$ is short for “predecessor of $\lambda$ on $L$”.
###### Definition 157.
For $\gamma\in\Gamma_{S}$, $\mathcal{L}(\gamma)\subset L(\gamma)$ s.t. if
$L\in\mathcal{L}(\gamma)$ then for all $\alpha\in\gamma$, $\lambda\in
L-\\{glb(L)\\}$, $[pred_{L}(\lambda),\lambda]$ is a space-segment of
$+\alpha[-\infty,pred_{L}(\lambda)]$.
$\gamma\in\Gamma_{S}^{s}$ if $\gamma\in\Gamma_{S}$ and
$\mathcal{L}(\gamma)\neq\emptyset$.
For quantum systems, the dynamics between successive measurements is always a
space-segment, and so when discussing measurements only the elements of
$\Gamma_{S}^{s}$ are of interest. For dynamic spaces,
$\Gamma_{S}^{s}=\Gamma_{S}=\daleth_{S}$.
### V.2 Partitions of Unity for Quantum Systems
A partition with total probability of $1$ will be given the fancy title of
being a _partition of unity_. In this section, we consider the quantum
partitions of unity.
When discussing quantum probabilities there are two cases to be considered:
the conditional case, where an initial state is known, and the non-conditional
case, where no initial state is known & everything about the system is
discovered via the experiment. We will start by considering the conditional
case.
#### V.2.1 The Conditional Case
###### Definition 158.
If $\gamma$ is a partition of dynamic set $S$ and $\bar{p}_{1},\bar{p}_{2}\in
S$, $\bar{p}_{1}$ and $\bar{p}_{2}$ are _co-located on $\gamma$_ if there’s a
$\alpha\in\gamma$ s.t. $\bar{p}_{1},\bar{p}_{2}\in\alpha$; $\bar{p}_{1}$ and
$\bar{p}_{2}$ _converge at $\lambda$_ if
$\bar{p}_{1}[\lambda,\infty]=\bar{p}_{2}[\lambda,\infty]$
The central object of study in this section will be the set of partitions
$q_{S}$:
###### Definition 159.
For dynamic set $S$, $\gamma\in q_{S}$ if $\gamma\in\Gamma_{S}^{s}$ and for
some $L\in\mathcal{L}(\gamma)$, all $\lambda_{1},\lambda_{2}\in L$ s.t.
$\lambda_{1}\equiv pred_{L}(\lambda_{2})$, all
$\bar{p}_{1}[-\infty,\lambda_{1}],\bar{p}_{2}[-\infty,\lambda_{1}]\in
S[-\infty,\lambda_{1}]$, either all
$\bar{p}\in+\bar{p}_{1}[-\infty,\lambda_{1}]$,
$\bar{p}^{\prime}\in+\bar{p}_{2}[-\infty,\lambda_{1}]$ that converge at
$\lambda_{2}$ are co-located on $\gamma$, or none are.
In the following claim, total probabilities of a set of outcomes are
“guaranteed to to equal $1$” if it’s known to be $1$ based only on the
structure of the outcomes, and independent of the details of the transition
probabilities, including the choice of initial state.
###### Claim 160.
For quantum probabilities in the conditional case, the sum over probabilities
of a set of outcomes is guaranteed to equal $1$ iff the the set of outcomes is
an element of $q_{S}$.
###### Remark.
It has already been argued that quantum outcomes are elements of
$\bigcup\Gamma_{S}^{s}$. The rest of the claim may be justified as follows.
Quantum outcomes are represented by time ordered products of projection
operators. Let’s take $L$ to be the set of times at which a projection
operator is applied, and define $\lambda_{1}$ to be the largest element of
$L$, $\lambda_{2}$ to be the largest element of $L-\\{\lambda_{0}\\}$, etc.
(Note that as the subscript increases, the time decreases.) An outcome is then
represented by
$\Pi_{i}=...\mathbb{P}(A_{ij};\lambda_{j})...\mathbb{P}(A_{i1};\lambda_{1})$,
where $A_{ij}\subset S(\lambda_{j})$, the “$i$” subscript identifies the
outcome, and the “$j$” subscript identifies the time. If the set of outcomes
form a partition then $\sum_{i}\Pi_{i}=\mathbb{I}$ ($\mathbb{I}$ being the
identity operator). This yields a total probability of $1$ irregardless of the
choice of the initial state iff
$\sum_{i}<\psi,\lambda_{0}|\Pi_{i}\Pi_{i}^{\dagger}|\psi,\lambda_{0}>=1$ for
all initial states $|\psi,\lambda_{0}>$, which holds iff
$\sum_{i}\Pi_{i}\Pi_{i}^{\dagger}=\mathbb{I}$.
For any $\lambda_{j}\in L$, define $\chi_{j}$ by
$(...,s_{k},...,s_{j})\in\chi_{j}$ if for all
$\lambda_{q}<\lambda_{r}\leq\lambda_{j}$ ($\lambda_{q},\lambda_{r}\in L$) the
transition $(s_{q},\lambda_{q})\Rightarrow(s_{r},\lambda_{r})$ is allowed;
Further, define $\Xi_{j}$ by
$(...,s_{k},s_{k}^{\prime},...,s_{j+1},s_{j+1}^{\prime},s_{j})\in\Xi_{j}$ if
1) $(...,s_{k},...,s_{j+1},s_{j})\in\chi_{j}$ and
$(...,s_{k}^{\prime},...,s_{j+1}^{\prime},s_{j})\in\chi_{j}$
2) For some outcome, $\Pi_{i}$, $s_{j}\in A_{ij}$ and for all $k>j$ (s.t.
$\lambda_{k},\in L$) $s_{k},s_{k}^{\prime}\in A_{ik}$.
$\sum_{i}\Pi_{i}\Pi_{i}^{\dagger}=\mathbb{I}$ can then be expanded to
Condition 1:
$\displaystyle\int_{\Xi_{1}}ds_{1}ds_{2}ds_{2}^{\prime}$
$\displaystyle...|s_{2},\lambda_{2}><s_{2},\lambda_{2}|s_{1},\lambda_{1}>$
$\displaystyle<s_{1},\lambda_{1}|s_{1}^{\prime},\lambda_{1}><s_{2}^{\prime},\lambda_{2}|...=\mathbb{I}$
Start by integrating over $s_{1}$; the above identity can not hold unless for
every $(...,s_{2},s_{2}^{\prime})=(\tilde{s},\tilde{s}^{\prime})\in
Dom(\Xi_{1})$ either
$<s_{2},\lambda_{2}|(\int_{(\tilde{s},\tilde{s}^{\prime},s_{n})\in\Xi_{1}}ds_{1}|s_{1},\lambda_{1}><s_{1},\lambda_{1}|)|s_{2}^{\prime},\lambda_{2}>=1$
or
$<s_{2},\lambda_{2}|(\int_{(\tilde{s},\tilde{s}^{\prime},s_{n})\in\Xi_{1}}ds_{1}|s_{1},\lambda_{1}><s_{1},\lambda_{1}|)|s_{1}^{\prime},\lambda_{2}>=0$.
When $\tilde{s}=\tilde{s}^{\prime}$ the first equality clearly holds. For
$\tilde{s}\neq\tilde{s}^{\prime}$, the only way that we could have an $s_{1}$
s.t. $(\tilde{s},s_{1})\in\chi_{1}$, and
$(\tilde{s}^{\prime},s_{1})\in\chi_{1}$, but
$(\tilde{s},\tilde{s}^{\prime},s_{1})\notin\Xi_{1}$, and still have one of
these equalities hold is if either $<s_{2},\lambda_{2}|s_{1},\lambda_{1}>=0$
or $<s_{2}^{\prime},\lambda_{2}|s_{1},\lambda_{1}>=0$ (this makes the claim of
$(\tilde{s},s_{1})\in\chi_{1}$, and $(\tilde{s}^{\prime},s_{1})\in\chi_{1}$
uncomfortable, but the situation can not be ruled out). Since we are
interested in the conditions under which the probabilities are guaranteed to
sum to $1$ regardless of the details of inner-products, we eliminate this last
case and are lead to:
Condition 2: If $(\tilde{s},\tilde{s}^{\prime})\in Dom(\Xi_{1})$ then for all
$s_{1}$ s.t. $(\tilde{s},s_{1})\in\chi_{1}$ and
$(\tilde{s}^{\prime},s_{1})\in\chi_{1}$,
$(\tilde{s},\tilde{s}^{\prime},s_{1})\in\Xi_{1}$.
Which is equivalent to applying the $q_{S}$ condition at $\lambda_{1}$.
Apply Condition 2 to Condition 1, Condition 1 is reduced to:
Condition 3:
$\displaystyle\int_{\Xi_{1}}ds_{2}ds_{3}ds_{3}^{\prime}$
$\displaystyle...|s_{3},\lambda_{3}><s_{3},\lambda_{3}|s_{2},\lambda_{2}>$
$\displaystyle<s_{2},\lambda_{2}|s_{3}^{\prime},\lambda_{3}><s_{3}^{\prime},\lambda_{3}|...=\mathbb{I}$
Which is identical in form to the Condition 1. Therefore, to satisfy Condition
$1$, it is sufficient to satisfy Condition 1, for all elements of $L$ to
satisfy Condition 2 (with the “$1$” subscript replaced with “$i$”). Note that
Condition 2 was imposed as a necessary condition for Condition 1 to hold; we
now see that it’s a necessary and sufficient condition. Finally, note that
requiring Condition 2 on all $\lambda_{i}$ is equivalent to requiring that the
set of outcomes is an element of $q_{S}$.
###### Theorem 161.
For dynamic set $S$
1) If $\gamma\in q_{S}$ then $\gamma$ is nearly compatible
2) If $S$ is interconnected, $\gamma\in\Gamma_{S}^{s}$, and $\gamma$ is nearly
compatible then $\gamma\in q_{S}$
###### Proof.
By Thm 150, for all $\gamma\in\Gamma_{S}^{s}$, all $\alpha\in\gamma$ are
companionable
1) For $\alpha,\beta\in\gamma$ assume
$\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}\neq\emptyset$,
and $\alpha\neq\beta$. $\lambda$ can not be an upper-bound on any $L(\gamma)$.
Take any
$\bar{p}[-\infty,\lambda]\in\alpha_{\rightarrow(\lambda,p)}\bigcap\beta_{\rightarrow(\lambda,p)}$,
$\bar{p}^{\prime}[-\infty,\lambda]\in\alpha_{\rightarrow(\lambda,p)}$,
$\bar{p}_{1}[\lambda,\infty]\in\alpha_{(\lambda,p)\rightarrow}$,
$\bar{p}_{2}[\lambda,\infty]\in\beta_{(\lambda,p)\rightarrow}$.
$\bar{p}_{1}\equiv\bar{p}[-\infty,\lambda]\circ\bar{p}_{1}[\lambda,\infty]\in\alpha$
$\bar{p}_{1}^{\prime}\equiv\bar{p}^{\prime}[-\infty,\lambda]\circ\bar{p}_{1}[\lambda,\infty]\in\alpha$
$\bar{p}_{2}\equiv\bar{p}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]\in\beta$
$\bar{p}_{2}^{\prime}\equiv\bar{p}^{\prime}[-\infty,\lambda]\circ\bar{p}_{2}[\lambda,\infty]$
(Exists by Thm 151)
Applying the definition of $q_{S}$ to any $\lambda^{\prime}\in L$ s.t.
$\lambda^{\prime}>\lambda$, it follows that $\bar{p}_{2}^{\prime}\in\beta$.
Therefore $\alpha_{\rightarrow(\lambda,p)}=\beta_{\rightarrow(\lambda,p)}$.
2) Assume $\gamma\in\Gamma_{S}^{s}$ is nearly compatible. Take any
$L\in\mathcal{L}(\gamma)$, any $\lambda_{1}\in L$ s.t. $\lambda_{1}\neq
glb(L)$, and define $\lambda_{0}\equiv pred_{L}(\lambda)$. Take any
$\alpha\in\gamma$ and any $\bar{p}_{1},\bar{p}_{2}\in\alpha$ that converge at
$\lambda_{1}$ (if they exist). Take $\bar{p}_{3},\bar{p}_{4}\in S$ such that
$\bar{p}_{3}[-\infty,\lambda_{0}]=\bar{p}_{1}[-\infty,\lambda_{0}]$,
$\bar{p}_{4}[-\infty,\lambda_{0}]=\bar{p}_{2}[-\infty,\lambda_{0}]$, and
$\bar{p}_{3}$ and $\bar{p}_{4}$ converge at $\lambda_{1}$. Further take
$\bar{p}_{3}\in\beta\in\gamma$. The theorem is proved if
$\bar{p}_{4}\in\beta$.
Because $(\bar{p}_{1},\bar{p}_{2},\bar{p}_{3},\bar{p}_{4})$ is an interconnect
on $[\lambda_{0},\lambda_{1}]$, and $S$ is interconnected, there’s a
$\lambda^{\prime}\in[\lambda_{0},\lambda_{1}]$ and
$\bar{p}_{1}^{\prime},\bar{p}_{2}^{\prime},\bar{p}_{3}^{\prime},\bar{p}_{4}^{\prime}\in
S$ s.t.
$\bar{p}_{i}^{\prime}[-\infty,\lambda_{0}]=\bar{p}_{i}[-\infty,\lambda_{0}]$,
$\bar{p}_{i}^{\prime}[\lambda_{1},\infty]=\bar{p}_{i}[\lambda_{1},\infty]$,
$\bar{p}_{1}^{\prime}[-\infty,\lambda^{\prime}]=\bar{p}_{3}^{\prime}[-\infty,\lambda^{\prime}]$,
$\bar{p}_{2}^{\prime}[-\infty,\lambda^{\prime}]=\bar{p}_{4}^{\prime}[-\infty,\lambda^{\prime}]$,
and all
$\bar{p}_{i}^{\prime}(\lambda^{\prime})=\bar{p}_{j}^{\prime}(\lambda^{\prime})=p$.
Since $\bar{p}_{i}^{\prime}$ and $\bar{p}_{i}$ are equal on $L$, and all
elements of $\gamma$ are determined on $L$, they are co-located on $\gamma$.
Therefore $\bar{p}_{1}^{\prime},\bar{p}_{2}^{\prime}\in\alpha$ and
$\bar{p}_{3}^{\prime}\in\beta$. Since
$\bar{p}_{1}^{\prime}[-\infty,\lambda^{\prime}]=\bar{p}_{3}^{\prime}[-\infty,\lambda^{\prime}]\in\alpha_{\rightarrow(\lambda^{\prime},p)}\bigcap\beta_{\rightarrow(\lambda^{\prime},p)}$
and $\gamma$ is nearly compatible,
$\alpha_{\rightarrow(\lambda^{\prime},p)}=\beta_{\rightarrow(\lambda^{\prime},p)}$.
Since
$\bar{p}_{4}^{\prime}[-\infty,\lambda^{\prime}]=\bar{p}_{2}^{\prime}[-\infty,\lambda^{\prime}]\in\alpha_{\rightarrow(\lambda^{\prime},p)}$,
$\bar{p}_{4}^{\prime}[-\infty,\lambda^{\prime}]\in\beta_{\rightarrow(\lambda^{\prime},p)}$.
Since
$\bar{p}_{4}^{\prime}[\lambda_{1},\infty]=\bar{p}_{4}[\lambda_{1},\infty]=\bar{p}_{3}[\lambda_{1},\infty]$,
for all $\lambda\in L$, $\bar{p}_{4}^{\prime}(\lambda)\in\beta(\lambda)$, so
$\bar{p}_{4}^{\prime}\in\beta$. Since $\bar{p}_{4}$ and $\bar{p}_{4}^{\prime}$
are equal on all $\lambda\in L$, $\bar{p}_{4}\in\beta$. ∎
#### V.2.2 The Non-Conditional Case
For the non-conditional case, no initial state is assumed. Once again
representing each outcome, $\Pi_{i}$, as a product of projection operators,
$\Pi_{i}=...\mathbb{P}(A_{ij};\lambda_{j})...\mathbb{P}(A_{i1};\lambda_{1})$,
the probability of a given outcome is
$P_{i}=\frac{1}{Tr(\mathbb{I})}Tr(\Pi_{i}\Pi_{i}^{\dagger})$ (ignoring
complications that arise if $Tr(\mathbb{I})$ is infinite). This can be arrived
at by assuming that, if the initial state is unknown, then the probability is
the average for all possible initial states: Given any initial state,
$(s,\lambda_{0})$, the probability for obtaining outcome $i$ is
$P_{i|(s,\lambda_{0})}=<s,\lambda_{0}|\Pi_{i}\Pi_{i}^{\dagger}|s,\lambda_{0}>$.
With $V$ the volume of $S(\lambda_{0})$, the average probability is then
$\frac{1}{V}\Sigma_{s\in
S(\lambda_{0})}<s,\lambda_{0}|\Pi_{i}\Pi_{i}^{\dagger}|s,\lambda_{0}>$. Since
$V=Tr(\mathbb{I})$ and $\Sigma_{s\in
S(\lambda_{0})}<s,\lambda_{0}|\Pi_{i}\Pi_{i}^{\dagger}|s,\lambda_{0}>=Tr(\Pi_{i}\Pi_{i}^{\dagger})$,
this average probability is equal to $P_{i}$ given above.
In the non-conditional case, the total probability is therefore guaranteed to
be $1$ iff $\sum_{i}Tr(\Pi_{i}\Pi_{i}^{\dagger})=Tr(\mathbb{I})$.
Note that the assumption that the non-conditional probability is equal to the
average conditional probabilities need not hold for a dps; there is nothing in
the nature of the dps that demands it. The assumption is equivalent to a
certain kind of additivity: Take $\alpha$ to be an outcome that’s $bb\lambda$,
define $x\equiv\alpha(\lambda)$, and select $x_{1},x_{2}$ s.t. $x_{1}\bigcup
x_{2}=x$ and $x_{1}\bigcap x_{2}=\emptyset$; define
$\alpha_{1}\equiv\alpha\bigcap S_{\rightarrow(\lambda,x_{1})}$ and
$\alpha_{2}\equiv\alpha\bigcap S_{\rightarrow(\lambda,x_{2})}$; the assumption
being made is that under such circumstances,
$P(\alpha)=P(\alpha_{1})+P(\alpha_{2})$.
We would expect this to hold if $\alpha$ is bounded from above by $\lambda$,
but not if it’s bounded from below. To see that it will hold when bounded from
above, assume $\alpha$ is $ba\lambda$ (so the measurement of
$\\{x_{1},x_{2}\\}$ would be the final measurement, rather than the initial
one), take $\gamma$ to be any ip s.t. $\alpha\in\gamma$, and define
$A\equiv\gamma-\\{\alpha\\}.$ $A\bigcup\\{\alpha_{1},\alpha_{2}\\}$ is an ip.
If $P(A)$ is unchanged by whether $A$ is paired with $\alpha$ or
$\\{\alpha_{1},\alpha_{2}\\}$ then $P(\alpha)=P(\alpha_{1})+P(\alpha_{2})$.
Call this property “additivity of final state”; call the bounded from below
case “additivity of initial state”.
Achieving additivity of initial state is not quite as straightforward as
additivity final of state. One obvious way to do so is to assume additivity of
initial state and reversibility. There are, however, other ways for the
behavior to be realized. (See Appendix C for the definition of reversibility
on a DPS.)
### V.3 Maximal Quantum Systems
The foundations of quantum physics may be thought of as being composed of
three interconnected parts: measurement theory, probability theory, and
probability dynamics. This article has largely been concerned with creating a
mathematical language sufficient for utilization in the measurement theory &
that portion of the probability theory implied by the measurement theory. One
does not expect a scientific theory to follow immediately from the
mathematical language that it uses, unless the theory is fairly trivial. For
this reason, Thm 161 is quite evocative.
Thm 161 says an interconnected dynamic system will posses the basic character
of a quantum system if the finite, discretely-determined portion of its dps
can be embedded in a $\Upsilon$-maximal dps. The question now is, under what
conditions is such an embedding possible?
All countable, discretely-determined ip’s will have consistent probabilities
if the system is countably additive in its final state. As mentioned earlier,
it may be assumed that a dynamic system has this property. To see why it is
sufficient, start by choosing any $\lambda$, and any countable partition of
$S(\lambda)$. By assumption, the total probability of the associated outcomes
will be $1$. For any of these outcome, optionally choose any
$\lambda^{\prime}>\lambda,$ and any countable partition of
$S(\lambda^{\prime})$; by assumption, the total probability of these new
outcomes will equal the probability of the original outcome. Continuing in
this way, one can create all countable, discretely-determined ip’s of the
dynamic space.
To understand when such a dps could be maximal, consider a simple case of a
set of outcomes that are nearly compatible, but not compatible. Assume there
are $p_{1,0},p_{1,1,}p_{1,2}\in S(\lambda_{1})$ and
$p_{2,0},p_{2,1,}p_{2,2}\in S(\lambda_{2})$ ($\lambda_{1}<\lambda_{2}$) s.t.
each of the $p_{2,j}$’s can be reached from $(\lambda_{1},p_{1,j})$ and
$(\lambda_{1},p{}_{1,(j+1)\,mod\,3})$, but not
$(\lambda_{1},p{}_{1,(j+2)\,mod\,3})$. For $p_{2,0}$ and $p_{2,1}$ form
outcomes consisting of both of the $p_{1,i}$ that can reach them:
$O_{0}=(\lambda_{1},\\{p{}_{1,0},p{}_{1,1}\\})\rightarrow(\lambda_{2},p_{2,0})$
and
$O_{1}=(\lambda_{1},\\{p{}_{1,1},p{}_{1,2}\\})\rightarrow(\lambda_{2},p_{2,1})$.
For $p_{2,2}$ form outcomes from $p_{1,2}$ and $p_{1,0}$ individually:
$O_{21}=(\lambda_{1},p{}_{1,2})\rightarrow(\lambda_{2},p_{2,2})$ and
$O_{22}=(\lambda_{1},p{}_{1,0})\rightarrow(\lambda_{2},p_{2,2})$. The
collection of these $4$ outcomes is nearly compatible, but not compatible.
That they are nearly compatible can be seen from the fact that they satisfy
the $q_{S}$ condition; that they are not compatible can be seen from the fact
that while the combination of the first two outcomes imply that only
$\\{p_{1,0},p_{1,1,}p_{1,2}\\}$ was measured at $\lambda_{1}$, the second two
imply that $p_{1,2}$ was distinguished from $p_{1,0}$ at $\lambda_{1}$.
The inclusion of such sets of outcomes into a t-algebra would make
probabilities more highly additive; in particular, it would weaken the non-
additivity of double-slit experiments. To see this, take the simplest case of
$S(\lambda_{i})=\\{p_{i,0},p_{i,1,}p_{i,2}\\}$; because
$\\{O_{0},O_{1},O_{21}\bigcup O_{22}\\}$ is compatible, and would certainly be
on the t-algebra, we’d then have $P(O_{21}\bigcup
O_{22})=P(O_{21})+P(O_{22})$, rendering the probabilities of this particular
double-slit experiment entirely additive. However, if there exists a $p\in
S(\lambda_{2})$ that can be reached by $p_{1,0}$, $p_{1,1}$, and $p_{1,2}$,
then the $q_{S}$ condition entails that there can be no element of $q_{S}$
that contains $O_{0}$, $O_{1}$, $O_{21}$, and $O_{22}$; in this case a maximal
t-algebra can not include $\\{O_{0},O_{1},O_{21},O_{22}\\}$, and so will not
demand $P(O_{21}\bigcup O_{22})=P(O_{21})+P(O_{22})$.
This illustrates how the finite & finitely determined portion of a dps can be
maximal even if its probabilities are highly non-additive. If the paths of a
dynamic system form a richly interlocking network, then the elements of
$q_{S}$ that are not ip’s will be limited, making it more likely that the
system will be maximal. A rich network of paths will not limit the kinds of
countable, discretely determined experiments that can be performed, it simply
limits the amount of probabilistic information that can be extracted from
them. A similar effect was seen previously when considering
interconnectedness.
For non-deterministic systems, it is difficult to distinguish sets of paths
that can not occur from sets of paths that occur with probability $0$. If the
approach to system dynamics is to include paths unless they can be ruled out
in principal and let the probability function handle the rest, then the
network of interlocking paths will be enriched. This will generally cause the
discretely determined portion of such systems to be maximal. Let’s see this in
detail.
###### Definition 162.
If $\gamma_{1}$ and $\gamma_{2}$ are partitions of some set, $S$,
$\gamma_{1}\prec\gamma_{2}$ if for all $\alpha\in\gamma_{1}$ there’s a
$\beta\in\gamma_{2}$ s.t. $\alpha\subset\beta$
As mentioned above, partitions with total probability of $1$ can be
iteratively created by taking any $\gamma\in\Gamma_{S}^{s}$ with probability
known to be $1$, and for each $\alpha\in\gamma$, select some $\lambda$ s.t.
$\alpha$ is $ba\lambda$, and slice up $\alpha$ by partitioning
$\alpha(\lambda)$. However, not all partitions can be formed in this manner.
If the partition is not compatible, and so can not be decided by an ideal
e-automata, it is allowed to forget at $\lambda$ some of what happened prior
to $\lambda$. In such cases, to form the new partition, you wouldn’t simply
take the elements of $\gamma$ and append measurements at $\lambda$, a further
step would also be possible: for each measurement outcome at $\lambda$,
multiple elements of $\gamma$ may be combined into a single outcome. More
precisely, select some countable set of $\gamma_{i}\succ\gamma$ such that each
$\gamma_{i}\in\Gamma_{S}^{s}$. For every $p\in S(\lambda_{n})$, select a
$\gamma_{i}$. For every $\gamma_{i}$, every $\alpha\in\gamma_{i}$, create a
countable partition of the set of $p\in S(\lambda_{n+1})$ that “selected”
$\gamma_{i}$, $X_{i,\alpha}$, and for every $A\in X_{i,\alpha}$ form the new
outcome $\alpha_{\rightarrow(\lambda_{n},A)\rightarrow}$. The set of all such
outcomes forms a new element of $\Gamma_{S}^{s}$. Under these general
conditions, we can not comfortably assume that our new partition has a total
probability of $1$.
The $q_{S}$ condition may be seen as a constraint on this formation process;
whenever a $\gamma_{i}\succ\gamma$ is selected for one $p\in S(\lambda)$, the
$q_{S}$ condition constrains what may happen at the other elements of
$S(\lambda)$. If this constraint implies that for all $\gamma\in q_{S}$, all
$\lambda$ s.t. $\gamma$ is $ba\lambda$, all $p\in S(\lambda)$ must select the
same $\gamma_{i}\succ\gamma$, and the selected $\gamma_{i}$ must itself be an
element of $q_{S}$, then we can expect all elements of $q_{S}$ to have a total
probability of $1$.
To understand the effect that combining outcomes at one $p\in S(\lambda)$ has
at the other points in $S(\lambda)$, it is sufficient to consider the
combination of two outcomes. Since the two outcomes, $\alpha,\beta\in\gamma$,
must form a discretely-determined set when combined into $\alpha\bigcup\beta$,
they must be chosen so that for all but one element of $L$,
$\alpha=\alpha_{\rightarrow(\lambda,\alpha(\lambda)\bigcup\beta(\lambda))\rightarrow}$
and
$\beta=\beta_{\rightarrow(\lambda,\alpha(\lambda)\bigcup\beta(\lambda))\rightarrow}$;
if $\alpha\neq\beta$, then $\alpha(\lambda)$ and $\beta(\lambda)$ must be
disjoint at the one remaining $\lambda\in L$.
Now to see the effect of the $q_{S}$ condition. Let’s start by considering a
pair of outcomes determined at a single $\lambda$,
$S_{\rightarrow(\lambda_{1},A)\rightarrow}$ and
$S_{\rightarrow(\lambda_{1},B)\rightarrow}$, and assume that at $p\in
S(\lambda_{2})$ these two are combined to form
$S_{\rightarrow(\lambda_{1},A\bigcup
B)\rightarrow(\lambda_{2},p)\rightarrow}$. The $q_{S}$ condition demands that
$A$ and $B$ must then also be combined for the outcomes at other elements of
$S(\lambda)$. To see which ones, some further definitions will be helpful.
(Note that $S_{(\lambda_{1},A)\rightarrow(\lambda_{2},p)}(\lambda_{1})$ is the
set of elements of $A$ that can reach $p$.)
###### Definition 163.
If $S$ is a dynamic set, $\lambda_{1}\leq\lambda_{2}$, $A\subset
S(\lambda_{1})$, $p\in S(\lambda_{2})$ then
$(A,\lambda_{1})\downarrow(p,\lambda_{2})\equiv
S_{(\lambda_{1},A)\rightarrow(\lambda_{2},p)}(\lambda_{1})$
For $X\subset S(\lambda_{2})$ ,
$(A,\lambda_{1})\downarrow(X,\lambda_{2})\equiv
S_{(\lambda_{1},A)\rightarrow(\lambda_{2},X)}(\lambda_{1})$.
$(A,\lambda_{1})\downarrow(p,\lambda_{2})$ may be thought of as $p$’s
footprint in $A$.
###### Definition 164.
$p^{\prime}\in[p,\lambda_{2};A,\lambda_{1}]$ if
$(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})\bigcap(A,\lambda_{1})\downarrow(p,\lambda_{2})\neq\emptyset$.
$p^{\prime}\in[p,\lambda_{2};A,\lambda_{1}]$ if $p$ and $p^{\prime}$’s
footprints overlap, so $p^{\prime}\in[p,\lambda_{2};A,\lambda_{1}]$ if there
exists an element of $A$ that can reach both $p$ and $p^{\prime}$. According
to the $q_{S}$ condition, if outcomes $A$ and $B$ combine at $p$, and
$p^{\prime}\in[p,\lambda_{2};A,\lambda_{1}]\bigcap[p,\lambda_{2};B,\lambda_{1}]$,
then they must combine at $p^{\prime}$. If
$p^{\prime\prime}\in[p^{\prime},\lambda_{2};A,\lambda_{1}]\bigcap[p^{\prime},\lambda_{2};B,\lambda_{1}]$,
they then must also combine at $p^{\prime\prime}$. This leads to:
###### Definition 165.
If $S$ is a dynamic set, $\lambda_{1}\leq\lambda_{2}$, $p\in S(\lambda_{2})$,
and $A$ and $B$ are disjoint subsets of $S(\lambda_{1})$:
$\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|_{0}\equiv[\lambda_{2},p;\lambda_{1},A]\bigcap[\lambda_{2},p;\lambda_{1},B]$
$\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|_{n+1}\equiv\bigcup_{p^{\prime}\in\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|_{n}}\left\|p^{\prime},\lambda_{2};A,B,\lambda_{1}\right\|_{0}$
$\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|\equiv\bigcup_{n\in\mathbb{N}}\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|_{n}$
If $A$ and $B$ combine at $p$, then the $q_{S}$ condition demands that they
must combine at all elements of
$\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$.
There may be cases of $p^{\prime}\in S(\lambda_{2})$ s.t. for some
$p_{1}\in\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$,
$(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})\bigcap(A,\lambda_{1})\downarrow(p_{1},\lambda_{2})\neq\emptyset$,
and for some $p_{2}\in\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$,
$(B,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})\bigcap(B,\lambda_{1})\downarrow(p_{2},\lambda_{2})\neq\emptyset$,
but there are no elements of $\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$
for which both hold, and so $p^{\prime}$ is not an element of
$\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$. This is a generalization of
the example seen earlier. If such cases occur, the $q_{S}$ condition will be
insufficient; they are eliminated if paths form a densely interlocking
network. More precisely:
1) Form the footprints of $\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$ in
$A$ and $B:$
$x\equiv(A,\lambda_{1})\downarrow(\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|,\lambda_{2})$
and
$y\equiv(B,\lambda_{1})\downarrow(\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|,\lambda_{2})$
2) Close the footprints with all intersecting footprints. That is, define:
$X_{0}\equiv x$
$X_{n+1}\equiv\bigcup\\{(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2}):(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})\bigcap
X_{n}\neq\emptyset\\}$
$X\equiv\bigcup_{n\in\mathbb{N}}X_{n}$
Similarly, starting with $y$, construct $Y$.
3) For the $q_{S}$ condition to be sufficient: For any
$p^{\prime}\notin\left\|p,\lambda_{2};A,B,\lambda_{1}\right\|$, if
$(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})$ is in $X$ then
$(B,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})$ must be disjoint from $Y$,
and if $(B,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})$ is in $Y$ then
$(A,\lambda_{1})\downarrow(p^{\prime},\lambda_{2})$ must be disjoint from $X$.
With $a=A-X$ and $b=B-Y$, combining
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{2},p)\rightarrow}$ and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{2},p)\rightarrow}$ under
the $q_{S}$ condition then has the effect of replacing $\\{A,B\\}$ with
$\\{a,b,X\bigcup Y\\}$; the resulting partition will therefore have a total
probability of $1$. It is interesting to note that statement (3) will be
satisfied if the paths are either quite dense or quite sparse; only the
intermediate case may cause difficulty.
Combining pairs of outcomes that are determined at multiple $\lambda$ lead to
similar conclusions. Once again, in order for the combination to form a
discretely-determined set, the two outcomes can only disagree at a single
$\lambda$. If the two outcomes disagree on the last measurement then the
analysis is little changed from above, except that we need only consider the
subset of points in $A$ & $B$ that can be reached from the prior measurements.
The trickier case is when a sequence of further measurements come after the
two being combined; for example, the case where
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$
and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$
are combined. To see what the $q_{S}$ condition demands of the other
$p^{\prime}\in S(\lambda_{3})$ in such cases, it will be helpful to expand
some the above definitions:
###### Definition 166.
If $S$ is a dynamic set, $\lambda_{1}\leq\lambda_{2}$, $A\subset
S(\lambda_{1})$, $\mathcal{Z}$ is a set of subsets of $S(\lambda_{2})$, and
$Z\in\mathcal{Z}$:
$Z^{\prime}\in[Z,\mathcal{Z},\lambda_{2};A,\lambda_{1}]$ if
$Z^{\prime}\in\mathcal{Z}$ and
$(A,\lambda_{1})\downarrow(Z^{\prime},\lambda_{2})\bigcap(A,\lambda_{1})\downarrow(Z,\lambda_{2})\neq\emptyset$
If, further, $A$ and $B$ are disjoint subsets of $S(\lambda_{0})$:
$\left\|Z,\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|_{0}\equiv[\lambda_{2},\mathcal{Z},Z;\lambda_{1},A]\bigcap[\lambda_{2},\mathcal{Z},Z;\lambda_{1},A]$
$\left\|Z,\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|_{n+1}\equiv\bigcup_{Z^{\prime}\in\left\|Z,\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|_{n}}\left\|Z^{\prime},\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|_{0}$
$\left\|Z,\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|\equiv\bigcup_{n\in\mathbb{N}^{+}}\left\|Z,\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|_{n}$
If for each $p\in S(\lambda_{3})$, $Z_{p}\equiv
S_{(\lambda_{1},A)\rightarrow(\lambda_{3},p)}(\lambda_{2})$, and
$\mathcal{Z}\equiv\\{Z_{p}:p\in S(\lambda_{3})\\}$ then
$Z_{p^{\prime}}\in[Z_{p},\mathcal{Z},\lambda_{2};A,\lambda_{1}]$ iff
$p^{\prime}\in[p,\lambda_{3};A,\lambda_{1}]$; from this it follows that
$p^{\prime}\in\left\|p,\lambda_{3};A,B,\lambda_{1}\right\|$ iff
$Z_{p^{\prime}}\in\left\|Z_{p},\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|$.
This allows us to take our earlier analysis on combining
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{3},p)\rightarrow}$ and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{3},p)\rightarrow}$, and
project it onto any $\lambda_{2}\in(\lambda_{1},\lambda_{3})$: if
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{3},p)\rightarrow}$ and
$S_{\rightarrow(\lambda_{10},B)\rightarrow(\lambda_{3},p)\rightarrow}$ are
combined, then
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{3},p^{\prime})\rightarrow}$
and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{3},p^{\prime})\rightarrow}$
must be combined if
$Z_{p^{\prime}}\in\left\|Z_{p},\mathcal{Z},\lambda_{2};A,B,\lambda_{1}\right\|$.
To extend this to combining
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$
and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$,
simply replace $Z_{p}$ with $Z_{p,C}\equiv Z_{p}\bigcap C$ and $\mathcal{Z}$
with $\mathcal{Z}_{C}\equiv\\{Z_{p,C}:p\in S(\lambda_{3})\\}$. If
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$
and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p)\rightarrow}$
are combined then
$S_{\rightarrow(\lambda_{1},A)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p^{\prime})\rightarrow}$
and
$S_{\rightarrow(\lambda_{1},B)\rightarrow(\lambda_{2},C)\rightarrow(\lambda_{3},p^{\prime})\rightarrow}$
must be combined if
$Z_{p^{\prime},C}\in\left\|Z_{p,C},\mathcal{Z}_{C},\lambda_{2};A,B,\lambda_{1}\right\|$.
This leads by the same reasoning to the same conclusion: if the network of
paths is sufficiently dense (or sparse), the finite and finitely determined
partitions that satisfy the $q_{S}$ condition can be expected to have a total
probability of $1$.
It follows that if the dynamic set in a dps follows the rule that paths are
included unless they can be excluded in principle, then we can reasonably
expect the discretely-determined portion of the dps to be maximal.
### V.4 Terminus & Exordium
A word or two is in order about dps’s, $(X,T,P)$, for which $X$ is not a
singleton. In such cases $T_{\mathbb{N}}^{S}$ may be larger than
$T_{S\mathbb{N}}$, and so admit partitions that are not nearly-compatible (and
therefore not elements of $q_{S}$). If the various $(\\{S\\},T_{S})$ are
copies of one another, in the sense of the reductions introduced in Sec.
IV.2.4, then $T_{\mathbb{N}}^{S}=T_{S\mathbb{N}}$. For quantum systems, if we
consider the various Hilbert space bases as generating the various elements of
$X$, we can partially assert that this is the case. From these bases, each
$S(\lambda)$ is equipped with a $\Sigma$-algebra, and outcomes are of the form
$S_{\rightarrow(\lambda_{1},\sigma_{1})\rightarrow...\rightarrow(\lambda_{n},\sigma_{n})\rightarrow}$
where the $\sigma$’s are elements of their respective $\Sigma$-algebra’s. For
any $S,S^{\prime}\in X$, $\lambda\in\Lambda$, the $\Sigma$-algebra’s on
$S(\lambda)$ and $S^{\prime}(\lambda)$ are isomorphic. This gets us part of
the way there. One further piece is missing: while the outcomes are
isomorphic, the dynamics may not be. In particular, it’s possible to have a
case where two points, $(\lambda,p_{1})$ and $(\lambda,p_{2})$, can transition
to $(\lambda^{\prime},p^{\prime})$, but under isomorphisms from
$(S(\lambda),\Sigma)$ to $(S^{\prime}(\lambda),\Sigma^{\prime})$ only one of
them can. It would then be possible to have $\gamma\in G_{TS^{\prime}}$, but
its isomorphic image not be an element of $G_{TS}$; $T_{\mathbb{N}}^{S}$ may
then be larger than $T_{S\mathbb{N}}$.888Then again it may not. For example,
if the transitions are different because the states are deterministic, or
conserved, in $S_{1}$ but not in $S_{2}$, this will not effect the existence
of a reduction. That is because, when states are deterministic, the order in
which measurements occur is irrelevant, so many different measurement
sequences will result in the same ip.
However, in quantum mechanics we would not expect any such differing dynamics
to effect the existence of a reduction between bases; for a given element of
$q_{S}$ we would expect to be able to select sequences of projection operators
s.t., under a given basis transformation, the transformed representation will
also be an element of $q_{S}$. More generally, the discretely determined
probabilities ought to be sufficient to determine all inner-products in a
given bases, which then entail the discretely determined probabilities in all
other bases. Dps’s in the various bases are therefore, in a very real sense,
copies of each other.
Moreover, it’s not clear that all bases are experimentally relevant;
naturally, if a basis is not experimentally relevant, then it can’t contribute
any information to $T_{\mathbb{N}}^{S}$. Unsurprisingly, given the subject
matter of physics, when a basis is given an experimental interpretation, it’s
generally in terms of paths though space. For example, when the conjugate-
coordinate basis is interpreted as momentum, it is necessarily being given a
spacial path interpretation; an interpretation predicated on the conjugate-
coordinate’s mathematical relation to aspects of spacial path probabilities
such as the probability flux. This connection between the conjugate-coordinate
basis and the particle’s spacial path is put on display when the value of the
conjugate-coordinate is determined by tracking particle paths in a detector.
However, because quantum systems satisfy all the dps axioms, even if
$T_{\mathbb{N}}^{S}$ is larger than $T_{S\mathbb{N}}$, this fact will have to
be reflected in the quantum probabilities. We don’t need results like Thm 161
to conclude that quantum systems are dps’s, we know that without them. What
such results do tell us is that, at least with regard to their measurement
theory and a central core of their probability theory, quantum systems appear
to be nothing more than fairly generic dps’s.
Indeed, one may start to suspect that quantum mechanics, as it currently
stands, is a phenomenological theory. It certainly is not holy writ, handed
down from on high, which we can’t possibly hope to truly understand, but which
we must none the less accept in all detail. Rather, it evolved over time by
trial and error as a means of calculating results to match newly discovered
experimental phenomenon. It has been quite successful in achieving that goal;
however it has also historically yielded little understanding of why its
calculational methods work. This lack of understanding has often led to claims
that we don’t understand these matters because we can’t understand them; they
lie outside the sphere of human comprehension. While we can not state with
certainty that such bold claims of necessary ignorance are false, we can say
that they are scientifically unfounded, and so ought to be approached with
skepticism. With all humility, it is hoped that this article has lent strength
to such skepticism, and shed light on some matters that have heretofore been
allowed to lie in darkness.
## Appendix A $\Delta$-Additivity
Quantum probabilities posses an interesting algebraic property that is not a
direct consequence of their t-algebra.
For any $t\in T$ s.t. $\\{\bigcup t\\}$ is also an element of $T$, define
$\Delta(t)\equiv P(t)-P(\\{\bigcup t\\})$. The $\Delta$ function measures the
additivity of a gps; if a gps has additive probabilities, then $\Delta(t)=0$
for all $t$. Quantum probabilities are not additive, but their $\Delta$
function is. For any $t\in T$ s.t. $\\{\bigcup t\\}\in T$ and for all pairs
$\alpha,\beta\in t$, $\\{\alpha,\beta\\},\\{\alpha\bigcup\beta\\}\in T$,
$\Delta(t)=\sum_{pairs\,of\,\alpha,\beta\in t}\Delta(\\{\alpha,\beta\\})$
This is an immediate consequence of quantum probabilities being calculated
from expressions of the form
$<\psi,\lambda_{0}|\Pi\Pi^{\dagger}|\psi,\lambda_{0}>$, where $\Pi$ is a
product of projection operators.
Any gps that obeys this rule is _$\Delta$ -additive_.
It follows from the definition of $\Delta$ that for disjoint $t_{1}$ and
$t_{2}$, $\Delta(t_{1}\cup t_{2})=\Delta(t_{1})+\Delta(\\{\bigcup t_{1}\\}\cup
t_{2})$. Applying this to the above formula, we get
$\Delta(\\{\alpha_{1}\cup\alpha_{2},\alpha_{3}\\})=\Delta(\\{\alpha_{1},\alpha_{3}\\})+\Delta(\\{\alpha_{2},\alpha_{3}\\})$,
which is directly analogous to the additivity seen in classic probabilities.
If $t$ has more than $3$ elements then
$\Delta(\\{\alpha_{1}\cup\alpha_{2},\alpha_{3},...\\})=\Delta(\\{\alpha_{1},\alpha_{3},...\\})+\Delta(\\{\alpha_{2},\alpha_{3},...\\})-\Delta(\\{\alpha_{3},...\\})$.
In the orthodox interpretation, $\Delta$-additivity is generally viewed as
being due to path interference possessing wave-like properties . An
interesting question is, under what conditions will a system obeying the
intuitive interpretation display $\Delta$-additivity?
Start by defining a _moment partition_ to be any partition, $\gamma$, s.t. for
any $\lambda$, either all $\alpha\in\gamma$ are $bb\lambda$, or all
$\alpha\in\gamma$ are $ba\lambda$. (See Defn 147 for the definition of
“moment”, and Thm 148 for proof that a partition composed of moments is an
ip). Moment partitions have some very useful properties. First, for any
$\alpha,\beta\in\gamma$,
$(\gamma-\\{\alpha,\beta\\})\bigcup(\alpha\bigcup\beta)$ is also a moment
partition. Second, if $\gamma$ is a moment partition, and $\nu\in\gamma$, then
$\gamma_{\nu}\equiv\\{\nu,S-\nu\\}$ is also a moment partition. Indeed if
$\gamma$ is an ip, and for all $\nu\in\gamma$, $\\{\nu,S-\nu\\}$ is also an
ip, then $\gamma$ is a moment partition.
Now take any $\alpha\in t\in T$, any moment partition, $\gamma$, and define
$\alpha/\gamma\equiv\\{\alpha\bigcap\beta:\beta\in\gamma\>and\>\alpha\bigcap\beta\neq\emptyset\\}$.
$\Delta(\alpha/\gamma)$ is the effect the measurement $\gamma$ has on the
probability that $\alpha$ occurs.
For $\beta\in\alpha/\gamma$ take $P_{o}(\beta)$ to be the probability that
$\beta$ occurs if all interactions required to measure $\gamma$ are
vanishingly small (while all the interactions for measuring $\alpha$ are
unchanged). $P_{o}(\beta)$ may be thought of as the omniscient probability: we
simply know which element of $\gamma$ occurs without having to perform a
measurement. (In the intuitive interpretation, some $\beta$ occurs even if the
measurement does not take place.) It follows that
$P(\alpha)=\Sigma_{\beta\in\alpha/\gamma}P_{o}(\beta)$. $P_{o}$ is, of course,
simply a conceptual construct; it can only be experimentally determined if
$P_{o}=P$.
Now define $\delta(\beta)\equiv P(\beta)-P_{o}(\beta)$; $\delta(\beta)$
represents the amount of deflection into/out of $\beta$ due to the measurement
of $\gamma$. Since $P(\alpha)=\Sigma_{\beta\in\alpha/\gamma}P_{o}(\beta)$,
$\Delta(\alpha/\gamma)=\sum_{\beta\in\alpha/\gamma}\delta(\beta)$. Finally,
for $\beta\in\alpha/\gamma$, define $\neg\beta\equiv\alpha-\beta$.
###### Theorem 167.
1) For any outcome, $\alpha$, any countable moment partition, $\gamma$,
$\alpha/\gamma$ is $\Delta$-additive iff
$\sum_{\beta\in\alpha/\gamma}\delta(\beta)=\sum_{\beta\in\alpha/\gamma}\delta(\neg\beta)$.
2) A discretely determined dps is $\Delta$-additive iff for every outcome,
$\alpha$, every moment partition, $\gamma$, s.t. $\alpha/\gamma$$\in T$,
$\alpha/\gamma$ is $\Delta$-additive.
###### Proof.
1) Assume $\alpha/\gamma$ has $N\geq 3$ elements ($N=2$ is trivial). Note that
since probabilities $P_{o}$ are fully additive,
$(N-1)\sum_{\beta\in\alpha/\gamma}P_{o}(\beta)=$$\sum_{\beta\in\alpha/\gamma}\sum_{\eta\in\alpha/\gamma-\\{\beta\\}}P_{o}(\eta)=$$\sum_{\beta\in\alpha/\gamma}P_{o}(\neg\beta)$.
Adding $(N-1)\sum_{\beta\in\alpha/\gamma}P_{o}(\beta)$ to the left side of
$\sum_{\beta\in\alpha/\gamma}\delta(\beta)=\sum_{\beta\in\alpha/\gamma}\delta(\neg\beta)$,
and $\sum_{\beta\in\alpha/\gamma}P_{o}(\neg\beta)$ to the right, yields
$(N-2)P(\alpha)+\sum_{\beta\in\alpha/\gamma}P(\beta)=\sum_{\beta\in\alpha/\gamma}P(\neg\beta)$.
Now note that
$(N-1)\sum_{\beta\in\alpha/\gamma}P(\beta)=$$\sum_{\beta\in\alpha/\gamma}P(\alpha/\gamma-\\{\beta\\})$.
Subtracting $(N-1)\sum_{\beta\in\alpha/\gamma}P(\beta)$ from the left side,
and $\sum_{\beta\in\alpha/\gamma}P(\alpha/\gamma-\\{\beta\\})$ from the left,
yields
$(N-2)\Delta(\alpha/\gamma)=\sum_{\beta\in\alpha/\gamma}\Delta(\alpha/\gamma-\\{\beta\\})$.
It remains to show that this is equivalent to
$\Delta(\alpha/\gamma)=\sum_{pairs\,of\,\beta,\eta\in\alpha/\gamma}\Delta(\\{\beta,\eta\\})$.
They are clearly equivalent when $N=3$. Assume they are equivalent for $N=M$,
for $N=M+1$
$\Delta(\alpha/\gamma)=\frac{1}{N-2}\sum_{\beta\in\alpha/t}\Delta(\alpha/\gamma-\\{\beta\\})$
$=\frac{1}{N-2}\sum_{\beta\in\alpha/t}\Delta((\alpha-\beta)/\gamma)$
$=\frac{1}{N-2}\sum_{\beta\in\alpha/\gamma}\sum_{pairs\,of\,\kappa,\eta\in\alpha/\gamma-\\{\beta\\}}\Delta(\\{\kappa,\eta\\})$
$=\sum_{pairs\,of\,\beta,\eta\in\alpha/\gamma}\Delta(\\{\beta,\eta\\})$
2) Let’s say that $T$ is discretely determined, and for $t\in T$, all
$\alpha,\beta\in t$, $\\{\alpha\bigcup\beta\\}\in T$. This means that for some
$L\subset\Lambda$ s.t $t$ is determined on $L$, at all elements of $L$ except
one, all $\alpha\in T$ correspond to the same outcome.
Take $\lambda$ to be the element of $L$ at which the various $\alpha\in t$
correspond to different outcomes. Taking $\nu=\bigcup t$, there exists a
moment partition at $\lambda$, $\gamma$, s.t. $t=\nu/\gamma$. ∎
So the probabilities of an intuitive system will have the algebraic properties
of quantum theory if for all outcomes, $\alpha$, all countable moment
partitions, $\gamma$,
$\sum_{\beta\in\alpha/\gamma}\delta(\beta)=\sum_{\beta\in\alpha/\gamma}\delta(\neg\beta)$.
Let’s now see how this requirement may hold.
First note that, since $\delta$ measures the effect of the environment on the
system, experimental methods should be chosen so as to minimize $\delta$. One
way to do this is to perform “passive measurements”, a type of measurement
that corresponds to how double-slit experiments are generally pictured. In a
passive measurement, to determine the probabilities for elements of
$\alpha/\gamma$, start with some $\gamma_{0}\in G_{T}$ such that
$\alpha\in\gamma_{0}$. Perform the measurement of $\gamma_{0}$ as before, but
for $\eta\in\gamma$ block $\neg\eta$ from occurring; if $\neg\eta$ would have
occurred, you get a null result. Now if $\alpha$ occurs it means that
$\alpha\bigcap\eta$ occurred. The proportion of $\alpha$’s to all trials run
(including null results) is then $P(\alpha\bigcap\eta)$. Doing this in turn
for each $\eta\in\gamma$ yields the probabilities for $\alpha/\gamma$.
Imagine that position measurements on particles are performed in this manner.
In the intuitive interpretation, if $\delta(\eta)\neq 0$ it’s because the
blocking off of $\neg\eta$ creates some minimum but non-vanishing ambient
field in $\eta$, which deflects the particles. The probabilities on
$\gamma_{0}/\gamma$ will sum to $1$ if these ambient fields cause the
particle’s path to be deflected among the outcomes of $\gamma_{0}$, but not
among outcomes of $\gamma$. Because $\gamma$ is a moment partition, this means
that paths are not deflected prior to the measurement taking place; this is
also a sufficient condition for additivity of final state to hold.
For these passive measurements,
$\sum_{\beta\in\alpha/\gamma}\delta(\beta)=\sum_{\beta\in\alpha/\gamma}\delta(\neg\beta)$
if the deflections in $\eta\in\gamma$ due to blocking off $\neg\eta$ are equal
to the sum of the deflections in $\eta$ caused by blocking off each element of
$\gamma-\\{\eta\\}$ individually. So the dps describing particle position will
be $\Delta$-additive if the ambient fields in $\eta$ caused by blocking
$\neg\eta$ is equal to the superposition of the ambient fields in $\eta$
caused by blocking the individual $\nu\in\gamma-\\{\eta\\}$. There are other
ways for $\Delta$-additivity to hold, but this one is conceptually simple, as
well as plausible.
## Appendix B Conditional Probabilities & Probability Dynamics
Conditional probabilities are a central concept of probability theory. They
are particularly interesting for dps’s because probability dynamics are
expressed in terms of conditional probabilities.
There are two ways in which a dps’s probabilities may be considered to be
dynamic. First, the probabilities of what will happen changes as our knowledge
of what has happened continues to unfold. Second, given that we measured the
system to be in state $s$ at time $\lambda_{0}$, we may be interested in the
probability of measuring the system in state as being in state $x$ at time
$\lambda$ as $\lambda$ varies. The Schrodinger equation deals with dynamics of
this second sort. Conditional probabilities are required for exploring both
kinds of dynamics.
The notion of conditional probability is the same in a gps as it is in a
classic probability space; essentially, if $P(B)\neq 0$, $P(A|B)=P(A\bigcap
B)/P(B)$. For a dps, the conditional probability of particular interest is the
probability that $t$ occurs given that, as of $\lambda$, the measurement is
consistent with $t$. In order to delineate this, a few preliminary definitions
will prove helpful.
###### Definition 168.
If $(X,T,P)$ is a dps, $t\in T$, and $t\subset\gamma\in G_{T}$,
$|t[-\infty,\lambda]|_{\gamma}\equiv\\{|\alpha[-\infty,\lambda]|_{\gamma}:\alpha\in
t\\}$
$(t)_{\lambda}^{\gamma}\equiv\bigcup_{\alpha\in t}(\alpha)_{\lambda}^{\gamma}$
$(X,T,P)$ is _$\Lambda$ -complete_ if for all $t\in T$, $\gamma\in G_{T}$ s.t.
$t\subset\gamma$, $\lambda\in\Lambda$, $(t)_{\lambda}^{\gamma}\in T$
Because $(t)_{\lambda}^{\gamma}\subset\gamma$, $\Lambda$-completeness places
no restrictions on the make-up of $G_{T}$, it only places a restriction on the
$\Sigma$-algebras of the constituent ip probability spaces. It is therefore a
fairly weak condition.
$P(t|(t)_{\lambda}^{\gamma})$ is the probability that $t$ occurs given that,
as of $\lambda$, the measurement is consistent with $t$. Because probabilities
of this type are of interest, it’s useful to extend the definition of
consistent probabilities (Defn 84) to insure that the they are independent of
$\gamma$.
###### Definition 169.
A dps, $(X,T,P)$, is _conditionally consistent_ if it is $\Lambda$-complete
and for all $t,t^{\prime}\in T$, all $\lambda\in\Lambda$, all
$t\subset\gamma\in G_{T}$, $t^{\prime}\subset\gamma^{\prime}\in G_{T}$ s.t.
$|t[-\infty,\lambda]|_{\gamma}=|t^{\prime}[-\infty,\lambda]|_{\gamma^{\prime}}$,
$P((t)_{\lambda}^{\gamma})=P((t^{\prime})_{\lambda}^{\gamma^{\prime}})$.
Conceptually, this demand is met if probabilities are consistent at all times
as the experiments unfold. Note that since
$|t[-\infty,\lambda]|_{\gamma}=|t^{\prime}[-\infty,\lambda]|_{\gamma^{\prime}}$,
the “all-reet nots” (introduced in Sec. III.5) of $(t)_{\lambda}^{\gamma}$ and
$(t^{\prime})_{\lambda}^{\gamma^{\prime}}$ are the same; as a result, if a
t-algebra is sufficiently rich, it ought to be conditionally consistent.
###### Definition 170.
If $(X,T,P)$ is a conditionally consistent dps, $t,t^{\prime}\in T$, and
$t^{\prime}\subset\gamma\in G_{T}$, then with
$y\equiv|t^{\prime}[-\infty,\lambda]|_{\gamma}$, $P(t||y)\equiv
P(t|(t^{\prime})_{\lambda}^{\gamma})$.
This is somewhat more intuitive notation for conditional probabilities. The
$P(t||y)$ allow us to see how dps probabilities unfold with time. Note that
$y$ does not have to be an element of $T$ in order for $P(t||y)$ to be
defined.
A common assumption with regard to conditional probabilities on stochastic
processes is that they satisfy the Markov property. The equivalent property
for dps is:
###### Definition 171.
A dps, $(X,T,P)$, is _point-Markovian_ if for all $t\in T$, $(\lambda,p)\in
Uni(\bigcup t)$, if $t=t_{\rightarrow(\lambda,p)}\circ
t_{(\lambda,p)\rightarrow}$ then
$+t_{\rightarrow(\lambda,p)},+t_{(\lambda,p)\rightarrow}\in T$ and
$P(t)=P(+t_{\rightarrow(\lambda,p)})\cdot P(+t_{(\lambda,p)\rightarrow})$.
(In the above definition, notation that has been used on sets of dynamic paths
have been applied to collections of sets. For example,
$t_{\rightarrow(\lambda,p)}$, which is understood to mean
$\\{\alpha_{\rightarrow(\lambda,p)}:\alpha\in t\\}$, and
$t_{\rightarrow(\lambda,p)}\circ t_{(\lambda,p)\rightarrow}$, which is
understood to mean
$\\{\alpha_{\rightarrow(\lambda,p)}\circ\beta_{(\lambda,p)\rightarrow}:\alpha,\beta\in
t,\>\alpha_{\rightarrow(\lambda,p)}\circ\beta_{(\lambda,p)\rightarrow}\neq\emptyset\\}$.
It is hoped that this notation has not caused confusion.)
When the probability function is not additive, this property looses much of
its power. None-the-less, the property can generally be assumed to hold, and
does have some interesting consequences.
One interesting property of point-Markovian dps’s with discrete parameters is
that, if the t-algebra is sufficiently rich, then their probabilities tend to
be additive. This can be seen for the case of $t\in T$ s.t. for some
$\lambda_{0}<\lambda$, $t$ is $sbb\lambda_{0}$, $wba\lambda$, and $(\bigcup
t)[\lambda_{0},\lambda]$ is finite.999Finite spin systems are point-Markovian,
and all their $t\in T$ satisfy these restrictions; however their parameters
are generally assumed to be non-discrete, which does allow them to have non-
additive probabilities. Start with any such $t\in T$ and define
$t_{1}\equiv\\{\alpha_{\rightarrow(\lambda-1,q)\rightarrow(\lambda,p)\rightarrow}:\alpha\in
t,\>q\in\alpha(\lambda-1),\>and\>p\in\alpha(\lambda)\\}$; since $t$ is
$wba\lambda$, for any $\gamma\in G_{T}$ s.t. $t\subset\gamma$,
$(\gamma-t)\bigcup t_{1}$ is an ip, so if it is an element of $G_{T}$ then
$P(t)=P(t_{1})$. If, further, the dps is point-Markovian, then
$P(\\{\alpha_{\rightarrow(\lambda-1,q)\rightarrow(\lambda,p)\rightarrow}\\})=P(\\{+\alpha_{\rightarrow(\lambda-1,q)}\\})\cdot
P(\\{+\alpha_{(\lambda-1,q)\rightarrow(\lambda,p)\rightarrow}\\})$. The same
manipulations can now be performed on the
$+\alpha_{\rightarrow(\lambda-1,q)}$’s. Defining $X_{\bigcup
t}\equiv\\{+\\{\bar{s}[\lambda_{0},\lambda]\\}:\bar{s}\in\bigcup t\\}$, these
iterations eventually yield $P(t)=P(X_{\bigcup t})$. Therefore, for any
$t^{\prime}$ s.t. $t^{\prime}$ is bounded by $\lambda_{0}$ & $\lambda$, and
$\bigcup t^{\prime}=\bigcup t$, $P(t^{\prime})=P(X_{\bigcup t})=P(t)$.
## Appendix C Invariance On Dynamic Sets and DPS’s
### C.1 Invariance On Dynamic Sets
Invariance is among the most fundamental concepts in the study of dynamic
systems. We start by establishing the concept for dynamic sets. (In what
follows, if $f$ & $g$ are functions, “$f\cdot g$” is the function s.t. $f\cdot
g(x)=f(g(x))$)
###### Definition 172.
A _global invariant_ , $I$, on a dynamic set, $S$, is a pair of functions,
$I_{\Lambda}:\Lambda_{S}\rightarrow\Lambda_{S}$ and
$I_{P}:\mathcal{P}_{S}\rightarrow\mathcal{P}_{S}$ s.t.
$\\{I_{\mathcal{P}}\cdot\bar{p}\cdot I_{\Lambda}:\bar{p}\in S\\}=S$.
For any $\bar{p}\in S$, $I(\bar{p})\equiv I_{\mathcal{P}}\cdot\bar{p}\cdot
I_{\Lambda}$; for $A\subset S$, $I[A]=\\{I(\bar{p})\,:\,\bar{p}\in A\\}$ (so
$I$ is an invariant iff $I[S]=S$).
Global invariants are overly restrictive when $\Lambda$ is bounded by $0$. To
handle that case equitably, the following definition of invariance will be
employed.
###### Definition 173.
If $S$ is a dynamic set and $\Lambda_{S}$ is unbounded from below, then $I$ is
an _invariant_ on $S$ if it is a global invariant on $S$.
If $\Lambda_{S}$ is bounded from below then $I$ is an _invariant_ on $S$ if
there exists a dynamic set, $S^{*}$ s.t. $\Lambda_{S^{*}}$ is unbounded from
below, $S=S^{*}[0,\infty]$, and $I$ is an invariant on $S^{*}$.
Unless stated otherwise, for the remainder of this section dynamic sets will
be assumed to be unbounded from below.
###### Theorem 174.
If $I$ is an invariant on $S$
1) $I_{\mathcal{P}}$ is a surjection
2) $I_{\Lambda}$ and $I_{\mathcal{P}}$ are invertible iff $I_{\Lambda}$ is an
injection and $I_{\mathcal{P}}$ is a bijection
3) $I_{\Lambda}$ and $I_{\mathcal{P}}$ are invertible and
$I^{-1}\equiv(I_{\Lambda}^{-1},I_{P}^{-1})$ is an invariant on $S$ iff
$I_{\Lambda}$ and $I_{\mathcal{P}}$ are bijections
###### Proof.
1) If it is not then $\mathcal{P}_{I[S]}\subset
Ran(I_{\mathcal{P}})\subsetneq\mathcal{P}_{S}$ and so $I[S]\neq S$.
2) Follows from (1) and that functions are invertible iff they are injections.
3) $\Rightarrow$ Follows from (2) and that for $I^{-1}$ to be an invariant,
the domain of $I_{\Lambda}^{-1}$ needs to be $\Lambda$.
$\Leftarrow$ Because $I_{\Lambda}$ and $I_{\mathcal{P}}$ are bijections, for
all $\bar{p}\in S$ $I^{-1}(I(\bar{p}))=I_{P}^{-1}\cdot
I_{\mathcal{P}}\cdot\bar{p}\cdot I_{\Lambda}\cdot I_{\Lambda}^{-1}=\bar{p}$.
Therefore $I^{-1}[I[S]]=S$. Since $I[S]=S$, $I^{-1}[S]=S$.∎
###### Definition 175.
Dynamic set $S$ is _weakly $\Lambda$-invariant_ if for all $\lambda\in\Lambda$
there’s an invariant, $L^{\lambda}$, s.t. for all
$\lambda^{\prime}\in\Lambda_{S}$,
$L_{\Lambda}^{\lambda}(\lambda^{\prime})=\lambda^{\prime}+\lambda$.
$S$ is _strongly $\Lambda$-invariant_ if it is weakly $\Lambda$-invariant and
for all $\lambda\in\Lambda$, $L_{\mathcal{P}}^{\lambda}$ is the identity on
$\mathcal{P}_{S}$.
###### Theorem 176.
A dynamic space $D$ is strongly $\Lambda$-invariant iff it homogeneous and all
of $\mathcal{P}_{S}$ is homogeneously realized
###### Proof.
When $\Lambda_{D}$ is unbounded from below, this is fairly clear.
If $\Lambda_{D}$ is bounded from below:
$\Rightarrow$ follows immediately from the unbounded case.
For $\Leftarrow$, it’s necessary to construct a homogeneous, homogeneously
realized, unbounded from below $D^{*}$ s.t. $D^{*}[0,\lambda]=D$. This is
relatively easy. Note that if $D$ is homogeneous & homogeneously realized then
for any $\triangle\lambda>0$, any $\lambda,\lambda^{\prime}\in\Lambda_{D}$,
$D[\lambda,\lambda+\triangle\lambda]$ and
$D[\lambda^{\prime},\lambda^{\prime}+\triangle\lambda]$ are copies of each
other, in that if you move
$D[\lambda^{\prime},\lambda^{\prime}+\triangle\lambda]$ to $\lambda$, the two
are equal. So to construct $D^{*}$, take any $D[0,\triangle\lambda]$, append
it to the beginning of $D$, then append it to the beginning of the resulting
set, and so forth. Only one set, $D^{*}$, will equal this construction for all
intervals, $[-n\triangle\lambda,\infty]$; $D^{*}$ is a homogeneous,
homogeneously dynamic space, and $\Lambda_{D^{*}}$ is unbounded from below, so
$D^{*}$ is strongly $\Lambda$-invariant.∎
###### Definition 177.
$S$ is _reversible_ if for all $\lambda\in\Lambda$ there’s and invariant,
$R^{\lambda}$, s.t. for all $\lambda^{\prime}\in\Lambda_{S}$,
$R_{\Lambda}^{\lambda}(\lambda^{\prime})=\lambda-\lambda^{\prime}$, and
$R_{\mathcal{P}}^{\lambda}\cdot R_{\mathcal{P}}^{\lambda}$ is the identity on
$\mathcal{P}_{S}$.
###### Theorem 178.
1) If $S$ is reversible then it is weakly $\Lambda$-invariant.
2) If $S$ is reversible and for all $\lambda_{1},\lambda_{2}\in\Lambda$,
$R_{\mathcal{P}}^{\lambda_{1}}=R_{\mathcal{P}}^{\lambda_{2}}$ then it is
strongly $\Lambda$-invariant.
###### Proof.
1) For every $\lambda\in\Lambda_{S}$, $L^{\lambda}=R^{\lambda}\cdot R^{0}$ is
an invariant (because the composite of any two invariants is an invariant;
$R^{0}$ being $R^{\lambda}$ with $\lambda=0$), and for all
$\lambda^{\prime}\in\Lambda$,
$L_{\Lambda}^{\lambda}(\lambda^{\prime})=R_{\Lambda}^{\lambda}(R_{\Lambda}^{0}(\lambda^{\prime}))=\lambda^{\prime}+\lambda$.
2) Again with $L^{\lambda}=R^{\lambda}\cdot R^{0}$, if for all
$\lambda_{1},\lambda_{2}\in\Lambda$,
$R_{\mathcal{P}}^{\lambda_{1}}=R_{\mathcal{P}}^{\lambda_{2}}$ then
$R_{\mathcal{P}}^{\lambda}\cdot R_{\mathcal{P}}^{0}=R_{\mathcal{P}}^{0}\cdot
R_{\mathcal{P}}^{0}$, which is the identity on $\mathcal{P}_{S}$. ∎
### C.2 Invariance on DPS’s
To apply the notion of invariance to dps’s, we’ll start by expanding the
definition to cover invariance on a collection of dynamic sets. (In the
interest of concision, previous considerations for the case where $\Lambda$ is
bounded by $0$ will not be explicitly mentioned, but they should be assumed to
continue to apply.)
###### Definition 179.
If $X$ is a collection of dynamic sets, $I$ is an _invariant_ on $X$ if it is
a pair of functions $I_{\Lambda}:\bigcup_{S\in
X}\Lambda_{S}\rightarrow\bigcup_{S\in X}\Lambda_{S}$ and
$I_{\mathcal{P}}:\bigcup_{S\in X}\mathcal{P}_{S}\rightarrow\bigcup_{S\in
X}\mathcal{P}_{S}$ s.t. for some bijection, $B_{I}:X\rightarrow X$, all $S\in
X$, $I[S]\equiv\\{\bar{p}\,:\,for\>some\>\bar{p}^{\prime}\in
S,\>\bar{p}=I_{\mathcal{P}}\cdot\bar{p}^{\prime}\cdot
I_{\Lambda}\\}=B_{I}(S)$.
In the case where $X=\\{S\\}$ this reduces to the prior definition of
invariance. When $B_{I}$ is simply the identity on $X$, $I$ represents a
collection of dynamic set invariants, one for each element of $X$. In the more
general case, where the dynamic sets in the collection are allowed to map onto
each other under the transformation, the collection can be invariant under the
transformation even when none of the individual dynamic sets are.
Generalizing types of invariance is straightforward. For example:
###### Definition 180.
$X$ is _reversible_ if for all $S,S^{\prime}\in X$,
$\Lambda_{S}=\Lambda_{S^{\prime}}=\Lambda$, and for all $\lambda\in\Lambda$
there’s and invariant on $X$, $R^{\lambda}$, s.t. for each $S\in X$,
$R_{\mathcal{P}}^{\lambda}\cdot R_{\mathcal{P}}^{\lambda}$ is the identity on
$\mathcal{P}_{S}$, and for all $\lambda^{\prime}\in\Lambda$,
$R_{\Lambda}^{\lambda}(\lambda^{\prime})=\lambda-\lambda^{\prime}$.
Invariance on a dps is now:
###### Definition 181.
If $(X,T,P)$ is a dps, and $I$ is an invariant on $X$, $I$ is an _invariant_
on $(X,T,P)$ if
1) If $\gamma\in G_{T}$ and $I[\gamma]$ is an ip on some $S\in X$ then
$I[\gamma]\in G_{T}$
2) If $t\in T$ and for some $\gamma\in G_{T}$, $I[t]\subset\gamma$ then
$I[t]\in T$
3) If $t\in T$ and $I[t]\in T$ then $P(t)=P(I[t])$
The dps versions of invariants such as reversibility follow immediately.
## Appendix D Parameter Theory
###### Definition 182.
A set, $\Lambda$, together with a binary relation on $\Lambda$, $<$, a binary
function on $\Lambda$, $+$, and a constant $0\in\Lambda$, is an _open
parameter_ if the structure, $(\Lambda,<,+,0)$, satisfies the following:
Total Ordering:
1) For all $\lambda\in\Lambda$, $\lambda\nless\lambda$
2) For all $\lambda_{1},\lambda_{2},\lambda_{3}\in\Lambda$, if
$\lambda_{1}<\lambda_{2}$ and $\lambda_{2}<\lambda_{3}$ then
$\lambda_{1}<\lambda_{3}$
3) For all $\lambda_{1},\lambda_{2}\in\Lambda$, either
$\lambda_{1}=\lambda_{2}$ or $\lambda_{1}<\lambda_{2}$ or
$\lambda_{2}<\lambda_{1}$
Addition:
4) For all $\lambda\in\Lambda$, $\lambda+0=\lambda$
5) For all $\lambda_{1},\lambda_{2}\in\Lambda$,
$\lambda_{1}+\lambda_{2}=\lambda_{2}+\lambda_{1}$
6) For all $\lambda_{1},\lambda_{2},\lambda_{3}\in\Lambda$,
$(\lambda_{1}+\lambda_{2})+\lambda_{3}=\lambda_{1}+(\lambda_{2}+\lambda_{3})$
The standard interrelationship between ordering and addition:
7) For all $\lambda_{1},\lambda_{2},\lambda_{3}\in\Lambda$,
$\lambda_{1}<\lambda_{2}$ iff
$\lambda_{1}+\lambda_{3}<\lambda_{2}+\lambda_{3}$
8) For all $\lambda_{1},\lambda_{2}\in\Lambda$, if $\lambda_{1}<\lambda_{2}$
then there’s a $\lambda_{3}\in\Lambda$ s.t.
$\lambda_{1}+\lambda_{3}=\lambda_{2}$
Possesses a positive element:
9) There exists a $\lambda\in\Lambda$ s.t $\lambda>0$
The enumerated statements in this definition will be referred to as the “open
parameter axioms”, and the individual statements will be referred to by
number: opa 1 referring to “For all $\lambda\in\Lambda$,
$\lambda\nless\lambda$”, etc.
Because in most cases $<$, $+$, and $0$ will be immediately apparent given the
set $\Lambda$, parameters will often be referred to by simply referring to
$\Lambda$.
###### Definition 183.
For $\lambda\in\Lambda$, $\lambda^{\prime}\in\Lambda$ is an _immediate
successor_ to $\lambda$ if $\lambda^{\prime}>\lambda$ and there does not exist
a $\lambda^{\prime\prime}\in\Lambda$ s.t.
$\lambda^{\prime}>\lambda^{\prime\prime}>\lambda$.
A parameter is _discrete_ if every $\lambda\in\Lambda$ has an immediate
successor.
A parameter is _dense_ if no $\lambda\in\Lambda$ has an immediate successor.
###### Theorem 184.
An open parameter is either discrete or dense
###### Proof.
Follows from opa 8 and opa 7.∎
###### Definition 185.
For a discrete open parameter, the immediate successor to $0$ is
$\boldsymbol{1}$.
###### Theorem 186.
If $\Lambda$ is discrete, then for every $\lambda\in\Lambda$, the immediate
successor to $\lambda$ is $\lambda+\boldsymbol{1}$.
###### Proof.
This too follows from opa 8 and opa 7.∎
###### Definition 187.
If $\chi\subset\Lambda$, $\lambda\in\Lambda$ is an _upper-bound_ of $\chi$ if
for every $\lambda^{\prime}\in\chi$, $\lambda\geq\lambda^{\prime}$; $\lambda$
is the _least upper-bound_ if it is an upper-bound, and given any other upper-
bound, $\lambda_{+}$, $\lambda\leq\lambda_{+}$. If $\chi$ has no upper-bound,
then $\chi$ is _unbounded from above_.
Similarly, $\lambda\in\Lambda$ is an _lower-bound_ of $\chi$ if for every
$\lambda^{\prime}\in\chi$, $\lambda\leq\lambda^{\prime}$; $\lambda$ is the
_greatest lower-bound_ if it is a lower-bound, and given any other lower-
bound, $\lambda_{-}$, $\lambda_{-}\leq\lambda$. If $\chi$ has no lower-bound,
then $\chi$ is _unbounded from below_.
$\lambda^{\prime}$ is the _additive inverse_ of $\lambda$ if
$\lambda+\lambda^{\prime}=0$
###### Theorem 188.
If $\Lambda$ is an open parameter
1) $\Lambda$ is unbounded from above.
2) If $\Lambda$ is bounded from below, it’s greatest lower bound is $0$.
3) If $\Lambda$ does not have a least element, every $\lambda\in\Lambda$ has
an additive inverse; if it does have a least element, only $0$ has an additive
inverse
###### Proof.
1) Follows from opa 9 and opa 7 (with help from opa 4)
2) Also follows from opa 7 with help from opa 4
3) First take the case where $\Lambda$ does not have a least element. If
$\lambda<0$ then by opa 8 there’s a $\lambda^{\prime}$ s.t.
$\lambda+\lambda^{\prime}=0$. If $\lambda>0$ take any $\lambda^{\prime}$ s.t
$\lambda+\lambda^{\prime}<0$ (such a $\lambda^{\prime}$ must exist because
$\Lambda$ is unbounded from below). As just established, there must be a
$\Delta\lambda$ s.t. $\lambda+\lambda^{\prime}+\Delta\lambda=0$, so
$\lambda^{\prime}+\Delta\lambda$ is the additive inverse of $\lambda$.
Now assume $\Lambda$ does have a least element. By opa 7 and pt. 2 of this
theorem, for any $\lambda,\lambda^{\prime}\in\Lambda$ s.t. $\lambda\neq 0$,
$\lambda+\lambda^{\prime}\geq\lambda>0$. ∎
Open parameters admit models which, under most interpretations of “parameter”,
would not be considered admissible. For example, infinite ordinals (with the
expected interpretations of $<$, $+$, and $0$) are open parameters, as are the
extended reals. Rational numbers are also open parameters, as are numbers
which, in decimal notation, have only a finite number of non-zero digits.
To eliminate these less-than-standard models, parameters will be defined as
open parameters that are finite (sometimes called Archimedean), and either
discrete or continuous (that is, parameters have the crucial property that all
limits which tend toward fixed, finite values exist). This will be
accomplished through the well known method of adding a “completeness axiom”.
###### Definition 189.
An open parameter, $\Lambda$, is a _parameter_ if every non-empty subset of
$\Lambda$ that is bounded from above has a least upper bound.
From here on out, it will be assumed that “$\Lambda$” refers to a parameter.
###### Definition 190.
$\lambda$ added to itself $n$ times will be denoted $n\lambda$ (for example
$3\lambda\equiv\lambda+\lambda+\lambda$). $0\lambda\equiv 0$.
$\lambda-\lambda^{\prime}\equiv\begin{cases}0&if\,\Lambda
is\,bounded\,by\,0\,and\,\lambda<\lambda^{\prime}\\\
\Delta\lambda&where\,\Delta\lambda+\lambda=\lambda^{\prime},\>otherwise\end{cases}$
###### Theorem 191.
If $\Lambda$ is a parameter then for every
$\lambda_{1},\lambda_{2}\in\Lambda$, $0<\lambda_{1}<\lambda_{2}$, there exists
an $n\in\mathbb{N}$ s.t. $n\lambda_{1}>\lambda_{2}$.
###### Proof.
Assume that for all $n\in\mathbb{N}$, $n\lambda_{1}<\lambda_{2}$. Then the set
$\chi=\\{x\in\Lambda\,:\,x=n\lambda_{1}\\}$, is bounded from above. Therefore
it has a least upper bound, $\lambda_{M}$. Since $\lambda_{1}>0$,
$\lambda_{M}-\lambda_{1}<\lambda_{M}$, and so $\lambda_{M}-\lambda_{1}$ can’t
an upper bound of $\chi$. Therefore for some $i\in\mathbb{N}$,
$i\lambda_{1}>\lambda_{M}-\lambda_{1}$. But then
$(i+2)\lambda_{1}>\lambda_{M}$, so $\lambda_{M}$ can not be an upper bound.
Therefore $\chi$ is unbounded, and so for some $n\in\mathbb{N}$,
$n\lambda_{1}>\lambda_{2}$. ∎
Thm 191 is equivalent to saying that for all $\lambda\in\Lambda$, $\lambda$ is
finite.
###### Theorem 192.
For any $n\in\mathbb{N}^{+}$
1) $\lambda_{1}>\lambda_{2}$ iff $n\lambda_{1}>n\lambda_{2}$
2) $\lambda_{1}=\lambda_{2}$ iff $n\lambda_{1}=n\lambda_{2}$
###### Proof.
1) Follows from opa 7, opa 2
2) Follows from (1) and opa 3.∎
###### Theorem 193.
If $\Lambda$ is a discrete parameter then given any $\lambda\in\Lambda$,
$\lambda>0$ there’s an $n\in\mathbb{N}$ s.t. $\lambda=n\boldsymbol{1}$.
###### Proof.
Follows from Thms 186 and 191∎
###### Theorem 194.
For any $n,m\in\mathbb{N}$
1) $n\boldsymbol{1}+m\boldsymbol{1}=(n+m)\boldsymbol{1}$
2) If $n>m$ then $n\boldsymbol{1}>m\boldsymbol{1}$
###### Proof.
1) Follows from opa 6
2) Take $k=n-m$; By (1) and opa 7, (2) holds iff $k\boldsymbol{1}>0$, which
follows from Thm 192.1. ∎
Thms 193 and 194, together with Thm 188.3, create a complete characterization
of discrete parameters. A similar characterization for dense parameters will
now be sketched.
###### Definition 195.
For $\lambda,\lambda^{\prime}\in\Lambda$, $m\in\mathbb{N}$,
$n\in\mathbb{N}^{+}$, $\lambda^{\prime}=\frac{m}{n}\lambda$ if
$n\lambda^{\prime}=m\lambda$.
###### Theorem 196.
If $\Lambda$ is dense then for all $\lambda\in\Lambda$, $m\in\mathbb{N}$,
$n\in\mathbb{N}^{+}$
1) $\frac{m}{n}\lambda\in\Lambda$
2) If $\lambda^{\prime}=\frac{m}{n}\lambda$ and
$\lambda^{\prime\prime}=\frac{m}{n}\lambda$ then
$\lambda^{\prime}=\lambda^{\prime\prime}$
###### Proof.
1) Take $\chi\equiv\\{\lambda^{\prime}\in\Lambda\,:\,n\lambda^{\prime}\leq
m\lambda\\}$. $\chi$ is bounded from above, so take $\lambda_{\chi}$ to be the
least upper bound. If $n\lambda_{\chi}$ is either greater or less than
$m\lambda$ then density provides an example that contradicts $\lambda_{\chi}$
being the least upper bound, so by opa 3 $n\lambda_{\chi}=m\lambda$.
2) $n\lambda^{\prime}=m\lambda$ and $n\lambda^{\prime\prime}=m\lambda$, so
$\lambda=\lambda^{\prime}$ by Thm 192.2.∎
###### Theorem 197.
If $\Lambda$ is dense then for all $\lambda\in\Lambda$
1) If $m_{1},m_{2}\in\mathbb{N}$, $n_{1},n_{2}\in\mathbb{N}^{+}$ and
$\frac{m_{1}}{n_{1}}=\frac{m_{2}}{n_{2}}$ then
$\frac{m_{1}}{n_{1}}\lambda=\frac{m_{2}}{n_{2}}\lambda$
2) For $q_{1},q_{2}\in\mathbb{Q}$,
$q_{1}\lambda+q_{2}\lambda=(q_{1}+q_{2})\lambda$
3) For $q_{1},q_{2}\in\mathbb{Q}$, $\lambda>0$, if $q_{1}>q_{2}$ then
$q_{1}\lambda>q_{2}\lambda$
###### Proof.
1) If $\frac{m_{1}}{n_{1}}=\frac{m_{2}}{n_{2}}$ then either there exist a
$m,n,k_{1},k_{2}$ s.t. $m_{1}=k_{1}m$, $n_{1}=k_{1}n$, $m_{2}=k_{2}m$, and
$n_{2}=k_{2}n$. It is sufficient to show that
$\frac{m_{1}}{n_{1}}\lambda=\frac{m}{n}\lambda$. Taking
$\lambda^{\prime}\equiv\frac{m_{1}}{n_{1}}\lambda$,
$(k_{1}n)\lambda^{\prime}=(k_{1}m)\lambda$. By Thm 194.1,
$k_{1}(n\lambda^{\prime})=k_{1}(m\lambda)$. By Thm 192.2
$n\lambda^{\prime}=m\lambda$.
2) For some $m_{1},m_{2}\in\mathbb{N}$, $n\in\mathbb{N}^{+}$,
$q_{1}=\frac{m_{1}}{n}$ and $q_{2}=\frac{m_{2}}{n}$. With
$\lambda_{1}\equiv\frac{m_{1}}{n}\lambda$ and
$\lambda_{2}\equiv\frac{m_{2}}{n}\lambda$,
$m_{1}\lambda+m_{2}\lambda=n\lambda_{1}+n\lambda_{2}$ and so by opa 6,
$n(\lambda_{1}+\lambda_{2})=(m_{1}+m_{2})\lambda$, which mean
$\lambda_{1}+\lambda_{2}=\frac{(m_{1}+m_{2})}{n}\lambda$.
3) Follows from (2) and opa 7, and the fact that $q_{3}\equiv q_{2}-q_{1}$ is
a positive rational number (note that if $\lambda>0$ and $q>0$ then
$q\lambda>0$).∎
###### Definition 198.
A parameter sequence, $(\lambda_{n})_{n\in\mathbb{N}}$, is _convergent_ if
there exists a $\lambda^{\prime}\in\Lambda$ s.t. for any $\Delta\lambda>0$
there’s an $n\in\mathbb{N}$ s.t. for all $i>n$,
$\lambda_{i}\in(\lambda^{\prime}-\Delta\lambda,\lambda^{\prime}+\Delta\lambda)$.
In this case we say $\lambda=\lim\lambda_{n}$.
$(\lambda_{n})_{n\in\mathbb{N}}$, is _Cauchy-convergent_ if for any
$\Delta\lambda>0$ there’s an $n\in\mathbb{N}$ s.t. for all $i,j>n$,
$\lambda_{j}\in(\lambda_{i}-\Delta\lambda,\lambda_{i}+\Delta\lambda)$.
$(\lambda_{n})_{n\in\mathbb{N}}$ is _monotonic_ if either for all an
$i\in\mathbb{N}$, $\lambda_{i+1}\geq\lambda_{i+1}$, or for all an
$i\in\mathbb{N}$, $\lambda_{i+1}\leq\lambda_{i+1}$.
###### Theorem 199.
If $\Lambda$ is a dense parameter and $(q_{n})_{n\in\mathbb{N}}$ is a
monotonic, Cauchy-convergent sequence of rational numbers, then
$(q_{n}\lambda)_{n\in\mathbb{N}}$ is a convergent parameter sequence (with the
understanding that if $\Lambda$ is bounded from below then all $q_{i}$ are
non-negative).
###### Proof.
A: If $(q_{n})_{n\in\mathbb{N}}$ is a Cauchy-convergent sequence of rational
numbers, then $(q_{n}\lambda)_{n\in\mathbb{N}}$ is a Cauchy-convergent.
\- Follows from Thm 197 and the fact that the set of rational numbers is
dense. -
B: If $\Lambda$ is a parameter, and $A\subset\Lambda$ is bounded from below,
then $A$ must have a greatest lower bond
\- Take $B$ to be the set of lower bounds of $A$. $B$ is bounded from above by
every element of $A$, so take $b$ to be the least upper bound of $B$. $b$ must
be a lower bound of $A$ because if for any $a\in A$, $a<b$ then $b$ can not be
the least upper bound of $B$. It also must be the greatest lower bound,
because if any $x>b$ is a lower bound of $A$, then $x\in B$, in which case $b$
would not be an upper bound of $B$. -
Assume $(q_{n})_{n\in\mathbb{N}}$ is monotonically increasing. Take
$\lambda^{\prime}$ to be the least upper bound of
$Ran((q_{n}\lambda)_{n\in\mathbb{N}})$. Take any $\Delta\lambda>0$. By (A)
there exists an $n\in\mathbb{N}$ s.t. for all $i,j>n$,
$q_{j}\lambda\in(q_{i}\lambda-\Delta\lambda,q_{i}\lambda+\Delta\lambda)$. If
follows that for all $i>n$,
$q_{i}\lambda\in(\lambda^{\prime}-\Delta\lambda,\lambda^{\prime}+\Delta\lambda)$.
By (B), the proof for monotonically decreasing sequences is similar. ∎
It is a foundational result of real analysis that all Cauchy-convergent
sequences of rational numbers converge to a real number, and for all real
numbers there exist monotonic, Cauchy-convergent sequences of rational numbers
that converge to it.
###### Theorem 200.
If $(q_{n})_{n\in\mathbb{N}}$ and $(q_{n}^{\prime})_{n\in\mathbb{N}}$ are two
monotonic sequences of rational numbers that converge to the same real number
then $\lim(q_{n}\lambda)=\lim(q_{n}^{\prime}\mathbf{\lambda})$.
###### Proof.
Assume $(q_{n})_{n\in\mathbb{N}}$ and $(q_{n}^{\prime})_{n\in\mathbb{N}}$ are
monotonically increasing. Because they converge to the same real number, they
have the same least upper bound. $(q_{n}\lambda)_{n\in\mathbb{N}}$ and
$(q_{n}^{\prime}\lambda)_{n\in\mathbb{N}}$ must then also have the same least
upper bound. By the proof of Thm 199, they have the same limit.
All other cases are similar.∎
###### Definition 201.
If $(q_{n})_{n\in\mathbb{N}}$ is a monotonic, Cauchy-convergent sequence of
rational numbers and $\lim q_{n}=r$ then for any $\lambda\in\Lambda$,
$r\lambda\equiv\lim(q_{n}\lambda)$.
By Thm 200 the above definition uniquely defines multiplication by a real
number.
###### Theorem 202.
If $\Lambda$ is a dense parameter and $\mathbf{1}$ any element of $\Lambda$
that greater than $0$
1) For any real number $r$, $r\mathbf{1}\in\Lambda$
2) For any $\lambda\in\Lambda$, there exists a real number, $r$, s.t.
$r\mathbf{1}=\lambda$
###### Proof.
1) Follow immediately from Thm 199.
2) We’ll take the case of $\lambda>0$; $\lambda<0$ is similar and $\lambda=0$
is trivial.
A: If $\lambda>0$ then there exists a rational number $q$ s.t
$0<q\mathbf{1}<\lambda$
\- The greatest lower bound of the set of $\frac{1}{2^{n}}\mathbf{1}$,
$n\in\mathbb{N}^{+}$, is $0$; since $\lambda>0$ there must be an $i$ s.t.
$\frac{1}{2^{i}}\mathbf{1}<\lambda$ -
Take $q$ to be any rational number s.t. $q\mathbf{1}<\lambda$. By Thm 191
there’s an $n\in\mathbb{N}^{+}$ s.t. $nq\mathbf{1}>\lambda$; take $m_{0}$ to
be the smallest such element of $\mathbb{N}^{+}$. Take
$\lambda_{0}=(m_{0}-1)q\mathbf{1}$; note that $\lambda_{0}\leq\lambda$ and
$\lambda-\lambda_{0}\leq q\mathbf{1}$. Similarly for each $i\in\mathbb{N}$
take $m_{i}$ to be the smallest element of $\mathbb{N}^{+}$ s.t.
$m_{i}(\frac{q}{2^{i}})\mathbf{1}>\lambda$. With
$\lambda_{i}\equiv(m_{i}-1)(\frac{q}{2^{i}})\mathbf{1}$ it follows that
$\lambda_{i}\leq\lambda_{i+1}\leq\lambda$ and
$\lambda_{i}-\lambda\leq(\frac{q}{2^{i}})\mathbf{1}$.
Consider the sequence
$((m_{i}-1)(\frac{q}{2^{i}})\mathbf{1})_{n\in\mathbb{N}}$; for any
$\Delta\lambda$ there’s an $n$ s.t. $(\frac{q}{2^{n}})<\Delta\lambda$. For all
$j>n$,
$\lambda-(m_{j}-1)(\frac{q}{2^{j}})\mathbf{1}<(\frac{q}{2^{n}})\mathbf{1}<\Delta\lambda$,
so $((m_{i}-1)(\frac{q}{2^{i}})\mathbf{1})_{n\in\mathbb{N}}$ converges to
$\lambda$. with $r=\lim(m_{i}-1)(\frac{q}{2^{i}})$, $\lambda=r\mathbf{1}$.∎
###### Theorem 203.
If $\Lambda$ is a dense parameter and $\mathbf{1}$ any element of $\Lambda$
that greater than $0$
1) For $r_{1},r_{2}\in\mathbb{R}$,
$r_{1}\lambda+r_{2}\lambda=(r_{1}+r_{2})\lambda$
2) For $r_{1},r_{2}\in\mathbb{R}$, $\lambda>0$, if $r_{1}>r_{2}$ then
$r_{1}\lambda>r_{2}\lambda$
###### Proof.
1) If $(q_{n})_{n\in\mathbb{N}}$ and $(q_{n}^{\prime})_{n\in\mathbb{N}}$ are
monotonically increasing sequences of rational numbers and $\lim q_{n}=r_{1}$
and $\lim q_{n}^{\prime}=r_{2}$ then $\lim(q_{n}+q_{n}^{\prime})=r_{1}+r_{2}$.
2) Follows from (1) and opa 7, and the fact that $r_{3}\equiv r_{2}-r_{1}$ is
a positive real number. ∎
Thms 202 & 203 create a complete characterization of dense parameters.
|
arxiv-papers
| 2011-06-11T16:10:42 |
2024-09-04T02:49:19.650403
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Eric Tesse",
"submitter": "Eric Tesse",
"url": "https://arxiv.org/abs/1106.2751"
}
|
1106.2792
|
# Algebraic codes for Slepian-Wolf code design
Shizheng Li and Aditya Ramamoorthy
This work was supported in part by NSF grant CCF-1018148. Department of
Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011
Email: {szli, adityar}@iastate.edu
###### Abstract
Practical constructions of lossless distributed source codes (for the Slepian-
Wolf problem) have been the subject of much investigation in the past decade.
In particular, near-capacity achieving code designs based on LDPC codes have
been presented for the case of two binary sources, with a binary-symmetric
correlation. However, constructing practical codes for the case of non-binary
sources with arbitrary correlation remains by and large open. From a practical
perspective it is also interesting to consider coding schemes whose
performance remains robust to uncertainties in the joint distribution of the
sources.
In this work we propose the usage of Reed-Solomon (RS) codes for the
asymmetric version of this problem. We show that algebraic soft-decision
decoding of RS codes can be used effectively under certain correlation
structures. In addition, RS codes offer natural rate adaptivity and
performance that remains constant across a family of correlation structures
with the same conditional entropy. The performance of RS codes is compared
with dedicated and rate adaptive multistage LDPC codes (Varodayan et al. '06),
where each LDPC code is used to compress the individual bit planes. Our
simulations show that in classical Slepian-Wolf scenario, RS codes outperform
both dedicated and rate-adaptive LDPC codes under $q$-ary symmetric
correlation, and are better than rate-adaptive LDPC codes in the case of
sparse correlation models, where the conditional distribution of the sources
has only a few dominant entries. In a feedback scenario, the performance of RS
codes is comparable with both designs of LDPC codes. Our simulations also
demonstrate that the performance of RS codes in the presence of inaccuracies
in the joint distribution of the sources is much better as compared to
multistage LDPC codes.
## I Introduction
We consider the problem of practical code design for the Slepian-Wolf problem.
Following the work of [1] that established the equivalence between the
Slepian-Wolf problem and channel coding, a lot of research work has addressed
this problem (see [2] and its references). However, by and large most of the
work considers the case of two binary sources that are related by an additive
error. In this paper, we propose a coding scheme for nonbinary sources using
Reed-Solomon codes that works under more general correlation models than an
additive error model. One previously proposed approach for compressing two
nonbinary sources is to use several LDPC codes, each for a bit level of the
binary image [3] along with multistage decoding. It requires the knowledge of
the joint distribution and the conditional distributions of the binary sources
that corresponding to the bit levels. It also requires the design of multiple
LDPC codes, multiple LDPC decodings at the terminal and may suffer from error
propagation. The multistage LDPC approach breaks down the symbol level
correlation to bit level correlations. When the correlation is essentially at
the symbol level, multistage LDPC may not be the most suitable approach. In
this paper we evaluate the performance of RS codes and multistage LDPC codes.
We note that very few simulation results of multistage LDPC codes for Slepian-
Wolf problem on large alphabet sizes have appeared in previous work. Turbo
code-based design of nonbinary SWC was proposed in [4] but only field size of
eight was considered. The work of [5] proposed algebraic codes for SWC using
list decoding. Our algorithm uses soft decoding and has better performance. In
addition, we provide simulations and comparisons with multistage LDPC codes.
In this paper, two scenarios are considered in our simulation. One is the
classical Slepian-Wolf scenario, where there is no feedback from the decoder
to the encoder. In the other scenario, there is feedback from the decoder to
the encoder that tells the encoder whether the decoding is successful. If the
decoding fails, the encoder will send more syndrome symbols. In this paper, we
consider two designs of multistage LDPC codes [3], (i) Dedicated codes for
each bit source. The degree distributions of the codes are optimized for AWGN
channels and the codes are generated by PEG algorithm [6]. These codes do not
offer rate adaptivity. (ii) The rate adaptive codes designed in [7]. The rate
adaptivity in Slepian-Wolf problem requires us to adapt the transmission rate
by adapting the syndrome length, rather than code length. If a low
transmission rate is not enough to decode the source, more syndrome symbols
are transmitted to the decoder and together with previously received syndrome,
the decoder attempts to decode. RS codes offer natural rate-adaptivity by
definition. Rate adaptive codes are useful in the feedback scenario. Our
simulations show that in the classical Slepian-Wolf coding scenario, under
$q$-ary symmetric correlation models, RS codes outperform both designs of
multistage LDPC codes. Under sparse correlation models, RS codes perform
better than rate adaptive LDPC codes when the correlation resembles $q$-ary
symmetric models. In the feedback scenario, the performance of rate-adaptive
LDPC codes and RS codes are comparable under $q$-ary symmetric channels but
under sparse correlation model, rate-adaptive LDPC codes perform better than
RS codes. Moreover, when the correlation given to the decoder is slightly
different from the true correlation model, RS codes suffer little but
multistage LDPC codes suffer significantly.
This paper is organized as follows. The preliminaries about RS codes and the
Koetter-Vardy decoding algorithm [8] are given in Section II. The RS code-
based asymmetric SWC schemes are described in Section III and the performance
comparisons with a single LDPC codes are presented in Section IV. In Section V
and Section VI the performance comparisons of RS codes and multistage LDPC
codes under two scenarios are presented respectively. Section VII concludes
the paper.
## II Preliminaries
Let $F_{q}$ be a finite field and $q$ be a power of two. A $(n,k)$ RS code can
be defined by its parity check matrix
$H_{ij}=(\alpha^{i})^{j-1},i=1,\ldots,n-k,j=1,\ldots,n$, where $\alpha$ is a
primitive element of $F_{q}$ and $n=q-1$. The code
$\mathcal{C}_{RS}=\\{\mathbf{c}\in F_{q}^{n}:H_{RS}\mathbf{c}=0\\}$. Suppose
$H_{1},H_{2}$ are the parity check matrices of two RS codes with rates
$k_{1}/n,k_{2}/n$ respectively and $k_{1}\geq k_{2}$, by definition, $H_{1}$
is a submatrix of $H_{2}$. As we shall see later, this allows rate adaptivity
for distributed source coding. An equivalent definition of an RS code is given
in terms of polynomial evaluation. Given a message vector $\mathbf{m}$ of
length $k$, the encoded codeword is obtained by evaluating the message
polynomial $f_{\mathbf{m}}(\mathcal{X})$ (of degree $k-1$) at $n$ points
$\\{1,\alpha,\alpha^{2},\ldots,\alpha^{n-1}\\}$. One only needs to specify the
code parameters $n$ and $k$ when designing codes.
Consider a channel coding scenario. A codeword $\mathbf{c}\in\mathcal{C}_{RS}$
is transmitted and the channel output is $\mathbf{r}$. Let
$\gamma_{1},\ldots,\gamma_{q}$ be a fixed ordering of the elements from
$F_{q}$. The receiver computes the $q$-by-$n$ reliability matrix
$\Pi=\\{\pi_{ij}=P(c_{j}=\gamma_{i}|r_{j})\\}$ based on the information from
the channel. The Koetter-Vardy soft decoding algorithm [8] first computes a
multiplicity matrix $M$ from $\Pi$. The simplest choice is
$M=\lfloor\lambda\Pi\rfloor$, where $\lambda$ is a positive real number. Next,
it constructs a bivariate polynomial $Q_{M}(\mathcal{X},\mathcal{Y})$ with
minimal weighted degree that passes through every point
$(\alpha^{j-1},\gamma_{i})$, $m_{ij}$ times. These algebraic constraints can
be given by $C(M)$ linear constraints, where
$C(M)=\frac{1}{2}\sum_{i=1}^{q}\sum_{j=1}^{q}m_{ij}(m_{ij}+1)$ is called the
cost of $M$. Finally it identifies all the factors of
$Q_{M}(\mathcal{X},\mathcal{Y})$ of type $\mathcal{Y}-f(\mathcal{X})$, where
$f(\mathcal{X})$ has degree no more than $k-1$. Among these, it picks the
candidate with the highest likelihood based on the channel pmf. Note that the
row index of $M$ can also be given by an element from the $F_{q}$, i.e.,
$m_{j}(\beta)=m_{ij}$ if $\beta=\gamma_{i}$. The score of a vector
$\mathbf{v}$ with respect to a multiplicity matrix $M$ is defined to be
$S_{M}(\mathbf{v})=\sum_{j=1}^{n}m_{j}(v_{j})$. If the entries in $M$
corresponding to the transmitted codeword $\mathbf{c}$ have large values, then
$\mathbf{c}$ has high score w.r.t. $M$. It has been shown [8] that as long as
the score of a codeword $S_{M}(\mathbf{c})\geq\Delta_{1,k-1}(C(M))$,
$\mathbf{c}$ will appear on the candidate list, i.e., the decoding is
successful. $\Delta_{1,k-1}(C(M))$ is defined in [8] and depends on $k$ and
$C(M)$ (increases with them).
## III RS codes for asymmetric SWC
Consider an asymmetric SWC scenario where source $X$ is available at the
terminal. If an RS code is used, the encoding for $\mathbf{y}$ is its syndrome
$\mathbf{s}=H\mathbf{y}$. The decoder needs to find the most probable
$\hat{\mathbf{y}}$ that belongs to the coset with syndrome $\mathbf{s}$. Upon
obtaining $\mathbf{x}$, the decoder finds the reliability matrix
$\Pi=\\{\pi_{ij}=P(Y_{j}=\gamma_{i}|X_{j}=x_{j})\\}$ based on the joint
distribution. Then, use the multiplicity algorithms to find a multiplicity
matrix $M$. The simplest choice is $M=\lfloor\lambda\Pi\rfloor$. If the RS
code is powerful enough to correct the errors introduced by the correlation
channel, the score $S_{M}(\mathbf{y})$ should satisfy the score condition. We
want to obtain $\mathbf{y}$ from the matrix $M$ by interpolation and
factorization. Note that $\mathbf{y}$ is not a codeword but belongs to a coset
with syndrome $\mathbf{s}$. This requires us to modify the KV algorithm
appropriately. An approach to modify Guruswami and Sudan's hard decision
decoding algorithm [9] to syndrome decoding was proposed in [10] and [5]
independently. Our approach is motivated by them. Find a $\mathbf{z}$
belonging to the coset with syndrome $\mathbf{s}$. This can be done by letting
any $k$ entries in $\mathbf{z}$ to be zero and solve $H\mathbf{z}=\mathbf{s}$.
The uniqueness of the solution is guaranteed by the MDS property of the RS
code. Construct a shifted multiplicity matrix $M^{\prime}$ from $M$ according
to $\mathbf{z}$, where $m^{\prime}_{j}(\gamma_{i})=m_{j}(\gamma_{i}+z_{j})$,
or, equivalently, $m^{\prime}_{j}(\gamma_{i}+z_{j})=m_{j}(\gamma_{i})$, for
$1\leq i\leq q,1\leq j\leq n$. Interpolate the
$Q_{M^{\prime}}(\mathcal{X},\mathcal{Y})$ according to $M^{\prime}$ as in KV
algorithm and find the list of candidate codewords $\mathcal{L}_{\mathbf{c}}$
by factorization. Adding $\mathbf{z}$ to each candidate codeword we obtain the
set of candidates $\mathcal{L}_{\mathbf{y}}$ for $\mathbf{y}$.
Claim: $\mathbf{y}\in\mathcal{L}_{\mathbf{y}}$ if $H\mathbf{y}=\mathbf{s}$ and
$S_{M}(\mathbf{y})\geq\Delta_{1,k-1}(C(M))$.
Proof: The interpolation and factorization ensure that if a codeword
$\mathbf{c}$ is such that
$S_{M^{\prime}}(\mathbf{c})\geq\Delta_{1,k-1}(C(M^{\prime}))$,
$\mathbf{c}\in\mathcal{L}_{\mathbf{c}}$. Note that each column of $M^{\prime}$
is just a permutation of the corresponding column of $M$, so
$C(M)=C(M^{\prime})$ and $\Delta_{1,k-1}(C(M^{\prime}))=\Delta_{1,k-1}(C(M))$.
If a vector $\mathbf{y}$ satisfies $H\mathbf{y}=\mathbf{s}$ and
$S_{M}(\mathbf{y})\geq\Delta_{1,k-1}(C(M))$, $\mathbf{y}+\mathbf{z}$ is a
codeword and
$S_{M^{\prime}}(\mathbf{y}+\mathbf{z})=\sum_{j=1}^{n}m^{\prime}_{j}(y_{j}+z_{j})=\sum_{j=1}^{n}m_{j}(y_{j})=S_{M}(\mathbf{y})\geq\Delta_{1,k-1}(C(M^{\prime}))$,
thus $\mathbf{y}+\mathbf{z}\in\mathcal{L}_{\mathbf{c}}$. So
$\mathbf{y}\in\mathcal{L}_{\mathbf{y}}$. $\blacksquare$
Next, the decoder performs ML decoding on $\mathcal{L}_{\mathbf{y}}$ based on
$\Pi$. It is shown in the simulations that this step is almost always correct.
Thus, if $\mathbf{y}$ satisfies the score condition, the decoding is
successful (with very high probability). The performance of the algorithm
depends on the multiplicity assignment, during which the correlation between
the sources is exploited.
Remark: 1) The soft information we used is the conditional pdf $P(Y|X)$. It
does not require the correlation model to be additive. So it is suitable for
more general correlation models.
2) RS codes enable rate adaptivity easily because of the structure of the
parity check matrix. Suppose a syndrome $H_{1}\mathbf{y}$ is available at the
decoder but the decoding fails. The terminal wants to know $H_{2}\mathbf{y}$,
where $H_{2}$ has $(n-k_{2})$ rows and $k_{1}\geq k_{2}$. We can transmit
additional inner products of $\mathbf{y}$ and newly added rows in $H_{2}$ and
together with the syndrome received previously, the decoder obtains the
syndrome $H_{2}\mathbf{y}$. Then the decoder works for a code with lower code
rate.
## IV Comparison with a single LDPC code
RS codes are Maximum Distance Separable (MDS) codes. However, it is well known
that RS codes are not capacity-achieving over probabilistic channels such as
the BSC and the $q$-ary symmetric channel. On the other hand, LDPC codes are
capacity-achieving under binary symmetric channels. It is expected and
observed in simulation that for binary correlated sources, LDPC codes have
better performance. However, we expect that RS codes could be a better fit for
sources over large alphabets, at least for the channels that resemble
deterministic channels, e.g., $q$-ary symmetric channels.
One simple way to use LDPC codes in nonbinary Slepian-Wolf coding is to use a
single LDPC code to encode the binary image of the nonbinary symbols. Consider
a correlation model for sources $X$ and $Y$ expressed as $X=Y+E$, where
$X,Y,E\in F_{512}$ such that $E$ is independent of $X$ and the agreement
probability $P_{a}=P(E=0)=1-p_{e},P(E=\gamma)=p_{e}/(q-1)$ for nonzero
$\gamma\in F_{512}$. $X$ and $Y$ are uniformly distributed. This is called
$q$-ary symmetric correlation model. RS codes are defined over $F_{512}$ with
length 511. The LDPC codes for comparison have length 4599 and a maximum
variable node degree of 30 and were generated using the PEG algorithm [6]. For
a given source pair, we use one LDPC code and encode for the binary image of
the source outputs and the initial bit level LLR for belief propagation
decoding is found by appropriate marginalization. We used three different code
rates. For each code, we increase $P_{a}$ (decrease $H(Y|X)$) until the frame
error rate was less than $10^{-3}$ and recorded the corresponding $H(Y|X)$ as
the maximum $H(Y|X)$ that allows us to perform near error-free compression.
The results are available in Table I.
TABLE I: Comparison of RS codes and LDPC codes $k/n$ | Tx Rate (bits/sym) | RS max $H(Y|X)$ | LDPC max $H(Y|X)$
---|---|---|---
0.2 | 7.2 | 5.3175 | 3.7855
0.3 | 6.3 | 4.3770 | 3.3740
0.5 | 4.5 | 2.8474 | 1.7271
We observe that LDPC has larger gap between the $H(Y|X)$ and the actual
transmission rate than RS codes. As expected, RS codes also have a gap to the
optimal rate. We also run the unique decoding algorithm for RS codes
(Berlekamp-Massey algorithm) and observe that the performance is better than
LDPC codes but worse than KVA.
## V Comparison with multistage LDPC codes: Classial Slepian-Wolf scenario
### V-A Multistage LDPC codes
Multistage LDPC codes have been proposed for Slepian-Wolf coding for nonbinary
alphabets in prior work [3]. To compress a source with alphabet size $q$, we
can view it as $r=\log_{2}q$ binary sources. Suppose $X$ is known at the
terminal and the source $Y$ is represented as bit sources
$Y_{b_{1}},Y_{b_{2}},\ldots,Y_{b_{r}}$. The source transmits the syndromes of
each bit source sequence,
$\mathbf{s}_{k}=H_{k}\mathbf{y}_{b_{k}},k=1,2,\ldots,r$, where $H_{k}$ is the
parity check matrix of a LDPC code. At the decoder, the side information $X$
is given, and to decode the $k$th bit source, the previous decoded bit sources
can also be used as side information, based on which the initial LLR is
computed. The decoding requires us to decode $r$ LDPC codes.
The design of optimized LDPC codes for our problem requires us to consider the
individual bit level channels and the distribution of the input LLRs at each
bit level. This is a somewhat complicated task and is part of ongoing work.
Here we use the following two designs for comparison.
#### V-A1 Dedicated LDPC codes
We optimize the degree distribution using density evolution for AWGN
channel111As explained before, ideally we should run density evolution for the
actual bit level channel broken down from the symbol level correlation
channel. This is part of ongoing work. In addition, we require a large number
of codes in order to match the required rates at the different bit levels.
Since AWGN optimized LDPC codes are known to have very good performance in
related channels such as the BSC, we chose to work with them for the
comparison.. Then, the code of length 512 is designed by PEG algorithm222We
need to choose a block length for each LDPC code so that the comparison with
the RS code of length 255 (8-bit symbols) is fair. We chose a length of 512,
that is approximately $2\times 255$. With higher LDPC block lengths, one can
expect better performance.. We design LDPC codes with rates
$0.02,0.04,0.06,\ldots,0.90$, a total of 45 codes. These codes are designed
separately and do not provide rate adaptivity.
#### V-A2 Rate-adaptive LDPC codes
Designed in [7], these irregular LDPC codes have length 6336 and the code rate
can be chosen among $\\{0/66,1/66,\ldots,64/66\\}$. The structure of their
parity check matrices allow us to use them in a rate-adaptive manner. Note
that these codes have a very high block length.
### V-B Simulation Setting
We consider classical SWC scenario. Given a correlation model, we gradually
increase the transmission rate until the frame error rate is less than
$10^{-3}$. The decoder attempts decoding only once. For LDPC codes, a frame is
in error if one of the decodings is in error. When we adjust the transmission
rate, we adjust the rate of the LDPC codes for each bit source, so that the
FER for each bit source are of the same order. To get the FER$<10^{-3}$ at
nonbinary symbol level, the FERs at the bit level are roughly $10^{-4}$. For
each rate configuration, we simulate until the number of error frame is at
least 100. The maximum iteration time of the belief propagation algorithm is
100. For RS codes, the field size $q=256$ and the length $n=255$.
$\lambda=100.99$ in the multiplicity assignment. We increase the transmission
rate until the FER $<10^{-3}$. The decoder attempts decoding only once.
### V-C $q$-ary symmetric correlation model
The simulation results for $q$-ary ($q=256$) symmetric correlation model under
different agreement probabilities are given in Fig. 1(solid lines). The gaps
between actual transmission rates and $H(Y|X)$ are presented. Larger gap
indicates worse performance. We observe that under $q$-ary symmetric
correlation models RS codes outperform both types of LDPC codes. This
coincides with our intuition since the $q$-ary symmetric is favorable for RS
codes. Note that RS codes performs better when the agreement probability
$P_{a}$ is very high or very low. For low $P_{a}$, a RS code with low rate is
used and it is observed before [8] that the Koetter-Vardy algorithm performs
better for low rate codes. When $P_{a}$ is very low, for multistage LDPC
codes, only a portion of bit sources can be compressed, several bit sources
need to be transmitted at rate one.
Figure 1: The gap between the transmission rate and $H(Y|X)$ for multistage
LDPC and RS codes under $q$-ary symmetric models. Solid line represents
classical SWC scenario and the dash-dot line represents feedback scenario.
### V-D Sparse correlation model
When the correlation model becomes more general, RS codes do not always
outperform LDPC codes. Under the correlation model where each column of the
conditional probability matrix $P(Y|X=j)$ contains a few dominant terms, it is
possible that RS codes still perform well. We call such kind of correlation
models to be sparse. We shall compare the performance of multistage LDPC codes
and RS codes under sparse correlation models defined as follows.
###### Definition 1
We say a conditional pdf $P(Y|X)$ is $(S,\epsilon)$-sparse if for every
$j=1,\ldots,q$, $P(Y=i|X=j),i=1,\ldots,q$ have $S$ entries that are greater
than $\epsilon$.
We are mostly interested in $(S,\epsilon)$-sparse conditional pdf $P(Y|X)$
with $S\ll q$ and $\epsilon\ll 1$, i.e., for each $j$, $P(Y=i|X=j)$ has few
dominant entries. For those entries with probability mass less than
$\epsilon$, we assume that the probabilities are the same. When $X$ is
uniformly distributed, the joint pdf is also sparse and we call such a
correlation model, a sparse correlation model. For a $(S,\epsilon)$-sparse
conditional pdf $P(Y=i|X=j)$, denote the vector of the $S$ dominant entries by
$D(j)$. We assume that the dominant entries are the same for all $j$ and
denote them by $D$. For example, for a $q$-ary symmetric correlation model
with $q=256$ and $P_{a}=0.8$, $D=[0.8]$ and it is $(1,10^{-3})$-sparse. For a
fixed $D$, there are a lot of choices of the locations of the dominant
entries. We consider the following dominant entry patterns.
The dominant entries can be put in the diagonal form, a generalization of
$q$-ary symmetric correlation model. The largest entries are on the diagonal
of the conditional pdf matrix and other entries are put around them. For
example, consider a joint pdf with $(3,10^{-3})$-sparse conditional
distribution and $D=[0.1~{}0.6~{}0.1]$. When it is placed in the diagonal
form, $P(Y=j|X=j)=0.6$ for all $j$, $P(Y=j-1|X=j)=0.1$ for all $j$ except
$j=1$, $P(Y=j+1|X=j)=0.1$ for all $j$ except $j=256$ and
$P(Y=256|X=1)=P(Y=1|X=256)=0.1$. All other entries are
$(1-0.1-0.6-0.1)/253<10^{-3}$. The dominant entries in a conditional pdf is
said to be in the random form if $D$ is uniformly randomly placed in the
column $P(Y|X=j)$. Note that this randomness only appear in the determination
of the pdf and it will be fixed during all transmissions. This correlation
model is a model $Y=X+E$ where $E$ depends on $X$ (data dependent model). Note
that different placements of probability masses in the columns of conditional
distribution do not change the conditional entropy $H(Y|X)$, and do not affect
the performance of KV algorithm for RS codes. But the performance of
multistage LDPC codes changes when the placement of probability masses
changes. In simulations, multistage LDPC codes performs better under diagonal
form conditional distribution than the random form.
Note that a dominant entry vector could have a number of forms. It is hard to
parameterize it using simple parameters. In our simulations, we fix the length
of $D$ to be three and there is one distinguished large value in the vector.
The vectors of dominant entries in conditional pdf are presented in Table II.
They are the same for different $j$ in $P(Y|X=j)$. Other than dominant
entries, other entries have the same probability. They are all
$(3,0.0015)$-sparse conditional pdfs. Source $X$ is uniformly distributed. For
a vector of dominant entries, we define peak factor to be the ratio between
the maximum entry and the minimum entry in the vector.
TABLE II: The $D$ vectors used in the simulations. $D$ | PF | D | PF
---|---|---|---
[0.15 0.6 0.15] | 4 | [0.1 0.6 0.1] | 6
[0.1 0.7 0.1] | 7 | [0.1 0.75 0.1] | 7.5
[0.1 0.79 0.1] | 7.9 | [0.05 0.6 0.05] | 12
[0.05 0.7 0.05] | 14 | [0.03 0.6 0.03] | 20
We show our simulation results in Fig. 2, in an ascending order of peak factor
(PF). The plots do not look as smooth as Fig. 1. This is because peak factor
is not a single parameter for the pdfs, e.g., for a fixed PF, there could be
multiple choices of the pdf and we choose one of them in our simulation. The
gaps between actual transmission rates and the conditional entropies are
presented. The alphabet size $q=256$. Both random form and diagonal form
conditional pdf are investigated. For RS codes, the performance is the same
under these two forms. We observe the following. The performance of RS codes
improves with the increase of the PF. RS codes perform better than rate-
adaptive LDPC codes under the correlation models with large PF, while rate-
adaptive LDPC codes perform better than RS codes under the correlation models
with small PF. However, dedicated LDPC codes outperform RS for most of PF
values.
We also investigate the situation where the decoder is given a slightly
different joint pdf. The actual pdf is in the diagonal form. The pdf provided
to the decoder has right locations for the largest dominant entries but wrong
(somewhat arbitrary) locations for another two smaller dominant entries in
$D$. In this case, the performance of LDPC codes suffer a lot and RS codes
suffer only a little. The results are also presented in Fig. 2. It is
important to note that in this situation, RS codes in fact perform better than
multistage LDPC codes. In a practical setting there may be situations where
there are modeling errors or incomplete knowledge about the joint pdf of the
sources. Our results indicate that RS codes are much more resilient to
inaccuracies in correlation models.
Figure 2: The gap between the actual transmission rate and the conditional
entropy for multistage LDPC codes and RS codes under sparse correlation
models. For RS codes, the performance under diagonal form conditional
distribution and random form conditional distribution are the same.
## VI Comparison with multistage LDPC codes: Feedback scenario
### VI-A Simulation setting
We consider the second scenario where the decoder feeds back some information
and the actual transmission rates are adapted such that the decoder is able to
decode. RS codes offer natural rate-adaptivity and we compare their
performance with the rate adaptive LDPC codes designed in [7]. For multistage
LDPC codes, after receiving the binary syndromes from the encoder, the decoder
tries to decode from the first bit source. If it fails, it requests more bits
from the source and tries to decode again. The decoder repeats this procedure
until the first bit source is decoded and then moves on to the second bit
source and works in a similar manner. It is guaranteed that the previously
decoded bits are always correct. Two rate-adaptive LDPC codes are used, with
length 6336 and 396, both designed in [7]. For RS codes, if the decoder fails
(there is no codeword on the candidate list), it requests more symbols from
the source and tries again. The decoder repeats this until the source sequence
is decoded. The amount of feedback is several bits per block for both LDPC
codes and RS codes, depending on the gap. But LDPC codes need more feedback
since the decoder needs to adjust rate for each bit source. We repeat this
experiment 500 times and record the minimum required transmission rates. The
simulation results are the average minimum required rates and their standard
deviation.
### VI-B $q$-ary symmetric correlation models
The gap of the average minimum transmission rate to the conditional entropy is
presented in Fig. 1 (dash-dot lines). RS outperform rate-adaptive LDPC codes
when the agreement probability is very high or very low. But for intermediate
$P_{a}$, multistage LDPC codes perform better. For LDPC codes with length
6336, the standard deviations of the required rates are in the range of 0.08
and 0.1, while LDPC codes with length 396, the standard deviation are between
0.19 and 0.30. The standard deviations of RS codes are between 0.13 and 0.32.
### VI-C Sparse correlation models
The gap of the average minimum transmission rate to $H(Y|X)$ is presented in
Fig. 3. RS performs worse than both multistage LDPC codes, although the
performance improves with the PF. The average rate performance is comparable
between LDPC codes with length 6336 and 396, and between diagonal form and
random form correlation models, but length 6336 codes are much more stable,
with standard deviation 0.06 to 0.1. RS codes have standard deviation between
0.24 and 0.30, and length 396 LDPC codes have standard deviation between 0.11
and 0.27. The results for the case where inaccurate pdfs are provided to the
decoder are also presented and we observe that RS codes are much more
resilient and perform better than LDPC codes with length 6336.
Figure 3: The gap between the average minimum transmission rate and $H(Y|X)$
for multistage LDPC and RS codes under sparse correlation models.
## VII conclusion
In this work we have proposed practical SW codes using RS codes. Compared to
multistage LDPC codes, RS codes are easy to design, offer natural rate-
adaptivity and allow for relatively fast performance analysis. Simulations
show that in classical SWC scenario, RS codes perform better than both designs
of multistage LDPC codes under $q$-ary symmetric model and better than rate-
adaptive LDPC codes under the sparse correlation model with high PF. In a
feedback scenario, the performance of RS codes and multistage LDPC codes are
similar under $q$-ary symmetric model but LDPC codes outperform RS codes under
sparse correlation model. An interesting conclusion is that RS codes are much
more resilient to inaccurate pdfs in both scenarios.
For symmetric Slepian-Wolf coding, if the correlation model is given by
additive error, i.e., $X=Y+E$, it is not hard to propose a scheme that first
recover the error vector $\mathbf{e}$ and then recover the source sequences.
The more interesting and challenging problem is to apply algebraic approaches
to more general correlation models, where the problem can not be mapped to a
simple channel decoding problem. The problem remains open and will be an
interesting future work.
## References
* [1] A. Wyner, ``Recent results in Shannon theory,'' _IEEE Trans. on Info. Theo._ , vol. 20, pp. 2–10, Jan. 1974.
* [2] Z. Xiong, A. Liveris, and S. Cheng, ``Distributed source coding for sensor networks,'' _Signal Processing Magazine, IEEE_ , vol. 21, no. 5, pp. 80–94, Sep. 2004.
* [3] Y. Yang, S. Cheng, Z. Xiong, and W. Zhao, ``Wyner-Ziv coding based on TCQ and LDPC codes,'' _Communications, IEEE Transactions on_ , vol. 57, no. 2, pp. 376–387, February 2009.
* [4] Y. Zhao and J. Garcia-Frias, ``Data compression of correlated non-binary sources using punctured turbo codes,'' in _Proceedings. DCC 2002_ , 2002, pp. 242 – 251.
* [5] M. Ali and M. Kuijper, ``Source coding with side information using list decoding,'' in _Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on_ , 2010, pp. 91 –95.
* [6] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, ``Progressive edge-growth tanner graphs,'' in _GLOBECOM '01. IEEE_ , vol. 2, 2001, pp. 995 –1001.
* [7] D. Varodayan, A. Aaron, and B. Girod, ``Rate-adaptive codes for distributed source coding,'' _Signal Process._ , vol. 86, no. 11, pp. 3123–3130, 2006\.
* [8] R. Koetter and A. Vardy, ``Algebraic soft-decision decoding of reed-solomon codes,'' _IEEE Trans. on Info. Theo._ , vol. 49, no. 11, pp. 2809–2825, Nov. 2003.
* [9] V. Guruswami and M. Sudan, ``Improved decoding of reed-solomon and algebraic-geometry codes,'' _Information Theory, IEEE Transactions on_ , vol. 45, no. 6, pp. 1757 –1767, Sep. 1999.
* [10] S. Li and A. Ramamoorthy, ``Improved compression of network coding vectors using erasure decoding and list decoding,'' _IEEE Communications Letters_ , vol. 14, no. 8, pp. 749–751, 2010.
|
arxiv-papers
| 2011-06-14T19:17:21 |
2024-09-04T02:49:19.681609
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shizheng Li and Aditya Ramamoorthy",
"submitter": "Shizheng Li",
"url": "https://arxiv.org/abs/1106.2792"
}
|
1106.2955
|
# Epitaxial interfaces between crystallographically mismatched materials
Steven C. Erwin Center for Computational Materials Science, Naval Research
Laboratory, Washington, DC 20375 Cunxu Gao Claudia Roder Jonas Lähnemann
Oliver Brandt Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz
5–7, 10117 Berlin, Germany
###### Abstract
We report an unexpected mechanism by which an epitaxial interface can form
between materials having strongly mismatched lattice constants. A simple model
is proposed in which one material tilts out of the interface plane to create a
coincidence-site lattice that balances two competing geometrical criteria—low
residual strain and short coincidence-lattice period. We apply this model,
along with complementary first-principles total-energy calculations, to the
interface formed by molecular-beam epitaxy of cubic Fe on hexagonal GaN and
find excellent agreement between theory and experiment.
###### pacs:
81.15.Aa, 81.15.Hi, 68.55.-a, 68.37.Lp
A fundamental goal of materials science is to elucidate and exploit the
physical principles that govern epitaxial growth Bauer et al. (1990);
Palmstrøm (1995). Some of these principles are well-established. For example,
if the lattice constants of the film and substrate are close but not identical
then a coherently strained film may grow up to a critical thickness, beyond
which misfit dislocations relieve the strain Matthews and Blakeslee (1974).
Alternatively, a film and substrate having lattice spacings close to an
integer ratio $m/n$ may form an epitaxial interface described by a coincidence
lattice Sutton and Balluffi (1987); Trampert (2002).
In this Letter we report a new and unexpected mechanism by which epitaxial
films can grow on substrates having, in principle, an arbitrary lattice
mismatch. We illustrate this mechanism experimentally by growing single-
crystal Fe on $M$-plane GaN. The Fe grows in an unusual crystallographic
orientation with a very high Miller index, Fe(205). We develop a simple
theoretical model which, when complemented with total-energy calculations,
correctly predicts this exact orientation as well as the single-domain nature
of the film. Finally, we use our model to propose a new strategy for growing
nonpolar epitaxial GaN films on high-index Si substrates.
Figure 1: (a) Cross-sectional transmission electron micrograph taken along the
$[\overline{1}\overline{1}20]$ zone axis of the Fe/GaN/4H-SiC structure under
investigation. The top 2 nm of the Fe film are oxidized. (b) Cross-sectional
high-resolution transmission electron micrograph of the
Fe/GaN$(1\overline{1}00)$ interface along the $[\overline{1}\overline{1}20]$
zone axis. The Fe($\overline{1}$01) and GaN(0001) lattice planes form an
angle, $\alpha$, of approximately $23^{\circ}$. Stacking faults in the GaN,
visible in both panels, do not affect the orientation of the Fe film.
The epitaxial growth of both GaN and Fe was performed in a custom-built
molecular-beam epitaxy system equipped with solid-source effusion cells for Ga
and Fe. Active nitrogen was provided by a radio-frequency N2 plasma source.
Nucleation and growth were monitored _in situ_ by reflection high-energy
electron diffraction. A 130-nm thick layer of $M$-plane GaN was first grown on
a 4H-SiC$(1\overline{1}00)$ substrate under Ga-stable conditions and a
temperature of 720∘C. After growth of the GaN layer, excess Ga was desorbed
prior to cooling down to 350∘C for the deposition of Fe Brandt et al. (2004).
The Fe film grew at this temperature at a rate of 0.13 nm/min to a final
thickness of 27 nm. The resulting Fe/GaN/SiC heterostructure was investigated
by cross-sectional transmission electron microscopy (TEM) and convergent-beam
electron diffraction (CBED) using a JEOL JEM-3010 operating at 300 keV.
Electron backscattering diffraction (EBSD) was carried out in a Zeiss Ultra-55
scanning electron microscope equipped with an EDAX-TSL EBSD system.
Figure 2: Schematic view of a commensurate interface between a film with
lattice constant $d_{f}$ and a substrate with lattice constant $d_{s}$. The
interface plane is $(h0\ell)$ with respect to the film, corresponding to a
tilting from (001) toward (101) by the angle $\theta=\tan^{-1}(h/\ell)$. The
condition for commensurability is that a coincidence-site lattice, defined by
a pair of integers $(m,n)$, exists such that
$md_{s}=n(h^{2}+\ell^{2})^{1/2}d_{f}$. The example depicted here is (205) with
$\theta=21.8^{\circ}$ (equivalent to the angle $\alpha=23.2^{\circ}$ as
defined in Fig. 1) and $(m,n)=(3,1)$.
Figure 1(a) shows a cross-sectional transmission electron micrograph of the
Fe/GaN/4H-SiC structure. Despite the structural and chemical differences of
the constituent materials, the respective layers are well defined and exhibit
abrupt interfaces. The GaN layer is seen to contain stacking faults due to the
stacking mismatch between 2H-GaN and 4H-SiC. Nevertheless, the high-resolution
detail of the Fe/GaN interfacial region shown in Fig. 1(b) reveals an
epitaxial relationship between the Fe film and the underlying
GaN$(1\overline{1}00)$ layer. The ($\overline{1}$01) lattice planes of the Fe
film are clearly resolved and are found to be well ordered, unaffected by the
stacking disorder in the GaN layer. Of special interest is the angle,
approximately 23∘, formed by the Fe ($\overline{1}$01) planes and the vertical
interface normal. This angle indicates that the Fe interface plane has a high
Miller index—an unexpected finding in light of the comparatively high surface
energies of high-index metal surfaces. We show now that precisely this
orientation is predicted by a simple, physically transparent model
(complemented with first-principles total-energy calculations) of epitaxial
interfaces between dissimilar materials.
Consider the formation of an interface between a film ($f$) and a substrate
($s)$ having different lattice constants $d_{f}$ and $d_{s}$. If the lattice
mismatch is sufficiently small then the strained film may grow coherently
until it reaches its critical thickness Matthews and Blakeslee (1974). For
much larger mismatch this scenario becomes unlikely. Epitaxial growth is
nevertheless possible by tilting the orientation of the film, as the TEM image
in Fig. 1(b) makes clear. Figure 2 illustrates how an arbitrary lattice
mismatch can be accommodated by allowing the film to have an orientation
between (001) and (101) given by the Miller indices $(h0\ell)$. Our goal below
is to predict the most stable film orientation given the lattice constants
$d_{f}$ and $d_{s}$.
The unit cell of a film with orientation $(h0\ell)$ has length
$L_{f}=(h^{2}+\ell^{2})^{1/2}d_{f}$. In order for the film and substrate to be
commensurate there must exist a coincidence-site lattice (CSL), defined by a
pair of integers $(m,n)$, such that $md_{s}=nL_{f}$. This condition is
unrealistically restrictive, however. In real systems the film will tolerate a
small compressive or tensile strain $\varepsilon_{xx}$ which relaxes the CSL
condition to $md_{s}=nL_{f}(1+\varepsilon_{xx})$.
Figure 3: Misfit strain in commensurate interfaces between bcc Fe and
GaN$(1\overline{1}00)$. The points represent all possible Fe$(h0\ell)$ planes
having $h\leq\ell\leq 75$. The misfit $\varepsilon_{xx}$ is the strain
component along the [0001] direction of the GaN. The size of each plot symbol
is inversely proportional to the period $m$ of the coincidence-site lattice
(CSL) that minimizes the misfit. Fe$(h0\ell)$ planes having both small misfit
strain and small CSL period are labeled.
We propose two geometrical criteria for identifying candidate orientations for
interfaces with low energy. (1) The misfit strain $\varepsilon_{xx}$ should be
as small as possible, and (2) the period $m$ of the CSL should be as small as
possible. The latter criterion is motivated by analogy to low-energy grain
boundaries between two identical materials, which often have a CSL with small
unit cell volume $\Sigma$ Sutton and Balluffi (1987). For interfaces between
different materials it is not generally possible to minimize the strain and
CSL period simultaneously. Nor is it obvious how to construct a single
objective function of both which could then be optimized. Instead, we apply
both criteria with the aim of selecting a small subset of candidate
orientations for subsequent study with a more quantitative method such as
density-functional theory (DFT).
To apply these criteria to the growth of Fe on the $M$-plane of GaN we equate
$d_{f}$ with the bcc Fe lattice constant, 2.866 Å Mao et al. (1967), and
$d_{s}$ with the GaN $c$ lattice parameter, 5.186 Å Leszczynski et al. (1996).
Figure 3 shows the resulting Fe misfit strain needed to satisfy the CSL
condition for a large number of hypothetical orientations of the Fe film. In
this plot each orientation $(h0\ell)$ is represented by its angle
$\theta=\tan^{-1}(h/\ell)$ relative to the (001) plane. The period of each CSL
is encoded by the size of the plot symbol, which is inversely proportional to
$m$. Only points with small strains, less than 3%, are displayed here.
Figure 4: (a) Pseudo-Kikuchi pattern of the Fe film on GaN$(1\overline{1}00)$
depicted in Fig. 1. The major low-index zone axes are indicated. (b) In-plane
orientation map of the Fe film. Variations in color encode local deviations
from the nominal orientation. The film has the same orientation over the
entire area within a tolerance of 1∘. (c) Model of the
Fe(205)/GaN$(1\overline{1}00)$ epitaxial interface. For clarity an ideal
geometry with arbitrary registry is shown here. Slightly less than one unit
cell along [0001] is depicted. A conventional unit cell of Fe is highlighted
in yellow.
The rich structure visible in Fig. 3 makes it clear that the misfit strain can
be made arbitrarily small for many different film orientations. Hence the
strain alone cannot provide a definitive criterion favoring a particular
growth plane. Moreover, the vast majority of these low-strain orientations
require a very large CSL period and hence do not constitute physically
meaningful commensurability. Only a very few orientations offer both a small
strain and small CSL period, namely (207), (205), and (203). The (205)
orientation corresponds to the angle $\theta=21.8^{\circ}$. This is equivalent
to $\alpha=45-\theta=23.2^{\circ}$ as defined in Fig. 1(b) and thus is in
excellent agreement with the measured angle, 23∘, obtained from TEM.
It is important to realize that our purely geometrical criteria do not
distinguish between the orientation (205) depicted in Fig. 2 and its symmetry-
equivalent counterpart, (20$\overline{5}$) ($\theta=-21.8^{\circ}$), created
by rotating the film by 180∘ about the substrate normal. Indeed, the pairs
[$(h0\ell)$, $(h0\overline{\ell})$] have the same strain and CSL and hence are
equivalent within this model. In the Fe/GaN system, however, the polarity of
the GaN wurtzite structure breaks this equivalence. The question that then
arises is whether the influence of the polarity is sufficiently strong to
select a single orientation, and if so, which one? The TEM image in Fig. 1
shows a single orientation but is limited to a nanometer-scale region of the
film. To characterize a much larger area we used EBSD Nolze et al. (2005).
Figure 4(a) shows the resulting pseudo-Kikuchi pattern of the Fe film in Fig.
1. The pattern exhibits sharp and well defined Kikuchi bands, reflecting high
crystal quality of the Fe film and allowing for a fast and reliable indexing
of the patterns recorded while scanning the electron beam over a large area (9
$\mu$m2). The resulting EBSD map shown in Fig. 4(b) visualizes the in-plane
orientation of the Fe film with a spatial and angular resolution of 20 nm and
1∘, respectively. The map reveals the complete absence of any domain
structure. Indeed, the film is single crystalline, and has the same
orientation as found by TEM throughout the mapped area.
To understand why a single orientational domain is found requires going beyond
a model based solely on interface geometry. Now the interface structure—its
precise atomic arrangement and chemical bonding—must be addressed. To do this
we used DFT to calculate the relative formation energies of finite Fe films
grown on the $M$-plane of GaN. We considered the six different orientations
predicted by the geometrical model to be favorable: (207), (205), (203), and
their rotated counterparts (20$\overline{7}$), (20$\overline{5}$),
(20$\overline{3}$). The different films were the same thickness, 6 Å,
equivalent to about four monolayers. Figure 4(c) depicts the
Fe(205)/GaN$(1\overline{1}00)$ interface as an example.
We have previously shown that at 3–4 monolayers the contribution of the Fe/GaN
interface formation energy to the full formation energy of the film is already
converged Gao et al. (2010). We also find that the Fe free surface energy
varies by less than 1 meV/Å2 among the three orientations we consider here
Erwin (unpublished). Therefore the relative formation energy of the finite
film closely mirrors, with good accuracy, the formation energy of the isolated
interface.
For each orientation the Fe film was slightly strained along the $x$
[GaN(0001)] direction according to its CSL as discussed above. There is also a
lattice mismatch in the $y$ [GaN$(\overline{1}\overline{1}20)$)] direction
because the Fe lattice constant and GaN $a$ lattice parameter differ by nearly
12%. This mismatch was accommodated by a single CSL, common to all
orientations, containing eight unit cells of Fe and seven of GaN. The GaN
substrate was represented by a slab of four atomic layers with fixed in-plane
equilibrium lattice parameters and a passivating bottom layer. Total energies
and forces were calculated within the PBE generalized-gradient approximation
Perdew et al. (1996) to DFT using projector-augmented-wave potentials as
implemented in vasp Kresse and Hafner (1993); Kresse and Furthmüller (1996).
All Fe and GaN atomic positions were fully relaxed except the bottom GaN
layer. For each Fe film orientation the formation energy depends strongly on
the choice of Fe-GaN interface registry. We systematically varied the registry
over a grid in both $x$ and $y$ to locate the global energy minimum for each
orientation. The plane-wave cutoff for all calculations was 400 eV.
The resulting formation energies are listed in Table 1. The most favorable
orientation is Fe(205). Of the candidates tested, this orientation has neither
the smallest possible strain nor the shortest possible CSL period, indicating
that the optimal interface structure is an important third criterion that must
supplement the two geometrical criteria. Note also that the small variation in
the formation energy of the Fe surface, which is included in the formation
energy of the film, is too small to affect the overall energy ordering of the
orientations.
Table 1: Formation energies calculated within density-functional theory of Fe/GaN$(1\overline{1}00)$ films for the Fe planes in Fig. 3 that have both a small misfit strain and a coincidence-site lattice (CSL) with small period $m$. The Fe film thickness is the same for all cases, approximately 6 Å. Formation energies are relative to the most favorable plane, Fe(205), in units of meV/Å2. Plane | Misfit strain | CSL period | Formation energy
---|---|---|---
Fe($205$) | $+0.008$ | 3 | 0
Fe($20\overline{5}$) | $+0.008$ | 3 | 4
Fe($203$) | $+0.004$ | 2 | 9
Fe($20\overline{3}$) | $+0.004$ | 2 | 16
Fe($207$) | $-0.006$ | 4 | 12
Fe($20\overline{7}$) | $-0.006$ | 4 | 13
To distinguish Fe(205) from Fe(20$\overline{5}$) experimentally requires
determining the absolute polarity of the GaN substrate. We did this by
recording CBED patterns with a beam spot size of approximately 15 nm under
two-beam conditions. Simulations of CBED patterns were performed using jems
sta to index the crystallographic directions observed in the experimental
patterns and thus to determine the polarity. The resulting absolute
orientation relationship is
Fe$(205)[0\overline{1}0]||$GaN$(1\overline{1}00)[\overline{1}\overline{1}20]$,
in agreement with the prediction of our theoretical model complemented with
the results of DFT calculations.
Our model suggests a new strategy for growing non-polar GaN films. The basic
idea is to turn Fig. 2 upside-down and consider the growth of $M$-plane GaN on
a suitable high-index substrate. For a given material this requires
identifying candidate orientations corresponding to small strain and small CSL
period. One very promising material is Si, which is already in widespread use
as a flat substrate for GaN/Si epitaxy despite the resulting high dislocation
densities Joblot et al. (2006). Many high-index Si substrates are readily
available, and some have already been used for GaN growths Ni et al. (2009);
Ravash et al. (2010). Calculations are in progress to identify promising high-
index Si orientations for growing nonpolar GaN with low strain Kutana and
Erwin (unpublished).
## References
* Bauer et al. (1990) E. G. Bauer, B. W. Dodson, D. J. Ehrlich, L. C. Feldman, C. P. Flynn, M. W. Geis, J. P. Harbison, R. J. Matyi, P. S. Peercy, P. M. Petroff, et al., J. Mater. Res. 5, 852 (1990).
* Palmstrøm (1995) C. J. Palmstrøm, Ann. Rev. Mat. Sci. 25, 389 (1995).
* Matthews and Blakeslee (1974) J. W. Matthews and A. E. Blakeslee, J. Crystal Growth 27, 118 (1974).
* Sutton and Balluffi (1987) A. P. Sutton and R. W. Balluffi, Acta Metall. 35, 2177 (1987).
* Trampert (2002) A. Trampert, Physica E 13, 1119 (2002).
* Brandt et al. (2004) O. Brandt, Y. J. Sun, L. Däweritz, and K. H. Ploog, Phys. Rev. B 69, 165326 (2004).
* Mao et al. (1967) H. K. Mao, W. A. Bassett, and T. Takahashi, J. Appl. Phys. 38, 272 (1967).
* Leszczynski et al. (1996) M. Leszczynski, H. Teisseyre, T. Suski, I. Grzegory, M. Bockowski, J. Jun, S. Porowski, K. Pakula, J. M. Baranowski, C. T. Foxon, et al., Appl. Phys. Lett. 69, 73 (1996).
* Nolze et al. (2005) G. Nolze, V. Geist, R. S. Neumann, and M. Buchheim, Cryst. Res. Technol. 40, 791 (2005).
* Gao et al. (2010) C. X. Gao, O. Brandt, S. C. Erwin, J. Lähnemann, U. Jahn, B. Jenichen, and H. P. Schönherr, Phys. Rev. B 82, 125415 (2010).
* Erwin (unpublished) S. C. Erwin (unpublished).
* Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
* Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
* Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
* (15) P. A. Stadelmann, JEMS electron microscopy software (Java version 3.5110U2010), CIME-EPFL, Switzerland, 1999–2010.
* Joblot et al. (2006) S. Joblot, Y. Cordier, F. Semond, S. Chenot, P. Vénnègues, O. Tottereau, P. Lorenzini, and J. Massies, Superlattices and Microstructures 40, 295 (2006).
* Ni et al. (2009) X. Ni, M. Wu, J. Lee, X. Li, A. A. Baski, Ü. Özgür, and H. Morkoç, Appl. Phys. Lett. 95, 111102 (2009).
* Ravash et al. (2010) R. Ravash, J. Blaesing, A. Dadgar, and A. Krost, Appl. Phys. Lett. 97, 142102 (2010).
* Kutana and Erwin (unpublished) A. Kutana and S. C. Erwin (unpublished).
|
arxiv-papers
| 2011-06-15T12:33:26 |
2024-09-04T02:49:19.691541
|
{
"license": "Public Domain",
"authors": "Steven C. Erwin, Cunxu Gao, Claudia Roder, Jonas L\\\"ahnemann, Oliver\n Brandt",
"submitter": "Steven C. Erwin",
"url": "https://arxiv.org/abs/1106.2955"
}
|
1106.3261
|
# LAGRANGIAN-HAMILTONIAN UNIFIED FORMALISM FOR AUTONOMOUS HIGHER-ORDER
DYNAMICAL SYSTEMS
Pedro Daniel Prieto-Martínez
Narciso Román-Roy
Departamento de Matemática Aplicada IV. Edificio C-3, Campus Norte UPC C/
Jordi Girona 1. 08034 Barcelona. Spain e-mail: peredaniel@ma4.upc.edue-mail:
nrr@ma4.upc.edu
###### Abstract
The Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk was
originally stated for autonomous dynamical systems in classical mechanics. It
has been generalized for non-autonomous first-order mechanical systems, as
well as for first-order and higher-order field theories. However, a complete
generalization to higher-order mechanical systems has yet to be described. In
this work, after reviewing the natural geometrical setting and the Lagrangian
and Hamiltonian formalisms for higher-order autonomous mechanical systems, we
develop a complete generalization of the Lagrangian-Hamiltonian unified
formalism for these kinds of systems, and we use it to analyze some physical
models from this new point of view.
Key words: Higher-order systems, Lagrangian and Hamiltonian formalisms,
Symplectic and presymplectic manifolds.
AMS s. c. (2000): 70H50, 53C80, 53C15
###### Contents
1. 1 Introduction
2. 2 Higher-order dynamical systems
1. 2.1 Geometric structures of higher-order tangent bundles
1. 2.1.1 Higher-order tangent bundles
2. 2.1.2 Higher-order canonical vector fields. Vertical endomorphisms and almost-tangent structures
3. 2.1.3 Vertical derivations and differentials. Tulczyjew’s derivation
4. 2.1.4 Higher-order semisprays
2. 2.2 Lagrangian formalism
3. 2.3 Hamiltonian formalism
3. 3 Skinner-Rusk unified formalism
1. 3.1 Unified phase space. Geometric and dynamical structures
2. 3.2 Dynamical vector fields
1. 3.2.1 Dynamics in ${\cal W}={\rm T}^{2k-1}Q\times_{{\rm T}^{k-1}Q}{\rm T}^{*}({\rm T}^{k-1}Q)$
2. 3.2.2 Dynamics in ${\rm T}^{2k-1}Q$
3. 3.2.3 Dynamics in ${\rm T}^{*}({\rm T}^{k-1}Q)$
3. 3.3 Integral curves
4. 4 Examples
1. 4.1 The Pais-Uhlenbeck oscillator
2. 4.2 The second-order relativistic particle
5. 5 Conclusions and outlook
## 1 Introduction
In recent decades, a strong development in the intrinsic study of a wide
variety of topics in theoretical physics, control theory and applied
mathematics has been done, using methods of differential geometry. Thus, the
intrinsic formulation of Lagrangian and Hamilonian formalisms has been
developed both for autonomous and non-autonomous systems. This study has been
carried out mainly for first-order dynamical systems; that is, those whose
Lagrangian or Hamiltonian functions depend on the generalized coordinates of
position and velocity (or momentum). From the geometric point of view, this
means that the phase space of the system is in most cases the tangent or
cotangent bundle of the smooth manifold representing the configuration space.
However, there are a significant number of relevant systems in which the
dynamics have explicit dependence on accelerations or higher-order derivatives
of the generalized coordinates of position. These systems, usually called
higher-order dynamical systems, can be modeled geometrically using higher-
order tangent bundles [17]. These models are typical of theoretical physics;
for example those describing the interaction of relativistic particles with
spin, string theories from Polyakov and others, Hilbert’s Lagrangian for
gravitation or Podolsky’s generalization of electromagnetism (see [8] and
references cited there). They also appear in a natural way in numerical models
arising from the discretization of first-order dynamical systems that preserve
their inherent geometric structures [16]. There are a lot of works devoted to
the development of the formalism of these kinds of theories and their
application to many models in mechanics and field theory (see, for instance,
[2], [3], [4], [7], [9], [12], [20], [25], [26], [35], [37], [38]).
Furthermore, a generalization of the Lagrangian and Hamiltonian formalisms
exists that compresses them into a single formalism. This is the so-called
Lagrangian-Hamiltonian unified formalism, or Skinner-Rusk formalism due to the
authors’ names of the original paper. It was originally developed for first-
order autonomous mechanical systems [39], and later generalized to non-
autonomous dynamical systems [6, 14], control systems [5], first-order
classical field theories [15, 18] and, more recently, to higher-order
classical field theories [10, 41]. Nevertheless, although the geometrization
of both higher-order Lagrangian and Hamiltonian formalisms was already
developed for autonomous mechanical systems [11, 17, 23, 29], a complete
generalization of the Skinner-Rusk formalism for higher-order mechanical
systems has yet to be developed. A first attempt was outlined in [13], with
the aim of providing a geometric model for studying optimal control of
underactuated systems, although a deep analysis of the model and its relation
with the standard Lagrangian and Hamiltonian formalisms was not performed.
Thus, the aim of this work is to provide a detailed and complete description
of the Lagrangian-Hamiltonian unified formalism for higher-order autonomous
mechanical systems. Our approach is different from that given in [13] (these
differences are commented on Section 5).
The paper is organized as follows:
Section 2 consists of a review of the basic definitions and the geometric
structures of higher-order tangent bundles, some of which are generalizations
of the geometric structures of tangent bundles; namely, the canonical vector
fields, the almost-tangent structures and semisprays; whereas others such as
the Tulczyjew derivation are needed for developing the Lagrangian and
Hamiltonian formalisms of higher-order mechanical systems, which are also
described in this section. In particular, higher-order regular and singular
systems are distinguished.
The main contribution of the work is found in Section 3, where the geometric
formulation of the Lagrangian-Hamiltonian unified formalism for higher-order
autonomous mechanical systems is described in detail, including the study of
how the Lagrangian and Hamiltonian formalisms are recovered from that
formalism.
Finally, in Section 4, two examples are analyzed in order to show the
application of the formalism; the first is a regular system, the so-called
Pais-Uhlenbeck oscillator, while the second is a singular one, the second-
order relativistic particle.
The paper concludes in Section 5 with a summary of results, discussion and
future research.
All the manifolds, the maps and the structures are smooth. In addition, all
the dynamical systems considered are autonomous. Summation over crossed
repeated indices is understood, although on some occasions the symbol of
summation is written explicitly in order to avoid confusion.
## 2 Higher-order dynamical systems
### 2.1 Geometric structures of higher-order tangent bundles
(See [17, 36, 23, 24, 27] for details).
#### 2.1.1 Higher-order tangent bundles
Let $Q$ be a $n$-dimensional differentiable manifold, and $k\in\mathbb{N}$.
The $k$th-order tangent bundle of $Q$, denoted by ${\rm T}^{k}Q$, is the
$(k+1)n$-dimensional manifold made of the $k$-jets with source at
$0\in\mathbb{R}$ and target $Q$; that is, ${\rm
T}^{k}Q=J_{0}^{k}(\mathbb{R},Q)$. It is a submanifold of
$J^{k}(\mathbb{R},Q)$.
We have the following canonical projections: if $r\leq k$,
$\begin{array}[]{rcclcrccl}\rho^{k}_{r}\colon&{\rm
T}^{k}Q&\longrightarrow&{\rm T}^{r}Q&,&\beta^{k}\colon&{\rm
T}^{k}Q&\longrightarrow&Q\\\ \\\
&\tilde{\sigma}^{k}(0)&\longmapsto&\tilde{\sigma}^{r}(0)&,&&\tilde{\sigma}^{k}(0)&\longmapsto&\sigma(0)\
,\end{array}$
where $\tilde{\sigma}^{k}(0)$ denotes a point in ${\rm T}^{k}Q$; that is, the
equivalence class of a curve $\sigma\colon I\subset\mathbb{R}\to Q$ by the
$k$-jet equivalence relation. Notice that $\rho^{k}_{0}=\beta^{k}$, where
${\rm T}^{0}Q$ is canonically identified with $Q$, and $\rho^{k}_{k}={\rm
Id}_{{\rm T}^{k}Q}$.
If $\left(U,\varphi\right)$ is a local chart in $Q$, with
$\varphi=\left(\varphi^{A}\right)$, $1\leq A\leq n$, and
$\sigma\colon\mathbb{R}\to Q$ is a curve in $Q$ such that $\sigma(0)\in U$; by
writing $\sigma^{A}=\varphi^{A}\circ\sigma$, the $k$-jet
$\tilde{\sigma}^{k}(0)$ is given in $\left(\beta^{k}\right)^{-1}(U)={\rm
T}^{k}U$ by $\left(q^{A},q^{A}_{1},\ldots,q^{A}_{k}\right)$, where
$q^{A}=\sigma^{A}(0)$ and $\displaystyle
q_{i}^{A}=\frac{d^{i}\sigma^{A}}{dt^{i}}(0)$ ($1\leq i\leq k$). Usually we
write $q_{0}^{A}$ instead of $q^{A}$, and so we have the local chart
$\left(\beta^{k}\right)^{-1}(U)$ in ${\rm T}^{k}Q$ with local coordinates
$\left(q_{0}^{A},q_{1}^{A},\ldots,q_{k}^{A}\right)$. Local coordinates in
${\rm T}({\rm T}^{k}Q)$ are denoted by
$\left(q_{0}^{A},q_{1}^{A},\ldots,q_{k}^{A};v_{0}^{A},v_{1}^{A},\ldots,v_{k}^{A}\right)$.
Using these coordinates, the local expression of the canonical projections are
$\rho^{k}_{r}\left(q_{0}^{A},q_{1}^{A},\ldots,q_{k}^{A}\right)=\left(q_{0}^{A},q_{1}^{A},\ldots,q_{r}^{A}\right)$,
and then for the tangent maps ${\rm T}\rho^{k}_{r}\colon{\rm T}({\rm
T}^{k}Q)\to{\rm T}({\rm T}^{r}Q)$, we have the local expression ${\rm
T}\rho^{k}_{r}\left(q_{0}^{A},q_{1}^{A},\ldots,q_{k}^{A},v_{0}^{A},v_{1}^{A},\ldots,v_{k}^{A}\right)=\left(q_{0}^{A},q_{1}^{A},\ldots,q_{r}^{A},v_{0}^{A},v_{1}^{A},\ldots,v_{r}^{A}\right)$.
If $\sigma\colon\mathbb{R}\to Q$ is a curve in $Q$, the canonical lifting of
$\sigma$ to ${\rm T}^{k}Q$ is the curve
$\tilde{\sigma}^{k}\colon\mathbb{R}\to{\rm T}^{k}Q$ defined as
$\tilde{\sigma}^{k}(t)=\tilde{\sigma}^{k}_{t}(0)$, where
$\sigma_{t}(s)=\sigma(s+t)$, (that is, the $k$-jet lifting of $\sigma$). If
$k=1$, we will write $\tilde{\sigma}^{1}\equiv\tilde{\sigma}$.
Let $V(\rho^{k}_{r-1})$ be the vertical sub-bundle of ${\rm T}^{k}Q$ in ${\rm
T}^{r-1}Q$. In the above coordinates, for every $p\in{\rm T}^{k}Q$ and $u\in
V_{p}(\rho^{k}_{r-1})$, we have that its components are
$u=\left(0,\ldots,0,v_{r}^{A},\ldots,v_{k}^{A}\right)$. Furthermore, if
$i_{k-r+1}\colon V(\rho^{k}_{r-1})\hookrightarrow{\rm T}({\rm T}^{k}Q)$ is the
canonical embedding, then
$i_{k-r+1}\left(q_{0}^{A},\ldots,q_{k}^{A},v_{r}^{A},\ldots,v_{k}^{A}\right)=\left(q_{0}^{A},\ldots,q_{k}^{A},0,\ldots,0,v_{r}^{A},\ldots,v_{k}^{A}\right)\
.$
Consider now the induced bundle of $\tau_{{\rm T}^{r-1}Q}\colon{\rm T}({\rm
T}^{r-1}Q)\to{\rm T}^{r-1}Q$ by the canonical projection $\rho^{k}_{r-1}$,
denoted by ${\rm T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm T}^{r-1}Q)$, which
is a vector bundle over ${\rm T}^{k}Q$. We have the following commutative
diagrams
$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
42.18335pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\\&\crcr}}}\ignorespaces{\hbox{\kern-42.18335pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
66.18335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.7311pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
66.18335pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
87.9216pt\raise-20.75443pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.17696pt\hbox{$\scriptstyle{\tau_{{\rm T}^{r-1}Q}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
87.9216pt\raise-29.8711pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-12.10971pt\raise-41.50885pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
36.52673pt\raise-35.6269pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.52084pt\hbox{$\scriptstyle{\rho_{r-1}^{k}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
73.68336pt\raise-41.50885pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.68336pt\raise-41.50885pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
T}^{r-1}Q}$}}}}}}}\ignorespaces}}}}\ignorespaces\quad,\quad\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
19.60973pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.21213pt\raise 6.07222pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 78.60973pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.49998pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.15695pt\hbox{$\scriptstyle{\tau_{{\rm
T}^{k}Q}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.22221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.60973pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\ }$}}}}}}}{\hbox{\kern 78.60973pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
100.34798pt\raise-20.49998pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.17696pt\hbox{$\scriptstyle{\tau_{{\rm T}^{r-1}Q}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
100.34798pt\raise-29.36221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-12.10971pt\raise-40.99997pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.26027pt\raise-35.11803pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
82.22084pt\raise-40.99997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.60973pt\raise-40.99997pt\hbox{\hbox{\kern 0.0pt\raise
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}$}}}}}}}{\hbox{\kern 82.22084pt\raise-40.99997pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm T}^{r-1}Q\
.}$}}}}}}}\ignorespaces}}}}\ignorespaces$
Then, there exists a unique bundle morphism $s_{k-r+1}\colon{\rm T}({\rm
T}^{k}Q)\to{\rm T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm T}^{r-1}Q)$ such
that the following diagram is commutative:
$\textstyle{{\rm T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{{\rm
T}^{k}Q}}$$\scriptstyle{s_{k-r+1}}$$\scriptstyle{{\rm T}\rho^{k}_{r-1}}$
$\textstyle{{\rm T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{{\rm
T}^{r-1}Q}}$ $\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{k}_{r-1}}$$\textstyle{{\rm
T}^{r-1}Q\ .}$
It is defined by $s_{k-r+1}(u)=\left(\tau_{{\rm T}^{k}Q}(u),{\rm
T}\rho^{k}_{r-1}(u)\right)$, for every $u\in{\rm T}({\rm T}^{k}Q)$. Its local
expression is
$s_{k-r+1}\left(q_{0}^{A},\ldots,q_{k}^{A},v_{0}^{A},\ldots,v_{k}^{A}\right)=\left(q_{0}^{A},\ldots,q_{r-1}^{A},q_{r}^{A},\ldots,q_{k}^{A},v_{0}^{A},\ldots,v_{r-1}^{A}\right)\
.$
As $s_{k-r+1}$ is a surjective map and ${\rm
Im}\,(i_{k-r+1})=\ker\,(s_{k-r+1})$, we have the exact sequence
$0\longrightarrow V(\rho^{k}_{r-1})\stackrel{{\scriptstyle
i_{k-r+1}}}{{\longrightarrow}}{\rm T}({\rm T}^{k}Q)\stackrel{{\scriptstyle
s_{k-r+1}}}{{\longrightarrow}}{\rm T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm
T}^{r-1}Q)\longrightarrow 0\ ,$
which is called the $(k-r+1)$-fundamental exact sequence. In local
coordinates, it is given by
$\displaystyle 0\longmapsto$
$\displaystyle\left(q_{0}^{A},\ldots,q_{k}^{A},v_{r}^{A},\ldots,v_{k}^{A}\right)\stackrel{{\scriptstyle
i_{k-r+1}}}{{\longmapsto}}\left(q_{0}^{A},\ldots,q_{k}^{A},0,\ldots,0,v_{r}^{A},\ldots,v_{k}^{A}\right)$
$\displaystyle\left(q_{0}^{A},\ldots,q_{k}^{A},v_{0}^{A},\ldots,v_{k}^{A}\right)\stackrel{{\scriptstyle
s_{k-r+1}}}{{\longmapsto}}\left(q_{0}^{A},\ldots,q_{k}^{A};q_{0}^{A},\ldots,q_{r-1}^{A},v_{0}^{A},\ldots,v_{r-1}^{A}\right)\longmapsto
0\ .$
Thus, we have $k$ exact sequences
$\displaystyle 1st\colon\ $ $\displaystyle 0\longrightarrow
V(\rho^{k}_{k-1})\stackrel{{\scriptstyle i_{1}}}{{\longrightarrow}}{\rm
T}({\rm T}^{k}Q)\stackrel{{\scriptstyle s_{1}}}{{\longrightarrow}}{\rm
T}^{k}Q\times_{{\rm T}^{k-1}Q}{\rm T}({\rm T}^{k-1}Q)\longrightarrow 0$
$\displaystyle\vdots$ $\displaystyle rth\colon\ $ $\displaystyle
0\longrightarrow V(\rho^{k}_{k-r})\stackrel{{\scriptstyle
i_{r}}}{{\longrightarrow}}{\rm T}({\rm T}^{k}Q)\stackrel{{\scriptstyle
s_{r}}}{{\longrightarrow}}{\rm T}^{k}Q\times_{{\rm T}^{k-r}Q}{\rm T}({\rm
T}^{k-r}Q)\longrightarrow 0$ $\displaystyle\vdots$ $\displaystyle kth\colon\ $
$\displaystyle 0\longrightarrow V(\beta^{k})\stackrel{{\scriptstyle
i_{k}}}{{\longrightarrow}}{\rm T}({\rm T}^{k}Q)\stackrel{{\scriptstyle
s_{k}}}{{\longrightarrow}}{\rm T}^{k}Q\times_{Q}{\rm T}Q\longrightarrow 0\ ,$
where $V(\beta^{k})\equiv V(\rho^{k}_{0})$ denotes the vertical subbundle of
${\rm T}^{k}Q$ on $Q$. These sequences can be connected by means of the
connecting maps
$h_{k-r+1}\colon{\rm T}^{k}Q\times_{{\rm T}^{k-r}Q}{\rm T}({\rm
T}^{k-r}Q)\longrightarrow V(\rho^{k}_{r-1})$
locally defined as
$h_{k-r+1}\left(q_{0}^{A},\ldots,q_{k}^{A},v_{0}^{A},\ldots,v_{k-r}^{A}\right)=\left(q_{0}^{A},\ldots,q_{k}^{A},0,\ldots,0,\frac{r!}{0!}v_{0}^{A},\frac{(r+1)!}{1!}v_{1}^{A},\ldots,\frac{k!}{(k-r)!}v_{k-r}^{A}\right)\
.$
These maps are globally well-defined and are vector bundle isomorphisms over
${\rm T}^{k}Q$. Then we have the following connection between exact sequences:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V(\rho^{k}_{k-r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{r}}$$\textstyle{{\rm
T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{r}}$$\textstyle{{\rm
T}^{k}Q\times_{{\rm T}^{k-r}Q}{\rm T}({\rm
T}^{k-r}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{k-r+1}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V(\rho^{k}_{r-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k-r+1}}$$\textstyle{{\rm
T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{k-r+1}}$$\textstyle{{\rm
T}^{k}Q\times_{{\rm T}^{r-1}Q}{\rm T}({\rm
T}^{r-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{r}}$$\textstyle{0\
.}$
#### 2.1.2 Higher-order canonical vector fields. Vertical endomorphisms and
almost-tangent structures
The canonical injection is the map
$\begin{array}[]{rccl}j_{r}\colon&{\rm T}^{k}Q&\longrightarrow&{\rm T}({\rm
T}^{r-1}Q)\\\
&\tilde{\sigma}^{k}(0)&\longmapsto&\tilde{\gamma}(0)\end{array}\quad,\quad(1\leq
r\leq k)\ ,$ (1)
where
$\begin{array}[]{rccl}\gamma\colon&\mathbb{R}&\longrightarrow&{\rm
T}^{r-1}Q\\\ &t&\longmapsto&\gamma(t)=\tilde{\sigma}_{t}^{r-1}(0)\
.\end{array}$
In local coordinates
$j_{r}\left(q_{0}^{A},\ldots,q_{k}^{A}\right)=\left(q_{0}^{A},\ldots,q_{r-1}^{A};q_{1}^{A},q_{2}^{A},\ldots,q_{r}^{A}\right)\
.$ (2)
Then, the following composition allows us to define a vector field
$\Delta_{r}\in{\mathfrak{X}}({\rm T}^{k}Q)$,
$\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Id}\times j_{k-r+1}}$$\scriptstyle{\Delta_{r}}$ $\textstyle{{\rm
T}^{k}Q\times_{{\rm T}^{k-r}Q}{\rm T}({\rm
T}^{k-r}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{k-r+1}}$
$\textstyle{V(\rho^{k}_{r-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k-r+1}}$
$\textstyle{{\rm T}({\rm T}^{k}Q)\ ;}$
that is, $\Delta_{r}=i_{k-r+1}\circ h_{k-r+1}\circ\left({\rm Id}\times
j_{k-r+1}\right)$. From the local expressions of $i_{k-r+1}$, $h_{k-r+1}$ and
$j_{k-r+1}$ we obtain that
$\Delta_{r}\left(q_{0}^{A},\ldots,q_{k}^{A}\right)=\left(q_{0}^{A},\ldots,q_{k}^{A},0,\ldots,0,r!\,q_{1}^{A},(r+1)!\,q_{2}^{A},\ldots,\frac{k!}{(k-r)!}q_{k-r+1}^{A}\right)$;
or what is equivalent,
$\Delta_{r}=\sum_{i=0}^{k-r}\frac{(r+i)!}{i!}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{r+i}^{A}}}=r!\,q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{r}^{A}}}+(r+1)!\,q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{r+1}^{A}}}+\ldots+\frac{k!}{(k-r)!}\,q_{k-r+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k}^{A}}}\
.$
In particular
$\Delta_{1}=\sum_{i=1}^{k}iq_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}=\sum_{i=0}^{k-1}(i+1)q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i+1}^{A}}}=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+2q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2}^{A}}}+\ldots+kq_{k}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k}^{A}}}\
.$
###### Definition 1
The vector field $\Delta_{r}$ is the $r$th-canonical vector field in ${\rm
T}^{k}Q$. In particular, $\Delta_{1}$ is called the Liouville vector field in
${\rm T}^{k}Q$.
Remember that, if $N$ is a $(k+1)n$-dimensional manifold, an almost-tangent
structure of order $k$ in $N$ is an endomorphism $J$ in ${\rm T}N$ such that
$J^{k+1}=0$ and ${\rm rank}\,J=kn$. Then, ${\rm T}^{k}Q$ is endowed with a
canonical almost-tangent structure. In fact:
###### Definition 2
For $1\leq r\leq k$, let $i_{k-r+1}$, $h_{k-r+1}$, $s_{r}$ be the morphisms of
the fundamental exact sequences introduced above. The map
$J_{r}=i_{k-r+1}\circ h_{k-r+1}\circ s_{r}\colon{\rm T}({\rm
T}^{k}Q)\longrightarrow{\rm T}({\rm T}^{k}Q)$
defined by the composition
$\textstyle{{\rm T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{r}}$$\scriptstyle{J_{r}}$
$\textstyle{{\rm T}^{k}Q\times_{{\rm T}^{k-r}Q}{\rm T}({\rm
T}^{k-r}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{k-r+1}}$
$\textstyle{V(\rho^{k}_{r-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{k-r+1}}$
$\textstyle{{\rm T}({\rm T}^{k}Q)}$
is called the $r$th-vertical endomorphism of ${\rm T}({\rm T}^{k}Q)$.
From the local expressions of $s_{r}$, $h_{k-r+1}$, $i_{k-r+1}$ we obtain that
$J_{r}\left(q_{0}^{A},\ldots,q_{k}^{A},v_{0}^{A},\ldots,v_{k}^{A}\right)=\left(q_{0}^{A},\ldots,q_{k}^{A},0,\ldots,0,r!\,v_{0}^{A},(r+1)!\,v_{1}^{A},\ldots,\frac{k!}{(k-r)!}\,v_{k-r}^{A}\right)\
;$
that is, $\displaystyle
J_{r}=\sum_{i=0}^{k-r}\frac{(r+i)!}{i!}\,dq_{i}^{A}\otimes\displaystyle\frac{\partial{}}{\partial{q_{r+i}^{A}}}$.
In particular, $\displaystyle
J_{1}=\sum_{i=0}^{k-1}(i+1)dq_{i}^{A}\otimes\displaystyle\frac{\partial{}}{\partial{q_{i+1}^{A}}}$.
The $r$th-vertical endomorphism $J_{r}$ has constant rank $(k-r+1)n$ and
satisfies that
$\left(J_{r}\right)^{s}=\begin{cases}0&\mbox{\rm if }rs\geqslant k+1\\\
J_{rs}&\mbox{\rm if }rs<k\end{cases}\ .$
As a consequence, the $1$st-vertical endomorphism $J_{1}$ defines an almost-
tangent structure of order $k$ in ${\rm T}^{k}Q$, which is called the
canonical almost-tangent structure of ${\rm T}^{k}Q$. Then, any other vertical
endomorphism $J_{r}$ is obtained by composing $J_{1}$ with itself $r$ times.
Furthermore, we have the following relation:
$J_{r}\circ\Delta_{s}=\begin{cases}0,&\mbox{\rm if }r+s\geqslant k+1\\\
\Delta_{r+s},&\mbox{\rm if }r+s<k+1\end{cases}$
As a consequence, starting from the Liouville vector field and the vertical
endomorphisms, we can recover all the canonical vector fields. However, as all
the vertical endomorphisms are obtained from $J_{1}$, we conclude that all the
canonical structures in ${\rm T}^{k}Q$ are obtained from the Liouville vector
field and the canonical almost-tangent structure.
Consider now the dual maps $J_{r}^{*}$ of $J_{r}$, $1\leqslant r\leqslant k$;
that is, the endomorphisms in ${\rm T}^{*}({\rm T}^{k}Q)$, and their natural
extensions to the exterior algebra $\bigwedge({\rm T}^{*}({\rm T}^{k}Q))$
(also denoted by $J_{r}^{*}$). Their action on the set of differential forms
is given by
$J_{r}^{*}\omega(X_{1},\ldots,X_{p})=\omega(J_{r}(X_{1}),\ldots,J_{r}(X_{p}))\
,$
for $\omega\in{\mit\Omega}^{p}({\rm T}^{k}Q)$ and
$X_{1},\ldots,X_{p}\in{\mathfrak{X}}({\rm T}^{k}Q)$, and for every $f\in{\rm
C}^{\infty}({\rm T}^{k}Q)$ we write $J_{r}^{*}(f)=f$. The endomorphism
$J_{r}^{*}\colon{\mit\Omega}({\rm T}^{k}Q)\to{\mit\Omega}({\rm T}^{k}Q)$,
$1\leq r\leq k$, is called the $r$th-vertical operator, and it is locally
given by
$\displaystyle J_{r}^{*}(f)=f\quad,\quad\mbox{\rm for every }\ f\in{\rm
C}^{\infty}({\rm T}^{k}Q)$ $\displaystyle J_{r}^{*}({\rm
d}q_{i}^{A})=\begin{cases}0,&\mbox{if }i<r\\\ \frac{i!}{(i-r)!}\,{\rm
d}q_{i-r}^{A},&\mbox{if }i\geq r\end{cases}\ .$
#### 2.1.3 Vertical derivations and differentials. Tulczyjew’s derivation
The inner contraction of the vertical endomorphisms $J_{r}$ with any
differential $p$-form $\omega\in{\mit\Omega}^{p}({\rm T}^{k}Q)$ is the
$p$-form $\mathop{i}\nolimits(J_{r})\omega$ defined as follows: for every
$X_{1},\ldots,X_{p}\in{\mathfrak{X}}({\rm T}^{k}Q)$
$\mathop{i}\nolimits(J_{r})\omega(X_{1},\ldots,X_{p})=\sum_{i=1}^{p}\omega(X_{1},\ldots,J_{r}(X_{i}),\ldots,X_{p})\
,$
and taking $\mathop{i}\nolimits(J_{r})f=0$, for every $f\in{\rm
C}^{\infty}({\rm T}^{k}Q)$, we can state:
###### Definition 3
The map
$\begin{array}[]{rcl}{\mit\Omega}({\rm
T}^{k}Q)&\longrightarrow&{\mit\Omega}({\rm T}^{k}Q)\\\
\omega&\longmapsto&\mathop{i}\nolimits(J_{r})\omega\end{array}$
is a derivation of degree $0$ in ${\mit\Omega}({\rm T}^{k}Q)$, which is called
the $r$th-vertical derivation in ${\mit\Omega}({\rm T}^{k}Q)$.
Its local expression is
$\mathop{i}\nolimits(J_{r})({\rm d}q_{i}^{A})=\begin{cases}0,&\mbox{\rm if
}i<r\\\ \frac{i!}{(i-r)!}\,{\rm d}q_{i-r}^{A},&\mbox{\rm if }i\geq
r\end{cases}\ .$
###### Definition 4
The operator $d_{J_{r}}=[\mathop{i}\nolimits(J_{r}),{\rm d}]$ is a skew-
derivation of degree $1$, which is called the $r$th-vertical differential.
Its local expression is given by
$\begin{array}[]{l}\displaystyle
d_{J_{r}}(f)=\sum_{i=r}^{k}\frac{i!}{(i-r)!}\displaystyle\frac{\partial{f}}{\partial{q_{i}^{A}}}{\rm
d}q_{i-r}^{A}\quad,\quad\mbox{\rm for every $f\in{\rm C}^{\infty}({\rm
T}^{k}Q)$}\\\ d_{J_{r}}({\rm d}q^{i})=0\end{array}\ .$
For $1\leq r,s\leq k$, we have that $d_{J_{r}}{\rm d}=-{\rm d}d_{J_{r}}$.
In the set $\oplus_{k\geqslant 0}{\mit\Omega}({\rm T}^{k}Q)$, we can define
the following equivalence relation: for $\omega\in{\mit\Omega}({\rm T}^{k}Q)$
and $\lambda\in{\mit\Omega}({\rm T}^{k^{\prime}}Q)$,
$\omega\sim\lambda\Longleftrightarrow\begin{cases}\omega=(\rho^{k}_{k^{\prime}})^{*}(\lambda),&\mbox{if
}k^{\prime}\leqslant k\\\
\lambda=(\rho^{k^{\prime}}_{k})^{*}(\omega),&\mbox{if }k^{\prime}\geqslant
k\end{cases}\ .$
Then we consider the quotient set
$\displaystyle\mit\Omega=\bigoplus_{k\geqslant 0}{\mit\Omega}({\rm
T}^{k}Q)/\sim$, which is a commutative graded algebra. In this set we can
define the so-called Tulczyjew’s derivation [40, 17], denoted by $d_{T}$, as
follows: for every $f\in{\rm C}^{\infty}({\rm T}^{k}Q)$ we construct the
function $d_{T}f\in{\rm C}^{\infty}({\rm T}^{k+1}Q)$ given by
$(d_{T}f)(\tilde{\sigma}^{k+1}(0))=(d_{\tilde{\sigma}^{k}(0)}f)(j_{k+1}(\tilde{\sigma}^{k+1}(0)))$
where $j_{k+1}\colon{\rm T}^{k+1}Q\to{\rm T}({\rm T}^{k}Q)$ is the canonical
injection introduced in the Section 2.1.2. From the coordinate expression for
$j_{k+1}$, we obtain that
$d_{T}f\left(q_{0}^{A},\ldots,q_{k+1}^{A}\right)=\sum_{i=0}^{k}q_{i+1}^{A}\displaystyle\frac{\partial{f}}{\partial{q_{i}^{A}}}(q_{0}^{A},\ldots,q_{k}^{A})\
.$
This map $d_{T}$ extends to a derivation of degree $0$ in $\mit\Omega$ and, as
$d_{T}{\rm d}={\rm d}d_{T}$, it is determined by its action on functions and
by the property $d_{T}({\rm d}q_{i}^{A})={\rm d}q_{i+1}^{A}$.
Furthermore, the maps $\mathop{i}\nolimits(J_{s})$, $d_{J_{s}}$,
$\mathop{i}\nolimits(\Delta_{s})$ and $\mathop{\rm L}\nolimits(\Delta_{s})$
extend to $\mit\Omega$ in a natural way.
#### 2.1.4 Higher-order semisprays
###### Definition 5
A vector field $X\in{\mathfrak{X}}({\rm T}^{k}Q)$ is a semispray of type $r$,
$1\leq r\leq k$, if for every integral curve $\sigma$ of $X$, we have that, if
$\gamma=\beta^{k}\circ\sigma$, then
$\tilde{\gamma}^{k-r+1}=\rho^{k}_{k-r+1}\circ\sigma$ (where
$\tilde{\gamma}^{k-r+1}$ is the canonical lifting of $\gamma$ to ${\rm
T}^{k-r+1}Q$).
$\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{k}_{k-r+1}}$$\scriptstyle{\beta^{k}}$$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\beta^{k}\circ\sigma}$$\scriptstyle{\rho^{k}_{k-r+1}\circ\sigma}$$\scriptstyle{\widetilde{\gamma}^{k-r+1}}$
$\textstyle{{\rm
T}^{k-r+1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Id}}$ $\textstyle{{\rm
T}^{k-r+1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{k-r+1}}$
$\textstyle{Q}$
In particular, $X\in{\mathfrak{X}}({\rm T}^{k}Q)$ is a semispray of type $1$
if for every integral curve $\sigma$ of $X$, we have that
$\gamma=\beta^{k}\circ\sigma$, then $\tilde{\gamma}^{k}=\sigma$.
The local expression of a semispray of type $r$ is
$X=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+\ldots+q_{k-r+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k-r}^{A}}}+X_{k-r+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k-r+1}^{A}}}+\ldots+X_{k}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k}^{A}}}$
###### Proposition 1
The following assertions are equivalent:
1. 1.
A vector field $X\in{\mathfrak{X}}({\rm T}^{k}Q)$ is a semispray of type $r$.
2. 2.
${\rm T}\rho^{k}_{k-r}\circ X=j_{k-r+1}$; that is, the following diagram
commutes
$\textstyle{{\rm T}({\rm
T}^{k}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
T}\rho^{k}_{k-r}}$$\textstyle{{\rm
T}^{k}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X}$$\scriptstyle{j_{k-r+1}}$
$\textstyle{{\rm T}({\rm T}^{k-r}Q)\ .}$
3. 3.
$J_{r}(X)=\Delta_{r}$.
Obviously, every semispray of type $r$ is a semispray of type $s$, for $s\geq
r$.
If $X\in{\mathfrak{X}}({\rm T}^{k}Q)$ is a semispray of type $r$, a curve
$\sigma$ in $Q$ is said to be a path or solution of $X$ if
$\tilde{\sigma}^{k}$ is an integral curve of $X$; that is,
$\widetilde{\tilde{\sigma}^{k}}=X\circ\tilde{\sigma}^{k}$, where
$\widetilde{\tilde{\sigma}^{k}}$ denotes the canonical lifting of
$\tilde{\sigma}^{k}$ from ${\rm T}^{k}Q$ to ${\rm T}({\rm T}^{k}Q)$. Then, in
coordinates, $\sigma$ verifies the following system of differential equations
of order $k+1$:
$\displaystyle\frac{d^{k-r+2}\sigma^{A}}{dt^{k-r+2}}$
$\displaystyle=X_{k-r+1}^{A}\left(\sigma,\frac{d\sigma}{dt},\ldots,\frac{d^{k}\sigma}{dt^{k}}\right)$
$\displaystyle\;\vdots$ $\displaystyle\frac{d^{k+1}\sigma^{A}}{dt^{k+1}}$
$\displaystyle=X_{k}^{A}\left(\sigma,\frac{d\sigma}{dt},\ldots,\frac{d^{k}\sigma}{dt^{k}}\right)$
Observe that, taking $k=1$, then $r=1$ and $\rho^{1}_{1-1+1}={\rm Id}_{{\rm
T}Q}$, we recover the definition of the holonomic vector field (sode in ${\rm
T}Q$). So, semisprays of type $1$ in ${\rm T}^{k}Q$ are the analogue to the
holonomic vector fields in ${\rm T}Q$; that is, they are the vector fields
whose integral curves are the canonical liftings to ${\rm T}^{k}Q$ of curves
on the basis $Q$. Their local expressions are
$X=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+\ldots+q_{k}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k-1}^{A}}}+X_{k}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k}^{A}}}\
.$
### 2.2 Lagrangian formalism
Let $Q$ be a $n$-dimensional differentiable manifold and ${\cal L}\in{\rm
C}^{\infty}({\rm T}^{k}Q)$. We say that ${\cal L}$ is a Lagrangian function of
order $k$.
###### Definition 6
The Lagrangian $1$-form $\theta_{\cal L}\in{\mit\Omega}^{1}({\rm T}^{2k-1}Q)$,
associated to ${\cal L}$ is defined as
$\theta_{\cal L}=\sum_{r=1}^{k}(-1)^{r-1}\frac{1}{r!}d_{T}^{r-1}d_{J_{r}}{\cal
L}\ .$
Then, the Lagrangian $2$-form, $\omega_{\cal L}\in{\mit\Omega}^{2}({\rm
T}^{2k-1}Q)$, associated to ${\cal L}$ is
$\omega_{\cal L}=-{\rm d}\theta_{\cal
L}=\sum_{r=1}^{k}(-1)^{r}\frac{1}{r!}d_{T}^{r-1}{\rm d}d_{J_{r}}{\cal L}\ .$
Observe that the Lagrangian $1$-form is a semibasic form of type $k$ in ${\rm
T}^{2k-1}Q$ .
We assume that $\omega_{\cal L}$ has constant rank (we refer to this fact by
saying that ${\cal L}$ is a geometrically admissible Lagrangian).
###### Definition 7
The Lagrangian energy, $E_{\cal L}\in{\rm C}^{\infty}({\rm T}^{2k-1}Q)$,
associated to ${\cal L}$ is defined as
$E_{\cal
L}=\left(\sum_{r=1}^{k}(-1)^{r-1}\frac{1}{r!}d_{T}^{r-1}(\Delta_{r}({\cal
L}))\right)-(\rho_{k}^{2k-1})^{*}{\cal L}$
It is usual to write ${\cal L}$ instead of $(\rho_{k}^{2k-1})^{*}{\cal L}$,
and we will do this in the sequel.
The coordinate expressions of these elements are
$\displaystyle\theta_{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{k}\sum_{i=0}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{L}}{\partial{q_{r+i}^{A}}}\right){\rm
d}q_{r-1}^{A}$ (3) $\displaystyle\omega_{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{k}\sum_{i=0}^{k-r}(-1)^{i+1}d_{T}^{i}\,{\rm
d}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r+i}^{A}}}\right)\wedge{\rm d}q_{r-1}^{A}$ $\displaystyle
E_{\cal L}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{k}q_{r}^{A}\sum_{i=0}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{L}}{\partial{q_{r+i}^{A}}}\right)-{\cal
L}\ .$ (4)
###### Definition 8
A Lagrangian function ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is said to
be regular if $\omega_{\cal L}$ is a symplectic form. Otherwise ${\cal L}$ is
a singular Lagrangian.
To say that ${\cal L}$ is a regular Lagrangian is locally equivalent to saying
that the Hessian matrix $\displaystyle\left(\frac{\partial^{2}{\cal
L}}{\partial q_{k}^{B}\partial q_{k}^{A}}\right)$ is regular at every point of
${\rm T}^{k}Q$.
###### Definition 9
A Lagrangian system of order $k$ is a couple $({\rm T}^{2k-1}Q,{\cal L})$,
where $Q$ represents the configuration space and ${\cal L}\in{\rm
C}^{\infty}({\rm T}^{k}Q)$ is the Lagrangian function. It is said to be a
regular (resp. singular) Lagrangian system if the Lagrangian function ${\cal
L}$ is regular (resp. singular).
Thus, in the Lagrangian formalism, ${\rm T}^{2k-1}Q$ represents the phase
space of the system. The dynamical trajectories of the system are the integral
curves of any vector field $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$
satisfying that:
1. 1.
It is a solution to the equation
$\mathop{i}\nolimits(X_{\cal L})\omega_{\cal L}={\rm d}E_{\cal L}$ (5)
2. 2.
It is a semispray of type $1$ in ${\rm T}^{2k-1}Q$.
Equation (5) is the higher-order Lagrangian equation, and a vector field
$X_{\cal L}$ solution to (5) (if it exists) is called a Lagrangian vector
field of order $k$. If, in addition, $X_{\cal L}$ satisfies condition 2, then
it is called an Euler-Lagrange vector field of order $k$, and its integral
curves on the base are solutions to the higher-order Euler-Lagrange equations.
In natural coordinates of ${\rm T}^{2k-1}Q$, if
$X_{\cal
L}=\sum_{i=0}^{2k-1}f_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}=f_{0}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+f_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+\ldots+f_{2k-1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2k-1}^{A}}}\
,$
as
${\rm d}E_{\cal
L}=\sum_{r=1}^{k}\sum_{i=0}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r+i}^{A}}}\right){\rm
d}q_{r}^{A}+\sum_{r=1}^{k}q_{r}^{A}\sum_{i=0}^{k-r}(-1)^{i}\sum_{j=0}^{k}d_{T}^{i}\left(\frac{\partial^{2}{\cal
L}}{\partial q_{j}^{B}\partial q_{r+i}^{A}}{\rm
d}q_{j}^{B}\right)-\sum_{r=0}^{k}\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r}^{A}}}{\rm d}q_{r}^{A}\ ,$
from (5) we obtain
$\begin{array}[]{l}\displaystyle\left(f_{0}^{B}-q_{1}^{B}\right)\frac{\partial^{2}{\cal
L}}{\partial q_{k}^{B}\partial q_{k}^{A}}=0\\\\[10.0pt]
\displaystyle\left(f_{1}^{B}-q_{2}^{B}\right)\frac{\partial^{2}{\cal
L}}{\partial q_{k}^{B}\partial
q_{k}^{A}}-\left(f_{0}^{B}-q_{1}^{B}\right)(\cdots\cdots)=0\\\
\qquad\qquad\qquad\qquad\vdots\\\
\displaystyle\left(f_{2k-2}^{B}-q_{2k-1}^{B}\right)\frac{\partial^{2}{\cal
L}}{\partial q_{k}^{B}\partial
q_{k}^{A}}-\sum_{i=0}^{2k-3}\left(f_{i}^{B}-q_{i+1}^{B}\right)(\cdots\cdots)=0\\\
\displaystyle(-1)^{k}\left(f_{2k-1}^{B}-d_{T}\left(q_{2k-1}^{B}\right)\right)\frac{\partial^{2}{\cal
L}}{\partial q_{k}^{B}\partial
q_{k}^{A}}+\sum_{i=0}^{k}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{L}}{\partial{q_{i}^{A}}}\right)-\sum_{i=0}^{2k-2}\left(f_{i}^{B}-q_{i+1}^{B}\right)(\cdots\cdots)=0\end{array}$
(6)
where the terms in brackets $(\cdots\cdots)$ contain relations involving
partial derivatives of the Lagrangian and applications of $d_{T}$, which for
simplicity are not written. These are the local expressions of the Lagrangian
equations for $X_{\cal L}$.
Now, if $\sigma\colon\mathbb{R}\to{\rm T}^{2k-1}Q$ is an integral curve of
$X_{\cal L}$, from (5) we obtain that $\sigma$ must satisfy the Euler-Lagrange
equation
$\mathop{i}\nolimits(\tilde{\sigma})(\omega_{\cal L}\circ\sigma)={\rm
d}E_{\cal L}\circ\sigma\ ,$ (7)
where $\tilde{\sigma}$ denotes the canonical lifting of $\sigma$ to ${\rm
T}({\rm T}^{2k-1}Q)$; and as $X_{\cal L}$ is a semispray of type $1$, we have
that $\sigma$ is the canonical lifting of a curve $\gamma\colon\mathbb{R}\to
Q$ to ${\rm T}^{2k-1}Q$; that is, $\sigma=\tilde{\gamma}^{2k-1}$.
Now, if ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular Lagrangian,
then $\omega_{\cal L}$ is a symplectic form in ${\rm T}^{2k-1}Q$, and as a
consequence we have that:
###### Theorem 1
Let $({\rm T}^{2k-1}Q,{\cal L})$ be a regular Lagrangian system of order $k$.
1. 1.
There exists a unique $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ which is
a solution to the Lagrangian equation (5) and is a semispray of type $1$ in
${\rm T}^{2k-1}Q$.
2. 2.
If $\gamma\colon\mathbb{R}\to Q$ is an integral curve of $X_{\cal L}$ then
$\sigma=\tilde{\gamma}^{2k-1}$ is a solution to the Euler-Lagrange equations:
$\displaystyle\frac{\partial{{\cal
L}}}{\partial{q^{0}}}\circ\tilde{\gamma}^{2k-1}-\frac{d}{dt}\displaystyle\frac{\partial{{\cal
L}}}{\partial{q^{1}}}\circ\tilde{\gamma}^{2k-1}+\frac{d^{2}}{dt^{2}}\displaystyle\frac{\partial{{\cal
L}}}{\partial{q^{2}}}\circ\tilde{\gamma}^{2k-1}+\ldots+(-1)^{k}\frac{d^{k}}{dt^{k}}\displaystyle\frac{\partial{{\cal
L}}}{\partial{q^{k}}}\circ\tilde{\gamma}^{2k-1}=0\ .$ (8)
If ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a singular Lagrangian, then
$\omega_{\cal L}$ is a presymplectic form, so the existence and uniqueness of
solutions to the Lagrangian equation (5) is not assured, except in special
cases (for instance, when $\omega_{\cal L}$ is a presymplectic horizontal
structure [17]). In general, in the most favourable cases, equation (5) has
solutions $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ in some submanifold
$S_{f}\hookrightarrow{\rm T}^{2k-1}Q$, for which these vector fields solution
are tangent. This submanifold is obtained by applying the well-known
constraint algorithms (see, for instance, [22, 21, 30]). Nevertheless, these
vector fields solution are not necessarily semisprays of type $1$ on $S_{f}$,
but only on the points of another submanifold $M_{f}\hookrightarrow
S_{f}\hookrightarrow{\rm T}^{2k-1}Q$ (see [21, 30]). On the points of this
last submanifold, the integral curves of $X_{\cal L}\in{\mathfrak{X}}({\rm
T}^{2k-1}Q)$ are solutions to the Euler-Lagrange equations (8).
A detailed study of higher-order singular Lagrangian systems can be found in
[23, 24].
### 2.3 Hamiltonian formalism
###### Definition 10
Let $({\rm T}^{2k-1}Q,{\cal L})$ be a Lagrangian system. The Legendre-
Ostrogradsky map (or generalized Legendre map) associated to ${\cal L}$ is the
map ${\cal FL}\colon{\rm T}^{2k-1}Q\to{\rm T}^{*}({\rm T}^{k-1}Q)$ defined as
follows: for every $u\in{\rm T}({\rm T}^{2k-1}Q)$,
$\theta_{\cal L}(u)=\left\langle{\rm T}\rho_{k-1}^{2k-1}(u),{\cal
FL}(\tau_{{\rm T}^{2k-1}Q}(u))\right\rangle$
This map verifies that $\pi_{{\rm T}^{k-1}Q}\circ{\cal FL}=\rho^{2k-1}_{k-1}$,
where $\pi_{{\rm T}^{k-1}Q}\colon{\rm T}^{*}({\rm T}^{k-1}Q)\to{\rm T}^{k-1}Q$
is the natural projection. Furthermore, if
$\theta_{k-1}\in{\mit\Omega}^{1}({\rm T}^{*}({\rm T}^{k-1}Q)$ and
$\omega_{k-1}=-{\rm d}\theta_{k-1}\in{\mit\Omega}^{2}({\rm T}^{*}({\rm
T}^{k-1}Q))$ are the canonical $1$ and $2$ forms of the cotangent bundle ${\rm
T}^{*}({\rm T}^{k-1}Q)$, we have that
${\cal FL}^{*}\theta_{k-1}=\theta_{\cal L}\quad,\quad{\cal
FL}^{*}\omega_{k-1}=\omega_{\cal L}\ .$
Given a local natural chart in ${\rm T}^{2k-1}Q$, we can define the following
local functions
$\hat{p}^{r-1}_{A}=\sum_{i=0}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{L}}{\partial{q_{r+i}^{A}}}\right)\
.$
Observe that
$\displaystyle\hat{p}^{r-1}_{A}-\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r}^{A}}}$
$\displaystyle=\sum_{i=0}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r+i}^{A}}}\right)-\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r}^{A}}}=\sum_{i=1}^{k-r}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r+i}^{A}}}\right)$
$\displaystyle=\sum_{i=0}^{k-r-1}(-1)^{i+1}d_{T}^{i+1}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r+i+1}^{A}}}\right)=-d_{T}\left(\sum_{i=0}^{k-(r+1)}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{(r+1)+i}^{A}}}\right)\right)=-d_{T}(\hat{p}^{r}_{A})$
and hence
$\hat{p}^{r-1}_{A}=\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{r}^{A}}}-d_{T}(\hat{p}^{r}_{A})\quad,\quad 1\leq r\leq k-1\
.$ (9)
Thus, bearing in mind the local expression (3) of the form $\theta_{\cal L}$,
we can write $\theta_{\cal L}=\sum_{r=1}^{k}\hat{p}^{r-1}_{A}{\rm
d}q_{r-1}^{A}$, and we obtain that the expression in natural coordinates of
the map ${\cal FL}$ is
${\cal
FL}\left(q_{0}^{A},q_{1}^{A},\ldots,q_{2k-1}^{A}\right)=\left(q_{0}^{A},q_{1}^{A},\ldots,q_{k-1}^{A},p^{0}_{A},p^{1}_{A},\ldots,p^{k-1}_{A}\right)\
,\ \mbox{\rm with $p^{i}_{A}\circ{\cal FL}=\hat{p}^{i}_{A}$}\ .$
${\cal L}$ is a regular Lagrangian if, and only if, ${\cal FL}\colon{\rm
T}^{2k-1}Q\to{\rm T}^{*}({\rm T}^{k-1}Q)$ is a local diffeomorphism.
As a consequence of this, we have that, if ${\cal L}$ is a regular Lagrangian,
then the set $(q_{i}^{A},\hat{p}^{i}_{A})$, $0\leq i\leq k-1$, is a set of
local coordinates in ${\rm T}^{2k-1}Q$, and $(\hat{p}^{i}_{A})$ are called the
Jacobi-Ostrogradsky momentum coordinates.
Observe that the relation (9) means that we can recover all the Jacobi-
Ostrogadsky momentum coordinates from the set $(\hat{p}^{k-1}_{A})$.
###### Definition 11
${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is said to be a hyperregular
Lagrangian of order $k$ if ${\cal FL}$ is a global diffeomorphism. Then,
$({\rm T}^{2k-1}Q,{\cal L})$ is a hyperregular Lagrangian system of order $k$.
As $\pi_{{\rm T}^{k-1}Q}\circ{\cal FL}=\rho^{2k-1}_{k-1}$, this condition is
equivalent to demanding that the restriction of $\rho^{2k-1}_{k-1}\colon{\rm
T}^{2k-1}Q\to{\rm T}^{k-1}Q$ to every fibre be one-to-one.
In order to explain the construction of the canonical Hamiltonian formalism of
a Lagrangian higher-order system, we first consider the case of hyperregular
systems (the regular case is the same, but restricting on the suitable open
submanifolds where ${\cal FL}$ is a local diffeomorphism).
So, $({\rm T}^{2k-1}Q,{\cal L})$ being a hyperregular Lagrangian system, since
${\cal FL}$ is a diffeomorphism, there exists a unique function $h\in{\rm
C}^{\infty}({\rm T}^{*}({\rm T}^{k-1}Q))$ such that ${\cal FL}^{*}h=E_{\cal
L}$, which is called the Hamiltonian function associated to this system. Then
the triad $({\rm T}^{*}({\rm T}^{k-1}Q),\omega_{k-1},h)$ is called the
canonical Hamiltonian system associated to the hyperregular Lagrangian system
$({\rm T}^{2k-1}Q,{\cal L})$. Thus, in the Hamiltonian formalism, ${\rm
T}^{*}({\rm T}^{k-1}Q)$ represents the phase space of the system.
The dynamical trajectories of the system are the integral curves of a vector
field $X_{h}\in{\mathfrak{X}}({\rm T}^{*}({\rm T}^{k-1}Q))$ which is a
solution to the Hamilton equation
$\mathop{i}\nolimits(X_{h})\omega_{k-1}={\rm d}h\ .$ (10)
As $\omega_{k-1}$ is symplectic, there is a unique vector field $X_{h}$
solution to this equation, and it is called the Hamiltonian vector field.
In natural coordinates of ${\rm T}^{*}({\rm T}^{k-1}Q)$,
$(q_{i}^{A},p^{i}_{A})$ (with $0\leq i\leq k-1$; $1\leq A\leq n$), taking
$\displaystyle
X_{h}=f_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+g^{i}_{A}\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}$,
as $\displaystyle{\rm
d}h=\displaystyle\frac{\partial{h}}{\partial{q_{i}^{A}}}{\rm
d}q_{i}^{A}+\displaystyle\frac{\partial{h}}{\partial{p^{i}_{A}}}{\rm
d}p^{i}_{A}$, and $\omega_{k-1}={\rm d}q_{i}^{A}\wedge{\rm d}p^{i}_{A}$, from
(10) we obtain that
$f_{i}^{A}=\displaystyle\frac{\partial{h}}{\partial{p^{i}_{A}}}\quad,\quad
g^{i}_{A}=-\displaystyle\frac{\partial{h}}{\partial{q_{i}^{A}}}\ .$
Now, if $\sigma\colon\mathbb{R}\to{\rm T}^{*}({\rm T}^{k-1}Q)$ is an integral
curve of $X_{h}$, we have that $\sigma$ must satisfy the Hamiltonian equation
$\mathop{i}\nolimits(\tilde{\sigma})(\omega_{k-1}\circ\sigma)={\rm
d}h\circ\sigma\ ,$
and, if $\sigma(t)=(q_{i}^{A}(t),p^{i}_{A}(t))$ in coordinates, it gives the
classical expression of the Hamilton equations:
$\frac{dq_{i}^{A}}{dt}=\displaystyle\frac{\partial{h}}{\partial{p^{i}_{A}}}\circ\sigma\quad,\quad\frac{dp^{i}_{A}}{dt}=-\displaystyle\frac{\partial{h}}{\partial{q_{i}^{A}}}\circ\sigma\
.$
For the case of singular higher-order Lagrangian systems, in general there is
no way to associate a canonical Hamiltonian formalism, unless some minimal
regularity condition are imposed [23]. In particular:
###### Definition 12
A Lagrangian ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is said to be an
almost-regular Lagrangian function of order $k$ if:
1. 1.
${\cal FL}({\rm T}^{2k-1}Q)=P_{o}$ is a closed submanifold of ${\rm
T}^{*}({\rm T}^{k-1}Q)$.
(We denote the natural embedding by $j_{P_{o}}\colon P_{o}\hookrightarrow{\rm
T}^{*}({\rm T}^{k-1}Q)$).
2. 2.
${\cal FL}$ is a surjective submersion on its image.
3. 3.
For every $p\in{\rm T}^{2k-1}Q$, the fibers ${\cal FL}^{-1}({\cal FL}(p))$ are
connected submanifolds of ${\rm T}^{2k-1}Q$.
Then $({\rm T}^{2k-1}Q,{\cal L})$ is an almost-regular Lagrangian system of
order $k$.
Denoting the map defined by ${\cal FL}=j_{P_{o}}\circ{\cal FL}_{o}$ by ${\cal
FL}_{o}\colon{\rm T}^{2k-1}Q\to P_{o}$, we have that the Lagrangian energy
$E_{\cal L}$ is a ${\cal FL}_{o}$-projectable function, and then there is a
unique function $h_{o}\in{\rm C}^{\infty}(P_{o})$ such that ${\cal
FL}_{o}^{*}h_{o}=E_{L}$ (see [23]).
This $h_{o}$ is the canonical Hamiltonian function of the almost-regular
Lagrangian system and, taking $\omega_{o}=j_{P_{o}}^{*}\omega_{k-1}$, the
triad $(P_{o},\omega_{o},h_{o})$ is the canonical Hamiltonian system
associated to the almost regular Lagrangian system $({\rm T}^{2k-1}Q,{\cal
L})$. For this system we have the Hamilton equation
$\mathop{i}\nolimits(X_{h_{o}})\omega_{o}={\rm d}h_{o}\quad,\quad
X_{h_{o}}\in{\mathfrak{X}}(P_{o})\ .$ (11)
As $\omega_{o}$ is, in general, a presymplectic form, in the best cases, this
equation has some vector field $X_{h_{o}}$ solution only on the points of some
submanifold $P_{f}\hookrightarrow P_{o}\hookrightarrow{\rm T}^{*}({\rm
T}^{k-1}Q)$, for which $X_{h_{o}}$ is tangent to $P_{f}$. This vector field is
not unique, in general. It can be proved that $P_{f}={\cal FL}(S_{f})$, where
$S_{f}\hookrightarrow{\rm T}^{2k-1}Q$ is the submanifold where there are
vector field solutions to the Lagrangian equation (5) which are tangent to
$S_{f}$ (see the above section). Furthermore, as ${\cal FL}_{o}$ is a
submersion, for every vector field $X_{h_{o}}\in{\mathfrak{X}}({\rm
T}^{*}({\rm T}^{k-1}Q))$ which is a solution to the Hamilton equation (11) on
$P_{f}$, and tangent to $P_{f}$, there exists some semispray of type $1$,
$X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$, which is a solution of the
Euler-Lagrange equation on $S_{f}$, and tangent to $S_{f}$, such that ${{\cal
FL}_{o}}_{*}X_{\cal L}=X_{h_{o}}$. This ${\cal FL}_{o}$-projectable semispray
of type $1$ could be defined only on the points of another submanifold
$M_{f}\hookrightarrow S_{f}$. (See [23, 24] for a detailed exposition of all
these topics).
## 3 Skinner-Rusk unified formalism
### 3.1 Unified phase space. Geometric and dynamical structures
Let ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ be the Lagrangian function of
order $k$ of the system. First we construct the unified phase space
${\cal W}={\rm T}^{2k-1}Q\times_{{\rm T}^{k-1}Q}{\rm T}^{*}({\rm T}^{k-1}Q)$
(the fiber product of the above bundles), which is endowed with the canonical
projections
$\operatorname{pr}_{1}\colon{\rm T}^{2k-1}Q\times_{{\rm T}^{k-1}Q}{\rm
T}^{*}({\rm T}^{k-1}Q)\to{\rm
T}^{2k-1}Q\quad;\quad\operatorname{pr}_{2}\colon{\rm T}^{2k-1}Q\times_{{\rm
T}^{k-1}Q}{\rm T}^{*}({\rm T}^{k-1}Q)\to{\rm T}^{*}({\rm T}^{k-1}Q)\ ,$
and also with the canonical projections onto ${\rm T}^{k-1}Q$. So we have the
diagram:
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\rm
T}^{2k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{2k-1}_{k-1}}$$\scriptstyle{\beta^{2k-1}}$
$\textstyle{{\rm T}^{*}({\rm
T}^{k-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\rm
T}^{k-1}Q}}$ $\textstyle{{\rm
T}^{k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{k-1}}$
$\textstyle{Q}$
If $(U,q_{0}^{A})$ is a local chart of coordinates in $Q$, denoting by
$((\beta^{2k-1})^{-1}(U);q_{0}^{A},q_{1}^{A},\ldots,q_{2k-1}^{A})$ and
$((\pi_{{\rm
T}^{k-1}Q}\circ\beta^{k-1})^{-1}(U);q_{0}^{A},q_{1}^{A},\ldots,q_{k-1}^{A},p^{0}_{A},p^{0}_{A},\ldots,p^{k-1}_{A})$
the induced charts in ${\rm T}^{2k-1}Q$ and in ${\rm T}^{*}({\rm T}^{k-1}Q)$,
respectively, we have that
$(q_{0}^{A},\ldots,q_{k-1}^{A};q_{k}^{A},\ldots,q_{2k-1}^{A};p^{0}_{A},\ldots,p^{k-1}_{A})$
are the natural coordinates in the suitable open domain $W\subset{\cal W}$.
Note that $\dim({\cal W})=3kn$.
The bundle ${\cal W}$ is endowed with some canonical geometric structures.
First, let $\omega_{k-1}\in{\mit\Omega}^{2}({\rm T}^{*}({\rm T}^{k-1}Q))$ be
the canonical symplectic form of ${\rm T}^{*}({\rm T}^{k-1}Q)$. Then we define
$\Omega=\operatorname{pr}_{2}^{*}\omega_{k-1}\in{\mit\Omega}^{2}({\cal W})\ ,$
which is a presymplectic form in ${\cal W}$, whose local expression is
$\Omega=\operatorname{pr}_{2}^{*}\omega_{k-1}=\operatorname{pr}_{2}^{*}\left({\rm
d}q_{i}^{A}\wedge{\rm d}p^{i}_{A}\right)={\rm d}q_{i}^{A}\wedge{\rm
d}p^{i}_{A}\ .$ (12)
Observe that
$\ker\,\Omega=\left\langle\displaystyle\frac{\partial{}}{\partial{q^{k}}},\ldots,\displaystyle\frac{\partial{}}{\partial{q^{2k-1}}}\right\rangle={\mathfrak{X}}^{V(\operatorname{pr}_{2})}({\cal
W})\ .$ (13)
The second relevant canonical structure in ${\cal W}$ is the following:
###### Definition 13
Let $p\in{\rm T}^{2k-1}Q$, its projection $q=\rho^{2k-1}_{k-1}(p)$ to ${\rm
T}^{k-1}Q$, and a covector $\alpha_{q}\in{\rm T}_{q}^{*}({\rm T}^{k-1}Q)$. The
coupling function ${\cal C}\in{\rm C}^{\infty}({\cal W})$ is defined as
follows:
$\begin{array}[]{rcl}{\cal C}\colon{\rm T}^{2k-1}Q\times_{{\rm T}^{k-1}Q}{\rm
T}^{*}({\rm T}^{k-1}Q)&\longrightarrow&\mathbb{R}\\\
(p,\alpha_{q})&\longmapsto&\langle\alpha_{q}\mid
j_{k}(p)_{q}\rangle\end{array}\ ,$ (14)
where $j_{k}\colon{\rm T}^{2k-1}Q\to{\rm T}({\rm T}^{k-1}Q)$ is the canonical
injection introduced in (1), $j_{k}(p)_{q}$ is the corresponding tangent
vector to ${\rm T}^{k-1}Q$ in $q$, and $\langle\alpha_{q}\mid
j_{k}(p)_{q}\rangle\equiv\alpha_{q}(j_{k}(p)_{q})$ denotes the canonical
pairing between vectors of ${\rm T}_{q}({\rm T}^{k-1}Q)$ and covectors of
${\rm T}^{*}_{q}({\rm T}^{k-1}Q)$.
Note that, in this case, $j_{k}\colon{\rm T}^{2k-1}Q\to{\rm T}({\rm
T}^{k-1}Q)$ is a diffeomorphism.
In local coordinates, if
$p=(q_{0}^{A},\ldots,q_{k-1}^{A},q_{k}^{A},\ldots,q_{2k-1}^{A})$, then
$q=\rho^{2k-1}_{k-1}(p)=(q_{0}^{A},\ldots,q_{k-1}^{A})$, and bearing in mind
the local expression (2) of $j_{k}$, we have
$j_{k}(p)=(q_{0}^{A},\ldots,q_{k-1}^{A},q_{1}^{A},\ldots,q_{k}^{A})$.
Therefore if $\displaystyle
j_{k}(p)_{q}=q_{i+1}^{A}\left.\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}\right|_{q}\in{\rm
T}_{q}({\rm T}^{k-1}Q)$, and if $\displaystyle\alpha_{q}=p^{i}_{A}\left.{\rm
d}q_{i}^{A}\right|_{q}$ we obtain the following local expression for the
coupling function ${\cal C}$
${\cal C}(p,\alpha_{q})=\langle\alpha_{q}\mid j_{k}(p)_{q}\rangle=\left\langle
p^{i}_{A}\left.dq_{i}^{A}\right|_{q}\bigg{|}\,q_{i+1}^{A}\left.\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}\right|_{q}\right\rangle=\left.p^{i}_{A}q_{i+1}^{A}\right|_{q}\
.$ (15)
Observe that, if $k=1$, the map $j_{1}\colon{\rm T}Q\to{\rm T}Q$ is the
identity on ${\rm T}Q$, and we recover the standard canonical coupling between
vectors in ${\rm T}_{p}Q$ and covectors in ${\rm T}^{*}_{p}Q$.
Using the coupling function, given a Lagrangian function ${\cal L}\in{\rm
C}^{\infty}({\rm T}^{k}Q)$, we can define the Hamiltonian function $H\in{\rm
C}^{\infty}({\cal W})$ as
$H={\cal C}-(\rho^{2k-1}_{k}\circ\operatorname{pr}_{1})^{*}{\cal L}\ ,$ (16)
whose coordinate expression is
$H=p^{i}_{A}q_{i+1}^{A}-{\cal L}(q_{0}^{A},\ldots,q_{k}^{A})\ .$ (17)
Now, $({\cal W},\Omega,H)$ is a presymplectic Hamiltonian system.
Finally, in order to give a complete description of the dynamics of higher-
order Lagrangian systems, we need to introduce the following concept:
###### Definition 14
A vector field $X\in{\mathfrak{X}}({\cal W})$ is said to be a semispray of
type $r$ in ${\cal W}$ if, for every integral curve $\sigma\colon
I\subset\mathbb{R}\to{\cal W}$ of $X$, the curve
$\sigma_{1}=\operatorname{pr}_{1}\circ\sigma\colon I\to{\rm T}^{2k-1}Q$
satisfies that, if $\gamma=\beta^{2k-1}\circ\sigma_{1}$,
$\tilde{\gamma}^{2k-r}=\rho^{2k-1}_{2k-r}\circ\sigma_{1}$.
In particular, $X\in{\mathfrak{X}}({\cal W})$ is a semispray of type $1$ if
$\tilde{\gamma}^{2k-1}=\sigma_{1}$.
The local expression of a semispray of type $r$ in ${\cal W}$ is
$X=\sum_{i=0}^{2k-1-r}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=2k-r}^{2k-1}X_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=0}^{k-1}G^{i}_{A}\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\
,$
and, in particular, for a semispray of type $1$ in ${\cal W}$ we have
$X=\sum_{i=0}^{2k-2}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+X_{2k-1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2k-1}^{A}}}+\sum_{i=0}^{k-1}G^{i}_{A}\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\
.$
### 3.2 Dynamical vector fields
#### 3.2.1 Dynamics in ${\cal W}={\rm T}^{2k-1}Q\times_{{\rm T}^{k-1}Q}{\rm
T}^{*}({\rm T}^{k-1}Q)$
As we know, the dynamical equation of the presymplectic Hamiltonian system
$({\cal W},\Omega,H)$ is geometrically written as
$\mathop{i}\nolimits(X)\Omega={\rm d}H\quad;\quad X\in{\mathfrak{X}}({\cal
W})\ .$ (18)
Then, according to [22] we have:
###### Proposition 2
Given the presymplectic Hamiltonian system $({\cal W},\Omega,H)$, a solution
$X\in{\mathfrak{X}}({\cal W})$ to equation (18) exists only on the points of
the submanifold ${\cal W}_{c}\hookrightarrow{\cal W}$ defined by
${\cal W}_{c}=\left\\{p\in{\cal
W}\colon\xi(p)\equiv(\mathop{i}\nolimits(Y){\rm d}H)(p)=0\ ,\
\forall\,Y\in\ker\,\Omega\right\\}\ .$ (19)
We have the following result:
###### Proposition 3
The submanifold ${\cal W}_{c}\hookrightarrow{\cal W}$ contains a submanifold
${\cal W}_{o}\hookrightarrow{\cal W}_{c}$ which is the graph of the Legendre-
Ostrogradsky map defined by ${\cal L}$; that is, ${\cal W}_{o}={\rm
graph}\,{\cal FL}$.
(Proof) As ${\cal W}_{c}$ is defined by (19), it suffices to prove that the
constraints defining ${\cal W}_{c}$ give rise to those defining the graph of
the Legendre-Ostrogradsky map associated to ${\cal L}$. We make this
calculation in coordinates. Taking the local expression (17) of the
Hamiltonian function $H\in{\rm C}^{\infty}({\cal W})$, we have
${\rm d}H=\sum_{i=0}^{k-1}(q_{i+1}^{A}{\rm d}p^{i}_{A}+p^{i}_{A}{\rm
d}q_{i+1}^{A})-\sum_{i=0}^{k}\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i}^{A}}}{\rm d}q_{i}^{A}\ ,$
and using the local basis of $\ker\,\Omega$ given in (13), we obtain that the
equations defining the submanifold ${\cal W}_{c}$ are
$\mathop{i}\nolimits(Y){\rm d}H=0\Longleftrightarrow
p^{k-1}_{A}-\displaystyle\frac{\partial{{\cal L}}}{\partial{q_{k}^{A}}}=0\ .$
Observe that these expressions relate the momentum coordinates $p^{k-1}_{A}$
with the Jacobi-Ostrogadsky functions
$\displaystyle\hat{p}^{k-1}_{A}=\partial{\cal L}/\partial q_{k}^{A}$, and so
we obtain the last group of equations of the Legendre-Ostrogradsky map.
Furthermore, in Section 2.3 we have seen that the other Jacobi-Ostrogradsky
functions $\hat{p}^{r-1}_{A}$ ($1\leq r\leq k-1$) satisfy the relations (9).
Thus we can consider that ${\cal W}_{c}$ contains a submanifold ${\cal W}_{o}$
which can be identified with the graph of a map
$\begin{array}[]{rcl}F\colon{\rm T}^{2k-1}Q&\longrightarrow&{\rm T}^{*}({\rm
T}^{k-1}Q)\\\
(q_{i}^{A})&\longmapsto&(q_{0}^{A},\ldots,q_{k-1}^{A},p^{0}_{A},\ldots,p^{k-1}_{A})\end{array}$
which we identify with the Legendre-Ostrogradsky map by making the
identification $p^{r-1}_{A}=\hat{p}^{r-1}_{A}$.
Remark: The submanifold ${\cal W}_{o}$ can be obtained from ${\cal W}_{c}$
using a constraint algorithm. Hence, ${\cal W}_{o}$ acts as the initial phase
space of the system.
We denote by $j_{o}\colon{\cal W}_{o}\hookrightarrow{\cal W}$ the natural
embedding and by ${\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ the set of vector
fields in ${\cal W}$ at support on ${\cal W}_{o}$. Hence, we look for vector
fields $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ which are solutions to
equation (18) at support on ${\cal W}_{o}$; that is
$\left.\left[\mathop{i}\nolimits(X)\Omega-{\rm d}H\right]\right|_{{\cal
W}_{o}}=0\ .$ (20)
In natural coordinates a generic vector field in ${\mathfrak{X}}({\cal W})$ is
$X=\sum_{i=0}^{k-1}f_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=0}^{k-1}G^{i}_{A}\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\
,$
bearing in mind the local expressions of $\Omega$ and ${\rm d}H$, from (18),
we obtain the following system of $(2k+1)n$ equations
$\displaystyle f_{i}^{A}=q_{i+1}^{A}\ ,$ (21) $\displaystyle
G^{0}_{A}=\displaystyle\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\quad,\quad
G^{i}_{A}=\displaystyle\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i}^{A}}}-p^{i-1}_{A}=d_{T}(p^{i}_{A})\ ,$ (22) $\displaystyle
p^{k-1}_{A}-\displaystyle\frac{\partial{{\cal L}}}{\partial{q_{k}^{A}}}=0\ ,$
(23)
where $0\leqslant i\leqslant k-1$ in (21) and $1\leqslant i\leqslant k-1$ in
(22). Therefore
$X=\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\
.$ (24)
We can observe that equations (23) are just a compatibility condition that,
together with the other conditions for the momenta, say that the vector fields
$X$ exist only with support on the submanifold defined by the graph of the
Legendre-Ostrogradsky map. So we recover, in coordinates, the result stated in
Propositions 2 and 3. Furthermore, this local expression shows that $X$ is a
semispray of type $k$ in ${\cal W}$.
The component functions $F_{i}^{A}$, $k\leqslant i\leqslant 2k-1$, are
undetermined. Nevertheless, we must study the tangency of $X$ to the
submanifold ${\cal W}_{o}$; that is, we have to impose that $\left.\mathop{\rm
L}\nolimits(X)\xi\right|_{{\cal W}_{o}}\equiv\left.X(\xi)\right|_{{\cal
W}_{o}}=0$, for every constraint function $\xi$ defining ${\cal W}_{o}$. So,
taking into account Prop. 3, these conditions lead to
$\displaystyle\left(\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\right)\left(p^{k-1}_{A}-\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{k}^{A}}}\right)=0$
$\displaystyle\left(\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\right)\left(p^{k-2}_{A}-\sum_{i=0}^{1}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{k-1+i}^{A}}}\right)\right)=0$
$\displaystyle\qquad\qquad\qquad\qquad\vdots$
$\displaystyle\left(\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\right)\left(p^{1}_{A}-\sum_{i=0}^{k-2}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2+i}^{A}}}\right)\right)=0$
$\displaystyle\left(\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\right)\left(p^{0}_{A}-\sum_{i=0}^{k-1}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1+i}^{A}}}\right)\right)=0\ ,$
and, from here, we obtain the following $kn$ equations
$\begin{array}[]{l}\displaystyle\left(F_{k}^{B}-q_{k+1}^{B}\right)\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}=0\\\\[10.0pt]
\displaystyle\left(F_{k+1}^{B}-q_{k+2}^{B}\right)\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}-\left(F_{k}^{B}-q_{k+1}^{B}\right)d_{T}\left(\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}\right)=0\\\
\qquad\qquad\qquad\qquad\vdots\\\
\displaystyle\left(F_{2k-2}^{B}-q_{2k-1}^{B}\right)\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}-\sum_{i=0}^{k-3}\left(F_{k+i}^{B}-q_{k+i+1}^{B}\right)(\cdots\cdots)=0\\\
\displaystyle(-1)^{k}\left(F_{2k-1}^{B}-d_{T}\left(q_{2k-1}^{B}\right)\right)\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}+\sum_{i=0}^{k}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i}^{A}}}\right)-\sum_{i=0}^{k-2}\left(F_{k+i}^{B}-q_{k+i+1}^{B}\right)(\cdots\cdots)=0\
,\end{array}$ (25)
where the terms in brackets $(\cdots\cdots)$ contain relations involving
partial derivatives of the Lagrangian and applications of $d_{T}$ which for
simplicity are not written. These are just the Lagrangian equations for the
components of $X$, as we have seen in (6). These equations can be compatible
or not, and a sufficient condition to ensure compatibility is the regularity
of the Lagrangian function. In particular, we have:
###### Proposition 4
If ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular Lagrangian
function, then there exists a unique vector field $X\in{\mathfrak{X}}_{{\cal
W}_{o}}({\cal W})$ which is a solution to equation (20); it is tangent to
${\cal W}_{o}$, and is a semispray of type $1$ in ${\cal W}$.
(Proof) As the Lagrangian function ${\cal L}$ is regular, the Hessian matrix
$\displaystyle\left(\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}\right)$ is regular at every
point, and this allows us to solve the above $k$ systems of $n$ equations (25)
determining all the functions $F_{i}^{A}$ uniquely, as follows
$\displaystyle F_{i}^{A}=q_{i+1}^{A}\quad,\quad(k\leqslant i\leqslant 2k-2)$
(26)
$\displaystyle(-1)^{k}\left(F_{2k-1}^{B}-d_{T}\left(q_{2k-1}^{B}\right)\right)\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{k}^{B}}\partial{q_{k}^{A}}}+\sum_{i=0}^{k}(-1)^{i}d_{T}^{i}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i}^{A}}}\right)=0\ .$
In this way, the tangency condition holds for $X$ at every point on ${\cal
W}_{o}$. Furthermore, the equalities (26) show that $X$ is a semispray of type
$1$ in ${\cal W}$
However, if ${\cal L}$ is not regular, the equations (25) can be compatible or
not. In the most favourable cases, there is a submanifold ${\cal
W}_{f}\hookrightarrow{\cal W}_{o}$ (it could be ${\cal W}_{f}={\cal W}_{o}$)
such that there exist vector fields $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal
W})$, tangent to ${\cal W}_{f}$, which are solutions to the equation
$\left.\left[\mathop{i}\nolimits(X)\Omega-dH\right]\right|_{{\cal W}_{f}}=0\
.$ (27)
In this case, the equations (25) are not compatible, and the compatibility
condition gives rise to new constraints.
#### 3.2.2 Dynamics in ${\rm T}^{2k-1}Q$
Now we study how to recover the Lagrangian dynamics from the dynamics in the
unified formalism, using the dynamical vector fields.
First we have the following results:
###### Proposition 5
The map $\overline{\operatorname{pr}}_{1}=\operatorname{pr}_{1}\circ
j_{o}\colon{\cal W}_{o}\to{\rm T}^{2k-1}Q$ is a diffeomorphism.
(Proof) As ${\cal W}_{o}={\rm graph}\,{\cal FL}$, we have that ${\rm
T}^{2k-1}Q\simeq{\cal W}_{o}$. Furthermore, $\overline{\operatorname{pr}}_{1}$
is a surjective submersion and, by the equality between dimensions, it is also
an injective immersion and hence it is a diffeomorphism.
###### Lemma 1
If $\omega_{k-1}\in{\mit\Omega}^{2}({\rm T}^{*}({\rm T}^{k-1}Q))$ is the
canonical symplectic $2$-form in ${\rm T}^{*}({\rm T}^{k-1}Q)$, and
$\omega_{{\cal L}}={\cal FL}^{*}\omega_{k-1}$ is the Lagrangian $2$-form, then
$\Omega=\operatorname{pr}_{1}^{*}\omega_{\cal L}$.
(Proof) In fact,
$\operatorname{pr}_{1}^{*}\omega_{{\cal L}}=\operatorname{pr}_{1}^{*}({\cal
FL}^{*}\omega_{k-1})=({\cal
FL}\circ\operatorname{pr}_{1})^{*}\omega_{k-1}=\operatorname{pr}_{2}^{*}\omega_{k-1}=\Omega\
.$
###### Lemma 2
There exists a unique function $E_{\cal L}\in{\rm C}^{\infty}({\rm
T}^{2k-1}Q)$ such that $\operatorname{pr}_{1}^{*}E_{\cal L}=H$. This function
$E_{\cal L}$ is the Lagrangian energy.
(Proof) As $\overline{\operatorname{pr}}_{1}$ is a diffeomorphism, we can
define the function $E_{\cal L}=(\overline{\operatorname{pr}}_{1}^{-1}\circ
j_{o})^{*}H\in{\rm C}^{\infty}({\rm T}^{2k-1}Q)$, which obviously verifies
that $\operatorname{pr}_{1}^{*}E_{\cal L}=H$.
In order to prove that $E_{\cal L}$ is the Lagrangian energy defined
previously, we calculate its local expression in coordinates. Thus, from (17)
we obtain that
$\overline{\operatorname{pr}}_{1}^{*}E_{\cal
L}=H=\sum_{i=0}^{k-1}p^{i}_{A}q_{i+1}^{A}-{\cal
L}(q_{0}^{A},\ldots,q_{k}^{A})\ ,$
but ${\cal W}_{o}\hookrightarrow{\cal W}$ is the graph of the Legendre-
Ostrogradsky map, and by Prop. 3 we have $\displaystyle
p^{i}_{A}=\sum_{j=0}^{k-i-1}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i+1+j}^{A}}}\right)$, and then
$\displaystyle\overline{\operatorname{pr}}_{1}^{*}E_{\cal L}$
$\displaystyle=\sum_{i=0}^{k-1}\sum_{j=0}^{k-i-1}q_{i+1}^{A}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i+1+j}^{A}}}\right)-{\cal L}(q_{0}^{A},\ldots,q_{k}^{A})$
$\displaystyle=\sum_{i=1}^{k}\sum_{j=0}^{k-i}q_{i}^{A}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i+j}^{A}}}\right)-{\cal L}(q_{0}^{A},\ldots,q_{k}^{A})\ .$
Now, as $\overline{\operatorname{pr}}_{1}=\operatorname{pr}_{1}\circ j_{o}$
and $\operatorname{pr}_{1}^{*}q_{i}^{A}=q_{i}^{A}$, we obtain finally
$E_{\cal
L}=\sum_{i=1}^{k}\sum_{j=0}^{k-i}q_{i}^{A}(-1)^{j}d_{T}^{j}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{i+j}^{A}}}\right)-{\cal L}(q_{0}^{A},\ldots,q_{k}^{A})$
which is the local expression (4) of the Lagrangian energy.
Using these results, we can recover an Euler-Lagrange vector field in ${\rm
T}^{2k-1}Q$ starting from a vector field $X\in{\mathfrak{X}}_{{\cal
W}_{0}}({\cal W})$ tangent to ${\cal W}_{o}$, a solution to (20). First we
have:
###### Lemma 3
Let $X\in{\mathfrak{X}}({\cal W})$ be a vector field tangent to ${\cal
W}_{o}$. Then there exists a unique vector field $X_{\cal
L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ such that $X_{\cal
L}\circ\operatorname{pr}_{1}\circ j_{o}={\rm T}\operatorname{pr}_{1}\circ
X\circ j_{o}$.
(Proof) As $X\in{\mathfrak{X}}({\cal W})$ is tangent to ${\cal W}_{o}$, there
exists a vector field $X_{o}\in{\mathfrak{X}}({\cal W}_{o})$ such that ${\rm
T}j_{o}\circ X_{o}=X\circ j_{o}$. Furthermore, as
$\overline{\operatorname{pr}}_{1}$ is a diffeomorphism, there is a unique
vector field $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ which is
$\overline{\operatorname{pr}}_{1}$-related with $X_{o}$; that is, $X_{\cal
L}\circ\overline{\operatorname{pr}}_{1}={\rm
T}\overline{\operatorname{pr}}_{1}\circ X_{o}$. Then
$X_{\cal L}\circ\operatorname{pr}_{1}\circ j_{o}=X_{\cal
L}\circ\overline{\operatorname{pr}_{1}}={\rm
T}\overline{\operatorname{pr}}_{1}\circ X_{o}={\rm
T}\operatorname{pr}_{1}\circ{\rm T}j_{o}\circ X_{o}={\rm
T}\operatorname{pr}_{1}\circ X\circ j_{o}$
And as a consequence we obtain:
###### Theorem 2
Let $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ be a vector field solution
to equation (20) and tangent to ${\cal W}_{o}$ (at least on the points of a
submanifold ${\cal W}_{f}\hookrightarrow{\cal W}_{o}$). Then there exists a
unique semispray of type $k$, $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$,
which is a solution to the equation
$\mathop{i}\nolimits(X_{\cal L})\omega_{\cal L}-dE_{\cal L}=0$ (28)
(at least on the points of $S_{f}=\operatorname{pr}_{1}({\cal W}_{f})$). In
addition, if ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular
Lagrangian, then $X_{\cal L}$ is a semispray of type $1$, and hence it is the
Euler-Lagrange vector field.
Conversely, if $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ is a semispray
of type $k$ (resp., of type $1$), which is a solution to equation (28) (at
least on the points of a submanifold $S_{f}\hookrightarrow{\rm T}^{2k-1}Q$),
then there exists a unique vector field $X\in{\mathfrak{X}}_{{\cal
W}_{o}}({\cal W})$ which is a solution to equation (20) (at least on ${\cal
W}_{f}=\overline{\operatorname{pr}}_{1}^{-1}(S_{f})\hookrightarrow{\cal
W}_{o}\hookrightarrow{\cal W}$), and it is a semispray of type $k$ in ${\cal
W}$ (resp., of type $1$).
(Proof) Applying Lemmas 1, 2, and 3, we have:
$0=\left.\left[\mathop{i}\nolimits(X)\Omega-{\rm d}H\right]\right|_{{\cal
W}_{o}}=\left.\left[\mathop{i}\nolimits(X)\operatorname{pr}_{1}^{*}\omega_{\cal
L}-{\rm d}\operatorname{pr}_{1}^{*}E_{\cal L}\right]\right|_{{\cal
W}_{o}}=\operatorname{pr}_{1}^{*}\left.\left[\mathop{i}\nolimits(X_{\cal
L})\omega_{\cal L}-{\rm d}E_{\cal L}\right]\right|_{{\cal W}_{o}}\ ,$
but, as $\operatorname{pr}_{1}$ is a surjective submersion, this is equivalent
to
$0=\left.\left[\mathop{i}\nolimits(X_{\cal L})\omega_{\cal L}-{\rm d}E_{\cal
L}\right]\right|_{\operatorname{pr}_{1}({\cal
W}_{o})}=\left.\left[\mathop{i}\nolimits(X_{\cal L})\omega_{\cal L}-{\rm
d}E_{\cal L}\right]\right|_{{\rm T}^{2k-1}Q}=0\ ,$
since $\operatorname{pr}_{1}({\cal W}_{o})={\rm T}^{2k-1}Q$. The converse is
immediate, reversing this reasoning.
In order to prove that $X_{\cal L}$ is a semispray of type $k$, we proceed in
coordinates. From the local expression (24) for the vector field $X$ (where
the functions $F_{i}^{A}$ are the solutions of the equations (25)), and using
Lemma 3, we obtain that the local expression of $X_{\cal
L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ is
$X_{\cal
L}=\sum_{i=0}^{k-1}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+\sum_{i=k}^{2k-1}F_{i}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}\
,$
and then
$J_{k}(X_{\cal
L})=\sum_{i=0}^{k-1}\frac{(k+i)!}{i!}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{k+i}^{A}}}=\Delta_{k}\
;$
so $X_{\cal L}$ is a semispray of type $k$ in ${\rm T}^{2k-1}Q$.
Finally, if ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular
Lagrangian, equations (25) become (26), and hence the local expression of $X$
is
$X=\sum_{i=0}^{2k-2}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+F_{2k-1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2k-1}^{A}}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\displaystyle\frac{\partial{}}{\partial{p^{0}_{A}}}+\sum_{i=1}^{k-1}d_{T}(p^{i}_{A})\displaystyle\frac{\partial{}}{\partial{p^{i}_{A}}}\
.$
Therefore
$X_{\cal
L}=\sum_{i=0}^{2k-2}q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i}^{A}}}+F_{2k-1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2k-1}^{A}}}\
,$
and then $\displaystyle J_{1}(X_{\cal
L})=\sum_{i=0}^{2k-2}(i+1)q_{i+1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{i+1}^{A}}}=\Delta_{1}$,
which shows that $X_{\cal L}$ is a semispray of type $1$ in ${\rm T}^{2k-1}Q$.
Remarks:
* •
It is important to point out that, if ${\cal L}$ is not a regular Lagrangian,
then $X$ is a semispray of type $k$ in ${\cal W}$, but not necessarily a
semispray of type $1$. This means that $X_{\cal L}$ is a Lagrangian vector
field, but it is not necessarily an Euler-Lagrange vector field (it is not a
semispray of type $1$, but just a semispray of type $k$). Thus, for singular
Lagrangians, this must be imposed as an additional condition in order that the
integral curves of $X_{\cal L}$ verify the Euler-Lagrange equations. This is a
different from the case of first-order dynamical systems ($k=1$), where this
condition ($X_{\cal L}$ is a semispray of type $1$; that is, a holonomic
vector field) is obtained straightforwardly in the unified formalism.
In general, only in the most interesting cases have we assured the existence
of a submanifold ${\cal W}_{f}\hookrightarrow{\cal W}_{o}$ and vector fields
$X\in{\mathfrak{X}}_{{\cal W}_{0}}({\cal W})$ tangent to ${\cal W}_{f}$ which
are solutions to the equation (27). Then, considering the submanifold
$S_{f}=\operatorname{pr}_{1}({\cal W}_{f})\hookrightarrow{\rm T}^{2k-1}Q$, in
the best cases (see [8, 23, 24]), we have that those Euler-Lagrange vector
fields $X_{\cal L}$ exist, perhaps on another submanifold
$M_{f}\hookrightarrow S_{f}$ where they are tangent, and are solutions to the
equation
$\left.\left[i_{X_{\cal L}}\omega_{\cal L}-{\rm d}E_{\cal
L}\right]\right|_{M_{f}}=0\ .$ (29)
* •
Observe also that Theorem 2 states that there is a one-to-one correspondence
between vector fields $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ which are
solutions to equation (20) and $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$
solutions to (28), but not uniqueness, unless ${\cal L}$ is regular. In fact:
###### Corollary 1
If ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular Lagrangian, then
there is a unique $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ tangent to
${\cal W}_{o}$ which is a solution to equation (20), and it is a semispray of
type $1$.
(Proof) As ${\cal L}$ is regular, by Proposition 1 there is a unique semispray
of type $1$, $X_{\cal L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$ which is a
solution to equation (28) on ${\rm T}^{2k-1}Q$. Then, by Theorem 2, there is a
unique $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$, tangent to ${\cal
W}_{o}$, which is a solution to (20) on ${\cal W}_{o}$.
#### 3.2.3 Dynamics in ${\rm T}^{*}({\rm T}^{k-1}Q)$
##### Hyperregular and regular Lagrangians
In order to recover the Hamiltonian formalism, we distinguish between the
regular and non-regular cases. We start with the regular case, although by
simplicity we analyze the hyperregular case (the regular case is recovered
from this by restriction on the corresponding open sets where the Legendre-
Ostrogadsky map is a local diffeomorphism). For this case we have the
following commutative diagram
$\textstyle{{\rm T}{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
T}\operatorname{pr}_{1}}$$\scriptstyle{{\rm T}\operatorname{pr}_{2}}$
$\textstyle{{\rm T}{\cal
W}_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
T}\overline{\operatorname{pr}}_{1}}$$\scriptstyle{{\rm
T}\overline{\operatorname{pr}}_{2}}$$\scriptstyle{{\rm T}j_{o}}$
$\textstyle{{\rm T}({\rm T}^{2k-1}Q)}$ $\textstyle{{\rm T}({\rm T}^{*}({\rm
T}^{k-1}Q))}$ $\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$$\scriptstyle{X}$
$\textstyle{{\cal W}_{o}={\rm graph}\,{{\cal
FL}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname{pr}}_{1}}$$\scriptstyle{\overline{\operatorname{pr}}_{2}}$$\scriptstyle{j_{o}}$$\scriptstyle{X_{o}}$
$\textstyle{{\rm
T}^{2k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{2k-1}_{k-1}}$$\scriptstyle{{\cal
FL}}$$\scriptstyle{X_{\cal L}}$$\scriptstyle{\beta^{2k-1}}$ $\textstyle{{\rm
T}^{*}({\rm
T}^{k-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\rm
T}^{k-1}Q}}$$\scriptstyle{X_{h}}$ $\textstyle{{\rm
T}^{k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{k-1}}$
$\textstyle{Q}$
###### Theorem 3
Let ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ be a hyperregular Lagrangian,
$h\in{\rm C}^{\infty}({\rm T}^{*}({\rm T}^{k-1}Q))$ the Hamiltonian function
such that ${\cal FL}^{*}h=E_{\cal L}$, and $X\in{\mathfrak{X}}_{{\cal
W}_{o}}({\cal W})$ the vector field solution to the equation (20), tangent to
${\cal W}_{o}$. Then, there exists a unique vector field $X_{h}={\cal
FL}_{*}X_{\cal L}\in{\mathfrak{X}}({\rm T}^{*}({\rm T}^{k-1}Q))$ which is a
solution to the equation
$\mathop{i}\nolimits(X_{h})\omega_{k-1}-{\rm d}h=0$ (30)
Conversely, if $X_{h}\in{\mathfrak{X}}({\rm T}^{*}({\rm T}^{k-1}Q))$ is a
solution to equation (30), then there exists a unique vector field
$X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$, tangent to ${\cal W}_{o}$,
which is a solution to equation (20).
(Proof) If ${\cal L}$ is hiperregular, then
$\overline{\operatorname{pr}}_{2}={\cal
FL}\circ\overline{\operatorname{pr}}_{1}$ is a diffeomorphism, since it is a
composition of diffeomorphisms; then there exists a unique vector field
$X_{o}\in{\mathfrak{X}}({\cal W}_{o})$ such that
${\overline{\operatorname{pr}}_{2}}_{*}X_{o}=X_{h}$, and there is a unique
$X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ such that
${j_{o}}_{*}X_{o}=\left.X\right|_{{\cal W}_{o}}$.
Now, as ${\cal FL}^{*}h=E_{\cal L}$, by applying Lemma 2 we have that
$\operatorname{pr}_{1}^{*}({\cal FL}^{*}(h))=\operatorname{pr}_{1}^{*}E_{\cal
L}=H$; but ${\cal FL}\circ\operatorname{pr}_{1}=\operatorname{pr}_{2}$, and
then $\operatorname{pr}_{2}^{*}h=H$. Therefore, by the definition of $\Omega$,
we have
$0=\left.\left[\mathop{i}\nolimits(X)\Omega-{\rm d}H\right]\right|_{{\cal
W}_{o}}=\left.\left[\mathop{i}\nolimits(X)\operatorname{pr}_{2}^{*}\omega_{k-1}-{\rm
d}\operatorname{pr}_{2}^{*}h\right]\right|_{{\cal
W}_{o}}=\operatorname{pr}_{2}^{*}\left.\left[\mathop{i}\nolimits(X_{h})\omega_{k-1}-{\rm
d}h\right]\right|_{{\cal W}_{o}}\ .$
However, as $\operatorname{pr}_{2}$ is a surjective submersion and
$\operatorname{pr}_{2}({\cal W}_{o})={\rm T}^{*}({\rm T}^{k-1}Q)$, we finally
obtain that
$\\\ 0=\left.\left[\mathop{i}\nolimits(X_{h})\omega_{k-1}-{\rm
d}h\right]\right|_{\operatorname{pr}_{2}({\cal
W}_{o})}=\left.\left[\mathop{i}\nolimits(X_{h})\omega_{k-1}-{\rm
d}h\right]\right|_{{\rm T}^{*}({\rm T}^{k-1}Q)}$
##### Singular (almost-regular) Lagrangians
Remember that, for almost-regular Lagrangians, only in the most interesting
cases have we assured the existence of a submanifold ${\cal
W}_{f}\hookrightarrow{\cal W}_{o}$ and vector fields
$X\in{\mathfrak{X}}_{{\cal W}_{0}}({\cal W})$ tangent to ${\cal W}_{f}$ which
are solutions to equation (27). In this case, the dynamical vector fields in
the Hamiltonian formalism cannot be obtained straightforwardly from the
solutions in the unified formalism, but rather by passing through the
Lagrangian formalism and using the Legendre-Ostrogadsky map.
Thus, we can consider the submanifolds $S_{f}=\operatorname{pr}_{1}({\cal
W}_{f})\hookrightarrow{\rm T}^{2k-1}Q$ and $P_{f}=\operatorname{pr}_{2}({\cal
W}_{f})={\cal FL}(S_{f})\hookrightarrow{\rm T}^{*}({\rm T}^{k-1}Q)$. Then,
using Theorem 2, from the vector fields $X\in{\mathfrak{X}}_{{\cal
W}_{o}}({\cal W})$ we obtain the corresponding $X_{\cal
L}\in{\mathfrak{X}}({\rm T}^{2k-1}Q)$, and from these the semisprays of type
$1$ (if they exist) which are perhaps defined on a submanifold
$M_{f}\hookrightarrow S_{f}$, are tangent to $M_{f}$ and are solutions to
equation (29). So we have the following commutative diagram
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\cal W}_{P}={\rm T}^{2k-1}Q\times_{{\rm
T}^{k-1}Q}P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{{\cal
W}_{P}}}$$\scriptstyle{\operatorname{pr}_{1,{\cal
W}_{P}}}$$\scriptstyle{\operatorname{pr}_{2,{\cal
W}_{P}}}$$\scriptstyle{\operatorname{pr}_{2,P_{o}}}$ $\textstyle{{\rm
T}^{2k-1}Q}$ $\textstyle{{\rm T}^{*}({\rm T}^{k-1}Q)}$ $\textstyle{{\cal
W}_{o}={\rm graph}\,({\cal
FL}_{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{o}}$$\scriptstyle{\overline{\operatorname{pr}}_{1,P_{o}}}$$\scriptstyle{\overline{\operatorname{pr}}_{2,P_{o}}}$$\textstyle{P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{o}}}$
$\textstyle{{\cal
W}_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{{\cal
W}_{f}}}$
$\textstyle{M_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{S_{f}}}$
$\textstyle{P_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{f}}}$
Now, it is proved ([24]) that there are Euler-Lagrange vector fields (perhaps
only on the points of another submanifold $\bar{M}_{f}\hookrightarrow M_{f}$),
which are ${\cal FL}$-projectable on $P_{f}={\cal FL}(S_{f})\hookrightarrow
P_{o}\hookrightarrow{\rm T}^{*}({\rm T}^{k-1}Q)$. These vector fields
$X_{h_{o}}={\cal FL}_{*}X_{{\cal L}}\in{\mathfrak{X}}({\rm T}^{*}({\rm
T}^{k-1}Q))$ are tangent to $P_{f}$ and are solutions to the equation
$\left.\left[i_{X_{h_{o}}}\omega_{o}-{\rm d}h_{o}\right]\right|_{P_{f}}=0\ .$
Conversely, as ${\cal FL}_{o}$ is a submersion, for every solution
$X_{h_{o}}\in{\mathfrak{X}}({\rm T}^{*}({\rm T}^{k-1}Q))$ to the last
equation, there is a semispray of type $1$, $X_{\cal L}\in{\mathfrak{X}}({\rm
T}^{2k-1}Q)$, such that ${{\cal FL}_{o}}_{*}X_{\cal L}=X_{h_{o}}$, and we can
recover solutions to equation (27) using Theorem 2.
### 3.3 Integral curves
After studying the vector fields which are solutions to the dynamical
equations, we analyze their integral curves, showing how to recover the
Lagrangian and Hamiltonian dynamical trajectories from the dynamical
trajectories in the unified formalism.
Let $X\in{\mathfrak{X}}_{{\cal W}_{o}}({\cal W})$ be a vector field tangent to
${\cal W}_{o}$ which is a solution to equation (20), and let $\sigma\colon
I\subset\mathbb{R}\to{\cal W}$ be an integral curve of $X$, on ${\cal W}_{o}$.
As $\tilde{\sigma}=X\circ\sigma$, this means that the following equation holds
$\mathop{i}\nolimits({\tilde{\sigma}})(\Omega\circ\sigma)={\rm d}H\circ\sigma\
.$ (31)
Furthermore, if $\sigma_{o}\colon I\to{\cal W}_{o}$ is a curve on ${\cal
W}_{o}$ such that $j_{o}\circ\sigma_{o}=\sigma$, we have that $\sigma_{o}$ is
an integral curve of the vector field $X_{o}\in{\mathfrak{X}}({\cal W}_{o})$
associated to $X$, and $\tilde{\sigma}_{o}=X_{o}\circ\sigma_{o}$.
In local coordinates, if $\sigma(t)=(q_{i}^{A}(t),p^{j}_{A}(t))$, we have that
$\displaystyle\dot{q}_{i}^{A}(t)=q_{i+1}^{A}\circ\sigma$
$\displaystyle\quad(0\leqslant i\leqslant k-1)\quad;\quad$
$\displaystyle\dot{q}_{i}^{A}(t)=F_{i}^{A}\circ\sigma\quad(k\leqslant
i\leqslant 2k-1)$
$\displaystyle\dot{p}^{0}_{A}(t)=\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}\circ\sigma$ $\displaystyle\qquad\qquad\qquad\qquad\
;\quad$
$\displaystyle\dot{p}^{i}_{A}(t)=d_{T}(p^{i}_{A})\circ\sigma\quad(1\leqslant
i\leqslant k-1)\ ,$
where $F_{i}^{A}$ are solutions to equations (25).
Now, for the Lagrangian dynamical trajectories we have the following result:
###### Proposition 6
Let $\sigma\colon I\subset\mathbb{R}\to{\cal W}$ be an integral curve of a
vector field $X$ solution to (20), on ${\cal W}_{o}$. Then the curve
$\sigma_{\cal L}=\operatorname{pr}_{1}\circ\sigma\colon I\to{\rm T}^{2k-1}Q$
is an integral curve of $X_{\cal L}$.
(Proof) As $\sigma=j_{o}\circ\sigma_{o}$, using that ${\rm T}j_{o}\circ
X_{o}=X\circ j_{o}$ and that ${\rm T}\operatorname{pr}_{1}\circ X=X_{\cal
L}\circ\operatorname{pr}_{1}$, we have
$\displaystyle\tilde{\sigma}_{\cal L}$ $\displaystyle=$
$\displaystyle\widetilde{\operatorname{pr}_{1}\circ\sigma}=\widetilde{\operatorname{pr}_{1}\circ
j_{o}\circ\sigma_{o}}={\rm T}\operatorname{pr}_{1}\circ{\rm
T}j_{o}\circ\tilde{\sigma}_{o}={\rm T}\operatorname{pr}_{1}\circ{\rm
T}j_{o}\circ X_{o}\circ\sigma_{o}$ $\displaystyle=$ $\displaystyle{\rm
T}\operatorname{pr}_{1}\circ X\circ j_{o}\circ\sigma_{o}=X_{\cal
L}\circ\operatorname{pr}_{1}\circ j_{o}\circ\sigma_{o}=X_{\cal
L}\circ\sigma_{\cal L}\ .$
###### Corollary 2
If ${\cal L}\in{\rm C}^{\infty}({\rm T}^{k}Q)$ is a regular Lagrangian, then
the curve $\sigma_{\cal L}=\operatorname{pr}_{1}\circ\sigma\colon I\to{\rm
T}^{2k-1}Q$ is the canonical lifting of a curve on $Q$; that is, there exists
$\gamma\colon I\subset\mathbb{R}\to Q$ such that $\sigma_{\cal
L}=\tilde{\gamma}^{2k-1}$.
(Proof) It is a straighforward consequence of Proposition 6 and Theorem 2.
And for the Hamiltonian trajectories, we have:
###### Proposition 7
Let $\sigma\colon I\subset\mathbb{R}\to{\cal W}$ be an integral curve of a
vector field $X$ solution to (20), on ${\cal W}_{o}$. Then the curve
$\sigma_{h}={\cal FL}\circ\sigma_{\cal L}\colon I\to{\rm T}^{*}({\rm
T}^{k-1}Q)$ is an integral curve of $X_{h}={\cal FL}_{*}(X_{\cal L})$.
(Proof) Given that $\sigma_{\cal L}$ is an integral curve of $X_{\cal L}$,
Proposition 6, and the definitions of $X_{h}$ and $\sigma_{h}$, we obtain
$\tilde{\sigma}_{h}=\widetilde{{\cal FL}\circ\sigma_{\cal L}}={\rm T}{\cal
FL}\circ\tilde{\sigma}_{\cal L}={\rm T}{\cal FL}\circ X_{\cal
L}\circ\sigma_{\cal L}=X_{h}\circ{\cal FL}\circ\sigma_{\cal
L}=X_{h}\circ\sigma_{h}\ .$
Thus $\sigma_{h}$ is an integral curve of $X_{h}$.
The relation among all these integral curves is summarized in the following
diagram
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\cal
W}_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\operatorname{pr}}_{1}}$$\scriptstyle{\overline{\operatorname{pr}}_{2}}$$\scriptstyle{j_{o}}$
$\textstyle{{\rm
T}^{2k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{2k-1}_{k-1}}$$\scriptstyle{\beta^{2k-1}}$$\scriptstyle{{\cal
FL}}$ $\textstyle{{\rm T}^{*}({\rm
T}^{k-1}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\rm
T}^{k-1}Q}}$
$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\sigma_{o}}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma_{\cal
L}}$$\scriptstyle{\sigma_{h}}$ $\textstyle{{\rm
T}^{k-1}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{k-1}}$
$\textstyle{Q}$
Remark: Observe that in Propositions 6 and 7 we make no assumption on the
regularity of the system. The only considerations in the almost-regular case
are that, in general, the curves are defined in some submanifolds which are
determined by the constraint algorithm, and that $\sigma_{\cal L}$ is not
necessarily the lifting of any curve in $Q$ and this condition must be
imposed. In particular:
* •
If the Lagrangian is regular (or hiperregular), then ${\rm
Im}\,(\sigma)\subset{\cal W}_{o}$, ${\rm Im}\,(\sigma_{\cal L})\subset{\rm
T}^{2k-1}Q$ and ${\rm Im}\,(\sigma_{h})\subset{\rm T}^{*}({\rm T}^{k-1}Q)$.
* •
If the Lagrangian is almost-regular, then ${\rm Im}\,(\sigma)\subset{\cal
W}_{f}\hookrightarrow{\cal W}_{o}$, ${\rm Im}\,(\sigma_{\cal L})\subset
S_{f}\hookrightarrow{\rm T}^{2k-1}Q$ and ${\rm Im}\,(\sigma_{h})\subset
P_{f}\hookrightarrow P_{o}\hookrightarrow{\rm T}^{*}({\rm T}^{k-1}Q)$.
## 4 Examples
### 4.1 The Pais-Uhlenbeck oscillator
The Pais-Uhlenbeck oscillator is one of the simplest (regular) systems that
can be used to explore the features of higher order dynamical systems, and has
been analyzed in detail in many publications [32, 28]. Here we study it using
the unified formalism.
The configuration space for this system is a $1$-dimensional smooth manifold
$Q$ with local coordinate $(q_{0})$. Taking natural coordinates in the higher-
order tangent bundles over $Q$, the second-order Lagrangian function ${\cal
L}\in{\rm C}^{\infty}({\rm T}^{2}Q)$ for this system is locally given by
${\cal
L}(q_{0},q_{1},q_{2})=\frac{1}{2}\left(q_{1}^{2}-\omega^{2}q_{0}^{2}-\gamma
q_{2}^{2}\right)$
where $\gamma$ is some nonzero real constant, and $\omega$ is a real constant.
${\cal L}$ is a regular Lagrangian function, since the Hessian matrix of
${\cal L}$ with respect to $q_{2}$ is
$\left(\displaystyle\frac{\partial^{2}{{\cal
L}}}{\partial{q_{2}}\partial{q_{2}}}\right)=-\gamma$
which has maximum rank, since we assume that $\gamma$ is nonzero. Notice that,
if we take $\gamma=0$, then ${\cal L}$ becomes a first-order regular
Lagrangian function, and thus it is a nonsense to study this system using the
higher-order unified formalism.
As this is a second-order dynamical system, the phase space that we consider
is
$\textstyle{{\cal W}={\rm T}^{3}Q\times_{{\rm T}Q}{\rm T}^{*}({\rm
T}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\rm
T}^{3}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{3}_{1}}$
$\textstyle{{\rm T}^{*}({\rm
T}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\rm
T}Q}}$ $\textstyle{{\rm T}Q}$$\textstyle{\ .}$
Denoting the canonical symplectic form by $\omega_{1}\in{\mit\Omega}^{2}({\rm
T}^{*}({\rm T}Q))$, we define the presymplectic form
$\Omega\in\operatorname{pr}_{2}^{*}\omega_{1}\in{\mit\Omega}^{2}({\cal W})$
with the local expression
$\Omega={\rm d}q_{0}\wedge{\rm d}p^{0}+{\rm d}q_{1}\wedge{\rm d}p^{1}\ ,$
The Hamiltonian function $H\in{\rm C}^{\infty}({\cal W})$ in the unified
formalism is $H={\cal C}-(\rho^{3}_{2}\circ\operatorname{pr}_{1})^{*}{\cal
L}$, where ${\cal C}$ is the coupling function, whose local expression is
${\cal C}(q_{0},q_{1},q_{2},q_{3},p^{0},p^{1})=p^{0}q_{1}+p^{1}q_{2}\ .$
and then the Hamiltonian function can be written locally
$H(q_{0},q_{1},q_{2},q_{3},p^{0},p^{1})=p^{0}q_{1}+p^{1}q_{2}-\frac{1}{2}\left(q_{1}^{2}-\omega^{2}q_{0}^{2}-\gamma
q_{2}^{2}\right)$
As stated in the above sections, we can describe the dynamics for this system
in terms of the integral curves of vector fields $X\in{\mathfrak{X}}({\cal
W})$ which are solutions to equation (18). If we take a generic vector field
$X$ in ${\cal W}$, given locally by
$X=f_{0}\displaystyle\frac{\partial{}}{\partial{q_{0}}}+f_{1}\displaystyle\frac{\partial{}}{\partial{q_{1}}}+F_{2}\displaystyle\frac{\partial{}}{\partial{q_{2}}}+F_{3}\displaystyle\frac{\partial{}}{\partial{q_{3}}}+G^{0}\displaystyle\frac{\partial{}}{\partial{p^{0}}}+G^{1}\displaystyle\frac{\partial{}}{\partial{p^{1}}},$
taking into acount that
${\rm d}H=\omega^{2}q_{0}{\rm d}q_{0}+(p^{0}-q_{1}){\rm d}q_{1}+(p^{1}+\gamma
q_{2}){\rm d}q_{2}+q_{1}{\rm d}p^{0}+q_{2}{\rm d}p^{1}\ ,$
from the dynamical equation $\mathop{i}\nolimits(X)\Omega={\rm d}H$, we obtain
the following system of linear equations for the coefficients of $X$
$\displaystyle f_{0}=q_{1}$ (32) $\displaystyle f_{1}=q_{2}$ (33)
$\displaystyle G^{0}=-\omega^{2}q_{0}$ (34) $\displaystyle G^{1}=q_{1}-p^{0}$
(35) $\displaystyle p^{1}+\gamma q_{2}=0$ (36)
Equations (32) and (33) give us the condition of semispray of type $2$ for the
vector field $X$. Furthermore, equation (36) is an algebraic relation stating
that the vector field $X$ is defined along a submanifold ${\cal W}_{o}$ that
can be identified with the graph of the Legendre-Ostrogradsky map, as we have
seen in Propositions 2 and 3. Thus, using (32), (33), (34) and (35), the
vector field $X$ is given locally by
$X=q_{1}\displaystyle\frac{\partial{}}{\partial{q_{0}}}+q_{2}\displaystyle\frac{\partial{}}{\partial{q_{1}}}+F_{2}\displaystyle\frac{\partial{}}{\partial{q_{2}}}+F_{3}\displaystyle\frac{\partial{}}{\partial{q_{3}}}-\omega^{2}q_{0}\displaystyle\frac{\partial{}}{\partial{p^{0}}}+\left(q_{1}-p^{0}\right)\displaystyle\frac{\partial{}}{\partial{p_{1}}}\
.$ (37)
As our goal is to recover the Lagrangian and Hamiltonian solutions from the
vector field $X$, we must require $X$ to be a semispray of type $1$.
Nevertheless, as ${\cal L}$ is a regular Lagrangian function, this condition
is naturally deduced from the formalism, as we have seen in (25).
Notice that the functions $F_{2}$ and $F_{3}$ in (37) are not determined until
the tangency of the vector field $X$ on ${\cal W}_{o}$ is required. Recall
that the Legendre-Ostrogradsky transformation is the map ${\cal FL}\colon{\rm
T}^{3}Q\longrightarrow{\rm T}^{*}({\rm T}Q)$ given in local coordinates by
$\displaystyle{\cal FL}^{*}p^{0}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1}}}-d_{T}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2}}}\right)\equiv\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1}}}-d_{T}\left(p^{1}\right)=q_{1}+\gamma q_{3}$
$\displaystyle{\cal FL}^{*}p^{1}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\partial{{\cal L}}}{\partial{q_{2}}}=-\gamma
q_{2}$
and, as $\gamma\neq 0$, we see that ${\cal L}$ is a regular Lagrangian since
${\cal FL}$ is a (local) diffeomorphism. Then, the submanifold ${\cal
W}_{o}={\rm graph}\,{\cal FL}$ is defined by
${\cal W}_{o}=\left\\{p\in{\cal W}\colon\xi_{0}(p)=\xi_{1}(p)=0\right\\}\ ,$
where $\xi_{r}=p^{r}-{\cal FL}^{*}p^{r}$, $r=1,2$. The diagram for this
situation is
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\cal W}_{o}={\rm graph}\,{\cal
FL}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{o}}$$\scriptstyle{\overline{\operatorname{pr}}_{1}}$$\scriptstyle{\overline{\operatorname{pr}}_{2}}$
$\textstyle{{\rm
T}^{3}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\cal
FL}}$ $\textstyle{{\rm T}^{*}({\rm T}Q)\ .}$
Next we compute the tangency condition for $X\in{\mathfrak{X}}({\cal W})$
given locally by (37) on the submanifold ${\cal W}_{o}\hookrightarrow{\cal
W}$, by checking if the following identities hold
$\mathop{\rm L}\nolimits(X)\xi_{0}|_{{\cal W}_{o}}=0\quad,\quad\mathop{\rm
L}\nolimits(X)\xi_{1}|_{{\cal W}_{o}}=0\ .$
As we have seen in Section 3.2.1, these equations give us the Lagrangian
equations for the vector field $X$; that is, on the points of ${\cal W}_{o}$
we obtain
$\displaystyle\mathop{\rm L}\nolimits(X)\xi_{0}=-\omega^{2}q_{0}-q_{2}-\gamma
F_{3}=0$ (38) $\displaystyle\mathop{\rm
L}\nolimits(X)\xi_{1}=\gamma\left(F_{2}-q_{3}\right)=0\ .$ (39)
Equation (39) gives us the condition of semispray of type $1$ for the vector
field $X$ (recall that $\gamma\neq 0$), and equation (38) is the Euler-
Lagrange equation for the vector field $X$. Notice that, as $\gamma$ is
nonzero, these equations give us a unique solution for $F_{2}$ and $F_{3}$.
Thus, there is a unique vector field $X\in{\mathfrak{X}}({\cal W})$ solution
to the equation $\left.\left[\mathop{i}\nolimits(X)\Omega-{\rm
d}H\right]\right|_{{\cal W}_{o}}=0$ which is tangent to the submanifold ${\cal
W}_{o}\hookrightarrow{\cal W}$, and it is given locally by
$X=q_{1}\displaystyle\frac{\partial{}}{\partial{q_{0}}}+q_{2}\displaystyle\frac{\partial{}}{\partial{q_{1}}}+q_{3}\displaystyle\frac{\partial{}}{\partial{q_{2}}}-\frac{1}{\gamma}\left(\omega^{2}q_{0}+q_{2}\right)\displaystyle\frac{\partial{}}{\partial{q_{3}}}-\omega^{2}q_{0}\displaystyle\frac{\partial{}}{\partial{p^{0}}}+\left(q_{1}-p^{0}\right)\displaystyle\frac{\partial{}}{\partial{p_{1}}}\
.$
Then, if $\sigma\colon\mathbb{R}\to{\cal W}$ is an integral curve of $X$
locally given by
$\sigma(t)=\left(q_{0}(t),q_{1}(t),q_{2}(t),q_{3}(t),p^{0}(t),p^{1}(t)\right)\
,$ (40)
and its component functions are solutions to the system
$\displaystyle\dot{q}_{0}(t)=q_{1}(t);$ (41)
$\displaystyle\dot{q}_{1}(t)=q_{2}(t);$ (42)
$\displaystyle\dot{q}_{2}(t)=q_{3}(t);$ (43)
$\displaystyle\dot{q}_{3}(t)=-\frac{1}{\gamma}\left(\omega^{2}q_{0}(t)+q_{2}(t)\right);$
(44) $\displaystyle\dot{p}^{0}(t)=-\omega^{2}q_{0}(t);$ (45)
$\displaystyle\dot{p}^{1}(t)=q_{1}(t)-p^{0}(t).$ (46)
Finally we recover the Lagrangian and Hamiltonian solutions for this system.
For the Lagrangian solutions, as we have shown in Lemma 3 and Theorem 2, the
Euler-Lagrange vector field is the unique semispray of type $1$, $X_{\cal
L}\in{\mathfrak{X}}({\rm T}^{3}Q)$, such that $X_{\cal
L}\circ\operatorname{pr}_{1}\circ j_{o}={\rm T}\operatorname{pr}_{1}\circ
X\circ j_{o}$. Thus this vector field $X_{\cal L}$ is locally given by
$X_{\cal
L}=q_{1}\displaystyle\frac{\partial{}}{\partial{q_{0}}}+q_{2}\displaystyle\frac{\partial{}}{\partial{q_{1}}}+q_{3}\displaystyle\frac{\partial{}}{\partial{q_{2}}}-\frac{1}{\gamma}\left(\omega^{2}q_{0}+q_{2}\right)\displaystyle\frac{\partial{}}{\partial{q_{3}}}\
.$
For the integral curves of $X_{\cal L}$ we know from Proposition 6 that if
$\sigma\colon\mathbb{R}\to{\cal W}$ is an integral curve of $X$, then
$\sigma_{\cal L}=\operatorname{pr}_{1}\circ\sigma$ is an integral curve of
$X_{\cal L}$. Thus, if $\sigma$ is given locally by (40), then $\sigma_{\cal
L}$ has the following local expression
$\sigma_{\cal L}(t)=\left(q_{0}(t),q_{1}(t),q_{2}(t),q_{3}(t)\right)\ ,$ (47)
and its components satisfy equations (41), (42), (43) and (44). Notice that
equations (41), (42) and (43) state that $\sigma_{\cal L}$ is the canonical
lifting of a curve in the basis, that is, there exists a curve
$\gamma\colon\mathbb{R}\to Q$ such that $\tilde{\gamma}^{3}=\sigma_{\cal L}$.
Furthermore, equation (44) is the Euler-Lagrange equation for this system.
Now, for the Hamiltonian solutions, as ${\cal L}$ is a regular Lagrangian,
Theorem 3 states that there exists a unique vector field $X_{h}={\cal
FL}_{*}X_{\cal L}\in{\mathfrak{X}}({\rm T}^{*}({\rm T}Q))$ which is a solution
to the Hamilton equation. Hence, it is given locally by
$X_{h}=q_{1}\displaystyle\frac{\partial{}}{\partial{q_{0}}}+q_{2}\displaystyle\frac{\partial{}}{\partial{q_{1}}}-\omega^{2}q_{0}\displaystyle\frac{\partial{}}{\partial{p^{0}}}+\left(q_{1}-p^{0}\right)\displaystyle\frac{\partial{}}{\partial{p_{1}}}$
For the integral curves of $X_{h}$, Proposition 7 states that if $\sigma_{\cal
L}\colon\mathbb{R}\to{\rm T}^{3}Q$ is an integral curve of $X_{\cal L}$ coming
from an integral curve $\sigma$ of $X$, then $\sigma_{h}={\cal
FL}\circ\sigma_{\cal L}$ is an integral curve of the vector field $X_{h}$.
Therefore, if $\sigma$ is given locally by (40), then $\sigma_{\cal L}$ is
given by (47) and so $\sigma_{h}$ can be locally written
$\sigma_{h}(t)=\left(q_{0}(t),q_{1}(t),p^{0}(t),p^{1}(t)\right)\ ,$
and its components must satisfy equations (41), (42), (45) and (46). Notice
that these equations are the standard Hamilton equations for this system.
### 4.2 The second-order relativistic particle
Let us consider a relativistic particle whose action is proportional to its
extrinsic curvature. This system was analyzed in [34, 33, 8, 31], and here we
study it using the Lagrangian-Hamiltonian unified formalism.
The configuration space is a $n$-dimensional smooth manifold $Q$ with local
coordinates $(q_{0}^{A})$, $1\leqslant A\leqslant n$. Then, if we take the
natural set of coordinates on the higher-order tangent bundles over $Q$, the
second-order Lagrangian function for this system, ${\cal L}\in{\rm
C}^{\infty}({\rm T}^{2}Q)$, can be written locally as
${\cal
L}(q_{0}^{i},q_{1}^{i},q_{2}^{i})=\frac{\alpha}{(q_{1}^{i})^{2}}\left[(q_{1}^{i})^{2}(q_{2}^{i})^{2}-(q_{1}^{i}q_{2}^{i})^{2}\right]^{1/2}\equiv\frac{\alpha}{(q_{1}^{i})^{2}}\sqrt{g}\
.$ (48)
where $\alpha$ is some nonzero constant. It is a singular Lagrangian, as we
can see by computing the Hessian matrix of ${\cal L}$ with respect to
$q_{2}^{A}$, which is
$\left(\frac{\partial^{2}{\cal L}}{\partial q_{2}^{B}\partial
q_{2}^{A}}\right)=\begin{cases}\displaystyle\frac{\alpha}{2(q_{1}^{i})^{2}\sqrt{g^{3}}}\left[\left((q_{1}^{i}q_{2}^{i})^{2}-2(q_{1}^{i})^{2}(q_{2}^{i})^{2}\right)q_{1}^{B}q_{1}^{A}\right.+(q_{1}^{i})^{2}(q_{1}^{i}q_{2}^{i})(q_{2}^{B}q_{1}^{A}-q_{1}^{B}q_{2}^{A})&\\\
\displaystyle\qquad\qquad\quad\left.-(q_{1}^{i})^{2}(q_{2}^{i})^{2}q_{2}^{B}q_{2}^{A}\right]&\mbox{
if }B\neq A\\\
\displaystyle\frac{\alpha}{\sqrt{g^{3}}}\left[g-(q_{2}^{i})^{2}q_{1}^{A}q_{1}^{A}+2(q_{1}^{i}q_{2}^{i})q_{1}^{A}q_{2}^{A}-(q_{1}^{i})^{2}q_{2}^{A}q_{2}^{A}\right]&\mbox{
if }B=A\ ,\end{cases}$
then after a long calculation we obtain that
$\displaystyle\det\left(\frac{\partial^{2}{\cal L}}{\partial q_{2}^{B}\partial
q_{2}^{A}}\right)=0$. In particular, ${\cal L}$ is an almost-regular
Lagrangian.
As this is a second-order dynamical system, the phase space that we consider
is
$\textstyle{{\cal W}={\rm T}^{3}Q\times_{{\rm T}Q}{\rm T}^{*}({\rm
T}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\rm
T}^{3}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{3}_{1}}$
$\textstyle{{\rm T}^{*}({\rm
T}Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\rm
T}Q}}$ $\textstyle{{\rm T}Q}$$\textstyle{\ .}$
As ${\cal L}$ is almost-regular, the “natural” phase space for this system
would be ${\rm T}^{3}Q\times_{{\rm T}Q}P_{o}$, where $P_{o}\hookrightarrow{\rm
T}^{*}({\rm T}Q)$ denotes the image of the Legendre-Ostrogradsky map. However,
as we have a set of natural coordinates defined in ${\cal W}$, it is easier to
work in ${\cal W}$ and then to obtain the constraints as a consequence of the
formalism.
If $\omega_{1}\in{\mit\Omega}^{2}({\rm T}^{*}({\rm T}Q))$ is the canonical
symplectic form, we define the presymplectic form
$\Omega=\operatorname{pr}_{2}^{*}\omega_{1}\in{\mit\Omega}^{2}({\cal W})$,
whose local expression is
$\Omega=dq_{0}^{i}\wedge dp_{i}^{0}+dq_{1}^{i}\wedge dp_{i}^{1}\ .$
The Hamiltonian function $H\in{\rm C}^{\infty}({\cal W})$ is $H={\cal
C}-(\rho^{3}_{2}\circ\operatorname{pr}_{1})^{*}{\cal L}$, where ${\cal C}$ is
the coupling function, whose local expression is ${\cal
C}\left(q_{0}^{i},q_{1}^{i},q_{2}^{i},q_{3}^{i},p_{i}^{0},p_{i}^{1}\right)=p_{i}^{0}q_{1}^{i}+p_{i}^{1}q_{2}^{i}$,
and then the Hamiltonian function can be written locally
$H\left(q_{0}^{i},q_{1}^{i},q_{2}^{i},q_{3}^{i},p_{i}^{0},p_{i}^{1}\right)=p_{i}^{0}q_{1}^{i}+p_{i}^{1}q_{2}^{i}-\frac{\alpha}{(q_{1}^{i})^{2}}\left[(q_{1}^{i})^{2}(q_{2}^{i})^{2}-(q_{1}^{i}q_{2}^{i})^{2}\right]^{1/2}\
.$
The dynamics for this system are described as the integral curves of vector
fields $X\in{\mathfrak{X}}({\cal W})$ which are solutions to equation (18). If
we take a generic vector field $X\in{\mathfrak{X}}({\cal W})$, given locally
by
$X=f_{0}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+f_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+F_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2}^{A}}}+F_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{3}^{A}}}+G_{A}^{0}\displaystyle\frac{\partial{}}{\partial{p_{A}^{0}}}+G_{A}^{1}\displaystyle\frac{\partial{}}{\partial{p_{A}^{1}}}\
,$
taking into account that
$\displaystyle{\rm d}H$ $\displaystyle=$ $\displaystyle\displaystyle
q_{1}^{A}{\rm d}p_{A}^{0}+q_{2}^{A}{\rm
d}p_{A}^{1}+\left[p^{0}_{A}+\frac{\alpha}{((q_{1}^{i})^{2})^{2}\sqrt{g}}\left[\left((q_{1}^{i})^{2}(q_{2}^{i})^{2}-2(q_{1}^{i}q_{2}^{i})^{2}\right)q_{1}^{A}+(q_{1}^{i}q_{2}^{i})(q_{1}^{i})^{2}q_{2}^{A}\right]\right]{\rm
d}q_{1}^{A}$
$\displaystyle+\left[p_{A}^{1}-\frac{\alpha}{(q_{1}^{i})^{2}\sqrt{g}}\left((q^{i}_{1})^{2}q_{2}^{A}-(q_{1}^{i}q_{2}^{i})q_{1}^{A}\right)\right]{\rm
d}q_{2}^{A}\ ;$
from the dynamical equation we obtain the following linear systems for the
coefficients of $X$
$\displaystyle f_{0}^{A}=q_{1}^{A}$ (49) $\displaystyle f_{1}^{A}=q_{2}^{A}$
(50) $\displaystyle G_{A}^{0}=0$ (51) $\displaystyle
G_{A}^{1}=-p^{0}_{A}-\frac{\alpha}{((q_{1}^{i})^{2})^{2}\sqrt{g}}\left[\left((q_{1}^{i})^{2}(q_{2}^{i})^{2}-2(q_{1}^{i}q_{2}^{i})^{2}\right)q_{1}^{A}+(q_{1}^{i}q_{2}^{i})(q_{1}^{i})^{2}q_{2}^{A}\right]$
(52) $\displaystyle
p_{A}^{1}-\frac{\alpha}{(q_{1}^{i})^{2}\sqrt{g}}\left((q^{i}_{1})^{2}q_{2}^{A}-(q_{1}^{i}q_{2}^{i})q_{1}^{A}\right)=0\
.$ (53)
Note that from equations (49) and (50) we obtain the condition of semispray of
type $2$ for $X$. Furthermore, equations (53) are algebraic relations between
the coordinates in ${\cal W}$ stating that the vector field $X$ is defined
along a submanifold ${\cal W}_{o}$ that is identified with the graph of the
Legendre-Ostrogradsky map, as we stated in Propositions 2 and 3. Thus, the
vector field $X$ is given locally by
$X=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+F_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2}^{A}}}+F_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{3}^{A}}}+G_{A}^{1}\displaystyle\frac{\partial{}}{\partial{p_{A}^{1}}}\
,$ (54)
where the functions $G_{A}^{1}$ are determined by (52). As we want to recover
the Lagrangian solutions from the vector field $X$, we must require $X$ to be
a semispray of type $1$. This condition reduces the set of vector fields
$X\in{\mathfrak{X}}({\cal W})$ given by (54) to the following ones
$X=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+q_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2}^{A}}}+F_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{3}^{A}}}+G_{A}^{1}\displaystyle\frac{\partial{}}{\partial{p_{A}^{1}}}\
.$ (55)
Notice that the functions $F_{3}^{A}$ are not determinated until the tangency
of the vector field $X$ on ${\cal W}_{o}$ is required. Now, the Legendre-
Ostrogradsky transformation is the map ${\cal FL}\colon{\rm
T}^{3}Q\longrightarrow{\rm T}^{*}({\rm T}Q)$ locally given by
$\displaystyle{\cal FL}^{*}(p^{0}_{A})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1}^{A}}}-d_{T}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2}^{A}}}\right)\equiv\displaystyle\frac{\partial{{\cal
L}}}{\partial{q^{A}_{1}}}-d_{T}\left(p_{A}^{1}\right)=$
$\displaystyle\frac{\alpha}{(q_{1}^{i})^{2}\sqrt{g^{3}}}\left[\left((q_{2}^{i})^{2}g+(q_{1}^{i})^{2}(q_{2}^{i})^{2}(q_{1}^{i}q_{3}^{i})-(q_{1}^{i})^{2}(q_{1}^{i}q_{2}^{i})(q_{2}^{i}q_{3}^{i})\right)q_{1}^{A}\right]+$
$\displaystyle\frac{\alpha}{(q_{1}^{i})^{2}\sqrt{g^{3}}}\left[\left(((q_{1}^{i})^{2})^{2}(q_{2}^{i}q_{3}^{i})-(q_{1}^{i})^{2}(q_{1}^{i}q_{2}^{i})(q_{1}^{i}q_{3}^{i})-(q_{1}^{i}q_{2}^{i})g\right)q_{2}^{A}-(q_{1}^{i})^{2}gq_{3}^{A}\right]$
$\displaystyle{\cal FL}^{*}(p^{1}_{A})$ $\displaystyle=$
$\displaystyle\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2}^{A}}}=\frac{\alpha}{(q^{i}_{1})^{2}\sqrt{g}}\left[(q^{i}_{1})^{2}q_{2}^{A}-(q^{i}_{1}q^{i}_{2})q^{A}_{1}\right]\
,$
and, in fact, ${\cal L}$ is an almost-regular Lagrangian. Thus, from the
expression in local coordinates of the map ${\cal FL}$, we obtain the
(primary) constraints that define the closed submanifold $P_{o}={\rm
Im}\,{\cal FL}$, which are
$\phi^{(0)}_{1}\equiv
p^{1}_{i}q_{1}^{i}=0\quad;\quad\phi^{(0)}_{2}\equiv(p_{i}^{1})^{2}-\frac{\alpha^{2}}{(q^{i}_{1})^{2}}=0\
;$ (56)
Let ${\cal FL}_{o}\colon{\rm T}^{3}Q\to P_{o}$. Then, the submanifold ${\cal
W}_{o}={\rm graph}\,{\cal FL}_{o}$ is defined by
${\cal W}_{o}=\left\\{p\in{\cal W}\ \colon\
\xi^{A}_{0}(p)=\xi^{A}_{1}(p)=\phi^{(0)}_{1}(p)=\phi^{(0)}_{2}(p)=0,\
1\leqslant A\leqslant\dim Q\right\\}$
where $\xi_{r}^{A}\equiv p_{A}^{r}-{\cal FL}^{*}p_{A}^{r}$. The diagram for
this situation is
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\cal W}_{P_{o}}={\rm T}^{3}Q\times_{{\rm
T}Q}P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{1,{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{2,{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{2,P_{o}}}$ $\textstyle{{\rm
T}^{3}Q}$ $\textstyle{{\rm T}^{*}({\rm T}Q)}$ $\textstyle{{\cal W}_{o}={\rm
graph}\,{\cal
FL}_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{o}}$$\scriptstyle{\overline{\operatorname{pr}}_{1,P_{o}}}$$\scriptstyle{\overline{\operatorname{pr}}_{2,P_{o}}}$$\textstyle{P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{o}}}$
Notice that ${\cal W}_{o}$ is a submanifold of ${\rm T}^{3}Q\times_{{\rm
T}Q}P_{o}$, and that ${\cal W}_{o}$ is the real phase space of the system,
where the dynamics take place.
Next we compute the tangency condition for $X\in{\mathfrak{X}}({\cal W})$
given locally by (55) on the submanifold ${\cal W}_{o}\hookrightarrow{\cal
W}_{P_{o}}\hookrightarrow{\cal W}$, by checking if the following identities
hold
$\displaystyle\mathop{\rm L}\nolimits(X)\xi^{A}_{0}|_{{\cal W}_{o}}=0$
$\displaystyle\ ,\ $ $\displaystyle\mathop{\rm
L}\nolimits(X)\xi^{A}_{1}|_{{\cal W}_{o}}=0$ (57) $\displaystyle\mathop{\rm
L}\nolimits(X)\phi^{(0)}_{1}|_{{\cal W}_{o}}=0$ $\displaystyle\ ,\ $
$\displaystyle\mathop{\rm L}\nolimits(X)\phi^{(0)}_{2}|_{{\cal W}_{o}}=0\ .$
(58)
As we have seen in Section 3.2.1, equations (57) give us the Lagrangian
equations for the vector field $X$. However, equations (58) do not hold since
$\mathop{\rm L}\nolimits(X)\phi^{(0)}_{1}=\mathop{\rm
L}\nolimits(X)(p^{1}_{i}q_{1}^{i})=-p_{i}^{0}q_{1}^{i}\quad,\quad\mathop{\rm
L}\nolimits(X)\phi^{(0)}_{2}=\mathop{\rm
L}\nolimits(X)((p^{1}_{i})^{2}-\alpha^{2}/(q_{1}^{i})^{2})=-2p_{i}^{0}q_{i}^{1}\
,$
and hence we obtain two first-generation secondary constraints
$\phi^{(1)}_{1}\equiv p_{i}^{0}q_{1}^{i}=0\quad,\quad\phi^{(1)}_{2}\equiv
p_{i}^{0}p_{i}^{1}=0$ (59)
that define a new submanifold ${\cal W}_{1}\hookrightarrow{\cal W}_{o}$. Now,
checking the tangency of the vector field $X$ to this new submanifold, we
obtain
$\mathop{\rm L}\nolimits(X)\phi^{(1)}_{1}=\mathop{\rm
L}\nolimits(X)(p_{i}^{0}q_{1}^{i})=0\quad,\quad\mathop{\rm
L}\nolimits(X)\phi^{(1)}_{2}=\mathop{\rm
L}\nolimits(X)(p_{i}^{0}p_{i}^{1})=-(p^{0}_{i})^{2}\ ,$
and a second-generation secondary constraint appears
$\phi^{(2)}\equiv(p^{0}_{i})^{2}=0\ ,$ (60)
which defines a new submanifold ${\cal W}_{2}\hookrightarrow{\cal W}_{1}$.
Finally, the tangency of the vector field $X$ on this submanifold gives no new
constraints, since
$\mathop{\rm L}\nolimits(X)\phi^{(2)}=\mathop{\rm
L}\nolimits(X)((p^{0}_{i})^{2})=0\ .$
So we have two primary constraints (56), two first-generation secondary
constraints (59), and a single second-generation secondary constraint (60).
Notice that these five constraints only depend on $q_{1}^{A}$, $p^{0}_{A}$ and
$p^{1}_{A}$, and so they are $\operatorname{pr}_{2}$-projectable. Thus, we
have the following diagram
$\textstyle{{\cal
W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{1}}$$\scriptstyle{\operatorname{pr}_{2}}$
$\textstyle{{\cal
W}_{P_{o}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{1,{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{2,{\cal
W}_{P_{o}}}}$$\scriptstyle{\operatorname{pr}_{2,P_{o}}}$ $\textstyle{{\rm
T}^{3}Q}$ $\textstyle{{\rm T}^{*}({\rm
T}Q)}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{S_{1}}}$
$\textstyle{{\cal
W}_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{o}}$$\scriptstyle{\overline{\operatorname{pr}}_{1,P_{o}}}$$\scriptstyle{\overline{\operatorname{pr}}_{2,P_{o}}}$
$\textstyle{P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{o}}}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{S_{2}}}$
$\textstyle{{\cal
W}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{1}}$$\scriptstyle{\overline{\operatorname{pr}}_{1,P_{1}}}$$\scriptstyle{\overline{\operatorname{pr}}_{2,P_{1}}}$
$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{1}}}$
$\textstyle{{\cal
W}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{2}}$$\scriptstyle{\overline{\operatorname{pr}}_{1,P_{2}}}$$\scriptstyle{\overline{\operatorname{pr}}_{2,P_{2}}}$
$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{2}}}$
where
$\displaystyle P_{1}=\left\\{p\in
P_{o}\colon\phi^{(1)}_{1}(p)=\phi^{(1)}_{2}(p)=0\right\\}=\operatorname{pr}_{2}({\cal
W}_{1})$ $\displaystyle P_{2}=\left\\{p\in
P_{o}\colon\phi^{(2)}(p)=0\right\\}=\operatorname{pr}_{2}({\cal W}_{2})$
$\displaystyle S_{1}={\cal FL}_{o}^{-1}(P_{1})=\operatorname{pr}_{1}({\cal
W}_{1})$ $\displaystyle S_{2}={\cal
FL}_{o}^{-1}(P_{2})=\operatorname{pr}_{1}({\cal W}_{2})\ .$
Focusing only on the Legendre-Ostrogradsky map, and ignoring the unified part
of the diagram, we have
$\textstyle{{\rm
T}^{3}Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\cal
FL}}$$\scriptstyle{{\cal FL}_{o}}$ $\textstyle{{\rm T}^{*}({\rm
T}Q)}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{S_{1}}}$$\scriptstyle{{\cal
FL}_{o}}$
$\textstyle{P_{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{o}}}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{S_{2}}}$$\scriptstyle{{\cal
FL}_{o}}$
$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{1}}}$
$\textstyle{P_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{P_{2}}}$
Notice that we still have to check (57). As we have seen in Section 3.2.1, we
will obtain the following equations
$\displaystyle\left(F_{3}^{B}-d_{T}\left(q_{3}^{B}\right)\right)\frac{\partial^{2}{\cal
L}}{\partial q_{2}^{B}\partial q_{2}^{A}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}-d_{T}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1}^{A}}}\right)+d_{T}^{2}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2}^{A}}}\right)+\left(F_{2}^{B}-q_{3}^{B}\right)d_{T}\left(\frac{\partial^{2}{\cal
L}}{\partial q_{2}^{B}\partial q_{2}^{A}}\right)=0$ (61)
$\displaystyle\left(F_{2}^{B}-q_{3}^{B}\right)\frac{\partial^{2}{\cal
L}}{\partial q_{2}^{B}\partial q_{2}^{A}}=0$ (62)
As we have already required the vector field $X$ to be a semispray of type
$1$, equations (62) are satisfied identically and equations (61) become
$\left(F_{3}^{B}-d_{T}\left(q_{3}^{B}\right)\right)\frac{\partial^{2}{\cal
L}}{\partial q_{2}^{B}\partial q_{2}^{A}}+\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{0}^{A}}}-d_{T}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{1}^{A}}}\right)+d_{T}^{2}\left(\displaystyle\frac{\partial{{\cal
L}}}{\partial{q_{2}^{A}}}\right)=0\ .$ (63)
A long calculation shows that this equation is compatible and so no new
constraints arise. Thus, we have no Lagrangian constraint appearing from the
semispray condition. If some constraint had appeared, it would not be ${\cal
FL}_{o}$-projectable (see [24])
Thus, the vector fields $X\in{\mathfrak{X}}({\cal W})$ given locally by (55)
which are solutions to the equation
$\left.\left[\mathop{i}\nolimits(X)\Omega-{\rm d}H\right]\right|_{{\cal
W}_{2}}=0\ ,$
are tangent to the submanifold ${\cal W}_{2}\hookrightarrow{\cal W}_{o}$.
Therefore, taking the vector fields $X_{o}\in{\mathfrak{X}}({\cal W}_{2})$
such that ${\rm T}j_{2}\circ X_{o}=X\circ j_{2}$, the form
$\Omega_{o}=(j_{{\cal W}_{P_{o}}}\circ j_{o}\circ j_{1}\circ
j_{2})^{*}\Omega$, and the canonical Hamiltonian function $H_{o}=(j_{{\cal
W}_{P_{o}}}\circ j_{o}\circ j_{1}\circ j_{2})^{*}H$, the above equation leads
to
$\mathop{i}\nolimits(X_{o})\Omega_{o}-{\rm d}H_{o}=0\ ,$
but a simple calculation in local coordinates shows that $H_{o}=0$, and thus
the last equation becomes $\mathop{i}\nolimits(X_{o})\Omega_{o}=0$.
One can easily check that, if the semispray condition is not required at the
beginning and we perform all this procedure with the vector field given by
(54), the final result is the same. This means that, in this case, the
semispray condition does not give any additional constraint.
As final results, we recover the Lagrangian and Hamiltonian vector fields from
the vector field $X\in{\mathfrak{X}}({\cal W})$. For the Lagrangian vector
field, by using Lemma 3 and Theorem 2 we obtain a semispray of type $2$,
$X_{\cal L}\in{\mathfrak{X}}({\rm T}^{3}Q)$, tangent to $S_{2}$. Thus,
requiring the condition of semispray of type $1$ to be satisfied (perhaps on
another submanifold $M_{2}\hookrightarrow S_{2}$), the local expression for
the vector field $X_{\cal L}$ is
$X_{\cal
L}=q_{1}^{A}\displaystyle\frac{\partial{}}{\partial{q_{0}^{A}}}+q_{2}^{A}\displaystyle\frac{\partial{}}{\partial{q_{1}^{A}}}+q_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{2}^{A}}}+F_{3}^{A}\displaystyle\frac{\partial{}}{\partial{q_{3}^{A}}}\
.$
where the functions $F_{3}^{A}$ are determined by (63). For the Hamiltonian
vector fields, recall that ${\cal L}$ is an almost-regular Lagrangian
function. Thus, we know that there are Euler-Lagrange vector fields which are
${\cal FL}_{o}$-projectable on $P_{2}$, tangent to $P_{2}$ and solutions to
the Hamilton equation.
## 5 Conclusions and outlook
After introducing the natural geometric structures needed for describing
higher-order autonomous dynamical systems, we review their Lagrangian and
Hamiltonian formalisms, following the exposition made in [17].
The main contribution of this work is that we develop the Lagrangian-
Hamiltonian unified formalism for higher-order dynamical systems, following
the ideas of the original article [39]. We pay special attention to showing
how the Lagrangian and Hamiltonian dynamics are recovered from this, both for
regular and singular systems.
A first consideration is to discuss the fundamental differences between the
first-order and the higher-order unified Lagrangian-Hamiltonian formalisms. In
particular:
* •
As there is no canonical pairing between the elements of ${\rm T}^{2k-1}_{q}Q$
and of ${\rm T}^{*}_{q}({\rm T}^{k-1}Q)$, in order to define the higher-order
coupling function ${\cal C}$ in an intrinsic way, we use the canonical
injection that transforms a point in ${\rm T}^{2k-1}Q$ into a tangent vector
along ${\rm T}^{k-1}Q$.
* •
When the equations that define the Legendre-Ostrogradsky map are recovered
from the unified formalism (both in the characterization of the compatibility
submanifold ${\cal W}_{o}$ as the graph of ${\cal FL}$, and in the equations
in local coordinates of the vector field $X\in{\mathfrak{X}}({\cal W})$
solution to the dynamical equations), the only equations that are recovered
are those that define the highest order momentum coordinates, and the
remaining equations that define the map must be recovered using the relations
between the momentum coordinates.
* •
The regularity of the Lagrangian function is more relevant in the higher-order
case, because the condition of semispray of type $1$ (the holonomy condition)
of the Lagrangian vector field cannot be deduced from the dynamical equations
if the Lagrangian is singular, unlike the first-order case, where this
holonomy condition is deduced straightforwardly from the equations
independently of the regularity of the Lagrangian function. When the
Lagrangian is singular, we can only ensure that the Lagrangian vector field is
a semispray of type $k$. It is therefore necessary, in general, to require the
condition of semispray of type $1$ as an additional condition.
Then, for regular Lagrangian systems, when the tangency condition of the
vector field $X\in{\mathfrak{X}}({\cal W})$ solution in the unified formalism
along the submanifold ${\cal W}_{o}$ is required, we obtain not only the
Euler-Lagrange equations for the vector field, but also the remaining $k-1$
systems of equations that the vector field must satisfy to be a semispray of
type $1$.
As we point out in the introduction, a previous and quick presentation of a
unified formalism for higher-order systems was outlined in [13]. Our formalism
differs from this one, since in that article the authors take ${\rm
T}^{k}Q\oplus_{{\rm T}^{k-1}Q}{\rm T}^{*}({\rm T}^{k-1}Q)$ as the phase space
in the unified formalism, instead of ours, which is ${\rm
T}^{2k-1}Q\oplus_{{\rm T}^{k-1}Q}{\rm T}^{*}({\rm T}^{k-1}Q)$. This is a
significant difference, since when we want to recover the dynamical solutions
of the Lagrangian formalism from the unified formalism, the Lagrangian phase
space is ${\rm T}^{2k-1}Q$, instead of ${\rm T}^{k}Q$, which is the bundle
where the Lagrangian function is defined. This fact makes it more natural to
obtain the Lagrangian dynamics as well as the Hamiltonian dynamics, which in
turn is obtained from the Lagrangian one using the Legendre-Ostrogradsky map.
By using any suitable generalization of some of the several formalisms for
first-order non-autonomous dynamical systems [1, 6, 19], a future avenue of
research consists in generalizing this unified formalism for higher-order non-
autonomous dynamical systems. This generalization should also be recovered as
a particular case of the corresponding unified formalism for higher-order
classical field theories. As regards this topic, a proposal for a unified
formalism for higher-order classical field theories has recently been made
[10, 41], which is based on the model presented in [13]. This formulation
allows us to improve some previous models for describing the Lagrangian and
Hamiltonian formalisms. Nevertheless, some ambiguities arise when considering
the solutions of the field equations. We hope that a suitable extension of our
formalism to field theories will enable these difficulties to be overcome and
complete the model given in [10, 41].
## Acknowledgments
We acknowledge the financial support of the Ministerio de Ciencia e Innovación
(Spain), projects MTM2008-00689 and MTM2009-08166-E. We also thank Mr. Jeff
Palmer for his assistance in preparing the English version of the manuscript.
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|
arxiv-papers
| 2011-06-16T16:07:30 |
2024-09-04T02:49:19.706712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pedro D. Prieto-Mart\\'inez, Narciso Rom\\'an-Roy",
"submitter": "Pedro D. Prieto-Mart\\'inez",
"url": "https://arxiv.org/abs/1106.3261"
}
|
1106.3270
|
# Chandra Phase-Resolved X-ray Spectroscopy of the Crab Pulsar II
Martin C. Weisskopf 11affiliation: Space Sciences Department, NASA Marshall
Space Flight Center, VP62, Huntsville, AL 35812 , Allyn F. Tennant
11affiliation: Space Sciences Department, NASA Marshall Space Flight Center,
VP62, Huntsville, AL 35812 , Dmitry G. Yakovlev 22affiliation: Ioffe Physical
Technical Institute, Politechnischeskaya 26, 194021 St. Petersburg, Russia &
St. Petersburg Polytechnical University, Politechnischeskaya 29, 195251 St.
Petersburg, Russia , Alice Harding 33affiliation: NASA Goddard Space Flight
Center, 8800, College Park, MD 20771 , Vyacheslav E. Zavlin 44affiliation:
Universities Space Research Association, NASA Marshall Space Flight Center,
VP62, Huntsville, AL 35812 , Stephen L. O’Dell 11affiliation: Space Sciences
Department, NASA Marshall Space Flight Center, VP62, Huntsville, AL 35812 ,
Ronald F. Elsner 11affiliation: Space Sciences Department, NASA Marshall Space
Flight Center, VP62, Huntsville, AL 35812 , and Werner Becker 55affiliation:
Max Planck Institut für Extraterrestrische Physik, 85740 Garching bei München,
Germany
###### Abstract
We present a new study of the X-ray spectral properties of the Crab Pulsar.
The superb angular resolution of the Chandra X-ray Observatory enables
distinguishing the pulsar from the surrounding nebulosity. Analysis of the
spectrum as a function of pulse phase allows the least-biased measure of
interstellar X-ray extinction due primarily to photoelectric absorption and
secondarily to scattering by dust grains in the direction of the Crab Nebula.
We modify previous findings that the line-of-sight to the Crab is under-
abundant in oxygen and provide measurements with improved accuracy and less
bias. Using the abundances and cross sections from Wilms, Allen & McCray
(2000) we find [O/H] = $(5.28\pm 0.28)\times 10^{-4}$ ($4.9\times 10^{-4}$ is
solar abundance). We also measure for the first time the impact of scattering
of flux out of the image by interstellar grains. We find $\tau_{\rm
scat}=0.147\pm 0.043$. Analysis of the spectrum as a function of pulse phase
also measures the X-ray spectral index even at pulse minimum — albeit with
increasing statistical uncertainty. The spectral variations are, by and large,
consistent with a sinusoidal variation. The only significant variation from
the sinusoid occurs over the same phase range as some rather abrupt behavior
in the optical polarization magnitude and position angle. We compare these
spectral variations to those observed in Gamma-rays and conclude that our
measurements are both a challenge and a guide to future modeling and will thus
eventually help us understand pair cascade processes in pulsar magnetospheres.
The data were also used to set new, and less biased, upper limits to the
surface temperature of the neutron star for different models of the neutron
star atmosphere. We discuss how such data are best connected to theoretical
models of neutron star cooling and neutron star interiors. The data restrict
the neutrino emission rate in the pulsar core and the amount of light elements
in the heat-blanketing envelope. The observations allow the pulsar,
irrespective of the composition of its envelope, to have a neutrino emission
rate higher than 1/6 of the standard rate of a non-superfluid star cooling via
the modified Urca process. The observations also allow the rate to be lower
but now with a limited amount of accreted matter in the envelope.
atomic processes — ISM: general — stars: individual: Crab Nebula — techniques:
spectroscopic — X-rays: stars
## 1 Introduction
In Weisskopf et al. (2004) — hereafter Paper I— we presented the first
Chandra-LETGS (Low Energy Transmission Grating Spectrometer) phase-averaged
and phase-resolved X-ray spectroscopy of the Crab Pulsar. In that paper we set
an upper limit to the thermal emission from the neutron star’s surface,
essentially unblemished by any contaminating signal from the pulsar’s wind
nebula. Here we present the results of new Chandra observations that made use
of a High Resolution Camera (HRC) shutter. The HRC serves as the readout for
the LETGS. Using the shutter, together with a factor of two for increased
observing time, allows for more high-time resolution data over our previous
observation by an order of magnitude and thus more meaningful phase-resolved
spectroscopy.
After briefly describing the observation and data reduction (§2), we discuss
the analysis of the measured spectra (§3). We update the work of Paper I
regarding the interstellar abundances in the line of sight to the pulsar and
other relevant parameters (§3.1). We do not repeat the discussion in Paper I
concerning the impacts on the spectroscopy from using different scattering
coefficients and cross-sections, nor do we repeat the comparison of our
results with certain other previous measurements. We present (§3.2) new, more
precise, measurements of the variation of the non-thermal spectral parameters
with pulse phase and the implications. We discuss constraints on the surface
temperature of the underlying neutron star (§3.3) assuming two different
models for the thermal emission and fitting the data as a function of pulse
phase allowing the power law component to vary. This approach yields a less-
biased approach to measuring, or setting upper limits to, the thermal spectral
parameters. We discuss ramifications of the temperature measurements (§3.3.3)
and summarize our findings (§4).
## 2 Observations and Data Reduction
In this paper we make use of data (ObsID 9765) we obtained in 2008, January.
As with our previous observation (Paper I) these data were taken using
Chandra’s Low-Energy Transmission Grating (LETG) and High-Resolution Camera
spectroscopy detector (HRC-S) — the LETGS. For this observation, however, we
also inserted one of the HRC’s shutters to occult most of the positive order,
a significant fraction of the zero order, and some of the negative order flux.
This was done deliberately to prevent the total counting rate from exceeding
the telemetry limit111http://asc.harvard.edu/proposer/POG/. Not exceeding the
telemetry limit, in turn, allows us to make full use of the HRC time
resolution without having to employ the severe filtering described in Paper I
which dramatically reduced the amount of detected events with time resolution
good enough to resolve the light curve of the pulsar. In addition, and also to
keep the telemetry rate low, the two outer HRC-S detectors were turned off.
We processed all data using Chandra X-ray Center (CXC) tools. Level 2 event
files were created using the CIAO script hrc_process_events with pixel
randomization off and CALDB 4.4.1. The program axbary was used to covert the
time of arrival of events to the solar system’s barycenter. Using LEXTRCT
(developed by one of us, AFT) we extracted the pulsar’s dispersed spectrum
from the images. The extraction used a $\pm 12.348$-pixel-wide (cross-
dispersive) region centered on the pulsar, in 2-pixel (dispersive) increments.
This extraction width is the standard extraction region for which the CIAO-
derived response functions apply for the central HRC-S detector. For the
LETG’s 0.9912-$\mu$m grating period and 8.638-m Rowland-circle radius, the
LETGS dispersion is 1.148 Å/mm. Consequently, the spectral resolution of the
binned data (two 6.4294-$\mu$m HRC-S pixels) is 0.01476 Å. Because of the
occultation of the HRC shutter, we only considered the minus-order dispersed
spectrum.
The occultation is a geometrical effect with the shutter occulting
progressively less and less of the negative order with the degree of
obscuration varying from about 75% at 3.8 keV to only about 10% at 0.3 keV.
The response function is the standard CIAO response but applying a correction
factor simply determined from the ratio of the minus-order ObsID 759 (Paper I)
flux to that seen in the ObsID 9765 data to account for the effect of
shuttering.
We restricted all data analysis to the (first-order) energy range 0.3 to 3.8
keV. The upper spectral limit avoids contamination from the zeroth-order
nebular image (see Fig. 1 in Paper I); the lower limit is placed at a level
where the first order flux effectively goes to zero.
Selecting the appropriate region of the data to be used as the background
estimator requires some care as one must deal with the dispersed flux from the
nebula which includes such bright features as the inner ring (Weisskopf et al.
2000). To determine the background we studied the flux at the deepest portion
of pulse minimum (phase range $0.65$ to $0.70$, Fig. 3.2) in the spectral
region corresponding to 0.3 to 3.8 keV in first order and projected onto the
cross-dispersion axis (Fig. 2). We chose pulse minimum for this study to avoid
having the pulsar dominate the projected data. Based on the information shown
in Fig. 2, and to estimate the background, we extract data from two 15-HRC-
pixel-wide bands starting 15 pixels to either side of the pulsar’s dispersed
image. Note that just beyond (more negative) $-15$ pixels in Fig. 2 there is a
slight rise traceable to the dispersed spectrum of features near the pulsar,
showing that extending the background region much wider would be a mistake. Of
course the asymmetric nature of the background in the cross-dispersion
direction due to features in the nebula makes the selection of regions to
either side of the dispersed spectrum a necessity, so that the average
provides an accurate background estimate.
The projected flux in the negative order from 0.3 to 3.8 keV versus position
in the cross-dispersion direction. These data cover the pulse-phase range
0.65$-$0.70. The two vertical lines indicate the regions selected for
background estimation. The sloping line (the fit to the background) indicates
that this choice of regions symmetrically, and accurately, estimates the
underlying background in the dispersed pulsar spectrum.
## 3 Analysis and Results
As in Paper I, whenever we fit data to a particular spectral model we allow
for interstellar absorption using cross-sections from vern, Verner et al.
(1996), and interstellar absorption tbvarabs, Wilms, Allen & McCray (2000,
references therein), allowing for absorption by interstellar grains. In
addition, owing to the small effective aperture of our observation of the
pulsar, we also allow for the effects of diffractive scattering by grains on
the interstellar extinction using a formula based on the Rayleigh–Gans
approximation (van der Hulst 1957; Overbeck 1965; Hayakawa 1970), valid when
the phase shift through a grain diameter is small. Details may be found in
Paper I. We analyzed the data using the XSPEC (v.11.3.2) spectral-fitting
package (Arnaud 1996). To ensure applicability of the $\chi^{2}$ statistic, we
merged spectral bins as needed to obtain at least 100 counts per fitting bin
(before background subtraction). Response functions were first generated using
the CIAO threads mkgarf and mkgrmf and then corrected for the occultation of
the shutter blade. The effective areas were generated using standard CIAO
tools and CALDB 4.4.3
### 3.1 Phase-Averaged Spectrum
We present here the results of our analysis of the phase-averaged spectrum,
although we will not use this spectrum to search for an additional thermal
component nor to measure phase-independent parameters. These tasks are more
appropriately accomplished using the phase-resolved spectrum (§ 3.2), a point
also emphasized by Jackson & Halpern (2005).
Fitting the phase-averaged data to a power law spectrum and allowing for
interstellar absorption, a variation in the abundance ratio of oxygen to
hydrogen, and small-angle scattering by intervening dust give an excellent
fit, $\chi^{2}$ of $1795$ for $1810$ degrees-of-freedom ($\nu$). The best-fit
spectrum is shown in Fig. 3.1.
In Table 1 we list the best-fit parameters, but now fix the dust scattering
factoring as was done in Paper I to make comparisons. As a convenience for the
reader, the first line in the table repeats the results from Paper I. The
second line shows how these results change using the more recent response
function, background extraction regions, etc. The reader will note that there
are differences between line 1 and 2 of the Table that are beginning to
approach statistical significance, primarily the change in the hydrogen
abundance is lower (3.68 as compared to 4.20). The differences are mostly a
consequence of the evolution of the calibration of the LETGS effective area.
The differences between the results for the same data (lines 1 and 2 of Table
1) emphasize that we have ignored (as many X-ray astronomers do) the
possibility that there are any errors in the response functions! The final
comparison is between our current results (line 3 of Table 1) and the updated
results for ObsID 759. As before, there are differences, but none of them at
the 3$\sigma$ level. The largest differences, in the normalization and in the
powerlaw index may be a reflection of inaccuracies at the 5%-level in our
correction to the response function for the insertion of the blade.
Fit of the phase-averaged data to a power law model allowing 5 variables: the
power law index, the normalization, $N_{\rm H}$, the oxygen to hydrogen ratio
$[O/H]$, and the dust-scattering constant. The upper curves compare the data
to the best-fit model folded through the response function. The lower curves
show the contributions to $\chi^{2}$. The plotted data were combined into
larger bins for visual clarity.
Table 1: Power law fita to the Chandra-LETGS phase-averaged spectrum of the Crab Pulsar in the band from 0.3 to 3.8 keV holding the dust-scattering factor constant at 0.15. ObsID | $\chi^{2}$/$\nu$ | $\Gamma_{\rm P}$ | $N_{\rm H}$ | $[O/H]$ | $norm$
---|---|---|---|---|---
| | | $10^{21}\;{\rm cm^{-2}}$ | $10^{-4}$ |
Paper Ib | $1539/1552^{c}$ | $1.587\pm 0.019^{d}$ | $4.20\pm 0.14$ | $3.33\pm 0.44$ | $0.506\pm 0.008$
759e | $1447/1476$ | $1.622\pm 0.023$ | $3.68\pm 0.13$ | $4.52\pm 0.42$ | $0.479\pm 0.009$
9765 | $1832/1811$ | $1.538\pm 0.023$ | $3.59\pm 0.11$ | $3.84\pm 0.32$ | $0.450\pm 0.008$
### 3.2 Spectral Variation with Pulse Phase
We were previously limited in our ability to study the spectrum as a function
of pulse phase (ObsID 759 and Paper I) because a HRC-S timing error assigns to
each event the time of the previous event, thus complicating the analysis for
this bright source when telemetry is saturated and events are dropped. In
Tennant et al. (2001) we discussed this problem and a method for maintaining
some timing accuracy under these conditions — albeit at significantly reduced
efficiency. The method filters the data, accepting only telemetered events
separated by no more than 2 ms, guaranteeing a timing accuracy never worse
than 2 ms and typically much better. Thus, although there were approximately
50 ksec of observing time for ObsID 759, only a small fraction (1/15) of the
data were useful for studying spectral variations with pulse phase. For the
current (ObsID 9765) observation, which lasted for about 100 ksec, this
problem does not exist since the observation was designed so that the
telemetry never saturated.
Jodrell Bank (Lyne, Pritchard, & Smith 1993) routinely observe the Crab Pulsar
(Wong, Backer & Lyne 2001) providing a period ephemeris
222http://www.jb.man.ac.uk/$\sim$pulsar/crab.html. Roberts & Kramer (2000,
2008, private communications) kindly prepared ephemerides matched to our
observation times.
In performing the phase-resolved spectral analysis, we allow the interstellar
absorption and dust scattering parameters to vary, but assume that they are
identical for each pulse phase bin. However, the spectral index and
normalization is allowed to vary. This removes any bias produced by assuming a
power law index that is independent of phase. The fit to the data was
excellent, $\chi^{2}$ was 3156 on 3207 degrees of freedom. The best-fit values
for the non-phase-varying parameters and their approximate (one-interesting-
parameter) uncertainties are given in Table 2. Note $\tau_{\rm scat}$, which
we previously (Paper I) postulated must be present and accounted for, has now
been detected at almost $4\sigma$. Table 3 and Figure 3.2 summarize the
results for the pulsar’s phase-resolved power law photon index, $\Gamma_{\rm
P}$.
Table 2: Power law fita to the Chandra-LETGS phase-resolved spectrum of the Crab Pulsar. ObsID | $\chi^{2}$/$\nu$ | $N_{\rm H}$ | $[O/H]$ | $\tau_{\rm scat}$
---|---|---|---|---
| | $10^{21}\;{\rm cm^{-2}}$ | $10^{-4}$ | $@1keV$
9765 | $3510/3546$ | $3.22\pm 0.12^{b}$ | $5.28\pm 0.28^{c}$ | $0.147\pm 0.043^{c}$
Table 3: Power law Index versus Pulse Phase. Phase Range $\times 1000$ | $\Gamma_{\rm P}$
---|---
001$-$017 | $1.627\pm 0.041^{a}$
017$-$031 | $1.582\pm 0.063$
031$-$051 | $1.462\pm 0.073$
051$-$075 | $1.461\pm 0.088$
075$-$120 | $1.449\pm 0.097$
120$-$330 | $1.341\pm 0.040$
330$-$370 | $1.471\pm 0.037$
370$-$390 | $1.499\pm 0.039$
390$-$400 | $1.535\pm 0.047$
400$-$410 | $1.621\pm 0.051$
410$-$430 | $1.604\pm 0.050$
430$-$470 | $1.655\pm 0.059$
470$-$650 | $1.698\pm 0.010$
650$-$830 | $1.886\pm 0.046$
830$-$950 | $1.502\pm 0.058$
950$-$960 | $1.733\pm 0.060$
960$-$970 | $1.611\pm 0.049$
970$-$980 | $1.649\pm 0.042$
980$-$984 | $1.651\pm 0.052$
984$-$992 | $1.627\pm 0.039$
992$-$001 | $1.594\pm 0.039$
Based upon a $\chi^{2}$ analysis of the distribution of best-fit photon
indices (Table 3, Figure 3.2), we reject, with high confidence, the hypothesis
that the spectral index is constant with phase. The error-weighted average of
the spectral indices is 1.563 and the value of $\chi^{2}$ was 71 on 20 degrees
of freedom. The variation of spectral index between pulse phases -$0.1$ and
$0.5$ is qualitatively similar in Chandra (Paper I, this paper), Beppo-SAX
(Massaro et al. 2000), and Rossi-XTE (Pravdo, Angelini, & Harding 1997)
measurements, with the index increasing (becoming softer) through the primary-
pulse maximum and decreasing (becoming harder) in the bridge between the
primary and secondary pulses. It is difficult to be more quantitative in this
comparison as the non-Chandra data were analyzed using different cross-
sections, different abundances, and covered different spectral ranges.
Moreover, Chandra provides the angular resolution needed to isolate the pulsar
from the nebula, something that is essential to measure the spectral index for
the pulse-phase range 0.5–0.9. Our analysis shows that the spectral index at
pulse minimum is consistent with an apparent continuation of the increase
(softening) of the spectral index until just before the onset of the primary
pulse. The spectral-index uncertainty near pulse minimum is, of course, large
because there are fewer counts. It is also interesting that the slope of the
spectral variations appears to be the highest (softest) during the peak of the
two pulses.
In Fig. 3.2 we also show the results of fitting a constant plus a sine wave to
the variation of the spectral index but excluding the three data points with
the largest uncertainty, i.e. those in the phase range from $0.47$ to $0.95$ —
pulse minimum and the points just before and just after pulse minimum. The fit
to the sine wave is excellent and is probably the type of behavior one might
expect as the orientation of the pulsar’s magnetic field varies with phase to
the distant observer. The phase of the variation is (probably) an indication
of the geometry. The point just before the rise to the pulse maximum, however,
clearly does not fit this simple picture.
The upper curve shows the measured variation of the powerlaw index. The solid
curve is the result of fitting a sine wave plus a constant to these data not
including the three points in and around pulse minimum which exhibit the
largest error in the powerlaw index. The X-ray light curve (background not
subtracted) is shown in the second panel from the top. The bottom two panels
show the variation of the optical degree of polarization and position angle.
The optical data are from Slowikowska et al. (2008). Slight differences, if
any, between the optical pulse phase and the X-ray pulse phase have been
ignored. One pulse cycle is repeated twice for clarity.
#### 3.2.1 Discussion
An interesting correlation is shown also in Fig. 3.2 in the bottom two panels
which plot optical polarization data kindly provided by G. Kanbach and A.
Slowikowska (Slowikowska et al. 2008). Here we emphasize the afore-mentioned
change of behavior in the X-ray power law index just before the rise of the
light curve to primary pulse maximum (phases 0.83 to 0.95) and the abrupt
change of optical polarization and position angle in this same phase interval.
These would appear to be correlated phenomena.
The variation of spectral index with phase shown in Fig. 3.2 and Table 3 is
also strikingly similar to the spectral index variations measured by the Fermi
Gamma-Ray Space Telescope above 100 MeV (Abdo et al. 2010a), with the hardest
(smallest) index occurring midway between the two peaks and rising
symmetrically through both peaks to reach maxima in the off-peak region. There
is also even a hint in the Fermi data of the small maximum preceding the first
peak. Indeed, the photon index variation is similar in other bright gamma-ray
pulsars, including Geminga (Abdo et al. 2010b) and Vela (Abdo et al. 2010c),
where the maximum preceding the first peak is even more pronounced. Yet, the
Crab broadband spectrum is very different from that of Vela or Geminga, being
one of very few pulsars (that include B1509$-$58) having equal or greater
power in the X-ray band as in the hard gamma-ray band. The multiwavelength
spectrum of the Crab pulsar (Kuiper et al. 2001) seems to comprise two
distinct components: one extending from UV to soft gamma-rays and one
extending from soft to hard gamma-rays (although the position of the division
varies somewhat with pulse phase). In the phase-resolved spectra, the spectral
indices of the two components tend to mirror each other, with the hardest
spectra in soft X-rays and hard gamma-rays occurring in the bridge region and
the softer spectra occurring in the peaks. The similarity of the Chandra and
Fermi spectral index behavior is consistent with this trend.
The fact that the soft X-ray and hard gamma-ray spectra are part of two
seemingly different radiation components, and most likely have different
emission mechanisms, raises the question of why their spectral index variation
with phase should be so similar. They must share a common property, such as
the same radiating particles or the same locations in the magnetosphere.
It is now widely agreed that the high energy emission from pulsars originates
in their outer magnetospheres, since the measurement of the cutoffs in their
gamma-ray spectra rules out attenuation due to magnetic pair production and
therefore emission near the polar caps (Abdo et al. 2009). Several different
outer-magnetosphere models, that advocate different emission mechanisms in the
X-ray range, make predictions for phase-resolved spectral variations. In outer
gap models (Cheng, Ho & Ruderman 1986, Romani 1996), particles are accelerated
in vacuum gaps that form along the last open magnetic field lines, from above
the null-charge surface to near the light cylinder. In slot gap models
(Muslimov & Harding 2004), particles are also accelerated along the last open
field lines, but in a charge-depleted layer from the neutron star surface to
near the light cylinder. In both models, the high-energy peaks in the light
curve are caustics, caused by cancellation of phase differences along the
trailing field lines (Morini 1983) or overlapping field lines near the light
cylinder.
Harding et al. (2008) presented a model for 3D acceleration and high-altitude
radiation from the slot gap, with application to the Crab pulsar. In this
model, emission in the optical to soft gamma-ray band is synchrotron radiation
from pairs outside the slot gap undergoing cyclotron resonant absorption of
radio photons. Hard gamma-rays come from primary electrons accelerating in the
slot gap and radiating curvature and synchrotron emission. The common element
for the X-ray and gamma-ray emission would then be the angles to the radio
photons. Since Harding et al. (2008) assumed that the pair spectrum was
constant throughout the open field volume, there was no spectral index
variation with phase. However, polar cap pair cascade simulations (the origin
of the X-ray emitting pairs in this model) show that there are large
variations in the pair spectrum across the polar cap (Arendt & Eilek 2002).
Thus the detailed measurements of X-ray spectral index variation presented in
this paper is mapping (and constraining) the variation in the pair spectrum
across the open field lines.
In recent studies of phase-resolved spectra of the Crab pulsar in the outer
gap model (Tang et al. 2008, Hirotani 2008), the optical to hard X-ray
spectrum comes from synchrotron radiation of secondary pairs produced in situ
in outer gap cascades while the gamma-rays come from inverse Compton radiation
of pairs and curvature radiation of primary particles. This model does not
match the observed X-ray spectral variations, although it does somewhat
produce the observed gamma-ray spectral variations. Thus, this model lacks the
essential physics that accounts for the X-ray spectral index variation and its
similarity to that in gamma-rays.
### 3.3 Temperature of the Neutron Star and Superfluidity
Here we investigate the hypothesis that there is a detectable underlying
thermal component in addition to the non-thermal flux that we see from the
pulsar. We fit the data as a function of pulse phase to spectral models that
allow both components. We first consider a power law together with a thermal
black-body and then a model with a spectrum of radiation emergent from the
hydrogen neutron star atmosphere. In XSPEC these models are the powerlaw,
bbodyrad, and nsa (Pavlov et al. 1995).
We examined two approaches for the analysis. In the first, we use the data
from all 21 pulse phase bins, in the second, we use the data from the 4 phase
bins that are at pulse minimum. In the latter case we use the data from the
other 17 phase bins to establish the values of the phase independent
parameters $N_{\rm H}$, $[O/H]$, and $\tau_{\rm scat}$ for fitting the data at
pulse minimum. We found that the sensitivity to a thermal component was
virtually the same in both approaches, however, the establishment of upper
limits was computationally much faster using the data at pulse minimum.
#### 3.3.1 Blackbody Model
Adding a phase-independent black-body model to the spectral fitting yields the
results listed in Table 4. The large uncertainties in both the normalization,
$\theta_{\infty}^{2}$, and the redshifted effective surface temperature,
$kT_{\infty}$, clearly point to the absence of a blackbody component within
statistics. $\theta_{\infty}$ is the angular size determined by a distant
observer, in XSPEC units — $\theta_{\infty}=(R_{\infty}/D_{10})$, with
$R_{\infty}$ the apparent stellar radius in km units and $D_{10}$ the source
distance in 10-kpc units. Fig. 3.3.1 shows the 2- and 3-$\sigma$ upper limits
to $kT_{\infty}$ for a range of values for $\theta_{\infty}^{2}$ that are
relevant to neutron star models with realistic equations of state.
In Paper I we found a somewhat lower 3-$\sigma$ upper limit and a higher
2-$\sigma$ upper limit to $kT_{\infty}$ than those shown in Figure 3.3.1.
However, these results were erroneous and should have been higher.
Unfortunately this error was only discovered while completing this paper. As
an example of the changes due to the error, the 2- and 3-$\sigma$ upper limits
at $\theta_{\infty}^{2}=6100$ should have been 0.195 and 0.209 keV
respectively. Using the newer response function and signal and background
extraction regions would have lowered these to 0.184 and 0.202 keV
respectively. Thus, the upper limits reported here, as expected, represent a
significant improvement over Paper I. (We note again that the difference in
upper limits using the old and new response functions ignores the possibility
that there are uncertainties associated with these responses.) Of course here
we analyze the data as a function of pulse phase. As discussed above, this is
a better approach as it removes any possible bias produced by averaging a
number of power laws, which, in turn won’t be a powerlaw and thus
inadvertently create a spurious thermal component.
Table 4: Best-fit values for the phase-independent parameters after analyzing the phase-resolved data using a bbodyrad plus powerlaw model. The latter is allowed to vary as a function of phase. Uncertainties are XSPEC estimates for the 1$\sigma$ statistical errors based on one interesting parameter ($\chi^{2}$ at minimum +1.0). Parameter |
---|---
$\chi^{2}/\nu$ | $3510/3544$
$N_{\rm H}(10^{21}cm^{-2})$ | $3.22\pm 0.13$
$[O/H]$ | $(4.28\pm 0.30)\times 10^{-4}$
$\tau_{\rm scat}\ at\ 1keV$ | $0.147\pm 0.045$
$kT_{\infty}(keV)$ | $0.1\pm 7.2$
$\theta_{\infty}^{2}$ | $44\pm 31000$
The 2- and 3$-\sigma$ upper limits to $kT_{\infty}$ derived (lower and upper
curves respectively) using the powerlaw+bbodyrad spectral model. The limits
are based on a single interesting parameter
($\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}=4.0$ (lower curve) and $9.0$
(upper curve) and plotted as a function of $\theta_{\infty}^{2}$. (See text
for details.)
#### 3.3.2 NSA Model
The NSA model requires a number of inputs. The normalization was set assuming
a distance to the Crab of 2 kpc. The surface magnetic-field parameter was set
at $1.0\times 10^{13}$ Gauss, although results are not terribly sensitive to
this choice. The model also requires the gravitational mass $M$ and the
circumferential radius $R$ of the neutron star. We examined a wide range of
$M$ from 1.0 to 2.5 $M_{\odot}$ and $R$ from 8 to 15 km in creating Fig. 3.3.2
which shows the 2- and 3-$\sigma$ upper limits to $kT_{\infty}$; hence the
reason for the multiple values for a given $\theta_{\infty}$.
The 2- and 3$-\sigma$ upper limits to $kT_{\infty}$ (lower and upper curves
respectively) fitting a powerlaw+NSA model. The limits are based on a single
interesting parameter ($\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}=4.0$ (lower
curve) and $9.0$ (upper curve) as a function of $\theta_{\infty}^{2}$.
Multiple values of $kT_{\infty}$ arise as different combinations of $M$ and
$R$ lead to the same (or similar) values of $\theta_{\infty}$.
The reader will note that powerlaw+NSA fits always yield a higher upper limit
for given values of $\theta_{\infty}$ than powerlaw+bbodyrad. Simulations with
fake data have shown this to be correct. We believe that this happens because
the NSA spectrum has a hard tail which makes it difficult to distinguish from
the power law.
#### 3.3.3 Implications
Here we apply cooling theory for neutron stars and formulate constraints on
the internal structure of the Crab pulsar that can be inferred from the
observational upper limits of its effective surface temperature $T$ (§3.3.1 &
§3.3.2).
Current cooling theories (e.g. Page, Geppert & Weber 2006, Page et al. 2009,
Yakovlev & Pethick 2004 , Yakovlev et al. 2008) state that any isolated
neutron star of the Crab pulsar age ($t\sim 10^{3}$ yr) should be at the
neutrino cooling stage and have an isothermal interior. The preceding cooling
stage of internal thermal relaxation, when the neutron star core is noticeably
colder than the crust (because of stronger neutrino cooling of the core and
slower thermal conduction in the crust), lasts no longer than $\sim 200$ yrs.
That stage should be over. The interior of the pulsar then should be highly
isothermal having the same internal temperature $\widetilde{T}_{i}(t)$, where
$\widetilde{T}_{i}$ is the internal temperature redshifted for a distant
observer (Thorne, 1977). The local (actual) temperature $T_{i}$ in the
isothermal interior is $\sim 10-30$% higher than $\widetilde{T}_{i}$, with the
stellar core being somewhat hotter than the crust. A noticeable temperature
gradient in a thermally relaxed star survives only near the surface, in the
outer heat-blanketing envelope (Gudmundsson, Pethick & Epstein 1983) with
thickness not higher than a few tens of meters. In the envelope, the
temperature drops from the temperature inside the star to the effective
surface temperature $T$. The temperature drop depends on the matter
composition and on the magnetic field in the envelope (which affects the
thermal conductivity – see Potekhin, Chabrier, & Yakovlev 1997, Potekhin et
al. 2003).
The cooling of the Crab pulsar (as of all isolated neutron stars of ages
$t\lesssim 10^{5}$ yrs) is driven by neutrino emission from its interior,
mainly from the superdense core. The decrease of the internal temperature
$\widetilde{T}_{i}(t)$ with time is determined by the physics of the core,
being insensitive at this cooling stage to the physics of the envelope.
Therefore, the internal cooling of a star with a given internal structure is
the same for any heat-blanketing envelope (looks the same from inside) but the
surface temperature is affected by the particular properties of the envelope
(looks different from outside).
There are numerous versions of current cooling theories as cited above and
they are still poorly constrained by observations. The theories comprise
different compositions and equations of state (EOSs) of neutron star cores,
different neutrino emission properties and models for superfluidity of baryons
(which affect heat capacity and neutrino emission). In spite of the large
number of scenarios, the cooling of isolated thermally relaxed neutron stars
with an isothermal interior is mostly regulated by the three factors 333This
statement is true as long as a noticeable temperature decrease $T_{s}(t)$ is
not observed for a given neutron star, which is so for all currently observed
isolated neutron stars except for the star in the Cas A supernova remnant
(Heinke & Ho 2010). In this case one can glean more information as to the
neutron star structure (Page et al. 2011, Shternin et al. 2011). (e.g.
Yakovlev et al. 2011): (i) the neutrino cooling rate; (ii) the properties of
the heat-blanketing envelope; and (iii) the stellar compactness.
The neutrino cooling rate $\ell$ [K s-1] is defined as
$\ell=L_{\nu}^{\infty}(\widetilde{T}_{i})/C(\widetilde{T}_{i}).$ (1)
Here, $L^{\infty}_{\nu}(\widetilde{T}_{i})$ is the neutrino luminosity of the
star (redshifted for a distant observer), and $C(\widetilde{T_{i}})$ is the
heat capacity (see, e.g. Eqs. (3) and (5) in Yakovlev et al. 2011). It is
instructive (Yakovlev et al. 2011) to introduce the normalized cooling rate
$f_{\ell}=\ell(\widetilde{T}_{i})/\ell_{\rm SC}(\widetilde{T}_{i}),$ (2)
where $\ell_{\rm SC}(\widetilde{T}_{i})$ is the neutrino cooling rate of the
standard neutrino candle, a neutron star with the same $M$, $R$,
$\widetilde{T}_{i}$, but with a non-superfluid nucleon core that is cooling
via the ordinary modified Urca process of neutrino emission. For isolated
neutron stars without any additional internal heat sources, physically
allowable values of $f_{\ell}$ may vary from $\sim 10^{-2}$ to $\sim 10^{6}$
(e.g. Yakovlev et al. 2011). If $f_{\ell}\sim 1$ this implies standard
neutrino cooling, $f_{\ell}\sim 10^{-2}$ very slow cooling (e.g. when the
modified Urca process is suppressed by superfluidity) and $f_{\ell}\sim
10^{2}-10^{6}$ fast cooling (accelerated by direct Urca processes, pion or
kaon condensates, or by neutrino emission due to Cooper pairing of neutrons).
We have employed the models of heat-blanketing envelopes of Potekhin,
Chabrier, & Yakovlev (1997) and Potekhin et al. (2003) which may contain some
mass $\Delta M$ of (light-element) accreted matter and have a dipole magnetic
field $B$ (with $B=3.8\times 10^{12}$ G at the magnetic equator for the Crab
pulsar). The effective temperature of a magnetized star varies over the
surface with the magnetic poles being hotter than the equator. Cooling theory
(e.g. Potekhin, Chabrier & Yakovlev 1997) suggests one use the effective
temperature averaged over the surface (it defines the thermal luminosity of
the star). The effect of the given magnetic field on the cooling is weak
although included in our calculations. The effect of the envelope on the
cooling is regulated then by $\Delta M$ which varies from $\Delta M=0$ for a
standard iron envelope to $\Delta M_{\mathrm{max}}\sim 10^{-7}\,M_{\odot}$ for
a fully accreted envelope. Larger $\Delta M$ are not realistic because light
elements transform into heavier ones at the bottom of the envelope through
electron capture and nuclear reactions.
The compactness of the star can be characterized by the parameter
$x=\frac{2GM}{Rc^{2}}\approx 2.95\;\frac{M}{M_{\odot}}\;\frac{1~{}{\rm
km}}{R}$ (3)
which is the ratio of the Schwarzschild radius to $R$. According to Yakovlev
et al. (2011), one can distinguish not very compact ($x\lesssim 0.5$) and very
compact ($x\gtrsim 0.5$) neutron stars. Values $x\geq 0.7$ are forbidden by
the causality principle (e.g. Haensel, Potekhin & Yakovlev 2007).
Thus, for the neutron star in the Crab, the cooling is mainly determined by
the three parameters, $f_{\ell}$, $\Delta M$, and $x$. The majority of
realistic models of neutron stars have $x\lesssim 0.5$ and we restrict
ourselves to these models. Their cooling weakly depends on $x$, so that we can
consider the effect of $f_{\ell}$ and $\Delta M$ but neglect the effect of $x$
(although we comment on the latter below).
Figure 1: Theoretical cooling curves for a 1.4 $M_{\odot}$ neutron star with
the APR EOS in the core compared with the 3$\sigma$ upper limit of
$T_{\infty}$ for the Crab pulsar (inferred from the blackbody fits) and with
measured or constrained $T_{\infty}$ for some other isolated neutron stars.
The solid curve is for the standard neutrino candle with an iron heat-
blanketing envelope. The long-dashed curve is for the same star but with a
fully accreted envelope. The dot-dashed curve corresponds to a star with
strong proton superfluidity in the core and an iron heat-blanketing envelope;
the short-dashed curve is for a star with the same core but for a fully
accreted envelope; the dotted curve is for the same core but for a partly
accreted envelope with $\Delta M=10^{-9}\,M_{\odot}$. (See text for details.)
Our new observational upper limits on $T$ are high, comparable to the highest
surface temperatures which a cooling neutron star can have. Thus these $T$
limits allow the Crab pulsar to have almost any theoretically possible
neutrino cooling rate (from very fast to very slow). It is nevertheless
instructive to compare the upper limits on $T$ for the Crab pulsar with
theoretical models of the warmest cooling neutron stars. This comparison is
illustrated in Fig. 1. The figure shows some theoretical cooling curves
($T_{\infty}(t)=T(t)\,\sqrt{1-x}$). The data for stars other than the Crab are
taken from the same references as in Shternin et al. (2011). The cooling
curves are calculated for a model of a neutron star whose core consists of
nucleons and has the EOS of Akmal, Pandharipande & Ravenhall (1998) (APR).
Specifically, we use a version of the APR EOS denoted as APR I in Gusakov et
al. (2005). The maximum mass of a stable neutron star with this EOS is
$M_{\mathrm{max}}=1.929\,M_{\odot}$; the powerful direct Urca process of
neutrino emission is allowed in stars with $M>1.829\,M_{\odot}$. In Fig. 1 we
take a star with $M=1.4\,M_{\odot}$ ($R=12.14$ km). The upper limit of
$T_{\infty}$ for the Crab pulsar ($\log T^{\rm
BB}_{\infty}(3\sigma)~{}{\rm[K]}=6.30$; $T_{\infty}^{\mathrm{BB}}\approx 2$
MK) is from the blackbody fits at the 3$\sigma$ level for this choice of $M$
and $R$ (§ 3.3.1).
The solid line in Fig. 1 is the basic cooling curve for a non-superfluid APR
$1.4\,M_{\odot}$ neutron star with the iron heat blanket ($\Delta M=0$). This
star cools via the modified Urca process, its $f_{\ell}=1$, and thus it
represents the standard neutrino candle. Its internal temperature at the Crab
age would be $\widetilde{T}_{i\rm SC}\approx 2.23\times 10^{8}$ K, with the
neutrino cooling rate $\ell_{\mathrm{SC}}\approx 3.7\times 10^{4}$ K yr-1. The
basic curve goes below the $T_{\infty}$ upper limit for the Crab pulsar.
Therefore, the assumption that the Crab pulsar is the standard neutrino candle
is compatible with the given upper limit. This conclusion can also be made
from previous work, e.g. Kaminker et al. (2006).
Next we consider two cooling regulators – the neutrino emission rate
$f_{\ell}$ in the stellar core (to change the internal temperature
$\widetilde{T}_{i}$) and the amount $\Delta M$ of light elements in the
envelope (to change $T_{\infty}$ for a given $\widetilde{T}_{i}$). The long-
dashed line in Fig. 1 shows cooling of the same standard candle but with a
fully accreted envelope. Light elements increase the thermal conductivity,
make the envelope more heat transparent, and increase $T_{\infty}$ (for a
given $\widetilde{T}_{i}$). The increase is substantial but cannot raise the
cooling curve above the $T_{\infty}$ upper limit for the Crab pulsar. Thus,
the Crab pulsar can well be the standard neutrino candle inside and have a
fully-accreted envelope outside.
The pulsar can also be warmed by reducing its neutrino emission below the
standard level. The dot-dashed line in Fig. 1shows the cooling of the same
star as in the previous paragraph with the iron heat-blanketing envelope, but
with strong proton superfluidity in the core. This superfluidity greatly
suppresses the modified Urca process of neutrino emission and the
bremsstrahlung emission of neutrino pairs in proton-proton and proton-neutron
collisions (and also suppresses proton heat capacity in the core). Under these
conditions, the star cools via neutrino bremsstrahlung emission in neutron-
neutron collisions. In this scenario, the normalized neutrino cooling rate is
small, $f_{\ell}\approx 0.01$, and the core warmer. Exact values of the
critical temperature for onset of proton superfluidity are unimportant here;
the critical temperature in the core should be higher than a few times of
$10^{9}$ K to establish this very slow cooling regime. It is thought to be one
of the slowest cooling regimes that can be realized in a cooling star (without
additional heat sources) and it produces stars with the hottest cores. The
dot-dashed line shows that this hottest star has about the same surface
temperature as the standard neutrino candle with a fully accreted envelope; it
is not forbidden by our observations.
The short-dashed curve in Fig. 1 displays cooling of the neutron star with the
same very slow cooling rate as previously but now with a fully-accreted
envelope. With respect to the basic solid cooling curve, its surface
temperature is increased by two factors: by proton superfluidity in the core
($f_{\ell}\approx 0.01$) and by accreted matter in the envelope ($\Delta
M=\Delta M_{\mathrm{max}}$). The large increase makes the surface
exceptionally warm, with $T_{\infty}$ higher than the upper limit for the Crab
pulsar. Our observational upper limit to $T_{\infty}$ does restrict this
particular model. Thus, if the Crab pulsar does have strong proton
superfluidity ($f_{\ell}\approx 0.01$) it can only have a partially accreted
envelope, with $\Delta M\lesssim 10^{-9}\,M_{\odot}$. This is demonstrated by
the dotted cooling curve calculated for $\Delta M=10^{-9}\,M_{\odot}$; it hits
the observational $T_{\infty}$ limit.
We also considered a sequence of cooling models with progressively weaker
proton superfluidity in the Crab pulsar core. The parameter $f_{\ell}$ varies
then from $f_{\ell}\approx 0.01$ for strongly superfluid to $f_{\ell}=1$ for a
non-superfluid star. At $f_{\ell}\gtrsim 0.15$ we find $T_{\infty}$ lower than
the observational upper limit for any $\Delta M$. All these models are
therefore allowed by the observations. At lower $f_{\ell}$ and $\Delta
M\gtrsim 10^{-9}\,M_{\odot}$ the pulsar surface would be warmer than the
observational limit in disagreement with our observations.
Our conclusions are fairly independent of neutron star mass. Indeed we have
considered a wide range of masses from 1 to 1.8 $M_{\odot}$ (for stars with
the APR EOS). The cooling curves stay essentially the same, and the
observational upper limits do not change much (Figs. 3.3.1 & 3.3.2). For
instance, $\log T^{\rm BB}_{\infty}(3\sigma)=6.31$ from the blackbody fits for
an $M=1.8\,M_{\odot}$ ($R=11.38$ km) star. Moreover, the cooling curves stay
nearly the same for a large variety of EOSs of dense nucleon matter. These
include the 9 original versions of the phenomenological PAL EOS (Prakash,
Ainsworth & Lattimer 1988), as well as 3 other versions of this EOS with the
symmetry energy of nuclear matter proposed by Page & Applegate (1992); and the
SLy EOS (Douchin & Haensel 2001). These neutron star models are all not very
compact ($x\lesssim 0.5$) which justifies our consideration of not very
compact stars (see above). The very weak dependence of the cooling curves on
$M$, $R$ and the EOS for standard candles and for very slowly cooling neutron
stars is not new and has been in the literature [starting from the paper by
Page & Applegate (1992); see, e.g. Yakovlev & Pethick (2004), for references].
Instead of the 3$\sigma$ upper limits of $T_{\infty}$ we could also use less
conservative 2$\sigma$ limits. For instance, we have $\log T^{\rm
BB}_{\infty}(2\sigma)=6.26$ ($T_{\infty}^{\mathrm{BB}}(2\sigma)\approx 1.8$
MK), for the $1.4\,M_{\odot}$ neutron star with the APR EOS. In this case we
obtain $f_{\ell}\gtrsim 0.013$ (very slow or any faster cooling) for an iron
envelope, $f_{\ell}\gtrsim 1$ (standard or any faster cooling) for a fully
accreted envelope. Also, the upper limits of $T_{\infty}$, inferred from
neutron star atmosphere fits (§3.3.2), are higher than for the black-body fits
[e.g. $\log T^{\rm NSA}_{\infty}(2\sigma)=6.44$ and $\log T^{\rm
NSA}_{\infty}(3\sigma)=6.45$ for the $1.4\,M_{\odot}$ APR star]; they would of
course be less restrictive.
For completeness we note that massive neutron stars may become especially
compact. If they were also slow neutrino coolers, their cooling curves would
depend on the compactness parameter $x$, Eq. (3), and their $T_{\infty}$ would
noticeably decrease with increasing $x$ (at $x\gtrsim 0.5$) (Yakovlev et al.
2011). Such (less realistic) models would be less restricted by the
observational $T_{\infty}$ limits.
Let us remark that according to the majority of current theories massive
neutron stars cool faster than the standard neutrino candles due to the onset
of fast neutrino emission in their cores. The mass range of stars which
demonstrate faster cooling is very uncertain. Our $T_{\infty}$ limits do not
constrain the parameters of the Crab pulsar if it is a rapidly cooling star.
We note that our model for proton superfluidity to suppress the neutrino
emission should be considered as an example. One may use a more general
approach (Yakovlev et al. 2011) introducing the normalized neutrino emission
rate $f_{\ell}$ without specifying a physical model of neutrino emission. The
cooling equation contains $f_{\ell}$, and it is $f_{\ell}$ that can be
constrained from the observations. Thus there could be several physical models
of stellar interior which give the same $f_{\ell}$, and the cooling theory
itself cannot discriminate between them.
The conclusions above follow from cooling calculations already available in
the literature (particularly, from the results of Kaminker et al. 2006). We
have repeated these calculations drawing special attention to the Crab pulsar;
our Fig. 1 is similar to the right panel of Fig. 2 of Kaminker et al. (2006).
Notice, that our short-dashed curve goes higher than the analogous curve in
Kaminker et al. (2006). This is both because we assume a fully-accreted heat-
blanketing envelope $\Delta M\sim 10^{-7}\,M_{\odot}$, while Kaminker et al.
(2006) took $\Delta M\sim 10^{-8}\,M_{\odot}$, and because we employ stronger
proton superfluidity in the core. This difference of cooling curves reflects
the uncertainty of the present cooling theory but does not affect our
principal conclusions. Finally, we emphasize that at the present stage it
would be better to use the model-independent formulation of the cooling theory
(Yakovlev et al. 2011), introducing $f_{\ell}$ instead of employing any
specific physical cooling model, especially when interpreting data.
## 4 Summary
We have obtained new Chandra data of the Crab Nebula and its pulsar. The new
data were collected in such a manner to prevent telemetry saturation of the
LETGS and thus enable efficient collection of high-time resolution data from
the pulsar. We have analyzed these data and re-analyzed our previous
observation of the phase averaged spectrum to update spectral parameters. The
updated phase-averaged spectral parameters no longer (Paper I) indicate that
the Crab line-of-sight is under abundant in oxygen given the abundances and
cross-sections employed in the spectral fitting. In all our analyses, we have
accounted for the contribution of scattering by interstellar dust to the
extinction of X rays in an aperture-limited measurement — a consideration
(often ignored) in spectral analysis of point sources observed with Chandra’s
exceptional angular resolution. Here we have measured, for the first time, the
magnitude of that extinction in the direction of the Crab pulsar.
In addition, we have measured with a high precision the spectrum of the Pulsar
as a function of pulse phase and at all pulse phases. We find highly
significant variation of the power law spectral index as a function of phase
and have discovered an unusual behavior of the spectral index as the pulse
rises out of pulse minimum on its approach towards the peak of the primary
pulse. Interestingly, this behavior appears to be connected to a similar
feature in the variation of the optical polarization as a function of pulse
phase as well as variations of the gamma-ray spectral index. In both slot- and
outer-gap models for phase-resolved radiation from the Crab, the X-ray
emission comes from synchrotron radiation of secondary pairs. The variations
in X-ray spectral index are thus mapping the variations in pair spectrum with
phase, although neither of these models currently includes the physical
elements that produce the observed spectral variations. Therefore, the more
accurate measurements presented in this paper will be a challenge to future
modeling, and they have the hope of helping us understand the pair cascade
processes in pulsar magnetospheres.
We also use the spectral data to obtain new and more precise upper limits to
the surface temperature of the neutron star for two different models of the
star’s atmosphere.
We have commented on the differences in measured parameters subsequent to
analyzing the same data with different releases of the response functions. Our
experience emphasizes the importance of accounting for the uncertainties in
the response functions when analyzing data. One might estimate the level of
those uncertainties by noting differences in spectral parameters using the old
(Paper I) and new (this paper) response functions, however, this might be too
extreme as the newer response functions are a product of several proven
refinements. Perhaps then these differences can serve as estimators of upper
limits to the variations. We urge the various observatories to provide users
with response functions with errors and the tools use them.
Finally, we clarify the means by which the observational data as to the
thermal emission may be connected to theories of neutron star cooling and
neutron star structure. Our principal conclusions, only slightly dependent on
the EOS of the pulsar core, pulsar mass, and pulsar radius, are:
* •
Our upper limits to the surface temperature $T_{\infty}$ of the Crab pulsar
weakly restrict the normalized neutrino emission rate $f_{\ell}$ (in units of
standard candles) in the pulsar core and the amount of light elements $\Delta
M$ in the heat-blanketing envelope.
* •
Our observations allow the pulsar to have a neutrino emission rate
$f_{\ell}\gtrsim 0.15$ (1/6 of the standard neutrino cooling or higher for the
fastest cooling) for any amount of light elements in the blanketing envelope.
For lower neutrino emission rates from $\sim 0.15$ to $0.01$ (the lowest rate
on physical grounds, e.g. due to strong proton superfluidity in the core), our
observations constrain the pulsar to have only a limited mass of accreted
material (with $\Delta M\lesssim 10^{-9}\,M_{\odot}$ at $f_{\ell}\sim 0.01$).
The absence of strong restrictions on the properties of the Crab pulsar follow
from the current 3$-\sigma$ upper limit on $T_{\infty}$. Nevertheless, this
state of affairs has its own advantage. There is still a chance that the Crab
pulsar is warm, with the surface temperature $T_{\infty}$ only slightly below
the present upper limit. While our upper limit is not very restrictive, a real
measurement of the surface temperature just below the present upper limit
would be more restrictive, indicating that the Crab pulsar is one of the
warmest neutron stars. For instance, if the temperature $T_{\infty}=1.6\times
10^{6}$ K ($\log T_{\infty}=6.20$) were measured, we would have
$f_{\ell}\approx 0.06$ (slow cooling) for an iron envelope, $f_{\ell}\approx
10$ (ten times faster than the standard cooling) for a fully accreted
envelope, and generally $0.06\lesssim f_{\ell}\lesssim 10$ for a partially
accreted envelope. Thus we would be able to constrain the neutrino emission
rate within two orders of magnitude. This large uncertainty comes from the
unknown mass of the accreted matter. If $\Delta M$ were known, $f_{\ell}$
would be even better constrained.
We acknowledge our tremendous debt to Leon Van Speybroeck for his remarkable
contributions to the development of the Chandra optics, to Harvey Tananbaum
for his superb stewardship of the Chandra X-ray Center, and to Steve Murray
and Michael Juda for their support in successfully configuring the HRC shutter
for this measurement. DY acknowledges support by: the Russian Foundation for
Basic Research, grant 11-02-00253a; Rosnauka, grant NSh 3769.2012.2; and
Ministry of Education and Science of the Russian Federation, contract
11.G34.31.001 with SPbSPU and Leading Scientist G. G. Pavlov.
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|
arxiv-papers
| 2011-06-16T16:38:46 |
2024-09-04T02:49:19.721398
|
{
"license": "Public Domain",
"authors": "Martin C. Weisskopf, Allyn F. Tennant, Dmitry G. Yakovlev, Alice\n Harding, Vyacheslav E. Zavlin, Stephen L. O'Dell, Ronald F. Elsner, and\n Werner Becker",
"submitter": "Martin C. Weisskopf",
"url": "https://arxiv.org/abs/1106.3270"
}
|
1106.3448
|
9(1:01)2013 1–27 Jun. 16, 2011 Feb. 14, 2013
# Type classes for efficient exact
real arithmetic in Coq
Robbert Krebbers and Bas Spitters Institute for Computing and Information
Sciences
Radboud University Nijmegen mail@robbertkrebbers.nl, spitters@cs.ru.nl
###### Abstract.
Floating point operations are fast, but require continuous effort by the user
to ensure correctness. This burden can be shifted to the machine by providing
a library of _exact_ analysis in which the computer handles the error
estimates. Previously, we provided a fast implementation of the exact real
numbers in the Coq proof assistant. This implementation incorporates various
optimizations to speed up the basic operations of O’Connor’s implementation by
a 100 times. We implemented these optimizations in a modular way using type
classes to define an abstract specification of the underlying dense set from
which the real numbers are built. This abstraction does not hurt the
efficiency.
This article is a substantially expanded version of (Krebbers/Spitters,
Calculemus 2011) in which the implementation is extended in the various ways.
First, we implement and verify the sine and cosine function. Secondly, we
create an additional implementation of the dense set based on Coq’s fast
rational numbers. Thirdly, we extend the hierarchy to capture order on
undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way.
Finally, we obtain another dramatic speed-up by avoiding evaluation of
termination proofs at runtime.
###### Key words and phrases:
Type classes, exact real arithmetic, type theory, Coq, verified computation
###### 1991 Mathematics Subject Classification:
F.1.m, F.4.1
The research leading to these results has received funding from the European
Union’s 7th Framework Programme under grant agreement nr. 243847 (ForMath).
[Theory of Computation]: Logic—Type theory & Constructive mathematics;
[Mathematics of Computing]: Mathematical analysis—Numerical
analysis—Arbitrary-precision arithmetic
## 1\. Introduction
There is a big gap between numerical algorithms in research papers, which
typically use concepts like Hilbert spaces and fixed point theorems from
functional analysis, and their actual implementation, which uses floating
point numbers111By floating points we mean numbers of the shape $n*b^{e}$,
where $n$ and $e$ are bounded integers and $b$ is the base for scaling
(typically 2, 10 or 16). The most widely used form of floating point
arithmetic is the IEEE 754 standard, which is present in many hardware and
software implementations. and matrix operations. This gap makes it difficult
to trust the code. Similarly, this gap is undesirable in proofs of theorems
(e.g. Kepler conjecture [Hal02], existence of the Lorentz attractor [Tuc02])
that rely on the execution of this code for numerical approximation. Finally,
from a purely software engineering point of view, the situation is
undesirable, because the gap between the (abstract mathematical) numerical
algorithms and the (concrete floating point) implemented program makes the
code difficult to maintain.
The challenge to close this gap has already been posed by Bishop in his
fundamental work on constructive analysis [Bis67]. Bishop proposed to use
constructive analysis to bridge this gap. Moreover, we can narrow this gap by
using $\bullet$
exact real numbers or intervals instead of floating point numbers;
functional programming instead of imperative programming;
dependent type theory which allows us to compute with complete metric spaces;
a proof assistant which allows us to verify the correctness proofs;
constructive mathematics to tightly connect mathematics with computations and
to avoid computationally impossible case distinctions. Separately, all these
tools have proved itself. By going to the limits of this proven technology we
should be able to come within a small constant factor of floating point
computations. In this way one would obtain a tool suitable for research and
education in numerical analysis that allows one to compute abstractly at the
level of functional analysis, e.g. to compute fixed points of operators on
Hilbert spaces. Like the development of Fortran and MATLAB this will require a
huge amount of work. In the present paper we focus on the performance of real
number computation in the Coq proof assistant.
Real numbers, being infinite objects, cannot be represented exactly in a
computer. Hence, in constructive analysis [Bis67] one uses functions which
when fed a desired precision approximate a real numbers by a rational, or a
dyadic number, to within that precision222One could argue that we capture only
the definable, or computable, real numbers in this way. These issues are
important and well-studied, see for instance [TvD88], but we will not go into
them here.. The real numbers are the completion of the rationals. This
completion construction can be organized in a monad, a familiar construct from
functional programming. The completion monad provides an efficient combination
of proving and computing [O’C07]. In this way, O’Connor [O’C08] implements
exact real numbers and the transcendental functions on them in Coq. A number
of possible improvements in this implementation were already suggested in
[OS10, O’C09].
1. (1)
Use Coq’s new machine integers instead of the integers built from ordinary
inductive data types;
2. (2)
use dyadic rationals (that are numbers of the shape $n*2^{e}$ for
$n,e\in\mathbbm{Z}$, also known as infinitary floats) instead of ordinary
rationals;
3. (3)
use approximate division to improve the implementation of power series.
Here we carry out all three optimizations. Unfortunately, changing O’Connor’s
implementation to use the new machine integers was far from trivial, as he
used a particular concrete representation of the rationals. To avoid this in
the future, we moreover provide an abstract specification of the dense set as
_approximate rationals_. Finally, we obtain another dramatic speed-up by
avoiding evaluation of termination proofs at runtime.
### Outline
Section 2 describes some aspects of the Coq proof assistant relevant for our
development. Section 3 describes metric spaces, monads, and O’Connor’s
implementation of the real numbers [O’C07]. Section 4 extends Spitters and van
der Weegen’s approach to abstract interfaces using type classes [SvdW11].
Section 5 describes the theory of approximate rationals, our implementation of
the real numbers, and deals with computing power series and the square root.
We finish with some benchmarks in Section 6 and conclusions in Section 7. The
sources of our developments can be found at https://github.com/c-corn/corn.
## 2\. The Coq-system
The Coq proof assistant is based on the calculus of inductive constructions
[CH88, CP90], a dependent type theory with (co)inductive types; see [Coq12,
BC04]. In true Curry-Howard fashion, it is both a pure functional programming
language with an expressive type system, and a language for mathematical
statements and proofs. We highlight some aspects of Coq relevant for our
development.
### 2.1. Notations
Coq has an extensible mechanism for defining complex notations. We use this
mechanism heavily, together with unicode symbols, to obtain notations that are
closer to common mathematical practice. However, due to conflicts with
standard Coq syntax, there are some small deviations. For example, we write
$\forall x,Px$ instead of $\forall x.Px$. In this paper we tried to stay as
close as possible to the notations in our Coq development.
### 2.2. Types and propositions.
Propositions in Coq are types [ML98, ML82], which themselves have types called
_sorts_. Coq features a distinguished sort called Prop that one may choose to
use as the sort for types representing propositions. The distinguishing
feature of the Prop sort is that terms of non-Prop type may not depend on the
values of inhabitants of Prop types (that is, proof terms). This regime of
discrimination establishes a weak form of proof irrelevance, in that changing
a proof can never affect the result of value computations. On a practical
level, this lets Coq safely erase all Prop components when extracting
certified programs to Ocaml or Haskell. We should note however, that in
practice, Coq’s extraction mechanism [Let08] is still very hard to use for
programs with the complexity, in terms of depth of definitions, that we are
interested in [CFS03, CFL06].
### 2.3. Constructive indefinite description
In spite of the restriction on Prop discussed in the previous paragraph, Coq
allows recursive functions to use a value of Prop type to ensure termination
[BC04, 14.2.3, 15.4]. In particular, this is used to prove _constructive
indefinite description_ , which states that given a decidable predicate over
the natural numbers, a Prop based existential can be converted into a Type
based one. Its formal statement can be found in the standard library:
⬇
Lemma constructive_indefinite_description_nat (P : nat $\to\ $ Prop) :
($\forall\ $ x : nat, {P x} + {$\neg\ $P x}) $\to\ $ ($\exists\ $ n : nat, P n) $\to\ $ {n : nat | P n}
Here the notation $\\{x:A\;|\;P\ x\\}$ for
${P:A\textrightarrow\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Prop}}}}}}$
denotes a Σ-type. This lemma can be seen as a form of Markov’s principle in
Coq. The algorithm does a bounded search for a new witness satisfying the
predicate. The witness from the Prop based existential is only used to prove
termination of the search. No information flows from the Prop universe to the
Type universe because the witness found for the Type based existential is
independent of the witness from the Prop based one.
### 2.4. Equality, setoids, and rewriting
Because the Coq type theory lacks quotient types (as it endangers the
decidability of type checking), one usually bases abstract structures on a
_setoid_ (‘Bishop set’): a type equipped with an equivalence relation [Bis67,
Hof97]. This leads to a naive set theory as described by Palmgren [Pal09].
When the user attempts to substitute a given (sub)term using an equality, the
system keeps track of, resolves, and combines proofs of equivalence [Soz09].
The ‘native’ notion of equality in Coq, _Leibniz equality_ , is that of terms
being convertible, naturally reified as a proposition by the inductive type
family eq with single constructor eq_refl : $\forall\ $ (T : Type) (x : T),
x$\equiv\ $x, where a$\equiv\ $b is notation for eq T a b. Since
convertibility is a congruence, a proof of a$\equiv\ $b lets us substitute b
for a anywhere inside a term without further conditions. Our interest is in
more complicated equalities, so we diverge from Coq tradition and reserve =
for setoid equality. Rewriting with = _does_ give rise to side conditions. For
instance, consider formal fractions of integers as a representation of
rationals. Rewriting a subterm using such an equality is permitted only if the
subterm is an argument of a function that has been proven to _respect_ the
equality. Such a function is called Proper, and that property must be proved
for each function in whose arguments we wish to enable rewriting.
### 2.5. Type classes
Type classes are a great success in the Haskell functional programming
language, as a means of organizing interfaces of abstract structures. Coq’s
type classes provide a superset of their functionality, but are implemented in
a different way.
In Haskell and Isabelle, type classes and their instances are second class.
They are handled as specialized syntactic constructs whose semantics are given
specifically by the type class apparatus. By contrast, the expressivity of
dependent types and inductive families as supported in Coq, combined with the
use of pre-existing technology in the system (namely proof search and implicit
arguments) enable a _first class_ type class implementation [SO08]: classes
are ordinary record types (‘dictionaries’), instances are ordinary constants
of these record types (registered as _hints_ with the proof search machinery),
class constraints are ordinary implicit arguments, and instance resolution is
achieved by augmenting the unification algorithm to invoke ordinary proof
search for implicit arguments of class type. Thus, type classes in Coq are
realized by relatively minor syntactic aids that bring together existing
facilities of the theory and the system into a coherent idiom, rather than by
introduction of a new category of qualitatively different definitions with
their own dedicated semantics.
We use the algebraic hierarchy based on type classes and its abstract
specification of $\mathbbm{N},\mathbbm{Z}$ and $\mathbbm{Q}$ described in
[SvdW11]. Unfortunately, we should note that we have clearly met the
efficiency problems connected to the current implementation of type classes in
Coq. Luckily, these efficiency problems are limited to instance resolution
which is only performed at compile time. Type classes effect the computation
time of type checked terms due to the absence of code inlining. In an
illustrative example the use of type classes caused only a 3% performance
penalty; see Section 6.
### 2.6. Virtual machine and machine integers
Coq includes a virtual machine [GL02], vm_compute, based on Ocaml’s virtual
machine to allow efficient evaluation. Both the abstract machine and its
compilation scheme have been proved correct, in Coq, with respect to the weak
reduction semantics. However, we still need to extend our trusted core to a
bigger kernel, as the _implementation_ has not been formally verified.
Machine integers were also added to the Coq system [AGST10]. The usual
evaluation inside Coq (compute) uses a special inductive type for cyclic
integers, but the virtual machine (vm_compute) uses actual machine integers.
The type bigZ of arbitrary precision integers is built from binary trees of
these cyclic integers. Primality tests in [Spi11] show a big speed-up compared
to the inductively defined integers. Our work confirms this big speed-up
gained by using machine integers. We pay for this speed-up, however, by having
to trust the virtual machine and its translation to actual machine integers.
## 3\. Metric spaces and the completion monad
Having completed our brief description of the Coq-system, we now turn to
O’Connor’s formalization of exact real numbers [O’C07]. Traditionally, a
metric space is defined as a set $X$ with a metric function $d:X\times
X\textrightarrow{\mathbbm{R}}_{\geq 0}$ satisfying certain axioms. We use a
more relaxed definition of a metric space that does not require the metric be
a function; see also [Ric08]. The metric is represented via a (respectful)
ball relation ${\mathbf{B}:{\mathbbm{Q}}_{>0}\textrightarrow X\textrightarrow
X\textrightarrow\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Prop}}}}}}$
satisfying:333We use the positive rational numbers ${\mathbbm{Q}}_{>0}$
instead of the non-negative relation numbers ${\mathbbm{Q}}_{\geq 0}$ as it
can be expressed that two points $x$ and $y$ are within $0$ of each other by
$∀ε,\ \mathbf{B}_{ε}\,x\ y$.
⬇
msp_refl : $$\forall\ $ x\,$\epsilon$,\ \ball {$\epsilon$} x x$
msp_sym : $$\forall\ $ x\,y\,$\epsilon$,\ \ball {$\epsilon$} x y $\to\ $ \ball
{$\epsilon$} y x$
msp_triangle : $$\forall\ $ x\,y\,z\,$\epsilon$_1\,$\epsilon$_2,\ \ball
{$\epsilon$_1} x y $\to\ $ \ball {$\epsilon$_2} y z $\to\ $ \ball
{$\epsilon$_1 + $\epsilon$_2} x z$
msp_closed : $$\forall\ $ x\,y\,$\epsilon$,\ ($\forall\ $ $\delta$,\ \ball
{$\epsilon$ + $\delta$} x y) $\to\ $ \ball {$\epsilon$} x y$
msp_eq : $$\forall\ $ x\,y,\ ($\forall\ $ $\epsilon$,\ \ball {$\epsilon$} x y)
$\to\ $ x = y$
The ball relation $\mathbf{B}_{ε}\,x\ y$ expresses that the points $x$ and $y$
are within $ε$ of each other. We call this a ball relationship because the
partially applied relation
${\mathbf{B}^{X}_{ε}\,x:X\textrightarrow\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Prop}}}}}}$
is a predicate that represents the closed ball of radius $ε$ around the point
$x$. For example, the ball relation on $\mathbbm{Q}$ is
$\mathbf{B}^{\mathbbm{Q}}_{ε}\,x\ y:=|x-y|≤ε$.
A metric space $X$ is a _prelength_ space if:
$∀a\,b\,ε\,δ_{1}\,δ_{2},\ ε<δ1+δ2\textrightarrow\mathbf{B}_{ε}\,a\
b\textrightarrow∃c,\ \mathbf{B}_{δ_{1}}\,a\ c\ ∧\ \mathbf{B}_{δ_{2}}\,c\ b.$
In particular, a prelength space has approximate midpoints: for any $δ>0$ we
can take $δ_{1}=δ_{2}=ε/2+δ$. Every complete prelength metric space is a
length metric space. The metric space $\mathbbm{Q}$ is a prelength space; see
[O’C07] for details.
We will introduce the completion of a metric space as a monad. In order to do
this we will first introduce monads.
### 3.1. Monads
Moggi [Mog89] recognized that many non-standard forms of computation may be
modeled by monads444In category theory one would speak about the Kleisli
category of a (strong) monad.. Wadler [Wad92] popularized their use in
functional programming. Monads are now an established tool to structure
computation with side-effects. For instance, programs with input $X$ and
output $Y$ which have access to a mutable state $S$ can be modeled as
functions of type $X\times S\textrightarrow Y\times S$, or equivalently
$X\textrightarrow(Y\times S)^{S}$. The type constructor
$\mathfrak{M}Y:=(Y\times S)^{S}$ is an example of a monad. Similarly, partial
functions may be modeled by maps $X\textrightarrow Y_{\bot}$, where
$Y_{\bot}:=Y+()$ is a monad.
The formal definition of a (strong) monad is a triple
$(\mathfrak{M},\operatorname{\mathsf{return}},\operatorname{\mathsf{bind}})$
consisting of a type constructor $\mathfrak{M}$ and two functions:
$\displaystyle\operatorname{\mathsf{return}}$
$\displaystyle:X\textrightarrow\mathfrak{M}X$
$\displaystyle\operatorname{\mathsf{bind}}$
$\displaystyle:(X\textrightarrow\mathfrak{M}Y)\textrightarrow\mathfrak{M}X\textrightarrow\mathfrak{M}Y$
We will denote $\operatorname{\mathsf{return}}x$ as $\hat{x}$, and
$\operatorname{\mathsf{bind}}f$ as $\check{f}$. These two operations must
satisfy:
$\displaystyle\operatorname{\mathsf{bind}}\ \operatorname{\mathsf{return}}\ a$
$\displaystyle=a$ $\displaystyle\check{f}\,\hat{a}$ $\displaystyle=f\,a$
$\displaystyle\check{f}\,(\check{g}\,a)$
$\displaystyle=\operatorname{\mathsf{bind}}\ (\check{f}∘g)\,a$
### 3.2. Completion monad
The completion of a metric space $X$ is defined by:
$\mathfrak{C}X:=\\{f:{\mathbbm{Q}}_{>0}\textrightarrow X\;|\;∀ε_{1}\,ε_{2},\
\mathbf{B}_{ε_{1}+ε_{2}}\,(f\,ε_{1})\ (f\,ε_{2})\\}.$
Given metric spaces $X$ and $Y$, a function $f:X\textrightarrow Y$ is
_uniformly continuous_ with _modulus_
$μ_{f}:{\mathbbm{Q}}_{>0}\textrightarrow{\mathbbm{Q}}_{>0}$ if:
$∀ε\,x_{1}\,x_{2},\ \mathbf{B}_{μ_{f}ε}\,x_{1}\
x_{2}\textrightarrow\mathbf{B}_{ε}\,(f\,x_{1})\ (f\,x_{2}).$
Completion is a monad on the category of metric spaces with uniformly
continuous functions. The function
$\operatorname{\mathsf{return}}:X\textrightarrow\mathfrak{C}X$ defined by
$λx\,ε,\,x$ is the embedding of a metric space in its completion. Moreover, a
uniformly continuous function $f:X\textrightarrow\mathfrak{C}Y$ with modulus
$μ_{f}$ can be lifted to operate on complete metric spaces as
$\operatorname{\mathsf{bind}}f:\mathfrak{C}X\textrightarrow\mathfrak{C}Y$
defined by $λx\,ε,\,f\,(x\,(µ_{f}\frac{ε}{2}))\,\frac{ε}{2}$. A restriction to
prelength spaces is essential for this efficient definition of
$\operatorname{\mathsf{bind}}$; see [O’C07] for details.
One advantage of this approach is that it helps us to work with simple
representations. Let $\mathbbm{R}:=\mathfrak{C}\mathbbm{Q}$. Then to specify a
function from $\mathbbm{R}\textrightarrow\mathbbm{R}$, we define a uniformly
continuous function $f:\mathbbm{Q}\textrightarrow\mathbbm{R}$, and obtain
$\check{f}:\mathbbm{R}\textrightarrow\mathbbm{R}$ as the required function.
Hence, the completion monad allows us to do in a structured way what was
already folklore in constructive mathematics: to work with simple, often
decidable, approximations to continuous objects.
## 4\. Abstract interfaces using type classes
An important part of this work is to further develop the algebraic hierarchy
based on type classes by Spitters and van der Weegen [SvdW11]. Especially, we
extend their hierarchy with constructive fields, order theory and interfaces
for mathematical operations, such as shift and power, common in programming
languages. This layer of abstraction makes both proof engineering and
programming more flexible: it avoids duplication of code, it introduces a
canonical way to refer to operations and properties, both by names and
notations, and it allows us to easily swap different implementations of number
representations and their operations. First we will briefly recap the design
decisions made in [SvdW11]. For a nice tutorial following this design see
[CS12]. More information on type classes and setoids in Coq can also be found
in the reference manual [Coq12].
Algebraic structures are expressed in terms of a number of carrier sets, a
number of relations and operations, and a number of laws that these operations
and relations must satisfy. One way of describing such a structure is by a
_bundled representation_ (as used in [GPWZ02] for example): one uses a
dependently typed record that contains the carrier, operations and laws. A
setoid can be represented as follows.
⬇
Record Setoid : Type := {
st_car :> Type;
st_equiv : st_car $\to\ $ st_car $\to\ $ Prop;
st_setoid : Equivalence st_eq }.
Infix ”=” := st_equiv : type_scope.
The notation :> registers the projection st_car : Setoid $\to\ $ Type as a
coercion. Using the above interface for Setoids, one can now define a
SemiGroup whose carrier is a setoid.
⬇
Record SemiGroup : Type := {
sg_car :> Setoid;
sg_op : sg_car $\to\ $ sg_car $\to\ $ sg_car;
sg_proper : Proper (st_equiv $\Longrightarrow\ $ st_equiv $\Longrightarrow\ $
st_equiv) sg_op;
sg_ass : $\forall\ $ x y z, sg_op x (sg_op y z) = sg_op (sg_op x y) z) }
The field sg_proper states that the operation sg_op respects the setoid
equality. Its definition expands to $\forall\ $ x${}_{1}\ $ x${}_{2}\ $,
x${}_{1}\ $ = x${}_{2}\ $ $\to\ $ $\forall\ $ y${}_{1}\ $ y${}_{2}\ $,
y${}_{1}\ $ = y${}_{2}\ $ $\to\ $ sg_op x${}_{1}\ $ y${}_{1}\ $ = sg_op
x${}_{2}\ $ y${}_{2}\ $.
However, this approach has some serious limitations, the most important one
being a lack of support for _sharing_ components. For example, suppose we want
to group together two CommutativeMonoids in order to create a SemiRing. Now
awkward hacks are necessary to establish equality between the carriers555An
elegant solution is proposed by Pollack [Pol02]. However, its implementation
requires simultaneous inductive recursive definitions which are currently not
supported in Coq.. A second problem is that if we stack up these records to
represent higher structures the projection paths become increasingly long. In
the above example, the projection path to obtain the carrier of a semigroup G
is st_car (sg_car G), but for fields, this path will be much longer.
Historically these problems have been an acceptable trade-off because
_unbundled representations_ , in which the carrier and operations are
parameterized, introduce even more problems. An unbundled representation of a
semigroup is as follows.
⬇
Record SemiGroup A (Ae : A $\to\ $ A $\to\ $ Prop) (Aop : A $\to\ $ A $\to\ $
A) : Prop := {
sg_setoid : Equivalence Ae;
sg_op_proper : Proper (Ae $\Longrightarrow\ $ Ae $\Longrightarrow\ $ Ae) Aop;
sg_ass : $\forall\ $ x y z, Ae (Aop x (Aop y z)) (Aop (Aop x y) z) }
There is nothing to bind notation to, no structure inference, and declaring
and passing requires too much manual bookkeeping. Spitters and van der Weegen
have proposed a use of Coq’s new type class machinery that resolves many of
the problems of unbundled representations. Our current experiment confirms
that this is a viable approach.
An alternative solution is provided by packed classes [GGMR09] which use an
alternative, and older, implementation of a variant of type classes: canonical
structures; see also Section 7. Yet another approach would be to use modules.
However, as these are not first class, we would be unable to define, e.g.
homomorphisms between algebraic structures.
The first step of the approach of Spitters and van der Weegen is to define an
_operational type class_ for each operation and relation.
⬇
Class Equiv A := equiv: relation A.
Infix ”=” := equiv: type_scope.
Notation ”(=)” := equiv (only parsing).
Class SgOp A := sg_op: A $\to\ $ A $\to\ $ A.
Infix ”&” := sg_op (at level 50, left associativity).
Notation ”(&)” := sg_op (only parsing).
Haskell-style notations (=) and (&) are defined so operations and relations
can easily be used in partially applied position.
Now an algebraic structure is just a type class living in Prop that is
parametrized by its carrier, relations and operations. This class contains all
laws that the operations should satisfy. The class for semigroups is as
follows666We sometimes use the @ prefix to bypass implicit arguments in order
to avoid ambiguity..
⬇
Class Setoid A {Ae : Equiv A} : Prop := setoid_eq :> Equivalence (@equiv A
Ae).
Class SemiGroup A {Ae : Equiv A} {Aop: SgOp A} : Prop := {
sg_setoid :> @Setoid A Ae;
sg_op_proper :> Proper ((=) $\Longrightarrow\ $ (=) $\Longrightarrow\ $ (=))
(&);
sg_ass :> Associative (&) }.
Since the operations are unbundled we can easily support sharing. First we
make classes for the semiring operations and show that these are in fact
special instances of the group operations. For example777We use (.*.) instead
of (*) due to conflicting notations with Coq’s comments.:
⬇
Class Mult A := mult: A $\to\ $ A $\to\ $ A.
Infix ”*” := mult.
Notation ”(.*.)” := mult (only parsing).
Instance mult_is_sg_op ‘{f : Mult A} : SgOp A := f.
The SemiRing class is then as follows.
⬇
Class SemiRing A {Ae : Equiv A} {Aplus : Plus A}
{Amult : Mult A} {Azero : Zero A} {Aone : One A} : Prop := {
semiplus_monoid :> @CommutativeMonoid A Ae plus_is_sg_op zero_is_mon_unit;
semimult_monoid :> @CommutativeMonoid A Ae mult_is_sg_op one_is_mon_unit;
semiring_distr :> LeftDistribute (.*.) (+);
semiring_left_absorb :> LeftAbsorb (.*.) 0 }.
The syntax :> in the definition of SemiRing declares certain fields as
substructures888This syntax should not be confused with the similar syntax for
coercions in records (e.g. in the bundled representation of a SemiGroup on
page 4)., so that in any context where $(A,=,+,*,0,1)$ is known to be a
SemiRing, $(A,=,+,0)$ and $(A,=,∗,1)$ are automatically known to be
CommutativeMonoids (and so on, transitively, because instance resolution is
recursive). In our hierarchy, these substructures by themselves establish the
inheritance diagram as in Figure 1.
Without type classes it would be cumbersome to manually carry around the
arguments of the class. However, because these arguments are type classes
themselves, the type class machinery will perform that job for us. Therefore,
all arguments, except the carrier A are declared as implicit using the syntax
{x : X}, so the user does not have to specify them.
Proving that an actual structure is an instance of the SemiRing interface is
straightforward. First we define instances of the operational type classes.
⬇
Instance nat_equiv: Equiv nat := eq.
Instance nat_plus: Plus nat := Peano.plus.
Instance nat_mult: Mult nat := Peano.mult.
Instance nat_0: Zero nat := 0%nat.
Instance nat_1: One nat := 1%nat.
Here we see that instances are just ordinary constants of the class types.
However, we use the Instance keyword instead of Definition to let the type
class machinery register the instance. Now, to prove that the Peano naturals
are in fact a semiring, we just write:
⬇
Instance: SemiRing nat.
Proof. $\ldots$ Qed.
The implicit arguments of SemiRing nat are automatically inferred by instance
search. In order to type check SemiRing nat, it has to solve @SemiRing nat ?1
?2 ?3 ?4 ?5 with obligations ?1 : Equiv nat, …, ?5 : One nat. Since we have
declared instances nat_equiv : Equiv nat, …, nat_1 : One nat, type class
search will trivially solve these obligations. Thus SemiRing nat is actually
@SemiRing nat nat_equiv nat_plus nat_mult nat_0 nat_1 with all type class
constraints resolved.
The SemiRing type class can be used as follows.
⬇
Lemma example ‘{SemiRing A} x : 1 * x = x + 0\.
The backtick instructs Coq to automatically insert implicit declarations,
namely Ae Aplus Amult Azero Aone. It also lets us omit a name for the SemiRing
A argument itself. All of these arguments will be given automatically
generated names that we will never refer to. Furthermore, instance resolution
will automatically find instances of the operational type classes for the
written notations. Thus the above is really:
⬇
Lemma example {A Ae Aplus Amult Azero Aone} {P : @SemiRing A Ae Aplus Amult
Azero Aone} (x : A) :
@equiv A Ae
(@mult A Amult (@one A Aone) x)
(@plus A Aplus x (@zero A Azero)).
This approach to interfaces proved useful to formalize a standard algebraic
hierarchy. Combined with category theory and universal algebra, $\mathbbm{N}$
and $\mathbbm{Z}$ are represented as interfaces specifying an initial semiring
and initial ring [SvdW11].
⬇
Class NaturalsToSemiRing (A : Type) :=
naturals_to_semiring : $\forall\ $ B ‘{Mult B} ‘{Plus B} ‘{One B} ‘{Zero B}, A
$\to\ $ B.
Class Naturals A {Ae Aplus Amult Azero Aone} ‘{U : NaturalsToSemiRing A} := {
naturals_ring :> @SemiRing A Ae Aplus Amult Azero Aone;
naturals_to_semiring_mor :> $\forall\ $ ‘{SemiRing B}, SemiRing_Morphism
(naturals_to_semiring A B);
naturals_initial :> Initial (semirings.object A) }.
These abstract interfaces for the naturals and integers make it easy to change
the concrete representation in the future. As fields are not algebraic, no
such algebraic specification exists for the rational numbers. Hence, we choose
to specify $\mathbbm{Q}$ as the field of fractions of $\mathbbm{Z}$. More
precisely, $\mathbbm{Q}$ is specified as a field containing $\mathbbm{Z}$ that
moreover can be embedded into the field of fractions of $\mathbbm{Z}$.
⬇
Inductive Frac A {Ae : Equiv A} {Azero : Zero A} : Type :=
frac { num : A; den : A; den_ne_0 : den $\neq\ $ 0 }.
Class RationalsToFrac (A : Type) := rationals_to_frac : $\forall\ $ B
‘{Integers B}, A $\to\ $ Frac B.
Class Rationals A {Ae Aplus Amult Azero Aone Aneg Arecip} ‘{U :
!RationalsToFrac A} : Prop := {
rationals_field :> @DecField A Ae Aplus Amult Azero Aone Aneg Arecip;
rationals_frac :> $\forall\ $ ‘{Integers Z}, Injective (rationals_to_frac A
Z);
rationals_frac_mor :> $\forall\ $ ‘{Integers Z}, SemiRing_Morphism
(rationals_to_frac A Z);
rationals_embed_ints :> $\forall\ $ ‘{Integers Z}, Injective (integers_to_ring
Z A) }.
In current versions of Coq, inference of substructures is based on _backward_
reasoning. In our semiring example that means, each time a CommutativeMonoid A
instance is needed, instance search may try to find a SemiRing A instance.
This style of instance search presents some problems, as the following example
illustrates.
⬇
Class Setoid_Morphism {A B} {Ae : Equiv A} {Be : Equiv B} (f : A $\to\ $ B) :=
{
setoidmor_a :> Setoid A;
setoidmor_b :> Setoid B;
sm_proper :> Proper ((=) $\Longrightarrow\ $ (=)) f }.
Each time we have to establish Setoid R for some R, instance search might try
to infer a Setoid_Morphism from an arbitrary S to R, or vice versa. Since this
search quickly results in a serious blow-up, we omit the substructure
declaration :>. Support for _forward_ reasoning may solve this problem. If we
would be in a context in which we know something to be a Setoid_Morphism, then
forward reasoning automatically infers that the source and target are Setoids.
Recently, an initial implementation of forward reasoning has been added to
Coq, but it suffers from some other performance problems.
### 4.1. Constructive fields and apartness
In constructive mathematics, the common notion of inequality as the negation
of equality is often too weak because a proof of a negation lacks
computational content. For example, in order to define the reciprocal of
$x\in\mathbbm{R}$, one needs a witness $ε\in{\mathbbm{Q}}_{>0}$ that $|x|\geq
ε$. Such a witness cannot be extracted from a proof of $x≠0$. To solve this
problem, one uses a setoid equipped with an apartness (irreflexive, asymmetric
and co-transitive) relation describing inequality [TvD88].
The algebraic hierarchy in the CoRN library [CFGW04] has been built on top of
such setoids. Unfortunately, this hierarchy is quite ‘heavy’ in practice.
First, for structures with decidable equality, the negation of equality is the
only _tight_ apartness. Hence, when working with decidable structures, an
apartness relation is unnecessary. Secondly, CoRN uses an _informative_ (that
is, Type based) apartness relation to facilitate extraction of witnesses.
However, Coq’s present implementation of setoid rewriting does not support
rewriting over relations in Type. So, it does not allow us to replace
equations in expressions involving CoRN’s informative apartness and thus many
proofs involve a lot of manual labor.
To remedy these issues we propose an alternative solution. We use a _non-
informative_ (that is, Prop-based) apartness relation to enable setoid
rewriting and include it just in the parts of the algebraic hierarchy where we
actually need it. The latter keeps our interfaces clean and easy to use and
should combine the best of two worlds. Type classes are of great help to
reduce bookkeeping and clutter in proofs.
Although using a non-informative apartness relation enables setoid rewriting,
it disables extraction of witnesses. Fortunately, in case of the reals, a
witness can be obtained inefficiently by bounded linear search (see Section
2.3 and 5.1). We think our approach is a reasonable trade-off since the amount
of reasoning exceeds the potential use of apartness in computation. In case we
need a witness for efficient computation, we just have to specify it
explicitly. This approach of specifying witnesses explicitly was already
preferred by O’Connor [O’C08], even when an informative apartness was
available.
Our interface for a setoid with apartness (henceforth StrongSetoid) is as
follows.
⬇
Class Apart A := apart: relation A.
Infix ”$\mathrel{>\\!\\!<}\ $” := apart (at level 70, no associativity) :
type_scope.
Class StrongSetoid A {Ae: Equiv A} {Aap : Apart A} : Prop := {
strong_setoid_irreflexive :> Irreflexive ($\mathrel{>\\!\\!<}$) ;
strong_setoid_symmetric :> Symmetric ($\mathrel{>\\!\\!<}$) ;
strong_setoid_cotrans :> CoTransitive ($\mathrel{>\\!\\!<}$) ;
tight_apart : $\forall\ $ x y, $\neg\ $x $\mathrel{>\\!\\!<}\ $ y
$\leftrightarrow\ $ x = y }.
This interface is equipped with a _tight_ equality. We prove that each
StrongSetoid is a Setoid. For decidable structures, we define the following
class to describe that the apartness relation is the negation of equality.
⬇
Class TrivialApart A ‘{Equiv A} {ap : Apart A} := trivial_apart : $\forall\ $
x y, x $\mathrel{>\\!\\!<}\ $ y $\leftrightarrow\ $ x $\neq\ $ y.
Given a setoid with decidable equality we can easily extend it to a
StrongSetoid.
⬇
Instance default_apart ‘{Equiv A} : Apart A | 20 := ($\neq\ $).
Instance default_apart_trivial ‘{Equiv A} : TrivialApart A
(ap:=default_apart).
Lemma dec_strong_setoid ‘{Setoid A} ‘{Apart A}
‘{!TrivialApart A} ‘{$\forall\ $ x y, Decision (x = y)} : StrongSetoid A.
Unfortunately, the type class mechanism is unable to detect simple loops.
Hence we define dec_strong_setoid as an ordinary Lemma instead of an Instance.
This trick prevents Coq from using it in instance search and therefore avoids
endless derivations of the form StrongSetoid A, Setoid A, StrongSetoid A, …
For ordinary setoids we want functions to be Proper, which means that they
respect equality. For setoids with apartness we need a stronger property,
_strong extensionality_.
⬇
Class StrongSetoid_Morphism {A B} {Ae : Equiv A} {Aap : Apart A}
{Be: Equiv B} {Bap : Apart B} (f : A $\to\ $ B):= {
strong_setoidmor_a: StrongSetoid A;
strong_setoidmor_b: StrongSetoid B;
strong_extensionality : $\forall\ $ x y, f x $\mathrel{>\\!\\!<}\ $ f y $\to\
$ x $\mathrel{>\\!\\!<}\ $ y }.
We prove that for each StrongSetoid_Morphism f we have Proper ((=)
$\Longrightarrow\ $ (=)) f. The only structures for which we actually need
apartness are implementations of the real numbers, hence we only base the
Field class on top of a StrongSetoid instead of the complete algebraic
hierarchy. Our class for fields is as follows. (The PropHolds class is
explained in the next subsection.)
⬇
Class Recip A ‘{Apart A} ‘{Zero A} := recip: { x : A | x $\mathrel{>\\!\\!<}\ $ 0 } $\to\ $ A.
Notation ”// x” := (recip x).
Notation ”(//)” := recip (only parsing).
Class Field A {Ae Aplus Amult Azero Aone Aneg} {Aap : Apart A} {Arecip : Recip
A} : Prop := {
field_ring :> @Ring A Ae Aplus Amult Azero Aone Aneg;
field_strongsetoid :> StrongSetoid A;
field_plus_ext :> StrongSetoid_BinaryMorphism (+);
field_mult_ext :> StrongSetoid_BinaryMorphism (.*.);
field_nontrivial :> PropHolds (1 $\mathrel{>\\!\\!<}\ $ 0);
recip_proper :> Setoid_Morphism (//);
recip_inverse : $\forall\ $ x, proj1_sig x // x = 1 }.
We do not include strong extensionality of the inverse and the reciprocal
because these properties can be derived.
For convenience, we define an additional class DecField for fields with
decidable equality and whose reciprocal function is total. This class
integrates nicely with Coq’s rational numbers Q and bigQ, and the field tactic
to solve field equations. This total reciprocal function should satisfy
$/0=0$, so properties as $f(/x)=/(fx)$, $/(/x)=x$ and $/x*/y=/(x*y)$ hold
without any additional premises. We proved that a DecField is also an instance
of our Field class. A diagram of our complete algebraic hierarchy is displayed
in Figure 1.
(a) The algebraic hierarchy (b) The order hierarchy
Figure 1. The algebraic and order hierarchy. Dotted lines denote derived
inheritance, filled nodes denote presence of apartness.
### 4.2. Order theory
Existing Coq libraries for ordered algebraic structures turn out to be too
limited to abstract from $\mathbbm{N}$, $\mathbbm{Z}$, $\mathbbm{Q}$ and
$\mathbbm{R}$ and their various implementations. The formalization of ordered
fields in the CoRN library [CFGW04] restricts to a very specific part of the
algebraic hierarchy (namely fields). Letouzey’s Numbers library, which is
included in recent versions of Coq trunk, only considers $\mathbbm{N}$ and
$\mathbbm{Z}$. The Ssreflect library presently restricts to decidable
structures with Leibniz equality. Moreover, even mathematically, the most
convenient abstraction is not entirely clear. Lešnik [Les10] provides a smooth
order theoretic characterization of these structures as so-called _streaks_.
We, however, prefer our theory below as it avoids partial functions.
In this work we present a library that captures the notion of order on a
variety of structures, including structures with undecidable equality. One of
the building blocks of our hierarchy is a pseudo order [Hey56], which is the
constructive variant of a total order.
⬇
Class PseudoOrder ‘{Ae : Equiv A} ‘{Aap : Apart A} (Alt : Lt A) : Prop := {
pseudo_order_setoid : StrongSetoid A;
pseudo_order_asym : $\forall\ $ x y, $\neg\ $(x < y $\land\ $ y < x);
pseudo_order_cotrans :> CoTransitive (<);
apart_iff_total_lt : $\forall\ $ x y, x $\mathrel{>\\!\\!<}\ $ y
$\leftrightarrow\ $ x < y $\lor\ $ y < x }.
In case equality is decidable, this interface is rather awkward to work with.
Therefore we present ways to go back and forth between the usual classical
notions and their constructive variants. For example, we use the type class
machinery to infer the classical trichotomy property in case apartness is just
the negation of equality.
⬇
Instance lt_trichotomy ‘{PseudoOrder A} ‘{!TrivialApart A} ‘{$\forall\ $ x y,
Decision (x = y)} : Trichotomy (<).
Also, we can go the other way around. If we have a StrictSetoidOrder (an
ordinary strict order built upon a setoid) satisfying the trichotomy property,
we obtain a pseudo order.
⬇
Lemma dec_strict_pseudo_order ‘{StrictSetoidOrder A} ‘{Apart A}
‘{!TrivialApart A} ‘{$\forall\ $ x y, Decision (x = y)} ‘{!Trichotomy (<)} :
PseudoOrder (<).
In order to avoid loops, we define the above as an ordinary Lemma instead of
an Instance. Next, one could extend a pseudo order to the standard notion of a
(pseudo) ring order.
⬇
Class PseudoRingOrder ‘{Equiv A} ‘{Apart A} ‘{Plus A}
‘{Mult A} ‘{Zero A} ‘{One A} ‘{Negate A} (Alt : Lt A) := {
pseudo_ringorder_spo :> PseudoOrder Alt;
pseudo_ringorder_ring : Ring A;
pseudo_ringorder_mult_ext :> StrongSetoid_BinaryMorphism (.*.);
pseudo_ringorder_plus :> $\forall\ $ z, StrictlyOrderPreserving (z +);
pseudo_ringorder_mult : $\forall\ $ x y, 0 < x $\to\ $ 0 < y $\to\ $ 0 < x * y
}.
However, we wish to use our library on ordered structures for implementations
of the natural numbers as well. Since the natural numbers do not form a ring,
but merely a semiring, we strengthen the above class with a partial
subtraction function (living in Prop, because we never use it for
computations) and require addition to be order reflecting. We call this,
apparently new notion, a PseudoSemiRingOrder.
⬇
Class PseudoSemiRingOrder ‘{Equiv A} ‘{Apart A} ‘{Plus A}
‘{Mult A} ‘{Zero A} ‘{One A} (Alt : Lt A) := {
pseudo_srorder_strict :> PseudoOrder Alt;
pseudo_srorder_semiring : SemiRing A;
pseudo_srorder_partial_minus : $\forall\ $ x y, $\neg\ $y < x $\to\ $
$\exists\ $ z, y = x + z;
pseudo_srorder_plus :> $\forall\ $ z, StrictOrderEmbedding (z +);
pseudo_srorder_mult_ext :> StrongSetoid_BinaryMorphism (.*.);
pseudo_srorder_pos_mult_compat : $\forall\ $ x y,
PropHolds (0 < x) $\to\ $ PropHolds (0 < y) $\to\ $ PropHolds (0 < x * y) }.
Instead of including the PseudoRingOrder class in our development, we include
a lemma to construct a PseudoSemiRingOrder from a ring satisfying the
PseudoRingOrder axioms.
Given a pseudo (semiring) order, one could define the non-strict order x
$\leq\ $ y in terms of the strict order, namely as $\neg\ $y < x. However,
this is quite inconvenient in practice, because we also want to talk about a
priori different non-strict orders such as those defined in the standard
library. Hence we introduce the following class.
⬇
Class FullPseudoSemiRingOrder ‘{Equiv A} ‘{Apart A} ‘{Plus A}
‘{Mult A} ‘{Zero A} ‘{One A} (Ale : Le A) (Alt : Lt A) := {
full_pseudo_srorder_pso :> PseudoSemiRingOrder Alt;
full_pseudo_srorder_le_iff_not_lt_flip : $\forall\ $ x y, x $\leq\ $ y
$\leftrightarrow\ $ $\neg\ $y < x }.
A diagram of our complete order hierarchy is displayed in Figure 1.
Our theory on abstract orders avoids duplication of theorems and proofs. For
example, the following lemmas apply to $\mathbbm{N}$, $\mathbbm{Z}$,
$\mathbbm{Q}$ and the dyadics, because all of these structures form a
FullPseudoSemiRingOrder.
⬇
Lemma plus_compat x${}_{1}\ $ y${}_{1}\ $ x${}_{2}\ $ y${}_{2}\ $ : x${}_{1}\
$ $\leq\ $ y${}_{1}\ $ $\to\ $ x${}_{2}\ $ $\leq\ $ y${}_{2}\ $ $\to\ $
x${}_{1}\ $ + x${}_{2}\ $ $\leq\ $ y${}_{1}\ $ + y${}_{2}\ $.
Lemma lt_1_2 : 1 < 2\.
Lemma square_nonneg x : 0 $\leq\ $ x * x.
To allow us to refer by canonical names to common properties, we introduce
classes like:
⬇
Class OrderPreserving {A B} {Ae : Equiv A} {Ale : Le A} {Be : Equiv B} {Ble :
Le B} (f : A $\to\ $ B) := {
order_preserving_morphism :> Order_Morphism;
order_preserving : $\forall\ $ x y, x $\leq\ $ y $\to\ $ f x $\leq\ $ f y }.
Class OrderReflecting {A B} {Ae : Equiv A} {Ale : Le A} {Be : Equiv B} {Ble :
Le B} (f : A $\to\ $ B) := {
order_preserving_back_morphism :> Order_Morphism;
order_preserving_back : $\forall\ $ x y, f x $\leq\ $ f y $\to\ $ x $\leq\ $ y
}.
Here, an Order_Morphism is just the factoring out of the common parts of both
classes; namely that f and $\leq\ $ respect equality. For the case of
multiplication these properties have additional premises, for example:
⬇
Global Instance: $\forall\ $ (z : A), PropHolds (0 < z) $\to\ $
OrderPreserving (z *.).
We introduce the PropHolds class to let the type class machinery prove these
properties automatically. For example consider:
⬇
Lemma example (n : N) (x y : A) : x $\leq\ $ y $\to\ $ (2 ^ n + 2) * x $\leq\
$ (2 ^ n + 2) * y.
Proof. intros. now apply (order_preserving (2 ^ n + 2 .*)). Qed.
In order to use order_preserving, we need a proof of PropHolds (0 < 2 ^ n +
2). Type class resolution is able to prove this in a fully automated way
because we have the following instances:
⬇
Instance: PropHolds (0 < 2);
Instance: $\forall\ $ x y : A, PropHolds (0 < x) $\to\ $ PropHolds (0 < y)
$\to\ $ PropHolds (0 < x + y)
Instance: $\forall\ $ (n : N) (x : A), PropHolds (0 < x) $\to\ $ PropHolds (0
< x ^ n)
This example shows that type class search is in fact very similar to proof
search by the auto tactic, but there is no need to call a tactic by hand.
For arbitrary instances of $\mathbbm{N}$, $\mathbbm{Z}$, $\mathbbm{Q}$ it is
easy to define an order satisfying these interfaces:
⬇
Instance nat_le ‘{Naturals N} : Le N | 10 := $\uplambda\ $ x y, $\exists\ $ z, y = x + z.
Instance nat_lt ‘{Naturals N} : Lt N | 10 := $\uplambda\ $ x y, x $\leq\ $ y $\land\ $ x $\neq\ $ y.
However, often we encounter an a priori different order on a structure, most
likely an order defined in Coq’s standard library (like Nle and Nlt on N).
Therefore we prove that a FullPseudoSemiRingOrder uniquely specifies the order
on $\mathbbm{N}$, $\mathbbm{Z}$ and $\mathbbm{Q}$. For example:
⬇
Context ‘{Naturals N} ‘{Naturals N2} {f : N $\to\ $ N2} ‘{!SemiRing_Morphism
f}
‘{Apart N} ‘{!TrivialApart N} ‘{!FullPseudoSemiRingOrder (A:=N) Nle Nlt}
‘{Apart N2} ‘{!TrivialApart N2} ‘{!FullPseudoSemiRingOrder (A:=N2) N2le N2lt}.
Global Instance: OrderEmbedding f.
Unfortunately Coq has no support to have an argument be ‘inferred if possible,
generalized otherwise’; see [SvdW11]. When declaring an argument of
FullPseudoSemiRingOrder, one is often in a context where most of its
components are already available. Here, only the additional arguments Le, Lt
and Apart have to be introduced. The current workaround in these cases (as
shown above) involves providing names for components that are then never
referred to, which is a bit awkward. In the above it would much nicer to
write:
⬇
Context ‘{Naturals N} ‘{Naturals N2} {f : N $\to\ $ N2} ‘{!SemiRing_Morphism
f}
‘{!TrivialApart N} ‘{!FullPseudoSemiRingOrder N} ‘{!TrivialApart N2}
‘{!FullPseudoSemiRingOrder N2}.
Global Instance: OrderEmbedding f.
### 4.3. Basic operations
The operation nat_pow is most commonly, but inefficiently, defined as repeated
multiplication and the operation shiftl is defined as repeated duplication.
Instead we specify the desired behavior of these operations. This approach
allows for different implementations for different number representations and
avoids definitions and proofs becoming implementation dependent.
We introduce interfaces that specify the behavior of the operations abs,
shiftl, nat_pow and int_pow. Again there are various ways of specifying these
interfaces: with $\Sigma$-types, bundled or unbundled. In general,
$\Sigma$-types are convenient for functions whose specification is easy, for
example:
⬇
Class Abs A ‘{Equiv A} ‘{Le A} ‘{Zero A} ‘{Negate A}
:= abs_sig: $\forall\ $ (x : A), { y : A | (0 $\leq\ $ x $\to\ $ y = x) $\land\ $ (x $\leq\ $ 0 $\to\ $ y = -x)}.
Definition abs ‘{Abs A} := $\uplambda\ $ x : A, proj1_sig (abs_sig x).
However, for more complex operations, such as shiftl, we follow the unbundled
approach by Spitters and van der Weegen [SvdW11].
⬇
Class ShiftL A B := shiftl: A $\to\ $ B $\to\ $ A.
Infix ”$\ll\ $” := shiftl (at level 33, left associativity).
Class ShiftLSpec A B (sl : ShiftL A B) ‘{Equiv A} ‘{Equiv B} ‘{One A}
‘{Plus A} ‘{Mult A} ‘{Zero B} ‘{One B} ‘{Plus B} := {
shiftl_proper : Proper ((=) $\Longrightarrow\ $ (=) $\Longrightarrow\ $ (=))
($\ll$) ;
shiftl_0 :> RightIdentity ($\ll$) 0;
shiftl_S : $\forall\ $ x n, x $\ll\ $ (1 + n) = 2 * x $\ll\ $ n }.
We do not specify shiftl as shiftl x n = x * 2 ^ n since on the dyadics we
cannot take a negative power while we can shift by a negative integer. Since
theory on shifting with exponents in $\mathbbm{N}$ and $\mathbbm{Z}$ is
similar we want to avoid duplication of theorems and proofs. To this end we
introduce a class describing the bi-induction principle.
⬇
Class Biinduction A ‘{Equiv A} ‘{Zero A} ‘{One A} ‘{Plus A} : Prop
:= biinduction (P: A $\to\ $ Prop) ‘{!Proper ((=) $\Longrightarrow\ $ iff) P}
: P 0 $\to\ $ ($\forall\ $n, P n $\leftrightarrow\ $ P (1 + n)) $\to\ $
$\forall\ $ n, P n.
Since this class is inhabited by any integer and natural implementation we can
parametrize theory on shiftl as follows.
⬇
Context ‘{SemiRing A} ‘{!LeftCancellation (.*.) (2:A)} ‘{SemiRing B}
‘{!Biinduction B} ‘{!ShiftLSpec A B sl}.
Lemma shiftl_base_plus x y n : (x + y) $\ll\ $ n = x $\ll\ $ n + y $\ll\ $ n.
Global Instance shiftl_inj: $\forall\ $ n, Injective ($$\ll\ $$n).
### 4.4. Decision procedures
The Decision type class by Spitters and van der Weegen collects decidable
propositions [SvdW11].
⬇
Class Decision P := decide: sumbool P ($\neg\ $P).
Using this type class we can declare a argument ‘{$\forall\ $ x y, Decision (x
= y)} to describe a decider for $=$ and say decide (x = y) to decide whether x
= y or not. This type class allows us to easily compose deciders, for example:
⬇
Instance prod_dec ‘(A_dec : $\forall\ $ x y : A, Decision (x = y))
‘(B_dec : $\forall\ $ x y : B, Decision (x = y)) : $\forall\ $ x y : A * B,
Decision (x = y).
We have to be careful however. Consider the definition of the order on the
dyadics.
⬇
Global Instance dy_le: Le Dyadic := $\uplambda\ $ x y : Dyadic,
ZtoQ (mant x) * 2 ^ (expo x) $\leq\ $ ZtoQ (mant y) * 2 ^ (expo y)
Global Instance dy_le_dec: $\forall\ $ (x y : Dyadic), Decision (x $\leq\ $
y).
Now, decide (x $\leq\ $ y) for x and y of type Dyadic is actually @decide (x
$\leq\ $ y)$\ $(dy_le_dec x y). This shows that the proposition x $\leq\ $ y
is just a phantom argument used for instance search only, whereas dy_le_dec is
the decision procedure doing the actual work. Due to eager evaluation of Coq’s
virtual machine, the term decide (x $\leq\ $ y) is expanded to
@decide (ZtoQ (mant x) * 2 ^ (expo x) $\leq\ $ ZtoQ (mant y) * 2 ^ (expo y))$\
$(dy_le_dec x y),
resulting in the phantom argument being evaluated first. In many cases
evaluation of such a phantom argument is cheap, but here it involves an
expensive conversion of x and y to Q. We avoid evaluation of this phantom
argument by wrapping it under a λ-abstraction.
⬇
Definition decide_rel ‘(R : relation A) {dec : $\forall\ $ x y, Decision (R x
y)}
(x y : A) : Decision (R x y) := dec x y.
Now, if we write decide_rel ($\leq$) x y, it expands to
@decide_rel ($\uplambda\ $ x y, ZtoQ (mant x) * 2 ^ (expo x) $\leq\ $ ZtoQ
(mant y) * 2 ^ (expo y))$\ $x y dy_le_dec,
where the definition of inequality is safely hidden under a λ-abstraction.
This problem would not appear if Coq’s virtual machine would evaluate
propositions lazily, as the phantom argument is just a proposition.
Unfortunately, lazy evaluation of propositions is not supported by its current
implementation.
### 4.5. Explicit type casts
The Cast type class collects (explicit) type casts.
⬇
Class Cast A B := cast: A $\to\ $ B.
Implicit Arguments cast [[Cast]].
Notation ”’ x” := (cast _ _ x) (at level 20).
Instance: Params (@cast) 3\.
This definition allows us to refer to a cast from x : A to B by using an
apostrophe, or writing cast A B x. An example of an instance of this class is:
⬇
Instance NonNeg_inject: Cast (A≥0) A := @proj1_sig A _.
Here, ${A}_{\geq 0}$ is a $\Sigma$-type describing the non-negative cone of an
ordered ring $A$. Contrary to Coq’s built-in coercion mechanism, our type
casts are explicit instead of implicit and type classes are used to register
them. Our approach has some advantages:
1. (1)
By using type classes to register casts, we are allowed to parametrize classes
with casts. An example is the AppRationals class, as defined in Section 5.
2. (2)
Implicit coercions often introduce ambiguity. Since our approach allows us to
refer to casts by a (canonical) name, e.g. cast B C (cast A B x), we can avoid
this ambiguity.
3. (3)
Casts can be put in partially applied position, e.g. order_preserving (cast Z
Q).
Coq’s coercion mechanism does not allow us to define a coercion from
${A}_{\geq 0}$ to $A$ nor a coercion from a ring to its polynomial ring. More
generally, it does not allow most forms of parametrized coercions nor non-
uniform coercions. An implementation that allows parametrized coercions like
NonNeg_inject has to avoid an infinite loop: to naively type check x : A, one
has to type check x : A≥0, x : (A≥0)≥0, … We suffer from such loops if we
compose our Cast classes automatically as well. Hence we refrain from adding:
⬇
Instance cast_comp_base ‘{f : Cast A B} : ComposedCast A B := f.
Instance cast_comp_step ‘{f : Cast B C} ‘{g : ComposedCast A B} : ComposedCast
A C := $\uplambda\ $ x, f (g x).
Matita [ASCTZ07] allows parametrized coercions and avoids the loop by not
applying coercions recursively, but instead building a well-chosen set of set
of composite coercions [Tas08]. Non-uniform coercions [ST11] are available in
Matita. They are implemented using unification hints, a feature similar to
type classes.
## 5\. The real numbers
To make our implementation of the reals independent of the underlying dense
set, we provide an abstract specification of _approximate rationals_ inspired
by the notion of _approximate fields_ — a field with approximate operations —
which is used in Bauer and Kavler’s RZ implementation of the exact reals
[BK08]; see also [BT09]. In particular, we provide an implementation of this
interface by dyadics based on Coq’s machine integers.
Our interface for approximate rationals describes an ordered ring containing Z
that is dense in Q. Here Z are the binary integers from Coq’s standard
library, and Q are the rationals based on these binary integers. We do not
parametrize by arbitrary integer and rational implementations because they are
hardly used for computation. For efficient computation we include the
operations: approximate division, normalization, an embedding of Z, absolute
value, power by N, shift by Z, and decision procedures for equality and order.
⬇
Class AppDiv AQ := app_div: AQ $\to\ $ AQ $\to\ $ Z $\to\ $ AQ.
Class AppApprox AQ := app_approx: AQ $\to\ $ Z $\to\ $ AQ.
Class AppRationals AQ {AQe AQplus AQmult AQzero AQone AQneg} ‘{Apart AQ} ‘{Le
AQ} ‘{Lt AQ}
{AQtoQ : Cast AQ Q_as_MetricSpace} ‘{!AppInverse AQtoQ} {ZtoAQ : Cast Z AQ}
‘{!AppDiv AQ} ‘{!AppApprox AQ} ‘{!Abs AQ} ‘{!Pow AQ N} ‘{!ShiftL AQ Z}
‘{$\forall\ $ x y : AQ, Decision (x = y)} ‘{$\forall\ $ x y : AQ, Decision (x
$\leq\ $ y)} : Prop := {
aq_ring :> @Ring AQ AQe AQplus AQmult AQzero AQone AQneg;
aq_trivial_apart :> TrivialApart AQ;
aq_order_embed :> OrderEmbedding AQtoQ;
aq_strict_order_embed :> StrictOrderEmbedding AQtoQ;
aq_ring_morphism :> SemiRing_Morphism AQtoQ;
aq_dense_embedding :> DenseEmbedding AQtoQ;
aq_div : $\forall\ $ x y k, ball (2 ^ k) (’app_div x y k) (’x / ’y);
aq_compress : $\forall\ $ x k, ball (2 ^ k) (’app_approx x k) (’x);
aq_shift :> ShiftLSpec AQ Z ($\ll$) ;
aq_nat_pow :> NatPowSpec AQ N (^);
aq_ints_mor :> SemiRing_Morphism ZtoAQ }.
We define the real numbers as the completion of the approximate rationals. To
create functions on the real numbers, we use the monadic operations
$\operatorname{\mathsf{bind}}$ or $\operatorname{\mathsf{map}}$. This approach
is convenient because equality and inequality are decidable on the approximate
rationals, whereas it is not on the real numbers. For binary functions, e.g.
addition and multiplication, we use the map2 function, as described in
[O’C07].
O’Connor [O’C07] keeps the size of the rational numbers small to avoid
efficiency problems. He introduces a function approx x $\epsilon$ that yields
the ‘simplest’ rational number between x - $\epsilon$ and x + $\epsilon$. We
modify the approx function slightly: app_approx x k yields an arbitrary
element between x - $2^k$ and x + $ 2^k$. Using this function we define the
compress operation on the real numbers: compress := bind ($\uplambda\ $ x
$\epsilon$, app_approx x (Qdlog2 $\epsilon$)) such that compress x = x.
In Section 5.4 we will explain our choice of using a power of 2 to specify the
precision of app_div and app_approx.
### 5.1. Order and apartness
Following [BB85, O’C09], we define non-negativity and the order on the real
numbers as follows.
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{NonNeg}}}}}}\
x$ $\displaystyle:=∀ε:{\mathbbm{Q}}_{>0},-ε≤x\ ε$ $\displaystyle x≤y$
${\displaystyle:=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{NonNeg}}}}}}\
(y-x)$
Bishop and Bridges [BB85] define positivity as the dual of non-negativity:
$∃ε:{\mathbbm{Q}}_{>0},\ ε<x\ ε$. O’Connor [O’C09] defines positivity and the
strict order differently so as to avoid a potentially expensive computation,
namely $x\ ε-ε$, to obtain a witness between $0$ and $x$.
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Pos}}}}}}\
x$ $\displaystyle:=\\{ε:{\mathbbm{Q}}_{>0}\;|\;ε≤x\\}$ $\displaystyle
x<_{\textbf{T}}y$
${\displaystyle:=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Pos}}}}}}\
(y-x)$
We use the T subscript to emphasize that the relation lives in Type. Next, we
define $x\mathrel{>\\!\\!<}_{\textbf{T}}y:=x<_{\textbf{T}}y∨y<_{\textbf{T}}x$.
Extraction of a witness $ε\in(0,x]$ from
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{Pos}}}}}}\
x$ allows us to define the reciprocal function of type
$∀x:\mathbbm{R},0\mathrel{>\\!\\!<}_{\textbf{T}}x\textrightarrow\mathbbm{R}$.
In order to use our type class based hierarchy we need a strict order and
apartness relation in Prop. We need this restriction because Coq’s present
implementation of setoid rewriting does not allow rewriting in Type-based
relations (see Section 4.1). Our definition is similar to Bishop and Bridges’
definition of positivity, but uses shifts instead.
$\displaystyle x<y$ $\displaystyle:=∃n:\mathbbm{N},\ 1≪-n<(y-x)\ (1≪(-n-1))$
$\displaystyle x\mathrel{>\\!\\!<}y$ $\displaystyle:=x<y∨y<x$
Using constructive indefinite description (see Section 2.3), it is an easy job
to prove that we indeed have $x<y↔x<_{\textbf{T}}y$ and
$x\mathrel{>\\!\\!<}y↔x\mathrel{>\\!\\!<}_{\textbf{T}}y$. Similar to O’Connor
[O’C09], we implement a tactic that automatically proves strict inequalities.
The tactic terminates iff the inequality holds and is similar to our use of
linear search to obtain $x<_{\textbf{T}}y$ from $x<y$.
### 5.2. Implementation using the dyadics
The dyadic rationals are numbers of the shape $n*2^{e}$ for
$n,e\in\mathbbm{Z}$. In order to remain independent of a specific
implementation of integers, we have defined most of the operations for
arbitrary integer implementations. Given such an implementation Int it is
straightforward to define the ring operations.
⬇
Notation ”x $\upharpoonright$ p” := (exist _ x p) (at level 20).
Record Dyadic := dyadic { mant : Int; expo : Int }.
Infix ”$\blacktriangledown\;$” := dyadic (at level 80).
Global Instance dy_inject: Cast Int Dyadic := $\uplambda\ $ x, x
$\blacktriangledown\;$ 0\.
Global Instance dy_negate: Negate Dyadic := $\uplambda\ $ x, -mant x
$\blacktriangledown\;$ expo x.
Global Instance dy_mult: Mult Dyadic := $\uplambda\ $ x y, mant x * mant y
$\blacktriangledown\;$ expo x + expo y.
Global Instance dy_0: Zero Dyadic := cast Int Dyadic 0\.
Global Instance dy_1: One Dyadic := cast Int Dyadic 1\.
Global Program Instance dy_plus: Plus Dyadic := $\uplambda\ $ x y,
if decide_rel ($\leq$) (expo x) (expo y)
then mant x + mant y $\ll\ $ (expo y - expo x) $\upharpoonright$ _
$\blacktriangledown\;$ min (expo x) (expo y)
else mant x $\ll\ $ (expo x - expo y) $\upharpoonright$ _ + mant y
$\blacktriangledown\;$ min (expo x) (expo y).
In this code ($\ll$) has type Int $\to\ $ Int≥0 $\to\ $ Int, where Int≥0 is a
$\Sigma$-type describing the non-negative cone of Int. Therefore, in the
definition of dy_plus we have to equip expo y - expo x with a proof that it is
in fact non-negative.
The operation of approximate division is not implemented in an abstract way as
we have not developed a type class and theory for right shifts yet. For our
implementation using Coq’s machine integers bigZ, we defined approximate
division concretely using the shift right function from the standard library.
### 5.3. Implementation using the rationals
Our development contains additional implementations of the AppRationals class
using Coq’s old rational numbers Q and the new rational numbers bigQ (which
are built from the machine integers bigZ). Although creating these
implementations is uninteresting from a performance point of view, it confirms
that it is trivial to change the underlying dense set from which our real
numbers are built.
To implement the app_approx function in an efficient manner, we use shifts on
the underlying integers. Furthermore, to keep the size of the results of the
division operation small, we incorporate the app_approx function.
⬇
Instance bigQ_div: AppDiv bigQ := $\uplambda\ $ x y, app_approx (x / y).
### 5.4. Power series
Elementary transcendental functions as exp, sin, ln and atan can be defined by
their power series. If the coefficients of a power series are alternating,
decreasing and have limit 0, then we obtain a fast converging sequence with an
easy termination proof. For $-1≤x≤0$,
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
x=\sum_{i=0}^{\infty}\frac{x^{i}}{i!}$
is of this form. To approximate
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
x$ with error $ε$ we take the partial sum until $\frac{x^{i}}{i!}≤ε$. In order
to implement this efficiently we use a stream representing the series and
define a function that sums the required number of elements. For example, the
series 1, a, a$^2$, a$^3$, $\ldots$ is defined by the following stream.
⬇
CoFixpoint powers_help (c : A) : Stream A := Cons c (powers_help (c * a)).
Definition powers : Stream A := powers_help 1\.
Streams in Coq, like lists in Haskell, are lazy. So, in the example the
multiplications are accumulated.
Since Coq only allows structural recursion (and guarded co-recursion) it
requires some work to convince Coq that our algorithm terminates. Intuitively,
one would describe the limit as an upperbound of the required number of
elements using the Exists predicate.
⬇
Inductive Exists A (P : Stream A $\to\ $ Prop) (x : Stream) : Prop :=
| Here : P x $\to\ $ Exists P x
| Further : Exists P (tl x) $\to\ $ Exists P x.
This approach leads to performance problems. The upperbound, encoded in unary
format, may become very large while generally only a few terms are necessary.
Due to vm_compute’s eager evaluation scheme, this unary number will be
computed before summing the series. Instead O’Connor [O’C09] uses LazyExists.
⬇
Inductive LazyExists A (P : Stream A $\to\ $ Prop) (x : Stream A) : Prop :=
| LazyHere : P x $\to\ $ LazyExists P x
| LazyFurther : (unit $\to\ $ LazyExists P (tl x)) $\to\ $ LazyExists P x.
Unfortunately, our experiments showed that the above still yields too much
overhead due unnecessary to reduction of proofs. To remedy this issue we
introduce the following function where Str_nth_tl $n$ $s$ takes the $n$-th
tail of the stream $s$.
⬇
Fixpoint LazyExists_inc ‘{P : Stream A $\to\ $ Prop}
(n : nat) s : LazyExists P (Str_nth_tl n s) $\to\ $ LazyExists P s :=
match n return LazyExists P (Str_nth_tl n s) $\to\ $ LazyExists P s with
| O $\,\Rightarrow\ $ $\uplambda\ $ x, x
| S n $\,\Rightarrow\ $ $\uplambda\ $ ex, LazyFurther ($\uplambda\ $ _,
LazyExists_inc n (tl s) ex)
end.
This function adds n additional LazyFurther constructors. When instantiated
with a big enough n, computation will suffer from the implementation limits of
Coq (e.g. a stack overflow) or runs out of memory, before it ever refers to
the actual proof. Using LazyExists_inc we are able to compute on average twice
the amount of decimals as we did before on examples such as the ones in Table
2.
O’Connor’s InfiniteAlternatingSum $s$ returns the real number represented by
the infinite alternating sum over $s$, where the stream $s$ is decreasing,
non-negative and has limit 0. We extend this in two ways. First, we generalize
various notions to abstract structures. Secondly, as we do not have exact
division on approximate rationals, we extend the algorithm to work with
approximate division. The latter requires changing InfiniteAlternatingSum $s$
to InfiniteAlternatingSum $n\ d$ which computes the infinite alternating sum
of the stream $λi,\frac{n_{i}}{d_{i}}$. This allows us to postpone divisions.
Also, we have to determine both the length of the partial sum and the required
precision of the divisions. To do so we find a $k$ such that:
${{\mathbf{B}_{\frac{ε}{2}}\,(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{app\textunderscore
div}}}}}}\ n_{k}\ d_{k}\
(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\frac{ε}{2k})+\frac{ε}{2k})\
0.$ (1)
Now $k$ is the length of the partial sum, and $\frac{ε}{2k}$ is the required
precision of division. Using O’Connor’s results we have verified that these
values are correct and such a $k$ indeed exists for a decreasing, non-negative
stream with limit 0.
As noted in Section 5, we have specified the precision of division in powers
of 2 instead of using a rational value. This allows us to replace (1) with:
${{{\mathbf{B}_{\frac{ε}{2}}\,(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{app\textunderscore
div}}}}}}\ n_{k}\ d_{k}\
(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\
ε-(k+1))+1≪(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\
ε-(k+1)))\ 0.$
Here $k$ is the length of the partial sum, and $2^{l}$, where
${l=\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\
ε-(k+1)$, is the required precision of division. This variant can be
implemented without any arithmetic on the rationals and is thus much more
efficient.
This method gives us a fast way to compute the infinite alternating sum, in
practice, only a few extra terms have to be computed and due to the
approximate division the auxiliary results are kept as small as possible.
Similarly, using this method to compute infinite alternating sums, we use the
following series to implement
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
x$ and
${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
x$ for $x\in[-1,1]$.
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
x$ $\displaystyle=\sum_{i=0}^{\infty}(-1)^{i}*\frac{x^{2i+1}}{(2i+1)!}$
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
x$ $\displaystyle=\sum_{i=0}^{\infty}(-1)^{i}*\frac{x^{2i+1}}{2i+1}$
We extend these functions to their complete domain by repeatedly applying the
following formulas [O’C09].
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
x$
${\displaystyle=(\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
(x≪1))^{2}$ (2)
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
x$
${\displaystyle=\frac{1}{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\
(-x)}$ (3)
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
x$
${{\displaystyle=3*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
\frac{x}{3}-4*\Big{(}\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
\frac{x}{3}\Big{)}^{3}$ (4)
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
x$
${\displaystyle=-\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
(-x)$ (5)
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
x$
${\displaystyle=\frac{\pi}{2}-\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
\frac{1}{x}\quad\textnormal{for $0<x$}$ (6)
${\displaystyle\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
x$
${\displaystyle=\frac{\pi}{4}-\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\
\Big{(}\frac{x-1}{x+1}\Big{)}\quad\textnormal{for $0<x$}$ (7)
Since we do not have exact division on the approximate rationals, we
parameterize infinite sums by two streams in Equation 4, 6 and 7.
The series described in this section converge faster for arguments closer to
0. We use Equation 2 and 4 repeatedly to reduce the input to a value
$|x|\in[0,2^{k})$. For $50≤k$, this yields nearly always major performance
improvements, for higher precisions setting it to $75≤k$ yields even better
results. Unfortunately, we are unaware of a similar trick for atan. We define
$\pi$ in terms of atan using the following Machin-like formula.
${{{{\pi:=176*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\frac{1}{57}+28*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\frac{1}{239}-48*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\frac{1}{682}+96*\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\frac{1}{12943}$
Again, here we notice the purpose of parameterizing infinite sums by two
streams. We define cos in terms of sin.
${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{cos}}}}}}\
x=1-2*\Big{(}\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\
\frac{x}{2}\Big{)}^{2}\\\ $
O’Connor [O’C07, O’C09] subtracts multiples of $2\pi$ to reduce the arguments
of sin and cos. In our tests this did not lead to performance improvements
because our implementation of $\pi$ turned out to be slower than the performed
range reductions.
### 5.5. Square root
We use Wolfram’s algorithm [Wol02, p.913] for computing the square root. Its
complexity is linear, in fact it provides a new binary digit in each step.
⬇
Context ‘(Pa : 1 $\leq\ $ a $\leq\ $ 4).
Fixpoint AQroot_loop (n : nat) : AQ * AQ :=
match n with
| O $\,\Rightarrow\ $ (a, 0)
| S n $\,\Rightarrow\ $
let (r, s) := AQroot_loop n in
if decide_rel ($\leq$) (s + 1) r
then ((r - (s + 1)) $\ll\ $ (2:Z), (s + 2) $\ll\ $ (1:Z))
else (r $\ll\ $ (2:Z), s $\ll\ $ (1:Z))
end.
We write $(r_{n},s_{n})$ for the $n$-th pair of approximations. By induction
we obtain:
$\displaystyle s_{n}^{2}+4r_{n}$ $\displaystyle=4^{n+1}a$ (8) $\displaystyle
r_{n}$ $\displaystyle≤2s_{n}+4$ (9) $\displaystyle 2^{m}s_{n}≤s_{n+m}$
$\displaystyle≤2^{m}(s_{n}+4)-4$ (10) $\displaystyle r_{n}$
$\displaystyle≤2^{3+n}$ (11)
By 8, $(2^{-(n+1)}s_{n})^{2}+2^{-2n}r_{n}=a$. By 11, $2^{-2n}r_{n}$ converges
to 0 as $n$ tends to $\infty$. Therefore, by 10, $2^{-(n+1)}s_{n}$ is a Cauchy
sequence which moreover converges to $\sqrt{a}$.
We extend the square root to its entire domain by repeatedly applying:
$\sqrt{x}=2*\sqrt{\frac{x}{4}}$
O’Connor’s Coq implementation [O’C08] includes the much faster Newton
iteration, whose complexity is logarithmic in the number of decimals. The
function to iterate is:
⬇
Definition f (r : Q) : Q := r / 2 + a / (2 * r).
Because of the absence of exact division on our approximate rationals we
cannot implement this function directly. However, we can implement it on our
real numbers. As the above definition does not use sharing, we have to define
this function on the reals by first defining:
⬇
Definition f (r : AQ) ($\epsilon$ : Qpos) : AQ := (r + approx_div (Qdlog2
$\epsilon$) a r) $\ll\ $ (-1).
and then showing that it gives rise to a continuous function
${{f:\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{AQ}}}}}}\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{AR}}}}}}$
which we finally lift to a function ${{\operatorname{\mathsf{bind}}\
f:\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{AR}}}}}}\to\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{AR}}}}}}$
on the reals. In this way we take care of sharing, division and intermediate
use of the approx function (see Section 5) all in one go. We hope the future
correctness proof to be quite smooth, since we work with _exact_ real numbers.
We have implemented this in Haskell and it performs really well.
## 6\. Benchmarks
The first step in this research was to create a Haskell prototype based on
O’Connor’s implementation of the real numbers in Haskell [O’C07]. The second
step was to implement and verify this prototype in Coq. Our Coq development
contains verified versions of: the field operations, exponentiation by a
natural, computation of power series, exp, atan, sin, cos, $\pi$ and the
square root.
In this section we present some benchmarks, taken from the ‘Many Digits’
friendly competition [NW09], comparing the old and the new implementation,
both in Haskell and Coq. All benchmarks have been carried out on an Intel Core
Quad 2.4 GHz with 8GB of memory running Debian GNU/Linux. The sources of our
developments can be found at https://github.com/c-corn/corn.
| Expression | Decimals | Old | New
---|---|---|---|---
P01 | ${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ 1))$ | 5.000 | 25s | 2.3s
P02 | $\sqrt{\pi}$ | 5.000 | 3.3s | 1.7s
P03 | sin e | 5.000 | 13s | 1.2s
P04 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ (\pi*\sqrt{163})$ | 5.000 | 22s | 2.0s
P05 | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ \textit{e})$ | 5.000 | 43s | 2.6s
P06 | ${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\ (1+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\ (1+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\ (1+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\ (1+\pi))))$ | 500 | 107s | 2.5s
P07 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ 1000$ | 20.000 | 1.1s | 0.7s
P08 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{cos}}}}}}\ (10^{50})$ | 20.000 | 6.7s | 1.4s
P09 | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ (3*\frac{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{log}}}}}}\ 640320}{\sqrt{163}})$ | 5.000 | 33s | 16s
P11 | ${{{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{tan}}}}}}\ \textit{e}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atan}}}}}}\ \textit{e}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{tanh}}}}}}\ \textit{e}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{atanh}}}}}}\ \frac{1}{\textit{e}}$ | 500 | 41s | 3.2s
P12 | ${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{asin}}}}}}\ \frac{1}{\textit{e}}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{cosh}}}}}}\ \textit{e}+\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{asinh}}}}}}\ \textit{e}$ | 500 | 99s | 3.2s
Table 1. Haskell, compiled with ghc version 6.12.1, using -O2. The column ‘old’ refers to the Haskell prototype of O’Connor, and the column ‘new’ to our Haskell prototype. | Expression | Decimals | Old | New | Decimals | New
---|---|---|---|---|---|---
P01 | ${{{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{sin}}}}}}\ 1))$ | 25 | 46s | 0.6s | 500 | 3.8s
P02 | $\sqrt{\pi}$ | 25 | 0.3s | 0.03s | 500 | 6.8s
P03 | sin e | 25 | 36s | 0.1s | 500 | 1.9s
P04 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ (\pi*\sqrt{163})$ | 10 | 214s | 0.1s | 500 | 3.7s
P05 | ${{\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ (\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ \textit{e})$ | 10 | 36s | 0.2s | 500 | 3.2s
P07 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{exp}}}}}}\ 1000$ | 10 | 2662s | 1.0s | 2.000 | 4.9s
P08 | ${\mbox{\leavevmode\lstinline{{\lst@@@set@language\lst@@@set@numbers\lst@@@set@frame\lst@@@set@rulecolor\lst@@@set@language{\@listingGroup{ltx_lst_identifier}{cos}}}}}}\ (10^{50})$ | 25 | 11s | 0.3s | 2.000 | 12s
Table 2. Coq trunk, revision 14023. The column ‘old’ refers to the Coq
implementation of O’Connor, and the column ‘new’ to our Coq implementation.
Computations using a higher precision did not terminate within a reasonable
time using O’Connor’s implementation, so these are omitted.
Table 1 shows some benchmarks in Haskell with compiler optimizations enabled
(-O2) and Table 2 compares our Coq implementation with O’Connor’s. More
extensive benchmarking shows that our Haskell implementation generally
benefits from a 15 times speed up while the speed up in Coq is generally more
than a 100 times for small examples already. This difference between the
comparison of the Haskell and the Coq implementation is explained by the fact
that O’Connor’s Haskell implementation already uses rational numbers built
from fast integers and incorporates various optimizations, while his Coq
implementation does not. The last column of Table 2 indicates that our new
implementation is able to compute an order of magnitude more decimals in the
same amount of time.
We also compared the new reals built from Coq’s fast rationals (Section 5.3)
and our dyadic rationals (Section 5.2). For exp, sin and cos we obtain quite
similar results due to the our range reductions to reduce the length of the
power series. In case of the square root, the dyadics rationals are much
faster because wolfram iteration is designed for an efficient shift. It is
interesting to notice that $\pi$ and atan benefit the least from our
improvements, as we are unaware of range reductions to reduce the length of
the series.
We conclude this section with a comparison between the performance of
Wolfram’s algorithm in Coq and Haskell. The Haskell prototype (without
compiler optimizations) is quite fast, computing 10,000 iterations (giving
3,010 decimals) of $\sqrt{2}$ takes 0.2s. In Coq it takes 7.4s using type
classes and 7.2s without type classes. Here we exclude the time spend on type
class resolution. Thus type classes cause only a 3% performance penalty on
computations, which is very acceptable for the modularity that they introduce.
Unfortunately, the Coq implementation is slow compared to Haskell. Laurent
Théry suggested that this is due to the representation of the fast integers,
which uses a tree with a fixed depth and when the size of the integer becomes
too big uses a less optimal representation. Increasing the size of the tree
representation and avoiding an inefficiency in the implementation of shifts
reduces this time to 5.4s.
## 7\. Conclusions and Related work
We have greatly improved the performance of real number computation in Coq
using Coq’s new machine integers. We produced highly structured and abstract
code using type classes with no apparent performance penalty. Moreover, Coq’s
notation mechanism combined with unicode characters gives nicely readable
statements and proofs. Type classes were a great help in our work. However,
the current implementation of instance resolution is still experimental and at
times too slow (at compile time).
Canonical structures provide an alternative, and partially complementary,
implementation of type classes [GZND11]. By choice, canonical structures
restrict to deterministic proof search, this makes them more efficient, but
also somewhat more intricate to use. The use of canonical structures by the
Ssreflect team [GGMR09] makes it plausible that with some effort we could have
used canonical structures for our work instead. However, the Ssreflect-library
is currently not suited for setoids which are crucial to us. The integration
of unification hints [ARCT09] into Coq may allow a tighter integration of type
classes and canonical structures.
We needed to adapt our correctness proofs to prevent the virtual machine from
eagerly evaluating them. Lazy evaluation for Prop would have allowed us to use
the original proofs. Moreover, setoid rewriting over relations in Type would
have made our work much easier.
The experimental native_compute by Boespflug, Dénès and Grégoire [BDG11]
performs evaluation by compilation to native Ocaml code. This approach uses
the Ocaml compiler available and is interesting for heavy compilation. Our
first experiments indicate an additional speed up of 3 times compared to
vm_compute.
The Flocq project [BM11] formalizes infinitary floating-points in Coq. It
provides a library of theorems on multi-radix multi-precision arithmetic and
supports efficient numerical computations inside Coq. However, the current
library is still too limited for our purposes, but in the future it should be
possible to show that they form an instance of our approximate rationals. This
may allow us to gain some speed by taking advantage of fine grained algorithms
instead of our more straightforward ones.
The encoding of real numbers as streams of ‘bits’ is potentially interesting.
However, currently there is a big difference in performance. The computation
of 37 decimals of the square root of 1/2 by Newton iteration [JP09], using the
framework described in [Ber07, Jul08], took 12s. This should be compared with
our use of the Wolfram iteration, which gives only linear convergence, but
with which we nevertheless obtain 3,000 decimals in a similar time. On the
other hand, the efficiency of $\pi$ in their framework is comparable with
ours. Berger [Ber09], too, uses co-induction for exact real computation.
The present work is part of a larger program to use constructive mathematics
based on type theory as a programming language for exact analysis. This should
culminate in a numerical ODE-solver. To do so we need to extend the current
technology to functional analysis. For instance we will build a type class
interface for metric spaces in order to treat various function spaces.
Cohen and Mahboubi [Coh12, CM12] provide a formalization of quantifier
elimination for the theory of decidable real closed fields, and an
implementation of the algebraic real numbers. Quantifier elimination will
automate many proofs in constructive analysis which only involve algebraic
real numbers. Conversely, our implementation could be used for efficiently
evaluating a Cauchy representation of an algebraic real number.
### Acknowledgements
We thank Eelis van der Weegen for many discussions and Pierre Letouzey and
Matthieu Sozeau for closing some of our bug reports. We are grateful to the
anonymous referees who helped to improve the presentation of the paper.
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|
arxiv-papers
| 2011-06-17T11:20:20 |
2024-09-04T02:49:19.737323
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Robbert Krebbers and Bas Spitters",
"submitter": "J\\\"urgen Koslowski",
"url": "https://arxiv.org/abs/1106.3448"
}
|
1106.3456
|
# Conditional and Unique Coloring of Graphs
P. Venkata Subba Reddy and K. Viswanathan Iyer
Dept. of Computer Science and Engg.
National Institute of Technology
Tiruchirapalli - 620015, India
E-mail: venkatpalagiri@gmail.com, vichuiyer@gmail.com
author for correspondence
###### Abstract
For integers $k,r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a
proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of
degree $d(v)$ in $G$ is adjacent to at least $\min\\{r,d(v)\\}$ differently
colored vertices. Given $r$, the smallest integer $k$ for which $G$ has a
conditional $(k,r)$-coloring is called the $r$th order conditional chromatic
number $\chi_{r}(G)$ of $G$. We give results (exact values or bounds for
$\chi_{r}(G)$, depending on $r$) related to the conditional coloring of some
graphs. We introduce _unique conditional colorability_ and give some related
results.
Keywords. cartesian product of graphs; conditional chromatic number; gear
graph; join of graphs; uniquely colorable graphs
Subject Classification 68R10, 05C15
## 1 Introduction
Let $G=(V(G),E(G))$ be a simple, connected, undirected graph. For a vertex
$v\in V(G)$, the neighborhood of $v$ in $G$ is defined as $N_{G}(v)$= {$u\in
V(G):(u,v)\in E(G)$}, and the degree of $v$ is denoted by $d(v)$=$|N_{G}(v)|$.
The closed neighborhood of $v$ is defined as $N_{G}[v]=N_{G}(v)\cup\\{v\\}$.
For an integer $k>0$, a proper $k$-coloring of a graph $G$ is a surjective
mapping $c\colon V(G)\to\\{1,2,\ldots,k\\}$ such that if $(u,v)\in E(G)$ then
$c(u)\neq c(v)$. The smallest $k$ such that $G$ has a proper $k$-coloring is
the chromatic number $\chi(G)$ of $G$. For a set $S\subseteq V(G)$ we define
$c(S)=\\{c(u):u\in S\\}$.
Definition 1 For integers $k,r>0$, a conditional $(k,r)$-coloring of $G$ is a
surjective mapping $c\colon V(G)\to\\{1,2,\ldots,k\\}$ such that both the
following conditions (C1) and (C2) hold:
> (C1) if $(u,v)\in E(G)$, then $c(u)\neq c(v)$.
> (C2) for any $v\in V(G)$, $|c(N_{G}(v))|\geq$ min {$d(v),r$}.
Given an integer $r>0$, the smallest integer $k$ such that $G$ has a proper
$(k,r)$-coloring is called the rth-order conditional chromatic number of $G$,
denoted by $\chi_{r}(G)$. It is proved in [4] that the conditional
$(k,r)$-coloring problem of a graph is $NP$-complete. Two of our earlier work
can be found in [5, 6]. We follow commonly used terminology and notations (see
for example [1, 2, 7]).
## 2 Unique conditional colorability
If $\chi(G)=k$ and every $k$-coloring of $G$ induces the same partition of
$V(G)$, then $G$ is called uniquely k-colorable. In a similar way we define
unique $(k,r)$-colorability of graphs.
Definition 2 If $\chi_{r}(G)=k$ and every conditional $(k,r)$-coloring of $G$
induces the same partition of $V(G)$, then $G$ will be called uniquely
$(k,r)$-_colorable_.
###### Proposition 1.
If $G$ is uniquely $n$-colorable and $r\leq n-1$ then $\chi_{r}(G)=n$.
###### Proof.
Since $G$ is uniquely $n$-colorable, let $c\colon V(G)\to\\{1,2,\ldots,n\\}$
be the proper coloring of $G$ and w.l.o.g. for $1\leq i\leq n$ let the color
class $C_{i}$ be defined as $C_{i}=\\{v:c(v)=i\\}$. For all $u\in V(G)$ if
$u\in C_{i}$ then for all $j\in\\{1,2,\ldots,n\\}$ there exists a $v\in
C_{j}\;(j\neq i)$ – this implies that for every $u\in V(G)$, $d(u)\geq n-1$
and $|c(N(u))|=n-1$. Note that $c$ is also a conditional $(n,r)$-coloring of
$G$ because (C2) is also satisfied as for every $u\in V(G),|c(N(u))|=n-1\geq$
min $\\{r,d(u)\\}$. ∎
The definition of conditional $(k,r)$-coloring of $G$ and Proposition 2.1
together imply:
###### Corollary 1.
Every uniquely $n$-colorable graph $G$ is also uniquely $(n,n-1)$-colorable.
###### Proposition 2.
For every $k\geq 3$, there exists a uniquely $(3,2)$-colorable graph $G_{k}$
with $k+2$ vertices.
###### Proof.
We take $G_{1}$ to be $C_{3}$. Suppose that $k\geq 1$ and assume that $G_{k}$
has been obtained. From $G_{k}$, we construct $G_{k+1}$ by introducing a new
vertex $w$. The vertex and edge sets of $G_{k+1}$ are defined thus:
$\displaystyle V(G_{k+1})$
$\displaystyle=V(G_{k})\cup\\{w\\},\text{where}\;w\notin V(G_{k}).$
$\displaystyle E(G_{k+1})$
$\displaystyle=E(G_{k})\cup\\{(u,w),(v,w)\\},\text{where}\ (u,v)\in E(G_{k}).$
Evidently, $|V(G_{k})|=k+2$. We need to show that for any $k$, $G_{k}$ is
uniquely $(3,2)$-colorable. We prove the result by induction on $k$. Clearly
$G_{1}$ is uniquely $(3,2)$-colorable. For some $k>1$, assume $G_{k}$ is
uniquely $(3,2)$-colorable. Then $G_{k+1}$ is also uniquely $(3,2)$-colorable
because the new vertex $w\in G_{k+1}$ is assigned a third color different from
its two neighbors which are adjacent; by the inductive hypothesis the result
follows. ∎
###### Proposition 3.
Every path $P_{n}\;(n\geq 3)$ is uniquely $(3,2)$-colorable.
###### Proof.
From [3] we get $\chi_{2}(P_{n})=3$ . Define a conditional $(3,2)$-coloring
$c\colon V(P_{n})\to\\{1,2,3\\}$ by
$\displaystyle c^{-1}(1)=C_{1}$ $\displaystyle=\\{v_{i}:i\bmod 3=1\\},$
$\displaystyle c^{-1}(2)=C_{2}$ $\displaystyle=\\{v_{i}:i\bmod 3=2\\},$
$\displaystyle c^{-1}(3)=C_{3}$ $\displaystyle=\\{v_{i}:i\bmod 3=0\\},$
where, $C_{1}$,$C_{2}$ and $C_{3}$ are the color classes. In the conditional
$(3,2)$-coloring of $P_{n}$, for any two vertices $v_{i},v_{j}\;(i\neq j)$ in
$V(P_{n})$ if $|i-j|\bmod 3=0$ then $v_{i}$ and $v_{j}$ must be colored same;
otherwise either (C1) will be violated at $v_{min\\{i,j\\}+1}$ or (C2) will be
violated at $v_{max\\{i,j\\}-1}$. Since $c$ is the only coloring wherein for
all $v_{i},v_{j}\in V(P_{n})$, $c(v_{i})=c(v_{j})$, if $|i-j|\bmod 3=0$,
$P_{n}$ is uniquely $(3,2)$-colorable. ∎
###### Proposition 4.
If $T$ ($\neq P_{n}$) is a rooted tree with $n$ verices and $k=\chi_{r}(T)$
then $T$ is not uniquely $(k,r)$-colorable unless $k=n$.
###### Proof.
Let the root of $T$ be a vertex of degree $\Delta(T)$. Let $c\colon
V(T)\to\\{1,2,\ldots,k\\}$ be a conditional $(k,r)$-coloring of $T$. We show
that a new conditional $(k,r)$-coloring of $T$ can be obtained based on $c$ if
$k\neq n$. We know that every tree $T$ has at least two vertices say, $u,v$
with degree less than two. Let $p(v)$ deonte the parent of $v$. We have the
following cases:
Case 1: $p(u)=p(v):$
Case 1.1: $c(u)=c(v):$ Since $r\geq 2$ assigning one of the colors from the
set $c(N_{T}(p(u)))\setminus\\{c(v)\\}$ to $u$ results in a new conditional
$(k,r)$-coloring.
Case 1.2: $c(u)\neq c(v):$
Case 1.2.1: There exists a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$ Interchanging the colors of $u$ and $v$ results in a different
induced partition of $V(T)$.
Case 1.2.2: There doesn’t exist a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$
Case 1.2.2.1: $r\geq\Delta:$ If $n\neq\Delta+1$ then there exists a vertex
$w^{\prime}\in N_{T}(p(u))$ such that $d(w^{\prime})\geq 2$; then
interchanging the colors of $u$ and $w^{\prime}$ results in a different
induced partition of $V(T)$ because the subtree rooted at $w^{\prime}$ doesn’t
contain any vertex colored $c(u)$. If $n=\Delta+1$ then $k=n$.
Case 1.2.2.2: $r<\Delta:$ There must exist at least two vertices
$w_{1},w_{2}\in N_{T}(p(u))$ such that $c(w_{1})=c(w_{2})$. Interchanging the
colors of $u$ and $w_{1}$ results in a different induced partition of $V(T)$
because the subtree rooted at $w_{1}$ doesn’t contain any vertex colored
$c(u)$.
Case 2: $p(u)\neq p(v):$
Case 2.1: $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or $d(p(v))<$ min
$\\{r,\Delta(T)\\}:$ Assigning to $u$ any color in the set $c(V(T))\setminus
c(N_{T}[p(u)])$ if $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or to $v$ any color in
the set $c(V(T))\setminus c(N_{T}[p(v)])$ if $d(p(v))<$ min
$\\{r,\Delta(T)\\}$ gives a new conditional $(k,r)$-coloring.
Case 2.2: $d(p(u))>$ min $\\{r,\Delta(T)\\}$ or $d(p(v))>$ min
$\\{r,\Delta(T)\\}:$ If $d(p(u))>$ min $\\{r,\Delta(T)\\}$ there must exist a
vertex $u^{\prime}\in N_{T}(p(u))$ such that $c(u)=c(u^{\prime})$ or two
vertices $u_{1},u_{2}\in N_{T}(p(u))$ such that $c(u_{1})=c(u_{2})\neq c(u)$.
Assigning to $u$ any of the color in the set
$c(V(T))\setminus\\{c(u^{\prime}),c(p(u))\\}$ or intechanging the colors of
$u$ and $u_{1}$ and the colors $c(u)$, $c(u_{1})$ in the subtree rooted at
$u_{1}$ gives a new conditional $(k,r)$-coloring in the former and latter
cases respectively. The case $d(p(v))>$ min $\\{r,\Delta(T)\\}$ is similar.
Case 2.3: $d(p(u))=d(p(v))=$ min $\\{r,\Delta(T)\\}:$
Case 2.3.1: min $\\{r,\Delta(T)\\}>2:$ There exists a $w\in V(T)$ such that
$c(u)\neq c(w)$ and $p(u)=p(w)$. Making the color of $u$ as $c(w)$ and in the
subtree rooted at $w$, swapping the colors $c(u)$ and $c(w)$ gives a new
conditional $(k,r)$-coloring.
Case 2.3.2: min $\\{r,\Delta(T)\\}=2:$ Since $T\neq P_{n}$ so $\Delta(T)>2$
and $r=2$. There exists an ancestor of $u$ and $v$ with degree $\geq 2$
because the root of $T$ has maximum degree. Let $w$ with $d(w)\geq 2$ be the
closest ancestor of $u$ and $v$. Then there exists two children of $w$ namely
$w_{1}$ and $w_{2}$ which are ancestors of $u$, $v$ respectively. If $w$ is
the root of $T$ and $c(w_{1})\neq c(w_{2})$, interchanging the colors
$c(w_{1})$ and $c(w_{2})$ in the subtree rooted at $w_{1}$ gives a new
conditional $(k,r)$-coloring. If $w$ is not the root of $T$ and $c(w_{1})\neq
c(w_{2})$, interchanging the colors $c(w_{1})$ and $c(w_{2})$ in the subtree
rooted at $w$ gives a new conditional $(k,r)$-coloring. Otherwise (i.e., if
$c(w_{1})=c(w_{2})$), there exists a $w_{3}\in N_{T}(w)$ such that
$c(w_{3})\neq c(w_{1})$ and interchanging the colors $c(w_{1})$ and $c(w_{2})$
in the subtree rooted at $w_{1}$ gives a new conditional $(k,r)$-coloring. ∎
## 3 Conditional colorability of some graphs
###### Theorem 1.
Let $G_{1}$ and $G_{2}$ be two graphs where $\chi(G_{1})=k_{1}$,
$\chi(G_{2})=k_{2}$ and w.l.o.g. let $k_{1}\leq k_{2}$. Then
$\chi_{r}(G_{1}+G_{2})$ = $\chi(G_{1}+G_{2})$ = $k_{1}+k_{2}$, where $r\leq
k_{1}+1$.
###### Proof.
In the graph $G_{1}+G_{2}$, $V(G_{2})\subset N_{G_{1}+G_{2}}(u)$ if $u\in
V(G_{1})$ or $V(G_{1})\subset N_{G_{1}+G_{2}}(u)$ if $u\in V(G_{2})$.
Therefore $c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, and in any proper
$k$-coloring of $G_{1}+G_{2}$, for all $u\in V(G_{1}+G_{2})$,
$|c(N_{G_{1}+G_{2}}(u))|\geq\min\\{d(u),r\\}$. This implies that every proper
$k$-coloring of $G_{1}+G_{2}$ is also a proper $(k,r)$-coloring – if not, we
get a contradiction: suppose that $\chi(G_{1}+G_{2})=k\neq k_{1}+k_{2}$; since
$c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, either $k_{1}\neq\chi(G_{1})$ or
$k_{2}\neq\chi(G_{2})$ which contradicts the given condition. ∎
###### Theorem 2.
Let $G(V_{1},V_{2},E)$ be a bipartite graph, $S_{1}=\bigcap_{u\in
V_{1}}N_{G}(u)$, $S_{2}=\bigcap_{v\in V_{2}}N_{G}(v)$ and w.l.o.g. let
$|S_{1}|\leq|S_{2}|$. Then $\chi_{r}(G)=2r$ where $r\leq|S_{1}|$ .
###### Proof.
In any proper coloring of $G$, from the given conditions $|c(V_{1})|\geq r$
and $|c(V_{2})|\geq r$ as $G$ is bipartite. Since $r\leq|S_{1}|$ and
$c(S_{1})\cap c(S_{2})=\emptyset$ we have $\chi_{r}(G)\geq 2r$. But there
exists a proper $2r$-coloring of $G$ such that $|c(S_{1})|=|c(S_{2})|=r$
because every bipartite graph is bicolorable and $r\geq 2$. This coloring also
satisfies (C2) as $S_{1}\subseteq V_{2}$ and $S_{2}\subseteq V_{1}$. Thus
$\chi_{r}(G)\leq 2r$. Hence $\chi_{r}(G)=2r$. ∎
###### Theorem 3.
Let $T_{1},T_{2}$ be two non trivial trees with $n_{1},n_{2}$ number of
vertices respectively and w.l.o.g. let $n_{1}\leq n_{2}$. Then
$\chi_{r}(T_{1}+T_{2})=2(r-1)$, where $4\leq r\leq\ n_{1}+1$.
###### Proof.
Every nontrivial tree has at least two vertices with degree one [2]. Therefore
there exist vertices $u,v$ where $u\in V(T_{1})$ and $v\in V(T_{2})$, such
that $d(u)=d(v)=1$. Therefore, $d_{u}(T_{1}+T_{2})=1+n_{2}$ and
$d_{v}(T_{1}+T_{2})=1+n_{1}$. If $\chi_{r}(T_{1}+T_{2})<2(r-1)$, then either
$|c(V(T_{1}))|<r-1$ or $|c(V(T_{2}))|<r-1$ or both because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$. Hence (C2) is violated at $u$ or $v$ or both.
Therefore, $\chi_{r}(T_{1}+T_{2})\geq 2(r-1)$. Since every tree is
$2$-colorable and $r\geq 4$, properly color $V(T_{1})$, $V(T_{2})$ in
$T_{1}+T_{2}$ using $r-1$ colors each such that $|c(V(T_{1}+T_{2}))|=2(r-1)$.
The resulting coloring is a conditional $(2(r-1),r)$-coloring of
$T_{1}+T_{2}$, as (C1) is satisfied because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$, and (C2) is satisfied because $c(V(T_{1}))\cap\
c(V(T_{2}))=\emptyset$ and for all $w\in V(T_{1}+T_{2})$,
$|c(N_{T_{1}+T_{2}}(w))|\geq(r-1)+1\geq$ min $\\{d(w),r\\}$. Hence the result.
∎
###### Theorem 4.
Given any two graphs $G_{1}$ and $G_{2}$, let $r_{1}$ and $r_{2}$ be such that
$r_{1}\geq\delta(G_{1})$ and $r_{2}\geq\delta(G_{2})$. Then $\chi_{r}(G_{1}\
\Box\ G_{2})\leq\chi_{r_{1}}(G_{1}).\chi_{r_{2}}(G_{2})$ where
$r\leq\delta(G_{1})+\delta(G_{2})$.
###### Proof.
Let $\chi_{r_{1}}(G_{1})=g_{1}$ and $\chi_{r_{2}}(G_{2})=g_{2}$. Let
$c_{G_{1}}$ (resp. $c_{G_{2}}$) be a proper $(g_{1},r_{1})$\- (resp.
$(g_{2},r_{2})$-) coloring of $G_{1}$ (resp. $G_{2}$). Then let $c_{G_{1}\Box
G_{2}}$ be a coloring of $G_{1}\Box G_{2}$ wherein we assign to any vertex
$(u_{1},u_{2})\in V(G_{1}\ \Box\ G_{2})$ the color denoted by the ordered pair
$(c_{g_{1}}(u_{1}),c_{g_{2}}(u_{2}))$. This coloring uses $g_{1}.g_{2}$ colors
and it defines a proper coloring of $G_{1}\ \Box\ G_{2}$. Therefore
$c_{G_{1}\Box G_{2}}$ satisfies (C1). Let $(u_{1},u_{2})\in V(G_{1}\ \Box\
G_{2})$ such that $u_{1}\in V(G_{1})$ and $u_{2}\in V(G_{2})$. Since
$c_{G_{1}}$ and $c_{G_{2}}$ satisfy (C2), by the definition of $G_{1}\Box
G_{2}$, a vertex $(u_{1},u_{2})$ has at least
$\min\\{r_{1},\delta(G_{1})\\}=\delta(G_{1})$ distinctly colored neighbors of
the form $(u^{\prime},u_{2})$ because $|c(N_{G_{1}}(u_{1}))|\geq\delta(G_{1})$
and at least $\min\\{r_{2},\delta(G_{2})\\}=\delta(G_{2})$ distinctly colored
neighbors of the form $(u_{1},u^{\prime\prime})$ because
$|c(N_{G_{2}}(u_{2}))|\geq\delta(G_{2})$. Therefore $|c(N_{G_{1}\ \Box\
G_{2}}((u_{1},u_{2}))|\geq\delta(G_{1})+\delta(G_{2})\geq r$. Hence
$c_{G_{1}\Box G_{2}}$ satisfies (C2) and the result follows. ∎
###### Theorem 5.
Let $L(T)$ be the line graph of complete $k$-ary tree $T$ with height $h\geq
2$. Then
$\chi_{r}(L(T))=\left\\{\begin{array}[]{l l}k+1,&\quad\mbox{if $r\leq k$.
{}}\\\ 2k+1,&\quad\mbox{if $r=\Delta$. {}}\\\ \end{array}\right.$
###### Proof.
Let $V(L(T))=\left\\{v_{1},v_{2},\dotsc,v_{e(h)}\right\\}$, where
$e(h)=\frac{k^{h+1}-1}{k-1}-1$. In $T$ we assume that the root is at level $0$
and for each $l$ ($1\leq l\leq h$), $v_{e(l-1)+1}$ to $v_{e(l)}$ represent the
edges between levels $l-1$ and $l$, numbered from ‘left’ to ‘right’. It can be
seen that $\Delta(L(T))=2k$ and $\omega(L(T))=k+1$. In the ordering
$v_{e(h)},\dotsc,v_{1}$ of the vertices of $L(T)$, for each $i$ ($1\leq i\leq
e(h)$), $v_{i}$ is a simplicial vertex in the subgraph induced by
$\\{v_{i},\dotsc,v_{1}\\}$. Hence the ordering is a p.e.o. and $L(T)$ is
chordal. As every chordal graph is perfect, $\chi(L(T))=\omega(L(T))=k+1$.
Since every vertex of $L(T)$ is in a $K_{k+1}$, we also have
$\chi_{r}(L(T))=k+1$ if $r\leq k$. Thus $\chi_{r}(L(T))=k+1$, if $r\leq k$.
From [3] we know $\chi_{r}(G)\geq\min\\{r,\Delta\\}+1$. Taking $G=L(T)$ we
have $\chi_{r}(L(T))\geq\min\\{r,\Delta\\}+1=2k+1$ if $r=\Delta$. Similar to
the greedy (vertex) coloring, color the vertices in the order
$v_{1},\dotsc,v_{e(h)}$ by assigning to each vertex the first available color
not already used for any of the lower indexed vertices within distance two. In
the assumed order, each vertex has at most $\Delta$ lower indexed vertices
within distance two; therefore $\chi_{\Delta}(L(T))\leq\Delta+1=2k+1$. Hence
$\chi_{r}(L(T))=2k+1$ if $r=\Delta$. ∎
Definition 3 The wheel graph $W_{n}$ consists of a $C_{n-1}$ together with a
center vertex $s$ that is adjacent to all the $n-1$ vertices in the $C_{n-1}$.
###### Theorem 6.
If $W_{n}$ denotes the wheel graph, for any $r\geq 3$
$\chi_{r}(W_{n})=\left\\{\begin{array}[]{l
l}\chi_{r}(C_{n-1})+1,&\quad\mbox{if $r\leq\chi_{r}(C_{n-1}).$ {}}\\\
\min\\{r,n-1\\}+1,&\quad\mbox{if $r>\chi_{r}(C_{n-1}).${}}\\\
\end{array}\right.$
###### Proof.
By definition $d(s)=n-1$. For any other $u\;(\neq s)\in V(W_{n}),\;d(u)=3$.
Since $r\geq 3$, every conditional $(k,r)$-coloring of $W_{n}$ is also a
conditional $(k,r)$-coloring of its induced subgraph $C_{n-1}$. There exists
conditional $(k,r)$-coloring of $G$ iff $k\geq\chi_{r}(G)$. Hence conditional
$(\chi_{r}(C_{n-1}),r)$-coloring of $C_{n-1}$ with a new color to $s$
satisfies (C1) and gives $|c(N_{W_{n}}(v))|\geq\min\\{r,d(v)\\}$, for all
$v\in V(W_{n})\backslash s$. If $r\leq\chi_{r}(C_{n-1})$, then
$|c(N_{W_{n}}(s))|\geq r$. Thus for each $v\in V(W_{n})$,
$|c(N_{W_{n}}(v))|\geq\min\\{r,d(v)\\}$ satisfying (C2) – in total
$\chi_{r}(C_{n-1})+1$ colors are used. If $r>\chi_{r}(C_{n-1})$, then
$|c(N_{W_{n}}(s))|<r$, and we need to give new colors to
$\min\\{r,n-1\\}-\chi_{r}(C_{n-1})$ vertices adjacent to $s$. In total
$\chi_{r}(C_{n-1})+1+\min\\{r,n-1\\}-\chi_{r}(C_{n-1})=\min\\{r,n-1\\}+1$
colors are used. Hence the result. ∎
Definition 4 The $n$-gear $G_{n}$ consists of a cycle $C_{2n}$ on $2n$
vertices where every other vertex on the cycle is adjacent to a $(2n+1)th$
center vertex labeled $v_{0}$. The vertices in the $C_{2n}$ are labeled
sequentially $v_{1},\ldots,v_{2n}$ such that for $1\leq i\leq 2n-1$, $v_{i}$
is adjacent to $v_{i+1}$, $v_{1}$ is adjacent to $v_{2n}$, and every vertex in
$V_{oddi}=\\{v_{i}|\;\text{i is odd and}\;1\leq i\leq 2n-1\\}$ is adjacent to
$v_{0}$.
###### Theorem 7.
If $G_{n}$ is the $n$-gear then for any $n\geq 3$
$\chi_{r}(G_{n})=\left\\{\begin{array}[]{l l}4,&\quad\mbox{if $r=2$. {}}\\\
\chi_{2}(C_{2n})+1,&\quad\mbox{if $r=3$.{}}\\\
min\\{r,\Delta\\}+1,&\quad\mbox{if $r\geq 4$. {}}\\\ \end{array}\right.$
###### Proof.
Let $k=\chi_{r}(G_{n})$. From [3] we know $\chi_{r}(G)\geq
min\\{r,\Delta\\}+1$. Taking $G=G_{n}$ we have $\chi_{r}(G_{n})\geq
min\\{r,\Delta\\}+1$. Let $k=\chi_{r}(G_{n})$.
Case 1: $r=2:$ We have $k\geq 3$. We assume that $k=3$. Let $c\colon
V(G_{n})\to\\{1,2,3\\}$ be a conditional $(3,2)$-coloring where $c(v_{i})=i$
for $i=1,2,3$. Then across $v_{1},v_{2},v_{3}$ (C1) must be true and in
particular (C2) must hold at $v_{2}$. To satisfy (C1) at $v_{3},c(v_{4})\neq
3$. We branch into two cases. If $c(v_{4})=1$, then by (C2) at $v_{4}$ we must
have $c(v_{5})\notin\\{1,3\\}$. Therefore we must have $c(v_{5})=2$. To
satisfy (C1), $c(v_{0})\notin\\{1,2,3\\}$. On the other hand if $c(v_{4})=2$
then to satisfy (C2) at $v_{4}$ we must have $c(v_{5})\notin\\{2,3\\}$. Hence
$c(v_{5})=1$. To satisfy (C2) at $v_{3}$ while preserving proper coloring,
$c(v_{0})\notin\\{1,2,3\\}$. Therefore $k\geq 4$. To show that $k=4$, it
suffices to construct a conditional $(4,2)$-coloring of $G_{n}$. Define
$c\colon V(G_{n})\to\\{1,2,3,4\\}$ as follows:
Case 1.1: $n\equiv 0\pmod{3}:$ Set $c(v_{0})=4$ and
$c(v_{i})=\left\\{\begin{array}[]{l l}1,&\quad\mbox{if \ $i\bmod 3=1$. {}}\\\
2,&\quad\mbox{if \ $i\bmod 3=2$.{}}\\\ 3,&\quad\mbox{if \ $i\bmod 3=0$. {}}\\\
\end{array}\right.$
Case 1.2: $n\equiv 2\pmod{3}:$ Modify $c$ in case (1.1) by the assignment
$c(v_{2n})=2$.
Case 1.3: $n\equiv 1\pmod{3}:$ Modify $c$ in case (1.1) by the assignments
$c(v_{2n-4})=2,c(v_{2n-3})=1,c(v_{2n-2})=3,c(v_{2n-1})=2$ and $c(v_{2n})=3$.
It can be verified that $c$ is a conditional $(4,2)$-coloring of $G_{n}$. Thus
$k\leq 4$, and hence $\chi_{2}(G_{n})=4$.
Case 2: $r=3:$ We have $k\geq 4$. By (C1) $c(v_{i})\neq c(v_{0})$ and by (C2)
at $v_{i}$, $c(v_{i+1})\neq c(v_{0})$ for all odd $i$ in the range $1\leq
i\leq 2n-1$. Hence $c(v_{i})\neq c(v_{0})$ for all $i\;(i\neq 0)$. Since
$d(v_{i})\leq 3$ for $1\leq i\leq 2n$ and $r=3$, a conditional
$(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$ gives a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$. In turn, a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$ with an additional color to
$v_{0}$ gives a conditional $(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$.
Case 3: $r\geq 4:$ Let $l=\min\\{r,\Delta(G_{n})\\}$. We know
$\chi_{r}(G_{n})\geq l+1$. Since $v_{0}$ is the only vertex with $d(v_{0})\geq
r$ and $\chi_{r}(C_{2n})\leq l$, conditional $(l,r)$-coloring of
$V(G_{n})\setminus\\{v_{0}\\}$ such that $|c(V_{oddi})|=l$, with $(l+1)$th
color assigned to $v_{0}$ results in a conditional $(l+1,r)$-coloring of
$V(G_{n})$. Thus $\chi_{r}(G_{n})\leq l+1$, and hence
$\chi_{r}(G_{n})=\min\\{r,\Delta\\}+1$. ∎
## References
* [1] M. C. Golumbic, Algorithmic graph theory and perfect graphs, Elsevier, North Holland, 2004.
* [2] F. Harary, Graph theory, Addison-Wesley, MA, 1969.
* [3] H. J. Lai, J. Lin, B. Montgomery, T. Shui, and S. Fan, Conditional colorings of graphs, Discrete Math. 306 (2006), pp. 1997-2004.
* [4] X. Li, X. Yao, W. Zhou, and H. Broersma, Complexity of conditional colorability of graphs, Appl. Math. Lett. 22 (2009), pp. 320-324.
* [5] P.V.S. Reddy, K.V.Iyer, “On conditional coloring of some graphs”, 76th Ann. Conf. Ind. Math. Soc.,Surat, Gujarat, Dec. 2010. (arxiv.org/abs/1011.5289)
* [6] P.V.S.Reddy, K.V.Iyer, “Conditional coloring of some parameterized graphs”, Submitted to Util. Math., May. 2010. (arxiv.org/abs/1012:2251)
* [7] D. B. West, Introduction to graph theory, Prentice Hall of India, 2003.
|
arxiv-papers
| 2011-06-17T11:57:12 |
2024-09-04T02:49:19.750227
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P.Venkata Subba Reddy and K.Viswanathan Iyer",
"submitter": "Iyer Viswanathan K.",
"url": "https://arxiv.org/abs/1106.3456"
}
|
1106.3478
|
# Conditional Elimination through Code Duplication
Joachim Breitner111e-mail: mail@joachim-breitner.de
###### Abstract
We propose an optimizing transformation which reduces program runtime at the
expense of program size by eliminating conditional jumps.
## 1 Preface
### 1.1 Motivation
In a variety of cases, code is written in a way that in one execution, a
conditional execution is evaluated several time. Situations where this may be
happening include the following:
* •
Repeated use of the ternary operator ($\cdot$?$\cdot$:$\cdot$) with a common
conditional expression.
* •
An if-then-else statement inside a loop, where the condition is loop
invariant.
* •
Use of macros or inlined functions provided by a library that include
conditional expression.
* •
Conditional jumps implicitly inserted by the compiler due to short-circuit
logic.
* •
Naive code mechanically generated from another source via tools such as parser
generators, or compilers of higher languages that compile to C and then invoke
a C compiler.
* •
Conditionals introduced by earlier compilation passes, such as the Partial
Dead Code pass conceived by Bodík and Gupta [2] are likely to make other
conditionals redundant. In fact, the PDE paper recommends a “branch
elimination” step without giving the details of this. CECD can serve as an
implementation of this step.
In some of these cases the programmer might be able to eliminate the redundant
conditional expression by himself, but often at the cost of less readable code
or repetition, such as two instances of the loop mentioned in the second
bullet. In other cases, such as the library-provided macros or the generated
code, it is not feasible to expect the source code to be free of redundant
conditionals. Therefore it is desirable that an optimizing compiler can
perform this transformation.
Furthermore, this transformation not only reduces execution time but can
enabled further optimizations: If the conditional expression is of the form
$v==c$ for a variable $v$ and a constant $c$, a constant propagation pass can
replace $v$ by $c$ in the then-branch, which has been enlarged by our
optimization. Also, modern computer architectures, due to long pipelines,
perform better if fewer conditional jumps occur in the code.
### 1.2 Outline
In the next section, we explain when a given region to duplicate is valid and
how to perform the conditional elimination. Aiming for a very clear, simple
and homogeneous presentation, we describe the algorithm in a very general
setting. This will possibly introduce dead code. An implementation would
either run a dead code elimination pass afterwards or refine the given
algorithm as required. The transformation is demonstrated by example.
Section 3 discusses which properties the region should satisfy for the
optimization to actually have a positive effect, and how to avoid useless code
duplication.
To decide whether to perform the optimization, we give a simple heuristic that
selects a region to be duplicated and decides whether the optimization should
be performed, weighting the (runtime) benefits weighted against the (code
size) cost in subsection 3.2. We also show that a slight more sophisticated
approach, which takes profiling information into account, becomes
$\mathcal{NP}$-hard.
Data flow equations for the properties discussed in the preceding two sections
are given in 4.
### 1.3 Acknowledgements
This paper was written for the group project of the CS614 “Advanced Compiler”
course at IIT Bombay under Prof. D. M. Dhamdhere. I have had fruitful
discussions with him and my fellow group members, Anup Agarwal, Yogesh Bagul
and Anup Naik, who subsequently implemented parts of this using the LLVM
compiler suite.
## 2 Conditional elimination
Let $e$ be an expression, which should occur as the condition for a
conditional branch in the control flow graph (CFG) of a program, and let
$v_{1},\,v_{2},\,\ldots$ be the operands of the expression.
Let $D$ be a region of the control flow graph, i.e. $D\subseteq BB$ where $BB$
is the set of basic blocks in the control flow graph.
The region $D$ is valid if and only if no basic block body in $D$ contains an
assignment to any of the operands $v_{1},\,v_{2},\,\ldots$ of $e$.
The parameters of the optimization are the conditional expression $e$ and any
valid set $D$. The transformation is performed in three steps, where the first
step is generic code duplication which does not yet consider the conditional
expression, the second step rewires some edges to make the other copies
reachable and the last step removes the redundant conditionals. Each step
preserves the meaning of the program.
1. 1.
(Code duplication) For every basic block $bb_{i}\in D$, create three
copies222Technically, this is triplication, not duplication.: the true copy
$bb_{i}^{t}$, the false copy $bb_{i}^{f}$ and the unknown copy $bb_{i}^{u}$.
The edges of the graph are modified as follows:
* •
An edge between $bb_{i}\notin D$ and $bb_{j}\notin D$ is left unchanged.
* •
An edge between $bb_{i}\in D$ and $bb_{j}\in D$ is reproduced by the three
edges $bb_{i}^{t}$ to $bb_{j}^{t}$, $bb_{i}^{f}$ to $bb_{j}^{f}$ and
$bb_{i}^{u}$ to $bb_{j}^{u}$.
* •
An edge between $bb_{i}\in D$ and $bb_{j}\notin D$ is reproduced by the three
edges $bb_{i}^{t}$ to $bb_{j}$, $bb_{i}^{f}$ to $bb_{j}$ to $bb_{i}^{u}$ to
$bb_{j}$.
* •
An edge between $bb_{i}\notin D$ and $bb_{j}\in D$ is changed to an edge from
$bb_{i}$ to $bb_{j}^{u}$.
2. 2.
(Conditional evaluation) For every conditional edge from $bb_{i}$ to $bb_{j}$
depending on $e$ being true (false) at the end of $bb_{i}$, where $bb_{j}$ is
a copy of a node in $D$, replace it by an edge $bb_{i}$ to $bb_{j}^{t}$
($bb_{j}^{f}$).
3. 3.
(Conditional elimination) For every basic block $bb_{i}\in D$ which has a
conditional branch depending on $e$ being true (false), remove the condition
in $bb_{i}^{t}$ ($bb_{i}^{f}$), unconditionally follow the true (false) case
and remove the other edge.
This algorithm is correct and safe. For correctness, consider an execution
path. If the path does not pass any node in $D$, it is not altered by the
above algorithm. If the path passes through $D$, but only through unknown
copies, it is also not altered. If the path eventually reaches a true (false)
copy of a node, it must be because of an edge altered in step 2. At that point
of execution, the value of $e$ is known to be true (false), and because $D$ is
valid, it remains so until the execution path leaves the region $D$. Any
conditional jump skipped because of step 3 is therefore behaving exactly as in
the original execution path. .
Safeness follows from the fact that we only copy nodes and remove the
evaluation of conditionals, so along no path new instructions are added.
### Example
if _…_ then
if _$e$_ then …;
else …;
else
$e\coloneqq\cdots$
end if
while _…_ do
if _$e$_ then …;
else …;
end while
…
Figure 1: Example code
Consider the code fragment in Figure 1 (leaving out any unrelated assignments
or expressions). The corresponding control flow graph is given in figure 2.
The largest valid region is marked, as well as the largest region if useful
nodes. Applying the algorithm with $D$ set to the region of useful nodes,
after step 1 we obtain the graph shown in figure 3. At this point, the true
and false copies are not reachable yet. Steps 2 and 3 modify the edges related
to conditional on $e$, and we reach figure 4. This contains a lot of dead
code. Removing this in a standard dead code removal pass, we reach the final
state 5. It can clearly be seen that on every path from entry to exit, the
conditional $e$ is evaluated at most once. Also the issue of a while-loop
occurrence (in contrast to the optimizer-friendly do-while-loop) is gracefully
taken care of.
$bb_{1}$$bb_{2}$$bb_{3}$$bb_{4}$$bb_{5}$$bb_{6}$$bb_{7}$$bb_{8}$$bb_{9}$$bb_{10}$$bb_{11}$$e\coloneqq\ldots$$e$$\neg
e$$e$$\neg e$validuseful Figure 2: Example control flow graph before CECD
$bb_{1}$$bb_{6}$$bb_{2}$$bb_{3}$$e\coloneqq\ldots$$bb_{11}$$bb_{4}^{t}$$bb_{5}^{t}$$bb_{7}^{t}$$bb_{8}^{t}$$bb_{9}^{t}$$bb_{10}^{t}$$bb_{4}^{f}$$bb_{5}^{f}$$bb_{7}^{f}$$bb_{8}^{f}$$bb_{9}^{f}$$bb_{10}^{f}$$bb_{4}^{u}$$bb_{5}^{u}$$bb_{7}^{u}$$bb_{8}^{u}$$bb_{9}^{u}$$bb_{10}^{u}$$e$$\neg
e$$e$$\neg e$$e$$\neg e$$e$$\neg e$ Figure 3: Example control flow graph after
code duplication of useful nodes
$bb_{1}$$bb_{6}$$bb_{2}$$bb_{3}$$e\coloneqq\ldots$$bb_{11}$$bb_{4}^{t}$$bb_{5}^{t}$$bb_{7}^{t}$$bb_{9}^{t}$$bb_{4}^{f}$$bb_{5}^{f}$$bb_{7}^{f}$$bb_{10}^{f}$$bb_{4}^{u}$$bb_{5}^{u}$$bb_{7}^{u}$$bb_{8}^{u}$$bb_{9}^{u}$$bb_{10}^{u}$$e$$\neg
e$$e$$\neg e$ Figure 4: Example control flow graph after conditional
evaluation and elimination
$bb_{1}$$bb_{6}$$bb_{2}$$bb_{3}$$e\coloneqq\ldots$$bb_{11}$$bb_{4}^{t}$$bb_{7}^{t}$$bb_{9}^{t}$$bb_{5}^{f}$$bb_{7}^{f}$$bb_{10}^{f}$$bb_{7}^{u}$$bb_{8}^{u}$$e$$\neg
e$$e$$\neg e$ Figure 5: Example control flow graph after conditional
evaluation and elimination and dead code elimination
## 3 The region of duplication
The above algorithm works for any valid region, and validity is a simple local
property that is easily checked. But not all valid regions are useful. For
example, entry nodes $bb_{i}$ of the region where no incoming edge depends on
$e$ would be duplicated, but only $bb_{i}^{u}$ would be reachable. Similarly,
exit nodes of the region that do not have a conditional evaluation of $e$
would be copied for no gain.
### 3.1 Usefulness
Therefore, we can define that a node $bb_{i}$ in a valid region $D$ to be
useless if
* •
on all paths leading to $bb_{i}$, there is no conditional evaluation of $e$
followed only by nodes in $D$ or
* •
no path originating from $bb_{i}$ reaches an conditional evaluation of $e$
before it leaves the region $D$.
A node $bb_{i}\in D$ that is not useless is useful.
Uselessness is, in contrast to validity, not a property of the basic block
alone but defined with respect to the chosen region $D$. A basic block may be
useless in $D$ but not so in a different region $D^{\prime}$. But the property
is monotonous: If $D^{\prime}\subseteq D$ and $D$ is useful in $D^{\prime}$,
then it is also useful in $D$.
### 3.2 Evaluation of a region
For a given conditional expression, there are many possible regions of
duplication, and even if we only consider fully useful regions, their number
might be exponential in the size of the graph. Therefore we need an heuristic
that selects a sensible region or decides that no region is good enough to
perform CECD. We split this decision into two independent steps: Region
Selection, where the the best region for a particular conditional, for some
meaning of “best” is chosen, and Region Evaluation, where it is decided
whether CECD should be performed for the selected region.
These decisions have to depend on the intended use of the code. Code for an
embedded system might have very tight size requirements and large regions of
duplication would be unsuitable, whereas code written for massive numerical
calculations may be allowed to grow quite a bit if it removes instructions
from the inner loops.
At this point, we suggest a very simple heuristic for Region Selection: To
cover as many executions paths as possible, we just pick the largest valid
region consisting of useful nodes. The heuristic for Region Evaluation expects
one parameter $k$, which is the number of additional expressions that the
program is allowed to grow for one conditional to be removed. Together, this
amounts to the following steps being taken:
1. 1.
Let $D$ be the largest valid region consisting only of useful nodes.
2. 2.
Let $R^{t}$, $R^{f}$ resp. $R^{u}$ the set of those basic blocks in $D$, whose
true, false respu̇nknown copy will be reachable after CECD.
3. 3.
Let $n$ be the number of basic blocks in $D$ that contain a conditional
evaluation of $e$, i.e. the number of redundant conditionals.
4. 4.
If
$\sum_{bb_{i}\in R^{t}}S(bb_{i})+\sum_{bb_{i}\in
R^{f}}S(bb_{i})+\sum_{bb_{i}\in R^{u}}S(bb_{i})-\sum_{bb_{i}\in
D}S(bb_{i})\leq n\cdot k,$
where $k$ is a user-defined parameter and $S(bb_{i})$ is the number of
instructions in the basic block $bb_{i}$, perform CECD on $D$, otherwise do
not perform CECD for this conditional expression.
A number of improvements to this scheme come to mind:
* •
The selection heuristic should consider subsets of the largest valid and
useful regions as well.
* •
It should give different weights to conditionals that are completely removed
and conditionals that are only partially removed.
* •
Removal of conditionals in inner loops should allow for a larger increase of
code size.
* •
Given sufficiently detailed execution traces, a more exact heuristic can be
implemented. In the next section we see that this easily leads to a
$\mathcal{NP}$-hard problem.
### 3.3 $\mathcal{NP}$-hardness of a profiling based Region Selection
heuristic
A straight forward extension of the above Region Selection heuristic that
takes profiling data in the form of execution traces into account, would
maximize the sum $\sum_{bb_{i}\in E}f(bb_{i})$, where $E$ is the set of of
basic blocks containing an eliminated conditional and $f(bb_{i})$ is the
number of paths in the execution traces where the conditional in $bb_{i}$
would be eliminated due to CECD. For simplicity, we assume that an occurrence
of a conditional expression does not contribute to the size $S(bb_{i})$ of a
basic block.
If we have an algorithm that selects the optimal region, we can solve the 0-1
knapsack problem, which is $\mathcal{NP}$-complete. The specification of this
problem is as follows:
> Given $n$ items with weight $w_{i}\in\mathbb{N}$ and value
> $v_{i}\in\mathbb{N}$, $i=1,\ldots,n$ and a bound $W\in\mathbb{N}$, find a
> selection of items $X\subseteq\\{1,\ldots,n\\}$ that maximizes the sum
> $\sum_{i\in X}v_{i}$ under the constraint $\sum_{i\in X}w_{i}\leq W$.
Given such a problem, we construct a control flow graph and profiling data as
follows:
* •
The entry node is $bb_{s}$, which contains a conditional expression $e$. Both
conditional branches point to the node $bb_{r}$.
* •
There is one exit node $bb_{e}$ with a conditional expression $e$.
* •
The node $bb_{r}$ is the root of a binary tree of basic blocks. The inner
nodes contain no instructions but conditional jumps with conditional
expressions that are pairwise distinct and distinct from $e$.
* •
The tree contains $n$ leaf nodes $bb_{l}^{i}$, $i=1,\ldots,n$. The node
$bb_{l}^{i}$ contains $w_{i}$ instructions, i.e. $S(bb_{l}^{i})=w_{i}$ and the
profiling data gives a frequency of $v_{i}$ for the execution path passing
through $bb_{l}^{i}$.
* •
The parameter $k$ is chosen to be $W$.
A valid and useful region of duplication $D$ in this CFG corresponds to a
subset of $X\in{1,..,n}$ and, if non-empty, includes $bb_{e}$, $bb_{l}^{i}$
for $i\in X$ and the nodes connecting $bb_{r}$ with those leaf nodes. Because
$bb_{s}$ dominates all nodes in $D$, no unknown copies will be generated, and
both true and false copies are reachable. The inner nodes of the binary tree
and $bb_{e}$ only contain conditional expressions and thus do not contribute
to the size of the duplicated region. Only one redundant conditional occurs,
hence $n=1$. The number of executions of $bb_{e}$ where the conditional is
eliminated is exactly the number of execution paths that pass through one of
the leaf nodes in $D$. Therefore, the constraint imposed by the Region
Evaluation heuristic becomes
$\displaystyle\sum_{bb_{i}\in R^{t}}S(bb_{i})+\sum_{bb_{i}\in
R^{f}}S(bb_{i})+\sum_{bb_{i}\in R^{u}}S(bb_{i})-\sum_{bb_{i}\in D}S(bb_{i})$
$\displaystyle\leq n\cdot k$ $\iff$ $\displaystyle\sum_{i\in
X}S(bb_{l}^{i})+\sum_{i\in X}S(bb_{l}^{i})+0-\sum_{i\in X}S(bb_{l}^{i})$
$\displaystyle\leq 1\cdot k$ $\iff$ $\displaystyle\sum_{i\in X}w_{i}$
$\displaystyle\leq W$
and the term to be optimized can be transformed as follows:
$\displaystyle\sum_{bb_{i}\in E}f(bb_{i})=\sum_{i\in
X}f(bb_{l}^{i})=\sum_{i\in X}v_{i}.$
This concludes the proof of $\mathcal{NP}$-hardness of this profiling-based
heuristic for CECD.
The assumption that conditional expressions do not contribute to the size of a
node is not critical: If they do contribute, then this result can still be
obtained by a technical modification: Increase $k$ by one and then scale $k$
and the number of instructions in the nodes $bb_{l}^{i}$ by a factor larger
than the number of all conditional expressions occurring.
## 4 Data Flow equations
Three properties of basic blocks have been defined so far: Validness,
usefulness and, for the heuristics, which copies of the block will be present
after dead code removal. The first one is a purely local property, while the
others can be obtained by standard data flow analyses. The defining equations
are given in this section. $\operatorname{succ}(i)$ is the set of successor
nodes of $bb_{i}$ in the control flow graph, $\operatorname{pred}(i)$ the set
of predecessors. We assume that nodes with a conditional jump have exactly two
successors, one for true and one for false.
Local properties:
* •
$\text{Valid}_{i}$: Basic block $bb_{i}$ does not contain an assignment to an
operator of $e$.
* •
$\text{TrueEdge}_{ij}$: An edge $bb_{i}\to bb_{j}$ exists and depends on $e$
being true.
* •
$\text{FalseEdge}_{ij}$: An edge $bb_{i}\to bb_{j}$ exists and depends on $e$
being false.
* •
$\text{Expr}_{i}=\sum_{j\in\operatorname{succ}(i)}\text{TrueEdge}_{ij}+\text{FalseEdge}_{ij}$:
$e$ is a conditional expression in $bb_{i}$
Determining the largest valid region $D$ of useful nodes:
* •
$\text{Live}_{i}=\text{Valid}_{i}\cdot\sum_{j\in\operatorname{pred}(i)}\text{Expr}_{j}+\text{Live}_{j}$
* •
$\text{Antic}_{i}=\text{Valid}_{i}\cdot(\text{Expr}_{i}+\sum_{j\in\operatorname{succ}(i)}\text{Antic}_{j})$
* •
$D_{i}=\text{Live}_{i}\cdot\text{Antic}_{i}$
Given a valid region $D$ (which may or may not be obtained using our suggested
simple heuristic), determining which copies of the nodes therein are
reachable:
* •
$R^{u}_{i}=D_{i}\cdot\sum_{j\in\operatorname{pred}(i)}\neg\text{Expr}_{j}\cdot(\neg
D_{j}+R^{u}_{j})$
* •
$R^{t}_{i}=D_{i}\cdot\sum_{j\in\operatorname{pred}(i)}R^{t}_{j}+\text{TrueEdge}_{ji}$
* •
$R^{f}_{i}=D_{i}\cdot\sum_{j\in\operatorname{pred}(i)}R^{f}_{j}+\text{FalseEdge}_{ji}$
All given data flow equations are any-path equations and therefore, the values
can be initialized to false before solving the equations using a standard
iterative round-robin or worklist approach.
## 5 Future work and conclusions
While the “how” of CECD is fully understood, the question of “where” and
“when”, i.e. coming up with good heuristics for the selection of the
conditional and region of duplication, needs much further investigation. Also,
experiments with real code have yet to be conducted to quantify the benefit
and suggest good values for the heuristics’ parameters. Another possible
improvement would be to not only consider syntactically equal conditions, but
also take algebraic identities into account.
The simplicity of the CECD transformation and the fact that it can easily
handle complex control flow indicate that it could be an optimization of
general interest.
## References
* [1]
* BG [97] Bodík, Rastislav ; Gupta, Rajiv: Partial dead code elimination using slicing transformations. In: _SIGPLAN Not._ 32 (1997), May, S. 159–170. http://dx.doi.org/10.1145/258916.258930. – DOI 10.1145/258916.258930. – ISSN 0362–1340
|
arxiv-papers
| 2011-06-15T08:27:04 |
2024-09-04T02:49:19.762740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joachim Breitner",
"submitter": "Joachim Breitner",
"url": "https://arxiv.org/abs/1106.3478"
}
|
1106.3549
|
# On the Origin of the $\log d$ Variation of the Electrostatic Force
Minimizing Voltage in Casimir Experiments
S. K. Lamoreaux steve.lamoreaux@yale.edu Yale University, Department of
Physics, P.O. Box 208120, New Haven, CT 06520-8120 A. O. Sushkov
alex.sushkov@yale.edu Yale University, Department of Physics, P.O. Box 208120,
New Haven, CT 06520-8120
###### Abstract
A number of experimental measurements of the Casimir force have observed a
logarithmic distance variation of the voltage that minimizes electrostatic
force between the plates in a sphere-plane geometry. We show that this
variation can be simply understood from a geometric averaging of surface
potential patches together with the Proximity Force Approximation.
A number of experimental measurements of the Casimir force have observed a
distance variation of the voltage applied between the plates that minimizes
the electrostatic potential.kim1 ; kim2 ; deman ; deman2 This distance
variation is of the approximate form
$V_{m}(d)=a+b\log d$ (1)
over a range of distances $d$ spanning up to nearly two orders of magnitude.
We have shown numerically that a variation in the minimizing voltage can
result from the geometrical averaging of patch potentials on the plate
surfaces, and have developed a heuristic explanation of the effect, as
described in kim . However the $\log d$ form of the variation was not
explained or derived in kim . Here we provide added details showing that in
the case of a spherical surface/plane surface geometry, the $\log d$ variation
arises quite naturally.
The electrostatic force between the plates is minimized when the free energy
is minimized. For a plate with a spherical surface curvature $R$ together with
a flat plate, both of diameter $2R_{m}$, the Proximity Force Approximation
(PFA) works well for distances $d<R_{m}^{2}/2R$; in addition, the
characteristic radius of the patches $r_{0}$ must satisfy $r_{0}>\sqrt{2Rd}$,
in which case we can consider each patch as only interacting with its own
image in the opposite plate. If this latter criterion is not met, the effects
calculated here will be reduced in magnitude, however the general conclusions
are otherwise unaltered.
For random patches on the surfaces, the electrostatic free energy is
$U=\sum{1\over 2}CV^{2}={\epsilon_{0}R\over
2}\int_{0}^{R_{m}}\int_{0}^{2\pi}(V_{a}-V_{p}(r,\phi))^{2}{(r/R)\ d\phi\
dr\over d+r^{2}/2R}$ (2)
where $V_{a}$ is a voltage applied between the plates, and $V_{p}(r,\phi)$
describes the random voltage patches on the plates’ surfaces. The value of
$V_{a}$ that minimizes the force (or electrostatic free energy) is referred to
as the minimizing potential, and is often called the contact potential.
$V_{m}$ is found by taking the derivative with respect to $V_{a}$:
${\partial U\over\partial
V_{a}}=0={\epsilon_{0}R}\int_{0}^{R_{m}}\int_{0}^{2\pi}(V_{a}-V_{p}(r,\phi)){(r/R)\
d\phi\ dr\over d+r^{2}/2R}\ \ {\rm for\ }V_{a}=V_{m}.$ (3)
Because the only $\phi$ dependence is in $V_{p}(r,\phi)$, the angular integral
of this term can be replaced with $V_{p}(r)$, with the understanding that this
replacement represents a sum of the individual patches, of typical radius
$r_{0}$, intersected by a circle of radius $r$. For homogeneous random
patches, this number is roughly $(2\pi r)/(2r_{0})$, for $r>r_{0}$, or 1 for
$r<r_{0}$. Then if $RMS$ magnitude of the patches is $V_{0}$ with zero
average, this sum will have magnitude proportional to the square root of the
number of patches in the sum times $\pm V_{0}$, or
$\int_{0}^{2\pi}(V_{m}-V_{p}(r,\phi))d\phi=2\pi\left[V_{m}+V_{p}(r)\right]$
(4) $\approx 2\pi
V_{m}\pm\left[\Delta\phi\left[\theta(r-r_{0})-\theta(r-R_{m})\right]\sqrt{2\pi
r\over 2r_{0}}+2\pi\left[\theta(r)-\theta(r-r_{0})\right]\right]V_{0}$ (5)
where $\theta(r-r_{0})$ is the Heaviside step function, $\Delta\phi=2\pi
r_{0}/r$ results from $d\phi$ in the integral, and a single patch near the
center of the plates is included. (The relative sign of these two terms on the
r.h.s. is random.) Thus, $V_{p}(r)\propto 1/\sqrt{r/r_{0}}$ for $r\gg r_{0}$.
Although we will not directly use this result in the subsequent discussion, it
is important to note that $|V_{p}(r)|\leq|V_{p}(0)|$ in the case of homogenous
random patches.
Equation (3) can be integrated by parts, yielding
$(V_{m}-V_{p}(r))\log(d+r^{2}/2R)|_{0}^{R_{m}}+\int_{0}^{R_{m}}\log(d+r^{2}/2R){dV_{p}(r)\over
dr}dr=$ (6) $=(V_{m}-V_{p}(R_{m}))\log(d+R_{m}^{2}/2R)-(V_{m}-V_{p}(0))\log
d-Q(d)=0$ (7)
where we have introduced a new function $Q(d)$. Then
$V_{m}(d)={V_{p}(R_{m})\log(d+R_{m}^{2}/2R)-V_{p}(0)\log
d+Q(d)\over\log(d+R_{m}^{2}/2R)-\log d}.$ (8)
There are three cases to consider.
Case 1: Close range: $d<<R_{m}^{2}/2R$, $|\log d|>>|\log R_{m}^{2}/2R|$ and
$d<<r_{0}^{2}/2R$
In this limit, one single patch at or near the center dominates in the
determination of $V_{m}$, and $dV_{p}(r)/dr\approx 0$ when $r^{2}/2R<d$ in the
integral defining $Q(d)$. The $\log d$ terms have the largest magnitude, and
thus we can neglect all terms not multiplied by $\log d$. In this case,
$V_{m}(d)=V_{p}(0).$ (9)
Case 2: Intermediate range: $d<<R_{m}^{2}/2R$, $|\log d|<|\log R_{m}^{2}/2R|$
and $d<r_{0}^{2}/2R$
In this case,
$\log(d+R_{m}^{2}/2r)\approx\log(R_{m}^{2}/2R)$ (10)
so
$Q(d)=Q_{0}\log(R_{m}^{2}/2R)$ (11)
because in this limit $Q(d)$ is nearly independent of $d$. This can be seen in
Eq. (5) where the $\log(d+r^{2}/2R)$ factor contributes a $d$ dependence only
when $r$ is small, but in that limit, $dV_{p}(r)/dr\approx 0$. Therefore,
$V_{m}(d)\approx{V_{p}(R_{m})\log(R_{m}^{2}/2R)-V_{p}(0)\log
d+Q_{0}\log(R_{m}^{2}/2R)\over\log(R_{m}^{2}/2R)-\log d}.$ (12)
The denominator can be Taylor expanded and we therefore arrive at
$V_{m}(d)\approx\left[V_{p}(R_{m})-{V_{p}(0)\log
d\over\log(R_{m}^{2}/2R)}+Q_{0}\right]\left[1+{\log
d\over\log(R_{m}^{2}/2R)}\right]\approx a+b\log d$ (13)
where $a$ and $b$ do not depend on $d$, and where terms only first order in
$\log d/\log(R_{m}^{2}/2R)$ are retained. In addition, there can be a
contribution to $a$ from external circuit contact potentials.
It should be noted that a single patch at large $r$ will generate a non-zero
$Q_{0}$. Depending on the size and magnitude of the single patch, it might
dominate the $\log d$ distance dependence of $V_{m}$.
Case 3: $d>R_{m}^{2}/2R$ and $d\ {\buildrel>\over{\sim}}\ r_{0}^{2}/2R$ (so
that the PFA remains valid)
For this case, it is easiest to go back to Eq. (3) and expand the denominator.
We have
$0=\int_{0}^{R_{m}}\int_{0}^{2\pi}(V_{m}-V_{p}(r,\phi))r(1-r^{2}/2Rd)drd\phi\approx\int_{0}^{R_{m}}\int_{0}^{2\pi}(V_{m}-V_{p}(r,\phi))rdrd\phi$
(14)
so that
$V_{m}={\int_{0}^{R_{m}}\int_{0}^{2\pi}V_{p}(r,\phi)rdrd\phi\over\pi
R_{m}^{2}}=\langle V_{p}(r,\phi)\rangle$ (15)
which is the surface average of $V_{p}(r,\phi)$.
Discussion
It is easy to understand the first and third cases where $d\rightarrow 0$ and
$d\rightarrow\infty$. In the first case, a single patch dominates the
electrostatic force, and its potential determines $V_{m}$. In the latter,
$V_{m}$ is simply the average surface potential, as the variation is the
distance between the surfaces due to the curvature is very small compared to
$d$.
The intermediate case is slightly more difficult to understand, but arises
from, with increasing distance, the loss in dominance (in magnitude) of $\log
d$, compared to $\log(d+R_{m}^{2}/2R)$.
Numerical calculations based on Eq. (3), using random patches specified on a
surface, are straightforward and fully support the essential conclusions
presented above. As an aside, there is a remaining question as to where it is
the energy or force that must be minimized. Numerical calculations based on
the minimization of the force produces no statistically significant
differences compared to the energy minimization, as might be expected.
Of course, this is a very simplistic model, particularly in the assumptions
that the patch potentials are randomly $\pm V_{0}$, that all patches have the
same radius and are circular, and that the radius of the positive patches is
equal to that of the negative patches. These assumptions are not essential to
the derivation of Eq. (8), which is the principal result reported here.
Further refinements will likely not significantly change the subsequent
conclusions presented here. For example, as discussed already, a single patch
at large $r$ will generate a non-zero $Q_{0}$. If this patch has a large area
and/or large potential (e.g., a charged speck of dust or a scratch), it could
easily be the dominant contribution to the $\log d$ distance dependence. For
such a patch, $V_{m}(d)$ will only vary slowly with a relative translational
repositioning of the plates.
This work was supported by the DARPA/MTOs Casimir Effect Enhancement project
under SPAWAR Contract No. N66001-09-1-2071.
## References
* (1) W.-J. Kim, M. Brown-Hayes, D.A.R. Dalvit, J.H. Brownell, and R. Onofrio, Phys. Rev. A 78, 020101(R) (2008).
* (2) W.-J. Kim, A.O. Sushkov, D.A.R. Dalvit, and S.K. Lamoreaux, Phys. Rev. Lett. 103, 060401 (2009).
* (3) S. de Man, K. Heeck, and D. Iannuzzi, Phys. Rev. A 79, 024102 (2009).
* (4) S. de Man, K. Heeck, R.J. Wijngaarden, and D. Iannuzzi, J. Vac. Sci. Tech. B 28, C4A25 (2010).
* (5) W.J. Kim, A.O. Sushkov, D.A.R. Dalvit, and S.K. Lamoreaux, Phys. Rev. A 81, 022505 (2010).
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arxiv-papers
| 2011-06-17T19:06:08 |
2024-09-04T02:49:19.769753
|
{
"license": "Public Domain",
"authors": "S.K. Lamoreaux and A.O. Sushkov",
"submitter": "Steve K. Lamoreaux",
"url": "https://arxiv.org/abs/1106.3549"
}
|
1106.3588
|
# Some Operators Associated to Rarita-Schwinger Type Operators
Junxia Li and John Ryan
_Department of Mathematics, University of Arkansas, Fayetteville, AR 72701,
USA_
###### Abstract
In this paper we study some operators associated to the Rarita-Schwinger
operators. They arise from the difference between the Dirac operator and the
Rarita-Schwinger operators. These operators are called remaining operators.
They are based on the Dirac operator and projection operators $I-P_{k}.$ The
fundamental solutions of these operators are harmonic polynomials, homogeneous
of degree $k$. First we study the remaining operators and their representation
theory in Euclidean space. Second, we can extend the remaining operators in
Euclidean space to the sphere under the Cayley transformation.
Keywords: Clifford algebra, Rarita-Schwinger operators, remaining operators,
Cayley transformation, Almansi-Fischer decomposition.
_This paper is dedicated to Michael Shapiro on the occasion of his 65th
birthday._
## 1 Introduction
Rarita-Schwinger operators in Clifford analysis arise in representation theory
for the covering groups of $SO(n)$ and $O(n)$. They are generalizations of the
Dirac operator. We denote a Rarita-Schwinger operator by $R_{k}$, where
$k=0,1,\cdots,m,\cdots.$ When $k=0$ it is the Dirac operator. The Rarita-
Schwinger operators $R_{k}$ in Euclidean space have been studied in [BSSV,
BSSV1, DLRV, Va1, Va2]. Rarita-Schwinger operators on the sphere denoted by
$R_{k}^{S}$ have also been studied in [LRV].
In this paper we study the remaining operators, $Q_{k}$, which are related to
the Rarita-Schwinger operators. In fact, The remaining operators are the
difference between the Dirac operator and the Rarita-Schwinger operators.
Let $\mathcal{H}$k be the space of harmonic polynomials homogeneous of degree
$k$ and $\mathcal{M}_{k}$, $\mathcal{M}_{k-1}$ be the spaces of $Cl_{n}-$
valued monogenic polynomials, homogeneous of degree $k$ and $k-1$
respectively. Instead of considering
$P_{k}:\mathcal{H}_{k}\rightarrow\mathcal{M}_{k}$ in [DLRV], we look at the
projection map, $I-P_{k}:\mathcal{H}_{k}\rightarrow u\mathcal{M}_{k-1},$ and
the Dirac operator, to define the $Q_{k}$ operators and construct their
fundamental solutions in $\mathbb{R}^{n}$. We introduce basic results of these
operators. This includes Stokes’ Theorem, Borel-Pompeiu Theorem, Cauchy’s
Integral Formula and a Cauchy Transform. In section 5, by considering the
Cayley transformation and its inverse, we can extend the results for the
remaining operators in $\mathbb{R}^{n}$ to the sphere, $\mathbb{S}^{n}.$ We
construct the fundamental solutions to the remaining operators by applying the
Cayley transformation to the fundamental solutions to the remaining operators
in $\mathbb{R}^{n}.$ We also obtain the intertwining operators for $Q_{k}$
operators and $Q_{k}^{S}$ operators. In turn, we establish the conformal
invariance of the remaining equations under the Cayley transformation and its
inverse. We conclude by giving some basic integral formulas with detailed
proofs, and pointing out that results obtained for Rarita-Schwinger operators
in [LRV] for real projective space readily carry over to the context presented
here.
## 2 Preliminaries
A Clifford algebra, $Cl_{n},$ can be generated from $\mathbb{R}^{n}$ by
considering the relationship
${x}^{2}=-\|{x}\|^{2}$
for each ${x}\in\mathbb{R}^{n}$. We have $\mathbb{R}^{n}\subseteq Cl_{n}$. If
$e_{1},\ldots,e_{n}$ is an orthonormal basis for $\mathbb{R}^{n}$, then
${x}^{2}=-\|{x}\|^{2}$ tells us that $e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij},$
where $\delta_{ij}$ is the Kronecker delta function. Let
$A=\\{j_{1},\cdots,j_{r}\\}\subset\\{1,2,\cdots,n\\}$ and $1\leq
j_{1}<j_{2}<\cdots<j_{r}\leq n$. An arbitrary element of the basis of the
Clifford algebra can be written as $e_{A}=e_{j_{1}}\cdots e_{j_{r}}.$ Hence
for any element $a\in Cl_{n}$, we have $a=\sum_{A}a_{A}e_{A},$ where
$a_{A}\in\mathbb{R}.$
We define the Clifford conjugation as the following:
$\bar{a}=\sum_{A}(-1)^{|A|(|A|+1)/2}a_{A}e_{A}$
satisfying $\overline{e_{j_{1}}\cdots e_{j_{r}}}=(-1)^{r}e_{j_{r}}\cdots
e_{j_{1}}$ and $\overline{ab}=\bar{b}\bar{a}$ for $a,b\in Cl_{n}.$
For each $a=a_{0}+\cdots+a_{1\cdots n}e_{1}\cdots e_{n}\in Cl_{n}$ the scalar
part of $\bar{a}a$ gives the square of the norm of $a,$ namely
$a_{0}^{2}+\cdots+a_{1\cdots n}^{2}$ .
The reversion is given by
$\tilde{a}=\sum_{A}(-1)^{|A|(|A|-1)/2}a_{A}e_{A},$
where $|A|$ is the cardinality of $A$. In particular,
$\widetilde{e_{j_{1}}\cdots e_{j_{r}}}=e_{j_{r}}\cdots e_{j_{1}}.$ Also
$\widetilde{ab}=\tilde{b}\tilde{a}$ for $a,b\in Cl_{n+1}.$
The Pin and Spin groups play an important role in Clifford analysis. The Pin
group can be defined as
$Pin(n):=\\{a\in Cl_{n}:a=y_{1}\ldots
y_{p}:{y_{1},\ldots,y_{p}}\in\mathbb{S}^{n-1},p\in\mathbb{N}\\}$
and is clearly a group under multiplication in $Cl_{n}$.
Now suppose that $y\in\mathbb{S}^{n-1}\subseteq\mathbb{R}^{n}$. Look at
$yxy=yx^{\parallel_{y}}y+yx^{\perp_{y}}y=-x^{\parallel_{y}}+x^{\perp_{y}}$
where $x^{\parallel_{y}}$ is the projection of $x$ onto $y$ and
$x^{\perp_{y}}$ is perpendicular to $y$. So $yxy$ gives a reflection of $x$ in
the $y$ direction. By the Cartan$-$Dieudonné Theorem each $O\in O(n)$ is the
composition of a finite number of reflections. If $a=y_{1}\ldots y_{p}\in
Pin(n)$, then $\tilde{a}:=y_{p}\ldots y_{1}$ and $ax\tilde{a}=O_{a}(x)$ for
some $O_{a}\in O(n).$ Choosing $y_{1},\ldots,y_{p}$ arbitrarily in
$\mathbb{S}^{n-1}$, we see that the group homomorphism
$\theta:Pin(n)\longrightarrow O(n):a\longmapsto O_{a}$
with $a=y_{1}\ldots y_{p}$ and $O_{a}(x)=ax\tilde{a}$ is surjective. Further
$-ax(-\tilde{a})=ax\tilde{a}$, so $1,-1\in ker(\theta)$. In fact
$ker(\theta)=\\{\pm 1\\}.$ The Spin group is defined as
$Spin(n):=\\{a\in Pin(n):a=y_{1}\ldots y_{p}\mbox{ and }p\mbox{ even}\\}$
and is a subgroup of $Pin(n)$. There is a group homomorphism
$\theta:Spin(n)\longrightarrow SO(n)$
which is surjective with kernel $\\{1,-1\\}$. See [P] for details.
The Dirac Operator in $\mathbb{R}^{n}$ is defined to be
$D:=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}.$
Note $D^{2}=-\Delta_{n},$ where $\Delta_{n}$ is the Laplacian in
$\mathbb{R}^{n}$.
Let $\mathcal{H}$k be the space of harmonic polynomials homogeneous of degree
$k.$ Let $\mathcal{M}_{k}$ denote the space of $Cl_{n}-$ valued polynomials,
homogeneous of degree $k$ and such that if $p_{k}\in$ $\mathcal{M}_{k}$ then
$Dp_{k}=0.$ Such a polynomial is called a left monogenic polynomial
homogeneous of degree $k$. Note if $h_{k}\in$ $\mathcal{H}_{k},$ the space of
$Cl_{n}-$ valued harmonic polynomials homogeneous of degree $k$, then
$Dh_{k}\in$ $\mathcal{M}$k-1. But $Dup_{k-1}(u)=(-n-2k+2)p_{k-1}(u),$ so
$\mathcal{H}_{k}=\mathcal{M}_{k}\bigoplus
u\mathcal{M}_{k-1},h_{k}=p_{k}+up_{k-1}.$
This is the so-called Almansi-Fischer decomposition of $\mathcal{H}$k, where
$\mathcal{M}_{k-1}$ is the space of $Cl_{n}-$ valued left monogenic
polynomials, homogeneous of degree $k-1$. See [BDS, R].
Note that if $Dg(u)=0$ then $\bar{g}(u)\bar{D}=-\bar{g}(u)D=0$. So we can talk
of right monogenic polynomials, homogeneous of degree $k$ and we obtain by
conjugation a right Almansi-Fisher decomposition,
$\mathcal{H}_{k}=\overline{\mathcal{M}_{k}}\bigoplus\overline{\mathcal{M}}_{k-1}u,$
where $\overline{\mathcal{M}_{k}}$ stands for the space of right monogenic
polynomials homogeneous of degree $k$.
Let $P_{k}$ be the left projection map
$P_{k}:\mathcal{H}_{k}\rightarrow\mathcal{M}_{k},$
then the left Rarita-Schwinger operator $R_{k}$ is defined by (see [BSSV,
BSSV1, DLRV, Va1, Va2])
$R_{k}g(x,u)=P_{k}D_{x}g(x,u),$
where $D_{x}$ is the Dirac operator with respect to $x$ and
$g(x,u):U\times\mathbb{R}^{n}\to Cl_{n}$ is a monogenic polynomial homogeneous
of degree $k$ in $u,$ and $U$ is a domain in $\mathbb{R}^{n}$. The left
Rarita-Schwinger equation is defined to be
$R_{k}g(x,u)=0.$
We also have a right projection
$P_{k,r}:\mathcal{H}_{k}\rightarrow\overline{\mathcal{M}_{k}},$ and a right
Rarita-Schwinger equation $g(x,u)D_{x}P_{k,r}=g(x,u)R_{k}=0.$
A Möbius transformation is a finite composition of orthogonal transformations,
inversions, dilations, and translations. Ahlfors [A] and Vahlen [V] show that
given a Möbius transformation $y=\phi(x)$ on
$\mathbb{R}^{n}\bigcup\\{\infty\\}$ it can be expressed as
$y=(ax+b)(cx+d)^{-1}$ where $a,b,c,d\in Cl_{n}$ and satisfy the following
conditions:
1. 1.
$a,b,c,d$ are all products of vectors in $\mathbb{R}^{n}.$
2. 2.
$a\tilde{b},c\tilde{d},\tilde{b}c,\tilde{d}a\in\mathbb{R}^{n}.$
3. 3.
$a\tilde{d}-b\tilde{c}=\pm 1.$
When $c=0,\phi(x)=(ax+b)(cx+d)^{-1}=axd^{-1}+bd^{-1}=\pm ax\tilde{a}+bd^{-1}.$
Now assume $c\neq 0,$ then
$\phi(x)=(ax+b)(cx+d)^{-1}=ac^{-1}\pm(cx\tilde{c}+d\tilde{c})^{-1}.$ These are
the so-called Iwasawa decompositions. Using this notation and the conformal
weights, $f(\phi(x))$ is changed to $J(\phi,x)f(\phi(x)),$ where
$J(\phi,x)=\displaystyle\frac{\widetilde{cx+d}}{\|cx+d\|^{n}}$. Note when
$\phi(x)=x+a$ then $J(\phi,x)\equiv 1.$
## 3 The $Q_{k}$ operators and their kernels
As
$I-P_{k}:\mathcal{H}_{k}\rightarrow u\mathcal{M}_{k-1},$
where $I$ is the identity map, then we can define the left remaining operators
$Q_{k}:=(I-P_{k})D_{x}:u\mathcal{M}_{k-1}\to u\mathcal{M}_{k-1}\quad
uf(x,u):\to(I-P_{k})D_{x}uf(x,u).$
See[BSSV].
The left remaining equation is defined to be $(I-P_{k})D_{x}uf(x,u)=0$ or
$Q_{k}uf(x,u)=0,$ for each $x$ and $(x,u)\in U\times\mathbb{R}^{n}$, where $U$
is a domain in $\mathbb{R}^{n}$ and $f(x,u)\in\mathcal{M}_{k-1}.$
We also have a right remaining operator
$Q_{k,r}:=D_{x}(I-P_{k,r}):\overline{\mathcal{M}}_{k-1}u\to\overline{\mathcal{M}}_{k-1}u\quad
g(x,u)u:\to g(x,u)uD_{x}(I-P_{k,r}),$
where $g(x,u)\in\overline{\mathcal{M}}_{k-1}$.
Consequently, the right remaining equation is $g(x,u)uD_{x}(I-P_{k,r})=0$ or
$g(x,u)uQ_{k,r}=0.$
Now let us establish the conformal invariance of the remaining equation
$Q_{k}uf(x,u)=0$.
It is easy to see that $I-P_{k}$ is conformally invariant under the Möbius
transformations, since the projection operator $P_{k}$ is conformally
invariant (see [BSSV, DLRV]). By considering orthogonal transformation,
inversion, dilation and translation and applying the same arguments in [DLRV]
used to establish the intertwining operators for Rarita-Schwinger operators,
we can easily obtain the intertwining operators for $Q_{k}$ operators:
###### Theorem 1.
$J_{-1}(\phi,x)Q_{k,u}uf(y,u)=Q_{k,w}wJ(\phi,x)f(\phi(x),\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^{2}}),$
where $Q_{k,u}$ and $Q_{k,w}$ are the remaining operators with respect to $u$
and $w$ respectively, $y=\phi(x)$ is the Möbius transformation,
$J(\phi,x)=\displaystyle\frac{\widetilde{cx+d}}{\|cx+d\|^{n}},J_{-1}(\phi,x)=\displaystyle\frac{cx+d}{\|cx+d\|^{n+2}},$
and $u=\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^{2}}$ for some
$w\in\mathbb{R}^{n}.$
Consequently, we have
$Q_{k,u}uf(x,u)=0$ implies
$Q_{k,w}wJ(\phi,x)f(\phi(x),\displaystyle\frac{\widetilde{(cx+d)}w(cx+d)}{\|cx+d\|^{2}})=0$.
This tells us that the remaining equation $Q_{k}uf(x,u)=0$ is conformally
invariant under Möbius transformations.
The reproducing kernel of $\mathcal{M}_{k}$ with respect to integration over
$\mathbb{S}^{n-1}$ is given by (see [BDS, DLRV])
$Z_{k}(u,v):=\displaystyle\sum_{\sigma}P_{\sigma}(u)V_{\sigma}(v)v,$
where
$P_{\sigma}(u)=\displaystyle\frac{1}{k!}\displaystyle\Sigma(u_{i_{1}}-u_{1}e_{1}^{-1}e_{i_{1}})\ldots(u_{i_{k}}-u_{1}e_{1}^{-1}e_{i_{k}}),V_{\sigma}(v)=\displaystyle\frac{\partial^{k}G(v)}{\partial
v_{2}^{j_{2}}\ldots\partial v_{n}^{j_{n}}}\,$
$j_{2}+\ldots+j_{n}=k,~{}~{}\mbox{and}~{}~{}i_{k}\in\\{2,\cdots,n\\}.$ Here
summation is taken over all permutations of the monomials without repetition.
This function is left monogenic in $u$ and right monogenic polynomial in $v$
and it is homogeneous of degree $k$. See [BDS] and elsewhere.
Let us consider the polynomial $uZ_{k-1}(u,v)v$ which is harmonic, homogeneous
degree of $k$ in both $u$ and $v$. Since $uZ_{k-1}(u,v)v$ does not depend on
$x$, $Q_{k}uZ_{k-1}(u,v)v=0$.
Now applying inversion from the left, we obtain
$H_{k}(x,u,v):=\displaystyle\frac{-1}{\omega_{n}c_{k}}u\displaystyle\frac{x}{\|x\|^{n}}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^{2}},v)v$
is a non-trivial solution to $Q_{k}uf(x,u)=0,$ where
$c_{k}=\displaystyle\frac{n-2}{n-2+2k}.$
Similarly, applying inversion from the right, we obtain
$\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{xvx}{\|x\|^{2}})\displaystyle\frac{x}{\|x\|^{n}}v$
is a non-trivial solution to $f(x,v)vQ_{k,r}=0.$ Using the similar arguments
in [DLRV], we can show that two representations of the solutions are equal.
The details are given in the following
$\begin{array}[]{ll}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{xvx}{\|x\|^{2}})\displaystyle\frac{x}{\|x\|^{n}}v\\\
\\\
=\displaystyle\frac{-1}{\omega_{n}c_{k}}u\displaystyle\frac{-x}{\|x\|}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^{2}},v)\displaystyle\frac{x}{\|x\|}\displaystyle\frac{x}{\|x\|^{n}}v=\displaystyle\frac{-1}{\omega_{n}c_{k}}u\displaystyle\frac{x}{\|x\|^{n}}Z_{k-1}(\displaystyle\frac{xux}{\|x\|^{2}},v)v.\end{array}$
In fact $H_{k}(x,u,v)$ is the fundamental solution to the $Q_{k}$ operator.
## 4 Some basic integral formulas related to $Q_{k}$ operators
In this section, we will establish some basic integral formulas associated
with $Q_{k}$ operators.
###### Definition 1.
[DLRV] For any $Cl_{n}-$valued polynomials $P(u),Q(u)$, the inner product
$(P(u),Q(u))_{u}$ with respect to $u$ is given by
$(P(u),Q(u))_{u}=\displaystyle\int_{\mathbb{S}^{n-1}}P(u)Q(u)ds(u).$
For any $p_{k}\in\mathcal{M}_{k},$ one obtains (see [BDS])
$p_{k}(u)=(Z_{k}(u,v),p_{k}(v))_{v}=\int_{\mathbb{S}^{n-1}}Z_{k}(u,v)p_{k}(v)ds(v).$
Now if we combine Stokes’ Theorems of the Dirac operator and the Rarita-
Schwinger operator, then we have two versions of Stokes’ Theorem for the
$Q_{k}$ operators .
###### Theorem 2.
(Stokes’ Theorem for $Q_{k}$ operators) Let $\Omega^{\prime}$ and $\Omega$ be
domains in $\mathbb{R}^{n}$ and suppose the closure of $\Omega$ lies in
$\Omega^{\prime}$. Further suppose the closure of $\Omega$ is compact and the
boundary of $\Omega,$ $\partial\Omega$, is piecewise smooth. Then for $f,g\in
C^{1}(\Omega^{\prime},$$\mathcal{M}_{k})$, we have version 1
$\begin{array}[]{ll}\displaystyle\int_{\Omega}[(g(x,u)Q_{k,r},f(x,u))_{u}+(g(x,u),Q_{k}f(x,u))_{u}]dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u),(I-P_{k})d\sigma_{x}f(x,u)\right)_{u}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u)d\sigma_{x}(I-P_{k,r}),f(x,u)\right)_{u}.\end{array}$
Then for $f,g\in C^{1}(\Omega^{\prime},$$\mathcal{M}_{k-1})$, we have version
2
$\begin{array}[]{ll}\displaystyle\int_{\Omega}[(g(x,u)uQ_{k,r}),uf(x,u))_{u}+(g(x,u)u,Q_{k}uf(x,u))_{u}]dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_{x}(I-P_{k,r}),uf(x,u)\right)_{u}.\end{array}$
Proof: It is easy to get version 1 of Stokes’ Theorem for the $Q_{k}$
operators by combining Stokes’ Theorems of the Dirac operator and the Rarita-
Schwinger operators.
Now we shall prove version 2 of Stokes’ Theorem.
First of all, we want to prove that
$\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}=\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_{x}(I-P_{k,r}),uf(x,u)\right)_{u}.$
Here $d\sigma_{x}=n(x)d\sigma(x)$. By the Almansi-Fischer decomposition, we
have
$g(x,u)un(x)uf(x,u)=g(x,u)u[f_{1}(x,u)+uf_{2}(x,u)]=[g_{1}(x,u)+g_{2}(x,u)u]uf(x,u),$
so
$\begin{array}[]{ll}g(x,u)ud\sigma_{x}uf(x,u)\\\ \\\
=g(x,u)u[f_{1}(x,u)+uf_{2}(x,u)]d\sigma(x)=[g_{1}(x,u)+g_{2}(x,u)u]uf(x,u)d\sigma(x),\end{array}$
where $f_{1}(x,u),f_{2}(x,u),g_{1}(x,u),g_{2}(x,u)$ are left or right
monogenic polynomials in $u.$ Now integrating the above formula over the unit
sphere in $\mathbb{R}^{n}$, one gets
$\begin{array}[]{ll}\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)ud\sigma_{x}uf(x,u)ds(u)\\\
\\\
=\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)uuf_{2}(x,u)d\sigma(x)ds(u)=\displaystyle\int_{\mathbb{S}^{n-1}}g_{2}(x,u)uuf(x,u)d\sigma(x)ds(u).\end{array}$
This follows from the fact that
$\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)uf_{1}(x,u)ds(u)=\displaystyle\int_{\mathbb{S}^{n-1}}g_{1}(x,u)uf(x,u)ds(u)=0.$
See [BDS].
Thus
$\displaystyle\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}g(x,u)u((I-P_{k})d\sigma_{x}uf(x,u))ds(u)$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}g(x,u)uuf_{2}(x,u)ds(u)d\sigma(x)$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}g_{2}(x,u)uuf(x,u)ds(u)d\sigma(x)$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}(g(x,u)d\sigma_{x}(I-P_{k,r}))uf(x,u)ds(u)$
$\displaystyle=$
$\displaystyle\displaystyle\int_{\partial\Omega}\left(g(x,u)ud\sigma_{x}(I-P_{k,r}),uf(x,u)\right)_{u}.$
Secondly, we need to show
$\begin{array}[]{ll}\displaystyle\int_{\Omega}[(g(x,u)uQ_{k,r},uf(x,u))_{u}+(g(x,u)u,Q_{k}uf(x,u))_{u}]dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}.\end{array}$
Consider the integral
$\displaystyle\displaystyle\int_{\Omega}[(g(x,u)uD_{x}P_{k,r},uf(x,u))_{u}+(g(x,u)u,P_{k}D_{x}uf(x,u))_{u}]dx^{n}$
$\displaystyle=\displaystyle\int_{\Omega}\int_{\mathbb{S}^{n-1}}[(g(x,u)uD_{x}P_{k,r})uf(x,u)+g(x,u)u(P_{k}D_{x}uf(x,u))]ds(u)dx^{n}.$
(1)
Since $g(x,u)uD_{x}P_{k,r},f(x,u),g(x,u)$ and $P_{k}D_{x}uf(x,u)$ are
monogenic functions in $u$,
$\int_{\mathbb{S}^{n-1}}(g(x,u)uD_{x}P_{k,r})uf(x,u)ds(u)=0=\int_{\mathbb{S}^{n-1}}g(x,u)u(P_{k}D_{x}uf(x,u))ds(u).$
Thus the previous integral (4) equals zero.
By Stokes’ Theorem for the Dirac operator, we have
$\begin{array}[]{lll}\displaystyle\int_{\Omega}[(g(x,u)uD_{x},uf(x,u))_{u}+(g(x,u)u,D_{x}uf(x,u))_{u}]dx^{n}\\\
\\\
=\displaystyle\int_{\Omega}\int_{\mathbb{S}^{n-1}}[(g(x,u)uD_{x})uf(x,u)+g(x,u)u(D_{x}uf(x,u))]ds(u)dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}[(g(x,u)ud\sigma_{x}uf(x,u))]ds(u)\\\
\\\
=\displaystyle\int_{\partial\Omega}(g(x,u)u,d\sigma_{x}uf(x,u))_{u}.\end{array}$
But
$\displaystyle\int_{\partial\Omega}(g(x,u)u,P_{k}d\sigma_{x}uf(x,u))_{u}=\displaystyle\int_{\partial\Omega}\int_{\mathbb{S}^{n-1}}g(x,u)u(P_{k}d\sigma_{x}uf(x,u))ds(u)=0,$
since
$\displaystyle\int_{\mathbb{S}^{n-1}}g(x,u)u(P_{k}d\sigma_{x}uf(x,u))ds(u)=0.$
Therefore we have shown
$\begin{array}[]{ll}\displaystyle\int_{\Omega}[(g(x,u)uQ_{k,r},uf(x,u))_{u}+(g(x,u)u,Q_{k}uf(x,u))_{u}]dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}.\quad\blacksquare\end{array}$
###### Remark 1.
In the proof of the previous theorem it is proved that
$\displaystyle\displaystyle\int_{\partial\Omega}\left(g(x,u)u,(I-P_{k})d\sigma_{x}uf(x,u)\right)_{u}=\displaystyle\int_{\partial\Omega}\left(g(x,u)u,d\sigma_{x}uf(x,u)\right)_{u}.$
(2)
###### Theorem 3.
(Borel-Pompeiu Theorem)Let $\Omega^{\prime}$ and $\Omega$ be as in the
previous Theorem. Then for $f\in C^{1}(\Omega^{\prime},$$\mathcal{M}_{k-1})$
and $y\in\Omega,$ we obtain
$\begin{array}[]{ll}uf(y,u)=\displaystyle\int_{\Omega}(H_{k}(x-y,u,v),Q_{k}vf(x,v))_{v}dx^{n}\\\
\\\
-\displaystyle\int_{\partial\Omega}\left(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(x,v)\right)_{v}.\end{array}$
Here we will use the representation
$H_{k}(x-y,u,v)=\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|^{n}}v$.
Proof: Consider a ball $B(y,r)$ centered at $y$ with radius $r$ such that
$\overline{B(y,r)}\subset\Omega$. We have
$\begin{array}[]{ll}\displaystyle\int_{\Omega}(H_{k}(x-y,u,v),Q_{k}vf(x,v))_{v}dx^{n}\\\
\\\
=\displaystyle\int_{\Omega\setminus{B(y,r)}}(H_{k}(x-y,u,v),Q_{k}vf(x,v))_{v}dx^{n}\\\
\\\
+\displaystyle\int_{B(y,r)}(H_{k}(x-y,u,v),Q_{k}vf(x,v))_{v}dx^{n}.\end{array}$
The last integral in the previous equation tends to zero as $r$ tends to zero.
This follows from the degree of homogeneity of $x-y$ in $H_{k}(x-y,u,v)$. Now
applying Stokes’ Theorem version 2 to the first integral, one gets
$\begin{array}[]{ll}\displaystyle\int_{\Omega\setminus{B(y,r)}}(H_{k}(x-y,u,v),Q_{k}vf(x,v))_{v}dx^{n}\\\
\\\
=\displaystyle\int_{\partial\Omega}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(x,v))_{v}-\displaystyle\int_{\partial{B(y,r)}}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(x,v))_{v}.\end{array}$
Now let us look at the integral
$\begin{array}[]{ll}\displaystyle\int_{\partial{B(y,r)}}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(x,v))_{v}dx^{n}\\\
\\\
=\displaystyle\int_{\partial{B(y,r)}}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(y,v))_{v}\\\
\\\
+\displaystyle\int_{\partial{B(y,r)}}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}v[f(x,v)-f(y,v)])_{v}.\end{array}$
Since the second integral on the right hand side tends to zero as $r$ goes to
zero because of the continuity of $f$, we only need to deal with the first
integral
$\begin{array}[]{llll}\displaystyle\int_{\partial{B(y,r)}}(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(y,v))_{v}\\\
\\\
=\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}H_{k}(x-y,u,v)(I-P_{k})d\sigma_{x}vf(y,v)ds(v)\\\
\\\
=\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}}\right)\displaystyle\frac{x-y}{\|x-y\|^{n}}v(I-P_{k})n(x)vf(y,v)ds(v)d\sigma(x),\end{array}$
where $n(x)$ is the unit outer normal vector and $d\sigma(x)$ is the scalar
measure on $\partial B(y,r).$ Now $n(x)$ here is
$\displaystyle\frac{y-x}{\|x-y\|}.$ Using equation (2) the previous integral
becomes
$\begin{array}[]{ll}\displaystyle\int_{\partial{B(y,r)}}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{1}{\omega_{n}c_{k}}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}}\right)\displaystyle\frac{x-y}{\|x-y\|^{n}}v\displaystyle\frac{x-y}{\|x-y\|}vf(y,v)ds(v)d\sigma(x)\\\
\\\
=\displaystyle\frac{1}{\omega_{n}c_{k}}\displaystyle\int_{\partial{B(y,r)}}\displaystyle\frac{1}{r^{n-1}}\int_{\mathbb{S}^{n-1}}uZ_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}}\right)\displaystyle\frac{x-y}{\|x-y\|}v\displaystyle\frac{x-y}{\|x-y\|}vf(y,v)ds(v)d\sigma(x)\end{array}$
Since
$Z_{k-1}\left(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}}\right)\displaystyle\frac{x-y}{\|x-y\|}v\displaystyle\frac{x-y}{\|x-y\|}$
is a harmonic polynomial with degree $k$ in $v$, we can apply Lemma $5$ in
[DLRV], then the integral is equal to
$\begin{array}[]{ll}\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,v)vvf(y,v)ds(v)d\sigma(x)\\\
\\\
=-u\displaystyle\int_{\mathbb{S}^{n-1}}Z_{k-1}(u,v)f(y,v)ds(v)=-uf(y,u),\end{array}$
Therefore, when the radius $r$ tends to zero, we obtain the desired result.
$\blacksquare$
Now if the function has compact support in $\Omega$, then by Borel-Pompeiu
Theorem we obtain:
###### Theorem 4.
$\displaystyle\iint_{\mathbb{R}^{n}}(H_{k}(x-y,u,v),Q_{k}v\phi(x,v))_{v}dx^{n}=u\phi(y,u)$
for each $\phi\in C_{0}^{\infty}(\mathbb{R}^{n})$.
Now suppose $vf(x,v)$ is a solution to the $Q_{k}$ operator, then using the
Borel-Pompeiu Theorem we have:
###### Theorem 5.
(Cauchy Integral Formula) If $Q_{k}vf(x,v)=0,$ then for $y\in\Omega,$
$\begin{array}[]{ll}uf(y,u)=-\displaystyle\int_{\partial\Omega}\left(H_{k}(x-y,u,v),(I-P_{k})d\sigma_{x}vf(x,v)\right)_{v}\\\
=-\displaystyle\int_{\partial\Omega}\left(H_{k}(x-y,u,v)d\sigma_{x}(I-P_{k,r}),vf(x,v)\right)_{v}.\qquad\blacksquare\end{array}$
We also can talk about a Cauchy transform for the $Q_{k}$ operators:
###### Definition 2.
For a domain $\Omega\subset\mathbb{R}^{n}$ and a function
$f:\Omega\times\mathbb{R}^{n}\longrightarrow Cl_{n},$ where $f(x,u)$ is
monogenic in $u$ with degree $k-1$, the Cauchy $($or $T_{k}$-transform$)$ of
$f$ is formally defined to be
$(T_{k}vf)(y,v)=-\iint_{\Omega}\left(H_{k}(x-y,u,v),uf(x,u)\right)_{u}dx^{n},\qquad
y\in\Omega.$
###### Theorem 6.
$Q_{k}\displaystyle\iint_{\mathbb{R}^{n}}\left(H_{k}(x-y,u,v),v\phi(x,v)\right)_{v}dx^{n}=u\phi(y,u),\mbox{for}~{}~{}\phi\in
C_{0}^{\infty}(\mathbb{R}^{n}).$
Here we use the representation
$H_{k}(x-y,u,v)=\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|^{n}}v.$
Proof: For each fixed $y\in\mathbb{R}^{n}$, we can construct a bounded
rectangle $R(y)$ centered at $y$ in $\mathbb{R}^{n}$.
Then
$\begin{array}[]{lll}Q_{k}\displaystyle\iint_{\mathbb{R}^{n}\setminus
R(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}dx^{n}\\\ \\\
=(I-P_{k})D_{y}\displaystyle\iint_{\mathbb{R}^{n}\setminus
R(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}dx^{n}=0.\end{array}$
Now consider
$\begin{array}[]{lll}\displaystyle\frac{\partial}{\partial
y_{i}}\displaystyle\iint_{R(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}dx^{n}\\\ \\\
=\lim_{\varepsilon\to
0}\displaystyle\frac{1}{\varepsilon}[\displaystyle\iint_{R(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}dx^{n}\\\
\\\ -\displaystyle\iint_{R(y+\varepsilon e_{i})}(H_{k}(x-y-\varepsilon
e_{i},u,v),v\phi(x,v))_{v}dx^{n}]\end{array}$
$\begin{array}[]{lll}=\displaystyle\iint_{R(y)}(H_{k}(x-y,u,v),\displaystyle\frac{\partial
v\phi(x,v)}{\partial x_{i}})_{v}dx^{n}\\\ \\\ +\displaystyle\int_{\partial
R_{1}(y)\cup\partial
R_{2}(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}d\sigma(x)\end{array}$
where $\partial R_{1}(y)$ and $\partial R_{2}(y)$ are the two faces of $R(y)$
with normal vectors $\pm e_{i}.$ So
$\begin{array}[]{ll}D_{y}\displaystyle\iint_{R(y)}(H_{k}(x-y,u,v),v\phi(x,v))_{v}dx^{n}\\\
\\\
=\displaystyle\iint_{R(y)}\sum_{i=1}^{n}e_{i}(H_{k}(x-y,u,v),\displaystyle\frac{\partial{v\phi(x,v)}}{\partial{x_{i}}})_{v}dx^{n}\\\
\\\ +\displaystyle\int_{\partial
R(y)}n(x)(H_{k}(x-y,u,v),v\phi(x,v))_{v}d\sigma(x).\end{array}$
The first integral tends to zero as the volume of $R(y)$ tends to zero. Thus
we will pay attention to the integral
$(I-P_{k})\displaystyle\int_{\partial
R(y)}n(x)(H_{k}(x-y,u,v),v\phi(x,v))_{v}d\sigma(x).$
This is equal to
$(I-P_{k})\displaystyle\int_{\partial
R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_{k}(x-y,u,v)v\phi(x,v)ds(v)d\sigma(x),$
which in turn is equal to
$\begin{array}[]{ll}(I-P_{k})\displaystyle\int_{\partial
R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_{k}(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x)\\\
\\\ +(I-P_{k})\displaystyle\int_{\partial
R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_{k}(x-y,u,v)v(\phi(x,v)-\phi(y,v))ds(v)d\sigma(x).\end{array}$
But the last integral on the right side of the above formula tends to zero as
the surface area of $\partial R(y)$ tends to zero because of the degree of the
homogeneity of $x-y$ in $H_{k}$ and the continuity of the function $\phi$.
Hence we are left with
$(I-P_{k})\displaystyle\int_{\partial
R(y)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_{k}(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x).$
By Stokes’ Theorem this is equal to
$(I-P_{k})\displaystyle\int_{\partial
B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}n(x)H_{k}(x-y,u,v)v\phi(y,v)ds(v)d\sigma(x).$
In turn this is equal to
$\begin{array}[]{ll}(I-P_{k})\displaystyle\int_{\partial
B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}-\displaystyle\frac{x-y}{\|x-y\|}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|^{n}}vv\phi(y,v)ds(v)d\sigma(x)\\\
\\\ =(I-P_{k})\displaystyle\int_{\partial
B(y,r)}\displaystyle\frac{-1}{\omega_{n}c_{k}}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}uZ_{k-1}(u,\displaystyle\frac{(x-y)v(x-y)}{\|x-y\|^{2}},v)\displaystyle\frac{x-y}{\|x-y\|^{n}}\phi(y,v)ds(v)d\sigma(x).\end{array}$
Since $Z_{k-1}(u,v)$ is the reproducing kernel of $\mathcal{M}_{k-1}$,
$\pm\tilde{a}Z_{k-1}(au\tilde{a},av\tilde{a})a$ is also the reproducing kernel
of $\mathcal{M}_{k-1}$ for each $a\in Pin(n)$. See [DLRV]. Now let
$a=\displaystyle\frac{x-y}{\|x-y\|},$ the previous integral equals
$\begin{array}[]{ll}(I-P_{k})\displaystyle\int_{\partial
B(y,r)}\displaystyle\frac{1}{\omega_{n}c_{k}}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}u\displaystyle\frac{x-y}{\|x-y\|}Z_{k-1}(\displaystyle\frac{(x-y)u(x-y)}{\|x-y\|^{2}},v)\displaystyle\frac{x-y}{\|x-y\|}\displaystyle\frac{x-y}{\|x-y\|^{n}}\\\
\\\ \phi(y,v)ds(v)d\sigma(x)\\\ \\\ =(I-P_{k})\displaystyle\int_{\partial
B(y,r)}\displaystyle\int_{\mathbb{S}^{n-1}}\displaystyle\frac{1}{\omega_{n}c_{k}}\displaystyle\frac{1}{r^{n-1}}\displaystyle\frac{x-y}{\|x-y\|}u\displaystyle\frac{x-y}{\|x-y\|}Z_{k-1}(\displaystyle\frac{(x-y)u(x-y)}{\|x-y\|^{2}},v)\phi(y,v)ds(v)d\sigma(x).\end{array}$
Applying Lemma 5 in [DLRV], the integral becomes
$(I-P_{k})\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,v)\phi(y,v)ds(v)=(I-P_{k})u\phi(y,u)=u\phi(y,u).\quad\blacksquare$
## 5 The $Q_{k}$ operators on the sphere
In this section, we will extend the results for the $Q_{k}$ operators in
$\mathbb{R}^{n}$ from the previous sections to the sphere.
Consider the Cayley transformation $C:\mathbb{R}^{n}\to\mathbb{S}^{n}$, where
$\mathbb{S}^{n}$ is the unit sphere in $\mathbb{R}^{n+1}$, defined by
$C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1}$, where
$x=x_{1}e_{1}+\cdots+x_{n}e_{n}\in\mathbb{R}^{n}$, and $e_{n+1}$ is a unit
vector in $\mathbb{R}^{n+1}$ which is orthogonal to $\mathbb{R}^{n}$. Now
$C(\mathbb{R}^{n})=\mathbb{S}^{n}\setminus\\{e_{n+1}\\}$. Suppose
$x_{s}\in\mathbb{S}^{n}$ and
$x_{s}=x_{s_{1}}e_{1}+\cdots+x_{s_{n}}e_{n}+x_{s_{n+1}}e_{n+1}$, then we have
$C^{-1}(x_{s})=(-e_{n+1}x_{s}+1)(x_{s}-e_{n+1})^{-1}$.
The Dirac operator over the $n$-sphere $\mathbb{S}^{n}$ has the form
$D_{s}=w(\Gamma+\frac{n}{2})$, where $w\in\mathbb{S}^{n}$ and
$\Gamma=\displaystyle\sum_{i<j,i=1}^{n}{e_{i}e_{j}(w_{i}\frac{\partial}{\partial
w_{j}}-w_{j}\frac{\partial}{\partial w_{i}})}$. See [CM, LR, R1, R2, Va3].
Let $U$ be a domain in $\mathbb{R}^{n}$. Consider a function
$f_{\star}:U\times\mathbb{R}^{n}\to Cl_{n+1}$ such that for each $x\in U$,
$f_{\star}(x,u)$ is a left monogenic polynomial homogeneous of degree $k-1$ in
$u$. This function reduces to $f(x_{s},u)$ on $C(U)\times\mathbb{R}^{n}$ and
$f(x_{s},u)$ takes its values in $Cl_{n+1},$ where $C(U)\subset\mathbb{S}^{n}$
and $f(x_{s},u)$ is a left monogenic polynomial homogeneous of degree $k-1$ in
$u$.
The left $n$-spherical remaining operator on the sphere is defined to be
$Q_{k}^{S}=:(I-P_{k})D_{s,x_{s}},$
where $D_{s,x_{s}}$ is the Dirac operator on the sphere with respect to
$x_{s}$.
Hence the left $n$-spherical remaining equation is defined to be
$Q_{k}^{S}uf(x_{s},u)=0.$
On the other hand, the right $n$-spherical remaining operator is defined to be
$Q_{k,r}^{S}:=D_{s,x_{s}}(I-P_{k,r}).$
The right $n$-spherical remaining equation is defined to be
$g(x_{s},v)vQ_{k,r}^{S}=0,$ where $g(x_{s},v)\in\overline{\mathcal{M}}_{k-1}.$
### 5.1 The intertwining operators for $Q_{k}^{S}$ and $Q_{k}$ operators and
the conformal invariance of $Q_{k}^{S}uf(x_{s},u)=0$
First let us recall that if $f(u)\in\mathcal{M}_{k-1}$ then it trivially
extends to $F(v)=f(u+u_{n+1}e_{n+1})$ with $u_{n+1}\in\mathbb{R}$ and
$F(v)=f(u)$ for all $u_{n+1}\in\mathbb{R}$. Consequently $D_{n+1}F(v)=0$ where
$D_{n+1}=\displaystyle\sum_{j=1}^{n+1}{e_{j}\frac{\partial}{\partial u_{j}}}$.
If $f(u)\in\mathcal{M}_{k-1}$ then for any boundary of a piecewise smooth
bounded domain $U\subseteq\mathbb{R}^{n}$ by Cauchy’s Theorem
$\displaystyle\int_{\partial U}{n(u)f(u)d\sigma(u)}=0.$ (3)
Suppose now $a\in\mbox{Pin}(n+1)$ and $u=aw\tilde{a}$ then although
$u\in\mathbb{R}^{n}$ in general $w$ belongs to the hyperplane
$a^{-1}\mathbb{R}^{n}\tilde{a}^{-1}$ in $\mathbb{R}^{n+1}$.
By applying a change of variable, up to a sign the integral (3) becomes
$\displaystyle\int_{a^{-1}\partial
U\tilde{a}^{-1}}{an(w)\tilde{a}F(aw\tilde{a})d\sigma(w)}=0.$ (4)
As $\partial U$ is arbitrary then on applying Stokes’ Theorem to (4) we see
that
$D_{a}\tilde{a}F(aw\tilde{a})=0,~{}~{}\mbox{where}~{}~{}D_{a}:=D_{n+1}\bigl{|}_{a^{-1}\mathbb{R}^{n}\tilde{a}^{-1}}.$
See [LRV].
From now on all functions on spheres take their values in $Cl_{n+1}.$
Now let $f(x_{s},u):U_{s}\times\mathbb{R}^{n}\to Cl_{n+1}$ be a monogenic
polynomial homogeneous of degree $k$ in $u$ for each $x_{s}\in U_{s}$, where
$U_{s}$ is a domain in $\mathbb{S}^{n}.$
It is known from section 3 that $I-P_{k}$ is conformally invariant under a
general Möbius transformation over $\mathbb{R}^{n}$. This trivially extends to
Möbius transformations on $\mathbb{R}^{n+1}$. It follows that if we restrict
$x_{s}$ to $\mathbb{S}^{n},$ then $I-P_{k}$ is also conformally invariant
under the Cayley transformation $C$ and its inverse $C^{-1},$ with
$x\in\mathbb{R}^{n}$.
We can use the intertwining formulas for $D_{x}$ and $D_{s,x_{s}}$ given in
[LR] to establish the intertwining formulas for $Q_{k}$ and $Q_{k}^{S}.$
###### Theorem 7.
$\begin{array}[]{ll}-J_{-1}(C^{-1},x_{s})Q_{k,u}uf(x,u)=Q_{k,w}^{S}wJ(C^{-1},x_{s})f(C^{-1}(x_{s}),\displaystyle\frac{(x_{s}-e_{n+1})w(x_{s}-e_{n+1})}{\|x_{s}-e_{n+1}\|^{2}}),\end{array}$
where $Q_{k,u}$ are the remaining operators with respect to
$u\in\mathbb{R}^{n}$, $Q_{k,w}^{S}$ are the remaining operators on the sphere
with respect to $w\in\mathbb{S}^{n}$,
$u=\displaystyle\frac{(x_{s}-e_{n+1})w(x_{s}-e_{n+1})}{||x_{s}-e_{n+1}||^{2}},$
$J(C^{-1},x_{s})=\displaystyle\frac{x_{s}-e_{n+1}}{\|x_{s}-e_{n+1}\|^{n}}$ is
the conformal weight for the inverse of the Cayley transformation and
$J_{-1}(C^{-1},x_{s})=\displaystyle\frac{x_{s}-e_{n+1}}{\|x_{s}-e_{n+1}\|^{n+2}}.$
Proof: In [LR] it is shown that
$D_{x}=J_{-1}(C^{-1},x_{s})^{-1}D_{s,x_{s}}J(C^{-1},x_{s}).$
Set
$u=\displaystyle\frac{J(C^{-1},x_{s})wJ(C^{-1},x_{s})}{\|J(C^{-1},x_{s})\|^{2}}$
for some $w\in\mathbb{R}^{n+1}$. Consequently,
$\begin{array}[]{ll}Q_{k,u}uf(x,u)=(I-P_{k,u})D_{x}uf(x,u)\\\ \\\
=(I-P_{k,u})J_{-1}(C^{-1},x_{s})^{-1}D_{s,x_{s}}J(C^{-1},x_{s})uf(C^{-1}(x_{s}),u)\\\
\\\
=J_{-1}(C^{-1},x_{s})^{-1}(I-P_{k,w})D_{s,x_{s}}J(C^{-1},x_{s})\displaystyle\frac{J(C^{-1},x_{s})wJ(C^{-1},x_{s})}{\|J(C^{-1},x_{s})\|^{2}}\\\
\\\
f(C^{-1}(x_{s}),\displaystyle\frac{J(C^{-1},x_{s})wJ(C^{-1},x_{s})}{\|J(C^{-1},x_{s})\|^{2}})\end{array}$
Since
$\displaystyle\frac{J(C^{-1},x_{s})wJ(C^{-1},x_{s})}{\|J(C^{-1},x_{s})\|^{2}}=\displaystyle\frac{(x_{s}-e_{n+1})w(x_{s}-e_{n+1})}{||x_{s}-e_{n+1}||^{2}},$
the previous equation becomes
$\begin{array}[]{ll}Q_{k,u}uf(x,u)=-J_{-1}(C^{-1},x_{s})^{-1}Q_{k,w}^{S}wJ(C^{-1},x_{s})f(C^{-1}(x_{s}),\displaystyle\frac{(x_{s}-e_{n+1})w(x_{s}-e_{n+1})}{\|(x_{s}-e_{n+1})\|^{2}}).\quad\blacksquare\end{array}$
Similarly, we have the following result for the remaining operators under the
Cayley transformation.
###### Theorem 8.
$-J_{-1}(C,x)Q_{k,u}^{S}ug(x_{s},u)=Q_{k,w}wJ(C,x)g(C(x),\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{\|x+e_{n+1}\|^{2}}),$
where $u=\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{||x+e_{n+1}||^{2}},$
$J(C,x)=\displaystyle\frac{x+e_{n+1}}{\|x+e_{n+1}\|^{n}}$ and
$J_{-1}(C,x)=\displaystyle\frac{x+e_{n+1}}{\|x+e_{n+1}\|^{n+2}}$ is the
conformal weight for the Cayley transformation.
As a consequence of two previous theorems we have the conformal invariance of
equation $Q_{k}^{S}uf(x_{s},u)=0$:
###### Theorem 9.
$Q_{k,u}uf(x,u)=0$ if and only if
$Q_{k,w}^{S}wJ(C^{-1},x_{s})f(C^{-1}(x_{s}),\displaystyle\frac{(x_{s}-e_{n+1})w(x_{s}-e_{n+1})}{\|x_{s}-e_{n+1}\|^{2}})=0$
and $Q_{k,u}^{S}ug(x_{s},u)=0$ if and only if
$Q_{k,w}wJ(C,x)g(C(x),\displaystyle\frac{(x+e_{n+1})w(x+e_{n+1})}{\|x+e_{n+1}\|^{2}})=0.$
### 5.2 A kernel for the $Q_{k}^{S}$ operator
Now consider the kernel in $\mathbb{R}^{n}$
$\begin{array}[]{ll}\displaystyle\frac{-1}{\omega_{n}c_{k}}w\displaystyle\frac{x-y}{\|x-y\|^{n}}Z_{k-1}(\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}},v)v\\\
\\\
=\displaystyle\frac{-1}{\omega_{n}c_{k}}\displaystyle\frac{J(C^{-1},x_{s})^{-1}uJ(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|^{2}}\\\
\\\
J(C^{-1},x_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}J(C^{-1},y_{s})^{-1}Z_{k-1}(\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}},v)v,\end{array}$
where
$w=\displaystyle\frac{J(C^{-1},x_{s})^{-1}uJ(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|^{2}}.$
Multiplying by $J(C^{-1},x_{s})$ and applying the Cayley transformation to the
above kernel, we obtain the kernel
$\displaystyle
H_{k}^{S}(x-y,u,v):=\displaystyle\frac{-1}{\omega_{n}c_{k}}u\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}J(C^{-1},y_{s})^{-1}Z_{k-1}(au\tilde{a},v)v,$
(5)
where
$a=a(x_{s},y_{s})=\displaystyle\frac{J(C^{-1},x_{s})^{-1}(x_{s}-y_{s})J(C^{-1},y_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|\|x_{s}-y_{s}\|\|J(C^{-1},y_{s})^{-1}\|}.$
This is a fundamental solution to $Q_{k}^{S}uf(x_{s},u)=0$ on
$\mathbb{S}^{n},$ for $x_{s},y_{s}\in\mathbb{S}^{n}.$
Similarly, we obtain that
$\displaystyle\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}v$
(6)
is a non trivial solution to $g(x_{s},v)vQ_{k,r}^{S}=0$.
We can see that the representations (5) and (6) are the same up to a
reflection by
$\begin{array}[]{ll}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}v\\\
\\\
=\displaystyle\frac{1}{\omega_{n}c_{k}}u\tilde{a}Z_{k}(au\tilde{a},v)aJ(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}v\\\
\\\
=\displaystyle\frac{-1}{\omega_{n}c_{k}}uJ(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}\displaystyle\frac{J(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|}Z_{k}(au\tilde{a},v)\displaystyle\frac{J(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|}v\\\
\\\
=u\displaystyle\frac{J(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|}\displaystyle\frac{-1}{\omega_{n}c_{k}}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}J(C^{-1},y_{s})^{-1}Z_{k}(au\tilde{a},v)\displaystyle\frac{J(C^{-1},x_{s})^{-1}}{\|J(C^{-1},x_{s})^{-1}\|}v.\end{array}$
### 5.3 Some basic integral formulas for the remaining operators on spheres
In this section we will study some basic integral formulas related to the
remaining operators on the sphere.
###### Theorem 10.
(Stokes’ Theorem for the $n$-spherical Dirac operator $D_{s}$) [LR]
Suppose $U_{s}$ is a domain on $\mathbb{S}^{n}$ and
$f,g:U_{s}\times\mathbb{R}^{n}\to Cl_{n+1}$ are $C^{1}$, then for a subdomain
$V_{s}$ of $U_{s}$, we have
$\begin{array}[]{ll}\displaystyle\int_{\partial
V_{s}}g(x_{s},u)n(x_{s})f(x_{s},u)d\Sigma(x_{s})\\\ \\\
=\displaystyle\int_{V_{s}}(g(x_{s},u)D_{s,x_{s}})f(x_{s},u)+g(x_{s},u)(D_{s,x_{s}}f(x_{s},u))dS(x_{s}),\end{array}$
where $dS(x_{s})$ is the $n$-dimensional area measure on $V_{s}$,
$d\Sigma(x_{s})$ is the $n-1$-dimensional scalar Lebesgue measure on $\partial
V_{s}$ and $n(x_{s})$ is the unit outward normal vector to $\partial V_{s}$ at
$x_{s}$.
Applying the similar arguments to prove the Stokes’ Theorem for $Q_{k}$
operators in section 4, we can obtain
###### Theorem 11.
(Stokes’ Theorem for the $Q_{k}^{S}$ operator )
Let $U_{s},V_{s},\partial V_{s}$ be as in the previous Theorem. Then for
$f,g\in C^{1}(U_{s}\times\mathbb{R}^{n},{\mathcal{M}}_{k})$, we have version 1
$\begin{array}[]{ll}\displaystyle\int_{V_{s}}[(g(x_{s},u)Q_{k,r}^{S},f(x_{s},u))_{u}+(g(x_{s},u),Q_{k}^{S}f(x_{s},u))_{u}]dS(x_{s})\\\
\\\ =\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u),(I-P_{k})n(x_{s})f(x_{s},u)\right)_{u}d\Sigma(x)\\\ \\\
=\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u)n(x_{s})(I-P_{k,r}),f(x_{s},u)\right)_{u}d\Sigma(x).\end{array}$
Then for $f,g\in C^{1}(U_{s}\times\mathbb{R}^{n},$$\mathcal{M}_{k-1})$, we
have version 2
$\begin{array}[]{ll}\displaystyle\int_{V_{s}}[(g(x_{s},u)uQ_{k,r}^{S},uf(x_{s},u))_{u}+(g(x_{s},u)u,Q_{k}^{S}uf(x_{s},u))_{u}]dS(x_{s})\\\
\\\ =\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u)u,(I-P_{k})n(x_{s})uf(x_{s},u)\right)_{u}d\Sigma(x)\\\
\\\ =\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u)un(x_{s})(I-P_{k,r}),uf(x_{s},u)\right)_{u}d\Sigma(x).\end{array}$
###### Remark 2.
Using the similar arguments to show the conformal invariance of Stokes’
Theorem for the Rarita-Schwinger operators in [LRV], we obtain that Stokes’
Theorem for the $Q_{k}$ operators is conformally invariant under the Cayley
transformation and the inverse of the Cayley transformation.
###### Remark 3.
We also have the following fact
$\displaystyle\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u)u,(I-P_{k})n(x_{s})uf(x_{s},u)\right)_{u}d\Sigma(x)=\displaystyle\int_{\partial
V_{s}}\left(g(x_{s},u)u,n(x_{s})uf(x_{s},u)\right)_{u}d\Sigma(x)$ (7)
###### Theorem 12.
(Borel-Pompeiu Theorem) Suppose $U_{s}$, $V_{s}$ and $\partial V_{s}$ are
stated as in Theorem $10$ and $y_{s}\in V_{s}.$ Then for $f\in
C^{1}(U_{s}\times\mathbb{R}^{n},{\mathcal{M}}_{k-1})$ we have
$\begin{array}[]{ll}u^{\prime}f(y_{s},u^{\prime})=J(C^{-1},y_{s})\displaystyle\int_{\partial
V_{s}}(H_{k}^{S}(x_{s}-y_{s},u,v),(I-P_{k})n(x_{s})vf(x_{s},v))_{v}d\Sigma(x_{s})\\\
\\\
-J(C^{-1},y_{s})\displaystyle\int_{V_{s}}(H_{k}^{S}(x_{s}-y_{s},u,v),Q_{k}^{S}vf(x_{s},v))_{v}dS(x_{s})\end{array}$
where
$u^{\prime}=\displaystyle\frac{(y_{s}-e_{n+1})u(y_{s}-e_{n+1})}{\|y_{s}-e_{n+1}\|^{2}},$
$dS(x_{s})$ is the $n$-dimensional area measure on
$V_{s}\subset\mathbb{S}^{n}$, $n(x_{s})$ and $d\Sigma(x_{s})$ as in Theorem
$10$.
Proof: In the proof we use the representation
$\begin{array}[]{ll}H_{k}^{S}(x-y,u,v)=\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}v.\end{array}$
Let $B_{s}(y_{s},\epsilon)$ be the ball centered at $y_{s}\in\mathbb{S}^{n}$
with radius $\epsilon$. We denote $C^{-1}(B_{s}(y_{s},\epsilon))$ by $B(y,r)$,
and $C^{-1}(\partial B_{s}(y_{s},\epsilon))$ by $\partial B(y,r),$ where
$y=C^{-1}(y_{s})\in\mathbb{R}^{n}$ and $r$ is the radius of $B(y,r)$ in
$\mathbb{R}^{n}$. Using the similar arguments in the proof of Theorem $2$, we
only deal with
$\begin{array}[]{ll}\displaystyle\int_{\partial
B_{s}(y_{s},\epsilon)}(H_{k}^{S}(x_{s}-y_{s},u,v),(I-P_{k})n(x_{s})vf(y_{s},v))_{v}d\Sigma(x_{s})\\\
\\\ =\displaystyle\int_{\partial
B_{s}(y_{s},\epsilon)}\int_{\mathbb{S}^{n-1}}H_{k}^{S}(x_{s}-y_{s},u,v)(I-P_{k})n(x_{s})vf(y_{s},v)ds(v)d\Sigma(x_{s}).\end{array}$
Now applying (7), the integral is equal to
$\begin{array}[]{lll}\displaystyle\int_{\partial
B_{s}(y_{s},\epsilon)}\int_{\mathbb{S}^{n-1}}H_{k}^{S}(x_{s}-y_{s},u,v)n(x_{s})vf(y_{s},v)ds(v)d\Sigma(x_{s})\\\
\\\ =\displaystyle\int_{\partial
B_{s}(y_{s},\epsilon)}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\tilde{a}va)J(C^{-1},y_{s})^{-1}\displaystyle\frac{x_{s}-y_{s}}{\|x_{s}-y_{s}\|^{n}}vn(x_{s})vf(y_{s},v)ds(v)d\Sigma(x_{s})\end{array}$
Applying the inverse of the Cayley transformation to the previous integral, we
have
$\begin{array}[]{ll}=\displaystyle\int_{\partial
B(y,r)}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}})J(C^{-1},y_{s})^{-1}J(C,y)^{-1}\displaystyle\frac{x-y}{\|x-y\|^{n}}J(C,x)^{-1}\\\
\\\
vJ(C,x)n(x)J(C,x)vf(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(v)d\sigma(x),\end{array}$
where $v=\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}}.$
Now if we replace $v$ with $\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}}$
in the previous integral and we also set $J(C,x)=(J(C,x)-J(C,y))+J(C,y)$, but
$J(C,x)-J(C,y)$ tends to zero as $x$ approaches $y$. Thus the previous
integral can be replaced by
$\begin{array}[]{ll}=\displaystyle\int_{\partial
B(y,r)}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|^{n}}J(C,y)^{-1}\\\
\\\
\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}}J(C,y)n(x)J(C,y)\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}}f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)d\sigma(x)\\\
\\\ =\displaystyle\int_{\partial
B(y,r)}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{-1}{\omega_{n}c_{k}}uZ_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|^{n}}wn(x)\\\
\\\
wJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)d\sigma(x)\\\
\\\ =\displaystyle\int_{\partial
B(y,r)}\int_{\mathbb{S}^{n-1}}\displaystyle\frac{1}{\omega_{n}c_{k}}\displaystyle\frac{1}{r^{n-1}}uZ_{k-1}(u,\displaystyle\frac{(x-y)w(x-y)}{\|x-y\|^{2}})\displaystyle\frac{x-y}{\|x-y\|}w\displaystyle\frac{x-y}{\|x-y\|}\\\
\\\
wJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)d\sigma(x).\end{array}$
Using Lemma $5$ in [DLRV], the previous integral becomes
$\displaystyle\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)wwJ(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)$
$\displaystyle=-\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)J(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)$
$\displaystyle=-uJ(C,y)f(C(y),\displaystyle\frac{J(C,y)uJ(C,y)}{\|J(C,y)\|^{2}}).$
(8)
If we set
$u^{\prime}=\displaystyle\frac{J(C,y)uJ(C,y)}{\|J(C,y)\|^{2}}=\displaystyle\frac{J(C^{-1},y_{s})^{-1}uJ(C^{-1},y_{s})^{-1}}{\|J(C^{-1},y_{s})^{-1}\|^{2}}=\displaystyle\frac{(y_{s}-e_{n+1})u(y_{s}-e_{n+1})}{\|y_{s}-e_{n+1}\|^{2}},$
then
$uJ(C,y)=uJ(C^{-1},y_{s})^{-1}=J(C^{-1},y_{s})\|J(C^{-1},y_{s})^{-1}\|^{2}u^{\prime}.$
Now if we multiply the both sides of equation (5.3) by
$\displaystyle\frac{J(C^{-1},y_{s})^{-1}}{\|J(C^{-1},y_{s})^{-1}\|^{2}}=-J(C^{-1},y_{s})$,
then we obtain
$\begin{array}[]{ll}J(C^{-1},y_{s})\displaystyle\int_{\mathbb{S}^{n-1}}uZ_{k-1}(u,w)J(C,y)f(C(y),\displaystyle\frac{J(C,y)wJ(C,y)}{\|J(C,y)\|^{2}})ds(w)\\\
\\\
=u^{\prime}f(C(y),u^{\prime})=u^{\prime}f(y_{s},u^{\prime}).\quad\blacksquare\end{array}$
###### Corollary 1.
Let $\psi$ be a function in $C^{\infty}(V_{s},\mathcal{M}_{k-1})$ and supp
$f\subset V_{s}$. Then
$u^{\prime}\psi(y_{s},u^{\prime})=-J(C^{-1},y_{s})\int_{V_{s}}(H_{k}^{S}(x_{s}-y_{s},u,v),Q_{k}^{S}v\psi(x_{s},v))_{v}dS(x_{s}),$
where
$u^{\prime}=\displaystyle\frac{(y_{s}-e_{n+1})u(y_{s}-e_{n+1})}{\|y_{s}-e_{n+1}\|^{2}}.$
###### Corollary 2.
(Cauchy Integral Formula for $Q_{k}^{S}$ operators)
If $Q_{k}^{S}vf(x_{s},v)=0$, then for $y_{s}\in V_{s}$ we have
$\displaystyle u^{\prime}f(y_{s},u^{\prime})$ $\displaystyle=$ $\displaystyle
J(C^{-1},y_{s})\int_{\partial
V_{s}}(H_{k}^{S}(x_{s}-y_{s},u,v),(I-P_{k})n(x_{s})vf(x_{s},v))_{v}d\Sigma(x_{s})$
$\displaystyle=$ $\displaystyle J(C^{-1},y_{s})\int_{\partial
V_{s}}(H_{k}^{S}(x_{s}-y_{s},u,v)n(x_{s})(I-P_{k,r}),vf(x_{s},v))_{v}d\Sigma(x_{s}),$
where
$u^{\prime}=\displaystyle\frac{(y_{s}-e_{n+1})u(y_{s}-e_{n+1})}{\|y_{s}-e_{n+1}\|^{2}}.$
###### Remark 4.
By factoring out $\mathbb{S}^{n}$ by the group $\mathbb{Z}_{2}=\\{\pm 1\\}$ we
obtain real projective space, $\mathbb{R}P^{n}$. Using the similar arguments
to obtain the results for Rarita-Schwinger operators on real projective space
in [LRV], we can easily extends the similar results for $Q_{k}$ operators to
real projective space.
## References
* [A] L. V. Ahlfors, _Old and new in Möbius groups,_ Ann. Acad. Sci. Fenn. Ser. A I Math., 9 (1984) 93-105.
* [BDS] F. Brackx, R. Delanghe, and F. Sommen, _Clifford Analysis,_ Pitman, London, 1982.
* [BSSV] J. Bureš, F. Sommen, V. Souček, P. Van Lancker, _Rarita-Schwinger Type Operators in Clifford Analysis,_ J. Funct. Annl. 185 (2001), No.2, 425-455.
* [BSSV1] J. Bureš, F. Sommen, V. Souček, P. Van Lancker, _Symmetric Analogues of Rarita-Schwinger Equations,_ Annals of Global Analysis and Geometry 21 (2002), 215-240.
* [CM] J. Cnops, H. Malonek, _An introduction to Clifford analysis,_ Textos de Matemática. Série B [Texts in Mathematics. Series B], 7. Universidade de Coimbra, Departamento de Matemática, Coimbra, 1995, vi+64 pp.
* [DLRV] C. Dunkl, J. Li, J. Ryan and P. Van Lancker, _Some Rarita-Schwinger Operators_ , Submitted, 2011. (http://arxiv.org/abs/1102.1205)
* [DX] C. Dunkl and Y. Xu, _Orthogonal Polynomials of Several Variables,_ Cambridge University Press, Cambridge, 2001.
* [LR] H. Liu, J. Ryan, Clifford Analysis Techniques for Spherical PDE, The Journal of Fourier Analysis and Applications, Vol. 8, Issue 6, 2002.
* [LRV] J. Li, J. Ryan and C. Vanegas, _Some Rarita-Schwinger Type Operators on Spheres and Real Projective Space_ , Submitted, 2011.
* [P] I. Porteous, _Clifford algebra and the classical groups,_ Cambridge University Press, Cambridge, 1995.
* [R] J. Ryan, _Iterated Dirac Operators in $C^{n}$,_ Z. Anal. Anwendungen, 9, 1990, 385-401.
* [R1] J. Ryan, _Dirac Operators on Spheres and Hyperbolae,_ Boletin de la Sociedad Matemática Mexicana, (3) 3 (1997), no. 2, 255-270.
* [R2] J. Ryan, _Clifford analysis on Spheres and Hyperbolae,_ Math. Methods Appl. Sci. 20 (1997), no. 18, 1617-1624.
* [Va1] P. Van Lancker, _Higher Spin Fields on Smooth Domains,_ in Clifford Analysis and Its Applications, Eds. F. Brackx, J.S.R. Chisholm and V. Souček, Kluwer, Dordrecht 2001, 389-398.
* [Va2] P. Van Lancker, _Rarita-Schwinger Fields in the Half Space,_ Complex Variables and Elliptic Equations, 51, 2006, 563-579.
* [Va3] P. Van Lancker, _Clifford Analysis on the Sphere,_ Clifford Algebra and their Applcation in Mathematical Physics (Aachen, 1996), 201-215, Fund. Theories Phys. 94, Kluwer Acad. Publ., Dordrecht, 1998.
* [V] K. Th. Vahlen, _Über Bewegungen und komplexe Zahlen,_ (German) Math. Ann., 55(1902), No.4 585-593.
Junxia Li Email: jxl004@uark.edu
John Ryan Email: jryan@uark.edu
|
arxiv-papers
| 2011-06-17T22:13:20 |
2024-09-04T02:49:19.777576
|
{
"license": "Public Domain",
"authors": "Junxia Li, John Ryan",
"submitter": "Junxia Li",
"url": "https://arxiv.org/abs/1106.3588"
}
|
1106.3592
|
# SLOCC determinant invariants of order $2^{n/2}$ for even $n$ qubits
Xiangrong Li1, Dafa Li2 1 Department of Mathematics, University of California,
Irvine, CA 92697-3875, USA
2 Department of mathematical sciences, Tsinghua University, Beijing 100084
CHINA
###### Abstract
In this paper, we study SLOCC determinant invariants of order $2^{n/2}$ for
any even $n$ qubits which satisfy the SLOCC determinant equations. The
determinant invariants can be constructed by a simple method and the set of
all these determinant invariants is complete with respect to permutations of
qubits. SLOCC entanglement classification can be achieved via the vanishing or
not of the determinant invariants. We exemplify the method for several even
number of qubits, with an emphasis on six qubits.
PACS Number: 03.67.Mn
Quantum entanglement is a key quantum mechanical resource in quantum
computation and information, such as quantum cryptography, quantum dense
coding and quantum teleportation Horodecki . Whereas bipartite entanglement
has been well understood, multipartite entanglement remains largely unexplored
due to the exponential growth of complexity with the number of qubits
involved.
Functions in the coefficients of pure states which are invariant under
stochastic local operations and classical communication (SLOCC) play a vital
role in the study of entanglement classification Dur ; Miyake ; Chterental ;
Cao ; LDFPLA ; LDF07a ; LDF07b ; LDFQIC09 ; Buniy ; Viehmann as well as
entanglement measures Luque ; Osterloh05 ; LDF09b . Invariants for four qubits
have been presented in Luque , and entanglement measures might be built from
the absolute values of these invariants. The three invariants of order 4,
denoted as $L$, $M$ and $N$, can be expressed in the form of determinants.
Invariants for five qubits have been highlighted in Luque06 ; Osterloh09 . To
date, very few attempts have been made toward the generalization to higher
number of qubits. The SLOCC equations of degree 2 for even $n$ qubits and of
degree 4 for odd $n$ qubits have been recently established for two states
equivalent under SLOCC LDF07a ; LDF09b . More recently, for even $n$ qubits,
the SLOCC determinant equations of degree $2^{n/2}$ has been established and
four determinant invariants of order $2^{n/2}$ have been obtained LDFJPA . In
light of the SLOCC determinant equations, several different genuine entangled
states of even $n$ qubits inequivalent to the $|GHZ\rangle$, $|W\rangle$, and
Dicke states have been constructed.
In this paper, we construct $\binom{n-1}{n/2-1}$ SLOCC determinant invariants
of order $2^{n/2}$ for any even $n$ qubits which satisfy the SLOCC determinant
equations. We also demonstrate the completeness of the set of all these
determinant invariants with respect to permutations of qubits. For six qubits,
we explicitly derive all the ten SLOCC determinants of order $8$. The
determinant invariants can be used for SLOCC classification of any even $n$
qubits. Finally, we illustrate the application of the equations and invariants
by proposing a genuine entangled state of even $n$ qubits and showing that it
is inequivalent to the $|GHZ\rangle$, $|W\rangle$, and Dicke states.
We write the state $|\psi^{\prime}\rangle$ of even $n$ qubits as
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$. We associate to
the state $|\psi^{\prime}\rangle$ a $2^{n/2}$ by $2^{n/2}$ coefficient matrix
$M(a,n)$ whose entries are the coefficients $a_{0},a_{1},\cdots,a_{2^{n}-1}$
arranged in ascending lexicographical order. To illustrate, we list $M(a,4)$
below as:
$M(a,4)=\left(\begin{array}[]{cccc}a_{0}&a_{1}&a_{2}&a_{3}\\\
a_{4}&a_{5}&a_{6}&a_{7}\\\ a_{8}&a_{9}&a_{10}&a_{11}\\\
a_{12}&a_{13}&a_{14}&a_{15}\end{array}\right).$ (1)
Let the state $|\psi\rangle$ of even $n$ qubits be
$|\psi\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle$. It is well known that the
states $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are equivalent under SLOCC
if and only if there exist local invertible operators $\mathcal{A}_{1}$,
$\mathcal{A}_{2},\cdots,\mathcal{A}_{n}$ such that Dur
$|\psi^{\prime}\rangle=\underbrace{\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\cdots\otimes\mathcal{A}_{n}}_{n}|\psi\rangle.$
(2)
Let $M(b,n)$ be obtained from $M(a,n)$ by replacing $a$ by $b$ . Then the
following SLOCC determinant equation holds LDFJPA :
$\displaystyle\det M(a,n)$ $\displaystyle=$ $\displaystyle\det
M(b,n)[\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})]^{2^{(n-2)/2}}.$ (3)
We refer to any determinant that satisfies Eq. (3) as a SLOCC determinant
invariant of order $2^{n/2}$ for even $n$ qubits. In particular, $\det M(a,n)$
is such a determinant invariant. Several other determinant invariants have
recently been obtained in LDFJPA . The aim of this paper is to construct all
the determinant invariants satisfying Eq. (3).
We write $|\psi^{\prime}\rangle$ in terms of an orthogonal basis as
$|\psi^{\prime}\rangle=\sum a_{i_{1}i_{2}\cdots i_{n}}|i_{1}i_{2}$ $\cdots
i_{n}\rangle$, where $i_{1}i_{2}\cdots i_{n}$ is the $n$-bit binary form of
the index $i$. Inspection of the structure of the matrix $M(a,n)$ reveals that
the coefficient $a_{i_{1}\cdots i_{n/2}i_{n/2+1}\cdots i_{n}}$ of the state
$|\psi^{\prime}\rangle$ is the entry in the $(i_{1}\cdots i_{n/2})$th row and
$(i_{n/2+1}$$\cdots$$i_{n})$th column of the matrix (define the topmost row as
the $0$th row and the leftmost column as the $0$th column). In other words,
bits $1,\cdots,{n/2}$ specify the row number, and the rest bits specify the
column number. We observe that using different bits to specify the row number
might result in matrices whose determinants are different from that of
$M(a,n)$. Since $n/2$ bits are needed to specify the row number for square
matrices, this amounts to $\binom{n}{n/2}$ different ways. But, as can easily
be verified, exchanging the row and column bits of a matrix is equivalent to
transposing the matrix. This gives a total of
$\frac{1}{2}\binom{n}{n/2}=\binom{n-1}{n/2-1}$ different determinants. As will
be seen later, these determinants satisfy the SLOCC determinant equations and
form a complete set of determinant invariants of order $2^{n/2}$ of even $n$
qubits with respect to permutations of qubits. We can construct the
determinants in the following way: use bit $n/2$ together with $n/2-1$ other
bits selected from the rest $n-1$ bits to specify the row number and the
remaining $n/2$ bits to specify the column number. We exemplify this for
$n=4$. Using bits 1 and 2 to specify the row number yields
$\displaystyle
D_{4}^{1}=\left|\begin{array}[]{*{4}{>{\centering\arraybackslash$}p{0.5cm}<{$}}}a_{0}$\@add@centering&a_{1}$\@add@centering&a_{2}$\@add@centering&a_{3}$\@add@centering\cr
a_{4}$\@add@centering&a_{5}$\@add@centering&a_{6}$\@add@centering&a_{7}$\@add@centering\cr
a_{8}$\@add@centering&a_{9}$\@add@centering&a_{10}$\@add@centering&a_{11}$\@add@centering\cr
a_{12}$\@add@centering&a_{13}$\@add@centering&a_{14}$\@add@centering&a_{15}$\@add@centering\end{array}\right|.$
(8)
Using bits 2 and 3 to specify the row number yields
$\displaystyle
D_{4}^{2}=\left|\begin{array}[]{*{4}{>{\centering\arraybackslash$}p{0.5cm}<{$}}}a_{0}$\@add@centering&a_{1}$\@add@centering&a_{8}$\@add@centering&a_{9}$\@add@centering\cr
a_{2}$\@add@centering&a_{3}$\@add@centering&a_{10}$\@add@centering&a_{11}$\@add@centering\cr
a_{4}$\@add@centering&a_{5}$\@add@centering&a_{12}$\@add@centering&a_{13}$\@add@centering\cr
a_{6}$\@add@centering&a_{7}$\@add@centering&a_{14}$\@add@centering&a_{15}$\@add@centering\end{array}\right|.$
(13)
Using bits 2 and 4 to specify the row number yields
$\displaystyle
D_{4}^{3}=\left|\begin{array}[]{*{4}{>{\centering\arraybackslash$}p{0.5cm}<{$}}}a_{0}$\@add@centering&a_{2}$\@add@centering&a_{8}$\@add@centering&a_{10}$\@add@centering\cr
a_{1}$\@add@centering&a_{3}$\@add@centering&a_{9}$\@add@centering&a_{11}$\@add@centering\cr
a_{4}$\@add@centering&a_{6}$\@add@centering&a_{12}$\@add@centering&a_{14}$\@add@centering\cr
a_{5}$\@add@centering&a_{7}$\@add@centering&a_{13}$\@add@centering&a_{15}$\@add@centering\end{array}\right|.$
(18)
Each of the above three determinants can be verified to satisfy Eq. (3) by
solving Eq. (2) for the coefficients of $|\psi^{\prime}\rangle$ and then
substituting the coefficients into the corresponding Eqs. (8)-(18), thereby
revealing that all the three determinants are SLOCC determinant invariants of
order 4 for four qubits LDFJPA . It is worth noting that the above three
determinants, ignoring the sign, turn out to be same as the ones given in
Luque .
While the determinants of order 4 for four qubits can be verified to satisfy
Eq. (3), this is an extremely difficult task for large number of qubits. To
solve this problem, we will resort to permutations. Suppose that we use bit
$n/2$ together with $n/2-1$ other bits $\ell_{1}$,
$\ell_{2},\cdots,\ell_{n/2-1}$ selected from the rest $n-1$ bits to specify
the row number and the remaining $n/2$ bits to specify the column number. This
gives a determinant of order $2^{n/2}$. We will show that this determinant can
be obtained by applying a permutation to $\det M(a,n)$. This can be seen as
follows. Let
$\displaystyle
C=\\{\ell_{1},\ell_{2},\cdots,\ell_{n/2-1}\\}\cap\\{1,2,\cdots,n/2-1\\},$ (19)
i.e. $C$ consists of those among the first $n/2-1$ bits which are used to
specify the row number. Consider the following two sets of bits:
$\displaystyle\\{t_{1},t_{2},\cdots,t_{k}\\}$
$\displaystyle=\\{\ell_{1},\ell_{2},\cdots,\ell_{n/2-1}\\}/C,$ (20)
$\displaystyle\\{r_{1},r_{2},\cdots,r_{k}\\}$
$\displaystyle=\\{1,2,\cdots,n/2-1\\}/C,$ (21)
for some $0\leq k\leq n/2-1$. Here $r_{1},\cdots,r_{k}$ are those among the
first $n/2$ bits which are used to specify the column number, and
$t_{1},\cdots,t_{k}$ are those among the last $n/2$ bits which are used to
specify the row number. Define the permutation
$\sigma=(r_{1},t_{1})(r_{2},t_{2})\cdots(r_{k},t_{k}).$ (22)
If $k=0$, we define $\sigma=I$. It is trivial to see that, ignoring the sign,
the determinant constructed above is equal to $\sigma\det M(a,n)$. To find all
the determinants of order $2^{n/2}$, we can simply exhaust all possible values
of $r_{1},\cdots,r_{k}$, $t_{1},\cdots,t_{k}$, and $k$, i.e. for all $1\leq
r_{1}<r_{2}<\cdots<r_{k}\leq n/2-1$, $n/2<t_{1}<t_{2}<\cdots<t_{k}\leq n$, and
$k$ varies from 0 to $n/2-1$. Inspection of the above condition yields
$\sum_{k=0}^{n/2-1}{\binom{n/2-1}{k}}{\binom{n/2}{k}}$ $={\binom{n-1}{n/2-1}}$
different permutations $\sigma$, which give rise to equally as many different
determinants $\sigma\det M(a,n)$ of order $2^{n/2}$.
Now, simply taking the permutations $\sigma$ to both sides of Eq. (3) yield
the following determinant equations:
$\displaystyle\sigma\det M(a,n)$ $\displaystyle=$ $\displaystyle\sigma\det
M(b,n)[\det(\mathcal{A}_{1})\cdots\det(\mathcal{A}_{n})]^{2^{(n-2)/2}}.$ (23)
It follows immediately from Eq. (23) that $\sigma\det M(a,n)$ are determinant
invariants of order $2^{n/2}$.
For the sake of completeness, we first do a simple manipulation of Eq. (22).
If we make use of the fact that
$\displaystyle(r_{k},t_{k})=(1,t_{k})(1,r_{k})(1,t_{k}),$ (24)
we are led to the following equation (ignoring the sign):
$\displaystyle(r_{i},t_{i})\det M(a,n)=(1,r_{i})(1,t_{i})\det M(a,n).$ (25)
That Eq. (25) holds is easily confirmed upon realizing that to take
transposition $(1,t_{i})$ to the determinant $(1,r_{i})(1,t_{i})\det M(a,n)$
is equivalent to interchanging two rows of the determinant. This leads to the
following expression for $\sigma$:
$\displaystyle\sigma=(1,r_{1})(1,t_{1})(1,r_{2})(1,t_{2})\cdots(1,r_{k})(1,t_{k}).$
(26)
Indeed, Eq. (26) is usually more convenient to use than Eq. (22). It can be
demonstrated that applying a transposition in the form $(1,i)$ with
$i=1,\cdots,n$ to the set formed by the determinant invariants constructed
above always yields the same set (ignoring the sign). Since any permutation
can be expressed as a product of transpositions in the form $(1,i)$, this
demonstrates the completeness of the set of all these determinant invariants.
This will be illustrated in discussing the cases $n=4$ and $n=6$ below.
We now proceed to present the determinant invariants for several even number
of qubits.
$n=2$: for two qubits, there is only one determinant invariant
$\left|\begin{array}[]{cc}a_{0}&a_{1}\\\ a_{2}&a_{3}\end{array}\right|$ of
order 2 (see also LDF07a ).
$n=4$: for four qubits, there are three determinant invariants of order 4,
namely, $D_{4}^{1}$, $D_{4}^{2}$, and $D_{4}^{3}$ (see Eqs. (8)-(18)). Note
that $D_{4}^{2}=(1,3)D_{4}^{1}$ and $D_{4}^{3}=(1,4)D_{4}^{1}$.
We now argue that the above three determinants form a complete set of
determinant invariants of order 4 for four qubits with respect to permutations
of qubits. This can be seen as follows. Applying any transposition $(1,i)$
with $i=1,\cdots,4$ to any one of the three determinant invariants always
yields a determinant invariant in the same set. This demonstrates the
completeness of the set formed by these determinant invariants. The results
are summarized in table 1.
Table 1: Completeness of the determinant invariants for four qubits with respect to permutations of qubits trans | determinant invariants |
---|---|---
$(1,1)$ | $D_{4}^{1}$ | $D_{4}^{2}$ | $D_{4}^{3}$
$(1,2)$ | $D_{4}^{1}$ | $D_{4}^{3}$ | $D_{4}^{2}$
$(1,3)$ | $D_{4}^{2}$ | $D_{4}^{1}$ | $D_{4}^{3}$
$(1,4)$ | $D_{4}^{3}$ | $D_{4}^{2}$ | $D_{4}^{1}$
Table 2: Ten determinant invariants for six qubits
$\sigma$ | $I$ | $(1,4)$ | $(1,5)$ | $(1,6)$ | $(1,2)(1,4)$ | $(1,2)(1,5)$ | $(1,2)(1,6)$ | $(1,4)(1,2)(1,5)^{a}$ | $(1,4)(1,2)(1,6)^{b}$ | $(1,5)(1,2)(1,6)^{c}$
---|---|---|---|---|---|---|---|---|---|---
$\det$ | $D_{6}^{1}$ | $D_{6}^{2}$ | $D_{6}^{3}$ | $D_{6}^{4}$ | $D_{6}^{5}$ | $D_{6}^{6}$ | $D_{6}^{7}$ | $D_{6}^{8}$ | $D_{6}^{9}$ | $D_{6}^{10}$
a $(1,4)(1,2)(1,5)D_{6}^{1}=(1,3)(1,6)D_{6}^{1}=D_{6}^{8}$.
b $(1,4)(1,2)(1,6)D_{6}^{1}=(1,3)(1,5)D_{6}^{1}=D_{6}^{9}$.
c $(1,5)(1,2)(1,6)D_{6}^{1}=(1,3)(1,4)D_{6}^{1}=D_{6}^{10}$.
$n=6$: for six qubits, there are ten determinant invariants of order 8. In
table 2, we list the permutations $\sigma$ and the corresponding determinant
invariants by virtue of Eq. (26). The determinant invariants are explicitly
given as follows:
$D_{6}^{1}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}&a_{7}\\\
a_{8}&a_{9}&a_{10}&a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\\
a_{16}&a_{17}&a_{18}&a_{19}&a_{20}&a_{21}&a_{22}&a_{23}\\\
a_{24}&a_{25}&a_{26}&a_{27}&a_{28}&a_{29}&a_{30}&a_{31}\\\
a_{32}&a_{33}&a_{34}&a_{35}&a_{36}&a_{37}&a_{38}&a_{39}\\\
a_{40}&a_{41}&a_{42}&a_{43}&a_{44}&a_{45}&a_{46}&a_{47}\\\
a_{48}&a_{49}&a_{50}&a_{51}&a_{52}&a_{53}&a_{54}&a_{55}\\\
a_{56}&a_{57}&a_{58}&a_{59}&a_{60}&a_{61}&a_{62}&a_{63}\end{array}\right|$
$D_{6}^{2}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{32}&a_{33}&a_{34}&a_{35}\\\
a_{4}&a_{5}&a_{6}&a_{7}&a_{36}&a_{37}&a_{38}&a_{39}\\\
a_{8}&a_{9}&a_{10}&a_{11}&a_{40}&a_{41}&a_{42}&a_{43}\\\
a_{12}&a_{13}&a_{14}&a_{15}&a_{44}&a_{45}&a_{46}&a_{47}\\\
a_{16}&a_{17}&a_{18}&a_{19}&a_{48}&a_{49}&a_{50}&a_{51}\\\
a_{20}&a_{21}&a_{22}&a_{23}&a_{52}&a_{53}&a_{54}&a_{55}\\\
a_{24}&a_{25}&a_{26}&a_{27}&a_{56}&a_{57}&a_{58}&a_{59}\\\
a_{28}&a_{29}&a_{30}&a_{31}&a_{60}&a_{61}&a_{62}&a_{63}\end{array}\right|$
$D_{6}^{3}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{4}&a_{5}&a_{32}&a_{33}&a_{36}&a_{37}\\\
a_{2}&a_{3}&a_{6}&a_{7}&a_{34}&a_{35}&a_{38}&a_{39}\\\
a_{8}&a_{9}&a_{12}&a_{13}&a_{40}&a_{41}&a_{44}&a_{45}\\\
a_{10}&a_{11}&a_{14}&a_{15}&a_{42}&a_{43}&a_{46}&a_{47}\\\
a_{16}&a_{17}&a_{20}&a_{21}&a_{48}&a_{49}&a_{52}&a_{53}\\\
a_{18}&a_{19}&a_{22}&a_{23}&a_{50}&a_{51}&a_{54}&a_{55}\\\
a_{24}&a_{25}&a_{28}&a_{29}&a_{56}&a_{57}&a_{60}&a_{61}\\\
a_{26}&a_{27}&a_{30}&a_{31}&a_{58}&a_{59}&a_{62}&a_{63}\end{array}\right|$
$D_{6}^{4}=\left|\begin{array}[]{cccccccc}a_{0}&a_{2}&a_{4}&a_{6}&a_{32}&a_{34}&a_{36}&a_{38}\\\
a_{8}&a_{10}&a_{12}&a_{14}&a_{40}&a_{42}&a_{44}&a_{46}\\\
a_{1}&a_{3}&a_{5}&a_{7}&a_{33}&a_{35}&a_{37}&a_{39}\\\
a_{9}&a_{11}&a_{13}&a_{15}&a_{41}&a_{43}&a_{45}&a_{47}\\\
a_{16}&a_{18}&a_{20}&a_{22}&a_{48}&a_{50}&a_{52}&a_{54}\\\
a_{24}&a_{26}&a_{28}&a_{30}&a_{56}&a_{58}&a_{60}&a_{62}\\\
a_{17}&a_{19}&a_{21}&a_{23}&a_{49}&a_{51}&a_{53}&a_{55}\\\
a_{25}&a_{27}&a_{29}&a_{31}&a_{57}&a_{59}&a_{61}&a_{63}\end{array}\right|$
$D_{6}^{5}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{16}&a_{17}&a_{18}&a_{19}\\\
a_{4}&a_{5}&a_{6}&a_{7}&a_{20}&a_{21}&a_{22}&a_{23}\\\
a_{8}&a_{9}&a_{10}&a_{11}&a_{24}&a_{25}&a_{26}&a_{27}\\\
a_{12}&a_{13}&a_{14}&a_{15}&a_{28}&a_{29}&a_{30}&a_{31}\\\
a_{32}&a_{33}&a_{34}&a_{35}&a_{48}&a_{49}&a_{50}&a_{51}\\\
a_{36}&a_{37}&a_{38}&a_{39}&a_{52}&a_{53}&a_{54}&a_{55}\\\
a_{40}&a_{41}&a_{42}&a_{43}&a_{56}&a_{57}&a_{58}&a_{59}\\\
a_{44}&a_{45}&a_{46}&a_{47}&a_{60}&a_{61}&a_{62}&a_{63}\end{array}\right|$
$D_{6}^{6}=\left|\begin{array}[]{cccccccc}a_{0}&a_{2}&a_{8}&a_{10}&a_{32}&a_{34}&a_{40}&a_{42}\\\
a_{1}&a_{3}&a_{9}&a_{11}&a_{33}&a_{35}&a_{41}&a_{43}\\\
a_{4}&a_{6}&a_{12}&a_{14}&a_{36}&a_{38}&a_{44}&a_{46}\\\
a_{5}&a_{7}&a_{13}&a_{15}&a_{37}&a_{39}&a_{45}&a_{47}\\\
a_{16}&a_{18}&a_{24}&a_{26}&a_{48}&a_{50}&a_{56}&a_{58}\\\
a_{17}&a_{19}&a_{25}&a_{27}&a_{49}&a_{51}&a_{57}&a_{59}\\\
a_{20}&a_{22}&a_{28}&a_{30}&a_{52}&a_{54}&a_{60}&a_{62}\\\
a_{21}&a_{23}&a_{29}&a_{31}&a_{53}&a_{55}&a_{61}&a_{63}\end{array}\right|$
$D_{6}^{7}=\left|\begin{array}[]{cccccccc}a_{0}&a_{2}&a_{4}&a_{6}&a_{16}&a_{18}&a_{20}&a_{22}\\\
a_{1}&a_{3}&a_{5}&a_{7}&a_{17}&a_{19}&a_{21}&a_{23}\\\
a_{8}&a_{10}&a_{12}&a_{14}&a_{24}&a_{26}&a_{28}&a_{30}\\\
a_{9}&a_{11}&a_{13}&a_{15}&a_{25}&a_{27}&a_{29}&a_{31}\\\
a_{32}&a_{34}&a_{36}&a_{38}&a_{48}&a_{50}&a_{52}&a_{54}\\\
a_{33}&a_{35}&a_{37}&a_{39}&a_{49}&a_{51}&a_{53}&a_{55}\\\
a_{40}&a_{42}&a_{44}&a_{46}&a_{56}&a_{58}&a_{60}&a_{62}\\\
a_{41}&a_{43}&a_{45}&a_{47}&a_{57}&a_{59}&a_{61}&a_{63}\end{array}\right|$
$D_{6}^{8}=\left|\begin{array}[]{cccccccc}a_{0}&a_{2}&a_{4}&a_{6}&a_{8}&a_{10}&a_{12}&a_{14}\\\
a_{1}&a_{3}&a_{5}&a_{7}&a_{9}&a_{11}&a_{13}&a_{15}\\\
a_{16}&a_{18}&a_{20}&a_{22}&a_{24}&a_{26}&a_{28}&a_{30}\\\
a_{17}&a_{19}&a_{21}&a_{23}&a_{25}&a_{27}&a_{29}&a_{31}\\\
a_{32}&a_{34}&a_{36}&a_{38}&a_{40}&a_{42}&a_{44}&a_{46}\\\
a_{33}&a_{35}&a_{37}&a_{39}&a_{41}&a_{43}&a_{45}&a_{47}\\\
a_{48}&a_{50}&a_{52}&a_{54}&a_{56}&a_{58}&a_{60}&a_{62}\\\
a_{49}&a_{51}&a_{53}&a_{55}&a_{57}&a_{59}&a_{61}&a_{63}\end{array}\right|$
$D_{6}^{9}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{4}&a_{5}&a_{8}&a_{9}&a_{12}&a_{13}\\\
a_{2}&a_{3}&a_{6}&a_{7}&a_{10}&a_{11}&a_{14}&a_{15}\\\
a_{16}&a_{17}&a_{20}&a_{21}&a_{24}&a_{25}&a_{28}&a_{29}\\\
a_{18}&a_{19}&a_{22}&a_{23}&a_{26}&a_{27}&a_{30}&a_{31}\\\
a_{32}&a_{33}&a_{36}&a_{37}&a_{40}&a_{41}&a_{44}&a_{45}\\\
a_{34}&a_{35}&a_{38}&a_{39}&a_{42}&a_{43}&a_{46}&a_{47}\\\
a_{48}&a_{49}&a_{52}&a_{53}&a_{56}&a_{57}&a_{60}&a_{61}\\\
a_{50}&a_{51}&a_{54}&a_{55}&a_{58}&a_{59}&a_{62}&a_{63}\end{array}\right|$
$D_{6}^{10}=\left|\begin{array}[]{cccccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{8}&a_{9}&a_{10}&a_{11}\\\
a_{4}&a_{5}&a_{6}&a_{7}&a_{12}&a_{13}&a_{14}&a_{15}\\\
a_{16}&a_{17}&a_{18}&a_{19}&a_{24}&a_{25}&a_{26}&a_{27}\\\
a_{20}&a_{21}&a_{22}&a_{23}&a_{28}&a_{29}&a_{30}&a_{31}\\\
a_{32}&a_{33}&a_{34}&a_{35}&a_{40}&a_{41}&a_{42}&a_{43}\\\
a_{36}&a_{37}&a_{38}&a_{39}&a_{44}&a_{45}&a_{46}&a_{47}\\\
a_{48}&a_{49}&a_{50}&a_{51}&a_{56}&a_{57}&a_{58}&a_{59}\\\
a_{52}&a_{53}&a_{54}&a_{55}&a_{60}&a_{61}&a_{62}&a_{63}\end{array}\right|$
An argument analogous to the one for $n=4$ establishes the completeness of the
set formed by the ten determinant invariants. The results are summarized in
table 3.
Table 3: Completeness of the determinant invariants for six qubits with respect to permutations of qubits trans | determinant invariants
---|---
$(1,1)$ | $D_{6}^{1}$ | $D_{6}^{2}$ | $D_{6}^{3}$ | $D_{6}^{4}$ | $D_{6}^{5}$ | $D_{6}^{6}$ | $D_{6}^{7}$ | $D_{6}^{8}$ | $D_{6}^{9}$ | $D_{6}^{10}$
$(1,2)$ | $D_{6}^{1}$ | $D_{6}^{5}$ | $D_{6}^{6}$ | $D_{6}^{7}$ | $D_{6}^{2}$ | $D_{6}^{3}$ | $D_{6}^{4}$ | $D_{6}^{8}$ | $D_{6}^{9}$ | $D_{6}^{10}$
$(1,3)$ | $D_{6}^{1}$ | $D_{6}^{10}$ | $D_{6}^{9}$ | $D_{6}^{8}$ | $D_{6}^{5}$ | $D_{6}^{6}$ | $D_{6}^{7}$ | $D_{6}^{4}$ | $D_{6}^{3}$ | $D_{6}^{2}$
$(1,4)$ | $D_{6}^{2}$ | $D_{6}^{1}$ | $D_{6}^{3}$ | $D_{6}^{4}$ | $D_{6}^{5}$ | $D_{6}^{8}$ | $D_{6}^{9}$ | $D_{6}^{6}$ | $D_{6}^{7}$ | $D_{6}^{10}$
$(1,5)$ | $D_{6}^{3}$ | $D_{6}^{2}$ | $D_{6}^{1}$ | $D_{6}^{4}$ | $D_{6}^{8}$ | $D_{6}^{6}$ | $D_{6}^{10}$ | $D_{6}^{5}$ | $D_{6}^{9}$ | $D_{6}^{7}$
$(1,6)$ | $D_{6}^{4}$ | $D_{6}^{2}$ | $D_{6}^{3}$ | $D_{6}^{1}$ | $D_{6}^{9}$ | $D_{6}^{10}$ | $D_{6}^{7}$ | $D_{6}^{8}$ | $D_{6}^{5}$ | $D_{6}^{6}$
$n=8$: for eight qubits, there are 35 determinant invariants of order 16 (not
presented here due to space limitations).
Finally, it follows from Eq. (23) that if two $n$-qubit states
$|\psi^{\prime}\rangle$ and $|\psi\rangle$ are SLOCC equivalent, then
$\sigma\det M(a,n)$ vanishes if and only if $\sigma\det M(b,n)$ vanishes. On
the other hand, if one of the determinants $\sigma\det M(a,n)$ and $\sigma\det
M(b,n)$ vanishes while the other does not, then the states
$|\psi^{\prime}\rangle$ and $|\psi\rangle$ belong to different SLOCC
equivalent classes. Thus, each determinant divides the space of the pure
states of even $n$ qubits into two inequivalent subspaces under SLOCC. Let
$c=\binom{n-1}{n/2-1}$. In total, $c$ determinants divide the space into
$2^{c}$ subspaces (or families) under SLOCC. Here each family
$F_{\delta_{1}\cdots\delta_{c}}$ is defined as
$F_{\delta_{1}\cdots\delta_{c}}=S_{\delta_{1}}^{(1)}\cap\cdots\cap
S_{\delta_{c}}^{(c)}$, where $\delta_{i}=0$ or $1$,
$S_{0}^{(i)}=\\{|\psi\rangle|D_{n}^{i}=0\\}$, and
$S_{1}^{(i)}=\\{|\psi\rangle|D_{n}^{i}\neq 0\\}$. Clearly, some families
include infinite SLOCC classes. It is straightforward to see that if two
states are SLOCC equivalent then they belong to the same family. However, the
converse does not hold, i.e. two states belonging to the same family are not
necessarily SLOCC equivalent.
As an application of the SLOCC determinant equations and invariants, consider,
for example, the following genuine entangled state for six qubits:
$|\chi\rangle=(1/\sqrt{8})(|0\rangle+|5\rangle+|18\rangle+|23\rangle+|40\rangle+|45\rangle+|58\rangle-|63\rangle).$
We observe that all the non-zero coefficients of $|\chi\rangle$ lie on the
diagonal of $D_{6}^{10}$. This leads to non-vanishing $D_{6}^{10}$ for
$|\chi\rangle$. However, $D_{6}^{10}$ vanishes for the $|GHZ\rangle$,
$|W\rangle$ and Dicke states. In light of Eq. (23), $|\chi\rangle$ is
inequivalent to the $|GHZ\rangle$, $|W\rangle$ and Dicke states under SLOCC.
For more examples, see LDFJPA .
In summary, we have constructed the set of all determinant invariants of order
$2^{n/2}$ for any even $n$ qubits and showed that the set is complete with
respect to permutations of qubits. We have presented a simple formula for
constructing the determinant invariants and given several examples for even
$n$. The determinant invariants can be used for SLOCC classification of any
even $n$ qubits and the absolute values of the determinant invariants can be
considered as entanglement measures. Finally, a more fundamental problem is
whether the determinant invariants are independent.
###### Acknowledgements.
The paper was supported by NSFC (Grant No.10875061) and Tsinghua National
Laboratory for Information Science and Technology.
## References
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* (10) R.V. Buniy and T.W. Kephart, arXiv:1012.2630 [quant-ph] (2010).
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|
arxiv-papers
| 2011-06-17T23:07:32 |
2024-09-04T02:49:19.785910
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangrong Li, Dafa Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/1106.3592"
}
|
1106.3708
|
# Information-Geometric Optimization Algorithms:
A Unifying Picture via Invariance Principles
Ludovic Arnold, Anne Auger, Nikolaus Hansen, Yann Ollivier
###### Abstract
We present a canonical way to turn any smooth parametric family of probability
distributions on an arbitrary search space $X$ into a continuous-time black-
box optimization method on $X$, the _information-geometric optimization_ (IGO)
method. Invariance as a major design principle keeps the number of arbitrary
choices to a minimum. The resulting method conducts a natural gradient ascent
using an adaptive, time-dependent transformation of the objective function,
and makes no particular assumptions on the objective function to be optimized.
The IGO method produces explicit IGO algorithms through time discretization.
The cross-entropy method is recovered in a particular case with a large time
step, and can be extended into a smoothed, parametrization-independent maximum
likelihood update.
When applied to specific families of distributions on discrete or continuous
spaces, the IGO framework allows to naturally recover versions of known
algorithms. From the family of Gaussian distributions on ${\mathbb{R}}^{d}$,
we arrive at a version of the well-known CMA-ES algorithm. From the family of
Bernoulli distributions on $\\{0,1\\}^{d}$, we recover the seminal PBIL
algorithm. From the distributions of restricted Boltzmann machines, we
naturally obtain a novel algorithm for discrete optimization on
$\\{0,1\\}^{d}$.
The IGO method achieves, thanks to its intrinsic formulation, maximal
invariance properties: invariance under reparametrization of the search space
$X$, under a change of parameters of the probability distribution, and under
increasing transformation of the function to be optimized. The latter is
achieved thanks to an adaptive formulation of the objective.
Theoretical considerations strongly suggest that IGO algorithms are
characterized by a minimal change of the distribution. Therefore they have
minimal loss in diversity through the course of optimization, provided the
initial diversity is high. First experiments using restricted Boltzmann
machines confirm this insight. As a simple consequence, IGO seems to provide,
from information theory, an elegant way to spontaneously explore several
valleys of a fitness landscape in a single run.
###### Contents
1. Introduction
2. 1 Algorithm description
1. 1.1 The natural gradient on parameter space
2. 1.2 IGO: Information-geometric optimization
3. 2 First properties of IGO
1. 2.1 Consistency of sampling
2. 2.2 Monotonicity: quantile improvement
3. 2.3 The IGO flow for exponential families
4. 2.4 Invariance properties
5. 2.5 Speed of the IGO flow
6. 2.6 Noisy objective function
7. 2.7 Implementation remarks
4. 3 IGO, maximum likelihood and the cross-entropy method
5. 4 CMA-ES, NES, EDAs and PBIL from the IGO framework
1. 4.1 IGO algorithm for Bernoulli measures and PBIL
2. 4.2 Multivariate normal distributions (Gaussians)
3. 4.3 Computing the IGO flow for some simple examples
6. 5 Multimodal optimization using restricted Boltzmann machines
1. 5.1 IGO for restricted Boltzmann machines
2. 5.2 Experimental results
1. 5.2.1 Convergence
2. 5.2.2 Diversity
3. 5.3 Convergence to the continuous-time limit
7. 6 Further discussion
8. Summary and conclusion
## Introduction
In this article, we consider an objective function $f:X\to{\mathbb{R}}$ to be
minimized over a search space $X$. No particular assumption on $X$ is needed:
it may be discrete or continuous, finite or infinite. We adopt a standard
scenario where we consider $f$ as a _black box_ that delivers values $f(x)$
for any desired input $x\in X$. The objective of black-box optimization is to
find solutions $x\in X$ with small value $f(x)$, using the least number of
calls to the black box. In this context, we design a stochastic optimization
method from sound theoretical principles.
We assume that we are given a family of probability distributions $P_{\theta}$
on $X$ depending on a continuous multicomponent parameter $\theta\in\Theta$. A
basic example is to take $X={\mathbb{R}}^{d}$ and to consider the family of
all Gaussian distributions $P_{\theta}$ on ${\mathbb{R}}^{d}$, with
$\theta=(m,C)$ the mean and covariance matrix. Another simple example is
$X=\\{0,1\\}^{d}$ equipped with the family of Bernoulli measures, i.e.
$\theta=(\theta_{i})_{1\leqslant i\leqslant d}$ and
$P_{\theta}(x)=\prod\theta_{i}^{x_{i}}(1-\theta_{i})^{1-x_{i}}$ for
$x=(x_{i})\in X$. The parameters $\theta$ of the family $P_{\theta}$ belong to
a space, $\Theta$, assumed to be a smooth manifold.
From this setting, we build in a natural way an optimization method, the
_information-geometric optimization_ (IGO). At each time $t$, we maintain a
value $\theta^{t}$ such that $P_{\theta^{t}}$ represents, loosely speaking,
the current belief about where the smallest values of the function $f$ may
lie. Over time, $P_{\theta^{t}}$ evolves and is expected to concentrate around
the minima of $f$. This general approach resembles the wide family of
_estimation of distribution algorithms_ (EDA) [LL02, BC95, PGL02]. However, we
deviate somewhat from the common EDA reasoning, as explained in the following.
The IGO method takes the form of a gradient ascent on $\theta^{t}$ in the
parameter space $\Theta$. We follow the gradient of a suitable transformation
of $f$, based on the _$P_{\theta^{t}}$ -quantiles_ of $f$. The gradient used
for $\theta$ is the _natural gradient_ defined from the Fisher information
metric [Rao45, Jef46, AN00], as is the case in various other optimization
strategies, for instance so-called _natural evolution strategies_ [WSPS08,
SWSS09, GSS+10]. Thus, we extend the scope of optimization strategies based on
this gradient to arbitrary search spaces.
The IGO method also has an equivalent description as an infinitesimal maximum
likelihood update; this reveals a new property of the natural gradient. This
also relates IGO to the _cross-entropy method_ [dBKMR05] in some situations.
When we instantiate IGO using the family of Gaussian distributions on
${\mathbb{R}}^{d}$, we naturally obtain variants of the well-known _covariance
matrix adaptation evolution strategy_ (CMA-ES) [HO01, HK04, JA06] and of
_natural evolution strategies_. With Bernoulli measures on the discrete cube
$\\{0,1\\}^{d}$, we recover the well-known _population based incremental
learning_ (PBIL) [BC95, Bal94]; this derivation of PBIL as a natural gradient
ascent appears to be new, and sheds some light on the common ground between
continuous and discrete optimization.
From the IGO framework, it is immediate to build new optimization algorithms
using more complex families of distributions than Gaussian or Bernoulli. As an
illustration, distributions associated with restricted Boltzmann machines
(RBMs) provide a new but natural algorithm for discrete optimization on
$\\{0,1\\}^{d}$, able to handle dependencies between the bits (see also
[Ber02]). The probability distributions associated with RBMs are multimodal;
combined with specific information-theoretic properties of IGO that guarantee
minimal loss of diversity over time, this allows IGO to reach multiple optima
at once very naturally, at least in a simple experimental setup.
Our method is built to achieve maximal _invariance properties_. First, it will
be invariant under reparametrization of the family of distributions
$P_{\theta}$, that is, at least for infinitesimally small steps, it only
depends on $P_{\theta}$ and not on the way we write the parameter $\theta$.
(For instance, for Gaussian measures it should not matter whether we use the
covariance matrix or its inverse or a Cholesky factor as the parameter.) This
limits the influence of encoding choices on the behavior of the algorithm.
Second, it will be invariant under a change of coordinates in the search space
$X$, provided that this change of coordinates globally preserves our family of
distributions $P_{\theta}$. (For Gaussian distributions on ${\mathbb{R}}^{d}$,
this includes all affine changes of coordinates.) This means that the
algorithm, apart from initialization, does not depend on the precise way the
data is presented. Last, the algorithm will be invariant under applying an
increasing function to $f$, so that it is indifferent whether we minimize e.g.
$f$, $f^{3}$ or $f\times|f|^{-2/3}$. This way some non-convex or non-smooth
functions can be as “easily” optimised as convex ones. Contrary to previous
formulations using natural gradients [WSPS08, GSS+10, ANOK10], this invariance
under increasing transformation of the objective function is achieved from the
start.
Invariance under $X$-reparametrization has been—we believe—one of the keys to
the success of the CMA-ES algorithm, which derives from a particular case of
ours. Invariance under $\theta$-reparametrization is the main idea behind
_information geometry_ [AN00]. Invariance under $f$-transformation is not
uncommon, e.g., for evolution strategies [Sch95] or pattern search methods
[HJ61, Tor97, NM65]; however it is not always recognized as an attractive
feature. Such invariance properties mean that we deal with _intrinsic_
properties of the objects themselves, and not with the way we encode them as
collections of numbers in ${\mathbb{R}}^{d}$. It also means, most importantly,
that we make a minimal number of arbitrary choices.
In Section 1, we define the IGO flow and the IGO algorithm. We begin with
standard facts about the definition and basic properties of the natural
gradient, and its connection with Kullback–Leibler divergence and diversity.
We then proceed to the detailed description of our algorithm.
In Section 2, we state some first mathematical properties of IGO. These
include monotone improvement of the objective function, invariance properties,
the form of IGO for exponential families of probability distributions, and the
case of noisy objective functions.
In Section 3 we explain the theoretical relationships between IGO, maximum
likelihood estimates and the cross-entropy method. In particular, IGO is
uniquely characterized by a weighted log-likelihood maximization property.
In Section 4, we derive several well-known optimization algorithms from the
IGO framework. These include PBIL, versions of CMA-ES and other Gaussian
evolutionary algorithms such as EMNA. This also illustrates how a large step
size results in more and more differing algorithms w.r.t. the continuous-time
IGO flow.
In Section 5, we illustrate how IGO can be used to design new optimization
algorithms. As a proof of concept, we derive the IGO algorithm associated with
restricted Boltzmann machines for discrete optimization, allowing for
multimodal optimization. We perform a preliminary experimental study of the
specific influence of the Fisher information matrix on the performance of the
algorithm and on diversity of the optima obtained.
In Section 6, we discuss related work, and in particular, IGO’s relationship
with and differences from various other optimization algorithms such as
natural evolution strategies or the cross-entropy method. We also sum up the
main contributions of the paper and the design philosophy of IGO.
## 1 Algorithm description
We now present the outline of our algorithms. Each step is described in more
detail in the sections below.
Our method can be seen as an _estimation of distribution algorithm_ : at each
time $t$, we maintain a probability distribution $P_{\theta^{t}}$ on the
search space $X$, where $\theta^{t}\in\Theta$. The value of $\theta^{t}$ will
evolve so that, over time, $P_{\theta^{t}}$ gives more weight to points $x$
with better values of the function $f(x)$ to optimize.
A straightforward way to proceed is to transfer $f$ from $x$-space to
$\theta$-space: define a function $F(\theta)$ as the $P_{\theta}$-average of
$f$ and then to do a gradient descent for $F(\theta)$ in space $\Theta$
[Ber00]. This way, $\theta$ will converge to a point such that $P_{\theta}$
yields a good average value of $f$. We depart from this approach in two ways:
* •
At each time, we replace $f$ with an adaptive transformation of $f$
representing how good or bad observed values of $f$ are relative to other
observations. This provides invariance under all monotone transformations of
$f$.
* •
Instead of the vanilla gradient for $\theta$, we use the so-called _natural
gradient_ given by the Fisher information matrix. This reflects the intrinsic
geometry of the space of probability distributions, as introduced by Rao and
Jeffreys [Rao45, Jef46] and later elaborated upon by Amari and others [AN00].
This provides invariance under reparametrization of $\theta$ and, importantly,
minimizes the change of diversity of $P_{\theta}$.
The algorithm is constructed in two steps: we first give an “ideal” version,
namely, a version in which time $t$ is continuous so that the evolution of
$\theta^{t}$ is given by an ordinary differential equation in $\Theta$.
Second, the actual algorithm is a time discretization using a finite time step
and Monte Carlo sampling instead of exact $P_{\theta}$-averages.
### 1.1 The natural gradient on parameter space
##### About gradients and the shortest path uphill.
Let $g$ be a smooth function from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}$, to be
maximized. We first present the interpretation of gradient ascent as “the
shortest path uphill”.
Let $y\in{\mathbb{R}}^{d}$. Define the vector $z$ by
$z=\lim_{\varepsilon\to 0}\operatorname*{arg\,max}_{z,\,\|z\|\leqslant
1}g(y+\varepsilon z).$ (1)
Then one can check that $z$ is the normalized gradient of $g$ at $y$:
$z_{i}=\frac{\partial g/\partial y_{i}}{\|\partial g/\partial y_{k}\|}$. (This
holds only at points $y$ where the gradient of $g$ does not vanish.)
This shows that, for small $\hskip 0.50003pt\delta t\hskip 0.50003pt$, the
well-known gradient ascent of $g$ given by
$y^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}_{i}=y^{t}_{i}+\hskip
0.50003pt\delta t\hskip 0.50003pt\,\tfrac{\partial g}{\partial y_{i}}$
realizes the largest increase in the value of $g$, for a given step size
$\|y^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}-y^{t}\|$.
The relation (1) depends on the choice of a norm $\|\\!\cdot\\!\|$ (the
gradient of $g$ is given by $\partial g/\partial y_{i}$ only in an orthonormal
basis). If we use, instead of the standard metric
$\|y-y^{\prime}\|=\sqrt{\sum(y_{i}-y^{\prime}_{i})^{2}}$ on
${\mathbb{R}}^{d}$, a metric $\|y-y^{\prime}\|_{A}=\sqrt{\sum
A_{ij}(y_{i}-y^{\prime}_{i})(y_{j}-y^{\prime}_{j})}$ defined by a positive
definite matrix $A_{ij}$, then the gradient of $g$ with respect to this metric
is given by $\sum_{j}A^{-1}_{ij}\frac{\partial g}{\partial y_{i}}$. (This
follows from the textbook definition of gradients by $g(y+\varepsilon
z)=g(y)+\varepsilon\langle\nabla g,z\rangle_{A}+O(\varepsilon^{2})$ with
$\langle\cdot,\cdot\rangle_{A}$ the scalar product associated with the matrix
$A_{ij}$ [Sch92].)
We can write the analogue of (1) using the $A$-norm. We get that the gradient
ascent associated with metric $A$, given by
$y^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=y^{t}+\hskip 0.50003pt\delta
t\hskip 0.50003pt\,A^{-1}\,\tfrac{\partial g}{\partial y_{i}},$
for small $\hskip 0.50003pt\delta t\hskip 0.50003pt$, maximizes the increment
of $g$ for a given $A$-distance $\|y^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}-y^{t}\|_{A}$—it realizes the steepest $A$-ascent. Maybe this
viewpoint clarifies the relationship between gradient and metric: this
steepest ascent property can actually be used as a definition of gradients.
In our setting we want to use a gradient ascent in the parameter space
$\Theta$ of our distributions $P_{\theta}$. The metric
$\|\theta-\theta^{\prime}\|=\sqrt{\sum(\theta_{i}-\theta^{\prime}_{i})^{2}}$
clearly depends on the choice of parametrization $\theta$, and thus is not
intrinsic. Therefore, we use a metric depending on $\theta$ only through the
distributions $P_{\theta}$, as follows.
##### Fisher information and the natural gradient on parameter space.
Let $\theta,\theta^{\prime}\in\Theta$ be two values of the distribution
parameter. The Kullback–Leibler divergence between $P_{\theta}$ and
$P_{\theta^{\prime}}$ is defined [Kul97] as
$\mathrm{KL}\\!\left(P_{\theta^{\prime}}\,||\,P_{\theta}\right)=\int_{x}\ln\frac{P_{\theta^{\prime}}(x)}{P_{\theta}(x)}\,P_{\theta^{\prime}}({\mathrm{d}}x).$
When $\theta^{\prime}=\theta+\delta\theta$ is close to $\theta$, under mild
smoothness assumptions we can expand the Kullback–Leibler divergence at second
order in $\delta\theta$. This expansion defines the Fisher information matrix
$I$ at $\theta$ [Kul97]:
$\mathrm{KL}\\!\left(P_{\theta+\delta\theta}\,||\,P_{\theta}\right)=\frac{1}{2}\sum
I_{ij}(\theta)\,\delta\theta_{i}\delta\theta_{j}+O(\delta\theta^{3}).$
An equivalent definition of the Fisher information matrix is by the usual
formulas [CT06]
$I_{ij}(\theta)=\int_{x}\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{i}}\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{j}}\,{\mathrm{d}}P_{\theta}(x)=-\int_{x}\frac{\partial^{2}\ln
P_{\theta}(x)}{\partial\theta_{i}\,\partial\theta_{j}}\,{\mathrm{d}}P_{\theta}(x).$
The Fisher information matrix defines a (Riemannian) metric on $\Theta$: the
distance, in this metric, between two very close values of $\theta$ is given
by the square root of twice the Kullback–Leibler divergence. Since the
Kullback–Leibler divergence depends only on $P_{\theta}$ and not on the
parametrization of $\theta$, this metric is intrinsic.
If $g:\Theta\to{\mathbb{R}}$ is a smooth function on the parameter space, its
_natural gradient_ at $\theta$ is defined in accordance with the Fisher metric
as
$(\widetilde{\nabla}_{\\!\theta}\,g)_{i}=\sum_{j}I^{-1}_{ij}(\theta)\,\frac{\partial
g(\theta)}{\partial\theta_{j}}$
or more synthetically
$\widetilde{\nabla}_{\\!\theta}\,g=I^{-1}\,\frac{\partial g}{\partial\theta}.$
From now on, we will use $\widetilde{\nabla}_{\\!\theta}$ to denote the
natural gradient and $\frac{\partial}{\partial\theta}$ to denote the vanilla
gradient.
By construction, the natural gradient descent is intrinsic: it does not depend
on the chosen parametrization $\theta$ of $P_{\theta}$, so that it makes sense
to speak of the natural gradient ascent of a function $g(P_{\theta})$.
Given that the Fisher metric comes from the Kullback–Leibler divergence, the
“shortest path uphill” property of gradients mentioned above translates as
follows (see also [Ama98, Theorem 1]):
###### Proposition 1.
The natural gradient ascent points in the direction $\delta\theta$ achieving
the largest change of the objective function, for a given distance between
$P_{\theta}$ and $P_{\theta+\delta\theta}$ in Kullback–Leibler divergence.
More precisely, let $g$ be a smooth function on the parameter space $\Theta$.
Let $\theta\in\Theta$ be a point where $\widetilde{\nabla}g(\theta)$ does not
vanish. Then
$\frac{\widetilde{\nabla}g(\theta)}{\rule{0.0pt}{8.61108pt}\|\widetilde{\nabla}g(\theta)\|}=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\operatorname*{arg\,max}_{\begin{subarray}{c}\delta\theta\text{
such that}\\\
\mathrm{KL}\\!\left(P_{\theta+\delta\theta}\,||\,P_{\theta}\right)\leqslant\varepsilon^{2}/2\end{subarray}}g(\theta+\delta\theta).$
Here we have implicitly assumed that the parameter space $\Theta$ is non-
degenerate and proper (that is, no two points $\theta\in\Theta$ define the
same probability distribution, and the mapping $P_{\theta}\mapsto\theta$ is
continuous).
##### Why use the Fisher metric gradient for optimization? Relationship to
diversity.
The first reason for using the natural gradient is its reparametrization
invariance, which makes it the only gradient available in a general abstract
setting [AN00]. Practically, this invariance also limits the influence of
encoding choices on the behavior of the algorithm. More prosaically, the
Fisher matrix can be also seen as an _adaptive learning rate_ for different
components of the parameter vector $\theta_{i}$: components $i$ with a high
impact on $P_{\theta}$ will be updated more cautiously.
Another advantage comes from the relationship with Kullback–Leibler distance
in view of the “shortest path uphill” (see also [Ama98]). To minimize the
value of some function $g(\theta)$ defined on the parameter space $\Theta$,
the naive approach follows a gradient descent for $g$ using the “vanilla”
gradient
$\theta_{i}^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta_{i}^{t}+\hskip
0.50003pt\delta t\hskip 0.50003pt\tfrac{\partial g}{\partial\theta_{i}}$
and, as explained above, this maximizes the increment of $g$ for a given
increment $\|\theta^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}-\theta^{t}\|$. On the other hand, the Fisher gradient
$\theta_{i}^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta_{i}^{t}+\hskip
0.50003pt\delta t\hskip 0.50003ptI^{-1}\tfrac{\partial g}{\partial\theta_{i}}$
maximizes the increment of $g$ for a given Kullback–Leibler distance
$\mathrm{KL}\\!\left(P_{\theta^{t+\hskip 0.25002pt\delta t\hskip
0.25002pt}}\,||\,P_{\theta^{t}}\right)$.
In particular, if we choose an initial value $\theta^{0}$ such that
$P_{\theta^{0}}$ covers a wide portion of the space $X$ uniformly, the
Kullback–Leibler divergence between $P_{\theta^{t}}$ and $P_{\theta^{0}}$
measures the loss of diversity of $P_{\theta^{t}}$. So the natural gradient
descent is a way to optimize the function $g$ with _minimal loss of diversity,
provided the initial diversity is large_. On the other hand the vanilla
gradient descent optimizes $g$ with minimal change in the numerical values of
the parameter $\theta$, which is of little interest.
So arguably this method realizes the best trade-off between optimization and
loss of diversity. (Though, as can be seen from the detailed algorithm
description below, maximization of diversity occurs only greedily at each
step, and so there is no guarantee that after a given time, IGO will provide
the highest possible diversity for a given objective function value.)
An experimental confirmation of the positive influence of the Fisher matrix on
diversity is given in Section 5 below. This may also provide a theoretical
explanation to the good performance of CMA-ES.
### 1.2 IGO: Information-geometric optimization
##### Quantile rewriting of $f$.
Our original problem is to minimize a function $f:X\to{\mathbb{R}}$. A simple
way to turn $f$ into a function on $\Theta$ is to use the expected value
$-\mathbb{E}_{P_{\theta}}f$ [Ber00, WSPS08], but expected values can be unduly
influenced by extreme values and using them is rather unstable [Whi89];
moreover $-\mathbb{E}_{P_{\theta}}f$ is not invariant under increasing
transformation of $f$ (this invariance implies we can only compare $f$-values,
not add them).
Instead, we take an adaptive, quantile-based approach by first replacing the
function $f$ with a monotone rewriting $W_{\theta}^{f}$ and then following the
gradient of $\mathbb{E}_{P_{\theta}}W_{\theta}^{f}$. A due choice of
$W_{\theta}^{f}$ allows to control the range of the resulting values and
achieves the desired invariance. Because the rewriting $W_{\theta}^{f}$
depends on $\theta$, it might be viewed as an _adaptive_ $f$-transformation.
The goal is that if $f(x)$ is “small” then $W_{\theta}^{f}(x)\in{\mathbb{R}}$
is “large” and vice versa, and that $W_{\theta}^{f}$ remains invariant under
monotone transformations of $f$. The meaning of “small” or “large” depends on
$\theta\in\Theta$ and is taken with respect to typical values of $f$ under the
current distribution $P_{\theta}$. This is measured by the
$P_{\theta}$-quantile in which the value of $f(x)$ lies. We write the lower
and upper $P_{\theta}$-$f$-quantiles of $x\in X$ as
$\displaystyle q_{\theta}^{-}(x)$ $\displaystyle=\Pr\nolimits_{x^{\prime}\sim
P_{\theta}}(f(x^{\prime})<f(x))$ (2) $\displaystyle q_{\theta}^{+}(x)$
$\displaystyle=\Pr\nolimits_{x^{\prime}\sim P_{\theta}}(f(x^{\prime})\leqslant
f(x))\enspace.$
These quantile functions reflect the probability to sample a better value than
$f(x)$. They are monotone in $f$ (if $f(x_{1})\leqslant f(x_{2})$ then
$q_{\theta}^{\pm}(x_{1})\leqslant q_{\theta}^{\pm}(x_{2})$) and invariant
under increasing transformations of $f$.
Given $q\in[0;1]$, we now choose a non-increasing function
$w:[0;1]\to{\mathbb{R}}$ (fixed once and for all). A typical choice for $w$ is
$w(q)=\mathbbm{1}_{q\leqslant q_{0}}$ for some fixed value $q_{0}$, the
_selection quantile_. The transform $W_{\theta}^{f}(x)$ is defined as a
function of the $P_{\theta}$-$f$-quantile of $x$ as
$W_{\theta}^{f}(x)=\begin{cases}w(q_{\theta}^{+}(x))&\text{if
}q_{\theta}^{+}(x)=q_{\theta}^{-}(x),\\\
\frac{1}{q_{\theta}^{+}(x)-q_{\theta}^{-}(x)}\int_{q=q_{\theta}^{-}(x)}^{q=q_{\theta}^{+}(x)}w(q)\,{\mathrm{d}}q&\text{otherwise.}\end{cases}$
(3)
As desired, the definition of $W_{\theta}^{f}$ is invariant under an
increasing transformation of $f$. For instance, the $P_{\theta}$-median of $f$
gets remapped to $w(\frac{1}{2})$.
Note that $\mathbb{E}_{P_{\theta}}W_{\theta}^{f}{}=\int_{0}^{1}w$ is
independent of $f$ and $\theta$: indeed, by definition, the quantile of a
random point under $P_{\theta}$ is uniformly distributed in $[0;1]$. In the
following, our objective will be to maximize the expected value of
$W_{\theta^{t}}^{f}$ in $\theta$, that is, to maximize
$\mathbb{E}_{P_{\theta}}W_{\theta^{t}}^{f}{}=\int
W_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x)\enspace$ (4)
over $\theta$, where ${\theta^{t}}$ is fixed at a given step but will adapt
over time.
Importantly, $W_{\theta}^{f}{(x)}$ can be estimated in practice: indeed, the
quantiles $\Pr_{x^{\prime}\sim P_{\theta}}(f(x^{\prime})<f(x))$ can be
estimated by taking samples of $P_{\theta}$ and ordering the samples according
to the value of $f$ (see below). The estimate remains invariant under
increasing $f$-transformations.
##### The IGO gradient flow.
At the most abstract level, IGO is a continuous-time gradient flow in the
parameter space $\Theta$, which we define now. In practice, discrete time
steps (a.k.a. iterations) are used, and $P_{\theta}$-integrals are
approximated through sampling, as described in the next section.
Let $\theta^{t}$ be the current value of the parameter at time $t$, and let
$\hskip 0.50003pt\delta t\hskip 0.50003pt\ll 1$. We define $\theta^{t+\hskip
0.35002pt\delta t\hskip 0.35002pt}$ in such a way as to increase the
$P_{\theta}$-weight of points where $f$ is small, while not going too far from
$P_{\theta^{t}}$ in Kullback–Leibler divergence. We use the adaptive weights
$W_{\theta^{t}}^{f}$ as a way to measure which points have large or small
values. In accordance with (4), this suggests taking the gradient ascent
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip 0.50003pt\,\widetilde{\nabla}_{\\!\theta}\int
W_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x)$ (5)
where the natural gradient is suggested by Proposition 1.
Note again that we use $W_{\theta^{t}}^{f}$ and not $W_{\theta}^{f}$ in the
integral. So the gradient $\widetilde{\nabla}_{\\!\theta}$ does not act on the
adaptive objective $W_{\theta^{t}}^{f}$. If we used $W_{\theta}^{f}$ instead,
we would face a paradox: right after a move, previously good points do not
seem so good any more since the distribution has improved. More precisely,
$\int W_{\theta}^{f}(x)\,P_{\theta}({\mathrm{d}}x)$ is constant and always
equal to the average weight $\int_{0}^{1}w$, and so the gradient would always
vanish.
Using the log-likelihood trick
$\widetilde{\nabla}P_{\theta}=P_{\theta}\,\widetilde{\nabla}\\!\ln P_{\theta}$
(assuming $P_{\theta}$ is smooth), we get an equivalent expression of the
update above as an integral under the current distribution $P_{\theta^{t}}$;
this is important for practical implementation. This leads to the following
definition.
###### Definition 2 (IGO flow).
The IGO flow is the set of continuous-time trajectories in space $\Theta$,
defined by the differential equation
$\displaystyle\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}$
$\displaystyle=\widetilde{\nabla}_{\\!\theta}\int
W_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x)$ (6) $\displaystyle=\int
W_{\theta^{t}}^{f}(x)\,\widetilde{\nabla}_{\\!\theta}\ln
P_{\theta}(x)\,P_{\theta^{t}}({\mathrm{d}}x)$ (7)
$\displaystyle=I^{-1}(\theta^{t})\,\int
W_{\theta^{t}}^{f}(x)\,\frac{\partial\ln
P_{\theta}(x)}{\partial\theta}\,P_{\theta^{t}}({\mathrm{d}}x).$ (8)
where the gradients are taken at point $\theta=\theta^{t}$, and $I$ is the
Fisher information matrix.
Natural evolution strategies (NES, [WSPS08, GSS+10, SWSS09]) feature a related
gradient _descent_ with $f(x)$ instead of $W_{\theta^{t}}^{f}(x)$. The
associated flow would read
$\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}=-\widetilde{\nabla}_{\\!\theta}\int
f(x)\,P_{\theta}({\mathrm{d}}x)\enspace,$ (9)
where the gradient is taken at $\theta^{t}$ (in the sequel when not explicitly
stated, gradients in $\theta$ are taken at $\theta=\theta^{t}$). However, in
the end NESs always implement algorithms using sample quantiles, as if derived
from the gradient ascent of $W_{\theta^{t}}^{f}(x)$.
The update (7) is a weighted average of “intrinsic moves” increasing the log-
likelihood of some points. We can slightly rearrange the update as
$\displaystyle\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}$
$\displaystyle=\int\overbrace{W_{\theta^{t}}^{f}(x)}^{\text{preference
weight\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}}\,\underbrace{\widetilde{\nabla}_{\\!\theta}\ln
P_{\theta}(x)}_{\text{intrinsic move to reinforce
$x$}}\,\overbrace{P_{\theta^{t}}({\mathrm{d}}x)}^{\text{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ current sample distribution}}$ (10)
$\displaystyle=\widetilde{\nabla}_{\\!\theta}\int\underbrace{W_{\theta^{t}}^{f}(x)\ln
P_{\theta}(x)}_{\text{weighted log-
likelihood}}\,P_{\theta^{t}}({\mathrm{d}}x).$ (11)
which provides an interpretation for the IGO gradient flow as a gradient
ascent optimization of the weighted log-likelihood of the “good points” of the
current distribution. In a precise sense, IGO is in fact the “best” way to
increase this log-likelihood (Theorem 13).
For exponential families of probability distributions, we will see later that
the IGO flow rewrites as a nice derivative-free expression (18).
##### The IGO algorithm. Time discretization and sampling.
The above is a mathematically well-defined continuous-time flow in the
parameter space. Its practical implementation involves three approximations
depending on two parameters $N$ and $\hskip 0.50003pt\delta t\hskip
0.50003pt$:
* •
the integral under $P_{\theta^{t}}$ is approximated using $N$ samples taken
from $P_{\theta^{t}}$;
* •
the value $W_{\theta^{t}}^{f}$ is approximated for each sample taken from
$P_{\theta^{t}}$;
* •
the time derivative $\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}$ is
approximated by a $\hskip 0.50003pt\delta t\hskip 0.50003pt$ time increment.
We also assume that the Fisher information matrix $I(\theta)$ and
$\frac{\partial\ln P_{\theta}(x)}{\partial\theta}$ can be computed (see
discussion below if $I(\theta)$ is unkown).
At each step, we pick $N$ samples $x_{1},\ldots,x_{N}$ under $P_{\theta^{t}}$.
To approximate the quantiles, we rank the samples according to the value of
$f$. Define $\mathrm{rk}(x_{i})=\\#\\{j,\,f(x_{j})<f(x_{i})\\}$ and let the
estimated weight of sample $x_{i}$ be
$\widehat{w}_{i}=\frac{1}{N}\,w\left(\frac{\mathrm{rk}(x_{i})+1/2}{N}\right),$
(12)
using the weighting scheme function $w$ introduced above. (This is assuming
there are no ties in our sample; in case several sample points have the same
value of $f$, we define $\widehat{w}_{i}$ by averaging the above over all
possible rankings of the ties111A mathematically neater but less intuitive
version would be
$\widehat{w}_{i}=\frac{1}{\mathrm{rk}^{+}(x_{i})-\mathrm{rk}^{-}(x_{i})}\int_{u=\mathrm{rk}^{-}(x_{i})/N}^{u=\mathrm{rk}^{+}(x_{i})/N}w(u){\mathrm{d}}u$
with $\mathrm{rk}^{-}(x_{i})=\\#\\{j,\,f(x_{j})<f(x_{i})\\}$ and
$\mathrm{rk}^{+}(x_{i})=\\#\\{j,\,f(x_{j})\leqslant f(x_{i})\\}$. .)
Then we can approximate the IGO flow as follows.
###### Definition 3 (IGO algorithm).
The _IGO algorithm_ with sample size $N$ and step size $\hskip 0.50003pt\delta
t\hskip 0.50003pt$ is the following update rule for the parameter
$\theta^{t}$. At each step, $N$ sample points $x_{1},\ldots,x_{N}$ are picked
according to the distribution $P_{\theta^{t}}$. The parameter is updated
according to
$\displaystyle\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}$
$\displaystyle=\theta^{t}+\hskip 0.50003pt\delta t\hskip
0.50003pt\sum_{i=1}^{N}\widehat{w}_{i}\,\left.\widetilde{\nabla}_{\theta}\ln
P_{\theta}(x_{i})\right|_{\theta=\theta^{t}}$ (13)
$\displaystyle=\theta^{t}+\hskip 0.50003pt\delta t\hskip
0.50003pt\,I^{-1}(\theta^{t})\,\sum_{i=1}^{N}\widehat{w}_{i}\,\left.\frac{\partial\ln
P_{\theta}(x_{i})}{\partial\theta}\right|_{\theta=\theta^{t}}$ (14)
where $\widehat{w}_{i}$ is the weight (12) obtained from the ranked values of
the objective function $f$.
Equivalently one can fix the weights
$w_{i}=\frac{1}{N}\,w\left(\frac{i-1/2}{N}\right)$ once and for all and
rewrite the update as
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip
0.50003pt\,I^{-1}(\theta^{t})\,\sum_{i=1}^{N}w_{i}\,\left.\frac{\partial\ln
P_{\theta}(x_{i:N})}{\partial\theta}\right|_{\theta=\theta^{t}}$ (15)
where $x_{i:N}$ denotes the $i^{\text{th}}$ sampled point ranked according to
$f$, i.e. $f(x_{1:N})<\ldots<f(x_{N:N})$ (assuming again there are no ties).
Note that $\\{x_{i:N}\\}=\\{x_{i}\\}$ and $\\{w_{i}\\}=\\{\widehat{w}_{i}\\}$.
As will be discussed in Section 4, this update applied to multivariate normal
distributions or Bernoulli measures allows to neatly recover versions of some
well-established algorithms, in particular CMA-ES and PBIL. Actually, in the
Gaussian context updates of the form (14) have already been introduced
[GSS+10, ANOK10], though not formally derived from a continuous-time flow with
quantiles.
When $N\to\infty$, the IGO algorithm using samples approximates the
continuous-time IGO gradient flow, see Theorem 4 below. Indeed, the IGO
algorithm, with $N=\infty$, is simply the Euler approximation scheme for the
ordinary differential equation defining the IGO flow (6). The latter result
thus provides a sound mathematical basis for currently used rank-based
updates.
##### IGO flow versus IGO algorithms.
The IGO _flow_ (6) is a well-defined continuous-time set of trajectories in
the space of probability distributions $P_{\theta}$, depending only on the
objective function $f$ and the chosen family of distributions. It does not
depend on the chosen parametrization for $\theta$ (Proposition 8).
On the other hand, there are several IGO _algorithms_ associated with this
flow. Each IGO algorithm approximates the IGO flow in a slightly different
way. An IGO algorithm depends on three further choices: a sample size $N$, a
time discretization step size $\hskip 0.50003pt\delta t\hskip 0.50003pt$, and
a choice of parametrization for $\theta$ in which to implement (14).
If $\hskip 0.50003pt\delta t\hskip 0.50003pt$ is small enough, and $N$ large
enough, the influence of the parametrization $\theta$ disappears and all IGO
algorithms are approximations of the “ideal” IGO flow trajectory. However, the
larger $\hskip 0.50003pt\delta t\hskip 0.50003pt$, the poorer the
approximation gets.
So for large $\hskip 0.50003pt\delta t\hskip 0.50003pt$, different IGO
algorithms for the same IGO flow may exhibit different behaviors. We will see
an instance of this phenomenon for Gaussian distributions: both CMA-ES and the
maximum likelihood update (EMNA) can be seen as IGO algorithms, but the latter
with $\hskip 0.50003pt\delta t\hskip 0.50003pt=1$ is known to exhibit
premature loss of diversity (Section 4.2).
Still, two IGO algorithms for the same IGO flow will differ less from each
other than from a non-IGO algorithm: at each step the difference is only
$O(\hskip 0.50003pt\delta t\hskip 0.50003pt^{2})$ (Section 2.4). On the other
hand, for instance, the difference between an IGO algorithm and the vanilla
gradient ascent is, generally, not smaller than $O(\hskip 0.50003pt\delta
t\hskip 0.50003pt)$ at each step, i.e. roughly as big as the steps themselves.
##### Unknown Fisher matrix.
The algorithm presented so far assumes that the Fisher matrix $I(\theta)$ is
known as a function of $\theta$. This is the case for Gaussian distributions
in CMA-ES and for Bernoulli distributions. However, for restricted Boltzmann
machines as considered below, no analytical form is known. Yet, provided the
quantity $\frac{\partial}{\partial\theta}\ln P_{\theta}(x)$ can be computed or
approximated, it is possible to approximate the integral
$I_{ij}(\theta)=\int_{x}\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{i}}\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{j}}\,P_{\theta}({\mathrm{d}}x)$
using $P_{\theta}$-Monte Carlo samples for $x$. These samples may or may not
be the same as those used in the IGO update (14): in particular, it is
possible to use as many Monte Carlo samples as necessary to approximate
$I_{ij}$, at no additional cost in terms of the number of calls to the black-
box function $f$ to optimize.
Note that each Monte Carlo sample $x$ will contribute $\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{i}}\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{j}}$ to the Fisher matrix approximation. This
is a rank-$1$ matrix. So, for the approximated Fisher matrix to be invertible,
the number of (distinct) samples $x$ needs to be at least equal to the number
of components of the parameter $\theta$ i.e.
$N_{\text{Fisher}}\geqslant\dim\Theta$.
For exponential families of distributions, the IGO update has a particular
form (18) which simplifies this matter somewhat. More details are given below
(see Section 5) for the concrete situation of restricted Boltzmann machines.
## 2 First properties of IGO
### 2.1 Consistency of sampling
The first property to check is that when $N\to\infty$, the update rule using
$N$ samples converges to the IGO update rule. This is _not_ a straightforward
application of the law of large numbers, because the estimated weights
$\widehat{w}_{i}$ depend (non-continuously) on the whole sample
$x_{1},\ldots,x_{N}$, and not only on $x_{i}$.
###### Theorem 4 (Consistency).
When $N\to\infty$, the $N$-sample IGO update rule (14):
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip
0.50003pt\,I^{-1}(\theta^{t})\,\sum_{i=1}^{N}\widehat{w}_{i}\,\left.\frac{\partial\ln
P_{\theta}(x_{i})}{\partial\theta}\right|_{\theta=\theta^{t}}$
converges with probability $1$ to the update rule (5):
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip 0.50003pt\,\widetilde{\nabla}_{\\!\theta}\int
W_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x).$
The proof is given in the Appendix, under mild regularity assumptions. In
particular we do not require that $w$ be continuous.
This theorem may clarify previous claims [WSPS08, ANOK10] where rank-based
updates similar to (5), such as in NES or CMA-ES, were derived from optimizing
the expected value $-\mathbb{E}_{P_{\theta}}f$. The rank-based weights
$\widehat{w}_{i}$ were then introduced somewhat arbitrarily. Theorem 4 shows
that, for large $N$, CMA-ES and NES actually follow the gradient flow of the
quantity $\mathbb{E}_{P_{\theta}}W_{\theta^{t}}^{f}$: the update can be
rigorously derived from optimizing the expected value of the quantile-
rewriting $W_{\theta^{t}}^{f}$.
### 2.2 Monotonicity: quantile improvement
Gradient descents come with a guarantee that the fitness value decreases over
time. Here, since we work with probability distributions on $X$, we need to
define the fitness of the distribution $P_{\theta^{t}}$. An obvious choice is
the expectation $\mathbb{E}_{P_{\theta^{t}}}f$, but it is not invariant under
$f$-transformation and moreover may be sensitive to extreme values.
It turns out that the monotonicity properties of the IGO gradient flow depend
on the choice of the weighting scheme $w$. For instance, if
$w(u)=\mathbbm{1}_{u\leqslant 1/2}$, then the median of $f$ improves over
time.
###### Proposition 5 (Quantile improvement).
Consider the IGO flow given by (6), with the weight
$w(u)=\mathbbm{1}_{u\leqslant q}$ where $0<q<1$ is fixed. Then the value of
the $q$-quantile of $f$ improves over time: if $t_{1}\leqslant t_{2}$ then
either $\theta^{t_{1}}=\theta^{t_{2}}$ or
$Q^{q}_{P_{\theta^{t_{2}}}}(f)<Q^{q}_{P_{\theta^{t_{1}}}}(f)$.
Here the $q$-quantile $Q^{q}_{P}(f)$ of $f$ under a probability distribution
$P$ is defined as any number $m$ such that $\Pr_{x\sim P}(f(x)\leqslant
m)\geqslant q$ and $\Pr_{x\sim P}(f(x)\geqslant m)\geqslant 1-q$.
The proof is given in the Appendix, together with the necessary regularity
assumptions.
Of course this holds only for the IGO gradient flow (6) with $N=\infty$ and
$\hskip 0.50003pt\delta t\hskip 0.50003pt\to 0$. For an IGO algorithm with
finite $N$, the dynamics is random and one cannot expect monotonicity. Still,
Theorem 4 ensures that, with high probability, trajectories of a large enough
finite population dynamics stay close to the infinite-population limit
trajectory.
### 2.3 The IGO flow for exponential families
The expressions for the IGO update simplify somewhat if the family
$P_{\theta}$ happens to be an exponential family of probability distributions
(see also [MMS08]). Suppose that $P_{\theta}$ can be written as
$P_{\theta}(x)=\frac{1}{Z(\theta)}\exp\left(\sum\theta_{i}T_{i}(x)\right)\,H({\mathrm{d}}x)$
where $T_{1},\ldots,T_{k}$ is a finite family of functions on $X$,
$H({\mathrm{d}}x)$ is an arbitrary reference measure on $X$, and $Z(\theta)$
is the normalization constant. It is well-known [AN00, (2.33)] that
$\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{i}}=T_{i}(x)-\mathbb{E}_{P_{\theta}}T_{i}$ (16)
so that [AN00, (3.59)]
$I_{ij}(\theta)=\operatorname{Cov}_{P_{\theta}}(T_{i},T_{j}).$ (17)
In the end we find:
###### Proposition 6.
Let $P_{\theta}$ be an exponential family parametrized by the natural
parameters $\theta$ as above. Then the IGO flow is given by
$\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}=\operatorname{Cov}_{P_{\theta}}(T,T)^{-1}\operatorname{Cov}_{P_{\theta}}(T,W_{\theta}^{f})$
(18)
where $\operatorname{Cov}_{P_{\theta}}(T,W_{\theta}^{f})$ denotes the vector
$(\operatorname{Cov}_{P_{\theta}}(T_{i},W_{\theta}^{f}))_{i}$, and
$\operatorname{Cov}_{P_{\theta}}(T,T)$ the matrix
$(\operatorname{Cov}_{P_{\theta}}(T_{i},T_{j}))_{ij}$.
Note that the right-hand side does not involve derivatives w.r.t. $\theta$ any
more. This result makes it easy to simulate the IGO flow using e.g. a Gibbs
sampler for $P_{\theta}$: both covariances in (18) may be approximated by
sampling, so that neither the Fisher matrix nor the gradient term need to be
known in advance, and no derivatives are involved.
The CMA-ES uses the family of all Gaussian distributions on
${\mathbb{R}}^{d}$. Then, the family $T_{i}$ is the family of all linear and
quadratic functions of the coordinates on ${\mathbb{R}}^{d}$. The expression
above is then a particularly concise rewriting of a CMA-ES update, see also
Section 4.2.
Moreover, the expected values $\bar{T_{i}}=\mathbb{E}T_{i}$ of $T_{i}$ satisfy
the simple evolution equation under the IGO flow
$\frac{{\mathrm{d}}\bar{T}_{i}}{{\mathrm{d}}t}=\operatorname{Cov}(T_{i},W_{\theta}^{f})=\mathbb{E}(T_{i}\,W_{\theta}^{f})-\bar{T}_{i}\,\mathbb{E}W_{\theta}^{f}.$
(19)
The proof is given in the Appendix, in the proof of Theorem 15.
The variables $\bar{T}_{i}$ can sometimes be used as an alternative
parametrization for an exponential family (e.g. for a one-dimensional
Gaussian, these are the mean $\mu$ and the second moment
$\mu^{2}+\sigma^{2}$). Then the IGO flow (7) may be rewritten using the
relation
$\widetilde{\nabla}_{\theta_{i}}=\frac{\partial}{\rule{0.0pt}{5.425pt}\partial\bar{T}_{i}}$
for the natural gradient of exponential families (Appendix, Proposition 22),
which sometimes results in simpler expressions. We shall further exploit this
fact in Section 3.
##### Exponential families with latent variables.
Similar formulas hold when the distribution $P_{\theta}(x)$ is the marginal of
an exponential distribution $P_{\theta}(x,h)$ over a “hidden” or “latent”
variable $h$, such as the restricted Boltzmann machines of Section 5.
Namely, with
$P_{\theta}(x)=\frac{1}{Z(\theta)}\sum_{h}\exp(\sum_{i}\theta_{i}T_{i}(x,h))\,H({\mathrm{d}}x,{\mathrm{d}}h)$
the Fisher matrix is
$I_{ij}(\theta)=\operatorname{Cov}_{P_{\theta}}(U_{i},U_{j})$ (20)
where $U_{i}(x)=\mathbb{E}_{P_{\theta}}(T_{i}(x,h)|x)$. Consequently, the IGO
flow takes the form
$\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}=\operatorname{Cov}_{P_{\theta}}(U,U)^{-1}\operatorname{Cov}_{P_{\theta}}(U,W_{\theta}^{f}).$
(21)
### 2.4 Invariance properties
Here we formally state the invariance properties of the IGO flow under various
reparametrizations. Since these results follow from the very construction of
the algorithm, the proofs are omitted.
###### Proposition 7 ($f$-invariance).
Let $\varphi:{\mathbb{R}}\to{\mathbb{R}}$ be an increasing function. Then the
trajectories of the IGO flow when optimizing the functions $f$ and
$\varphi(f)$ are the same.
The same is true for the discretized algorithm with population size $N$ and
step size $\hskip 0.50003pt\delta t\hskip 0.50003pt>0$.
###### Proposition 8 ($\theta$-invariance).
Let $\theta^{\prime}=\varphi(\theta)$ be a one-to-one function of $\theta$ and
let $P^{\prime}_{\theta^{\prime}}=P_{\varphi^{-1}(\theta)}$. Let $\theta^{t}$
be the trajectory of the IGO flow when optimizing a function $f$ using the
distributions $P_{\theta}$, initialized at $\theta^{0}$. Then the IGO flow
trajectory $(\theta^{\prime})^{t}$ obtained from the optimization of the
function $f$ using the distributions $P^{\prime}_{\theta^{\prime}}$,
initialized at $(\theta^{\prime})^{0}=\varphi(\theta^{0})$, is the same,
namely $(\theta^{\prime})^{t}=\varphi(\theta^{t})$.
For the algorithm with finite $N$ and $\hskip 0.50003pt\delta t\hskip
0.50003pt>0$, invariance under $\theta$-reparametrization is only true
approximately, in the limit when $\hskip 0.50003pt\delta t\hskip 0.50003pt\to
0$. As mentioned above, the IGO update (14), with $N=\infty$, is simply the
Euler approximation scheme for the ordinary differential equation (6) defining
the IGO flow. At each step, the Euler scheme is known to make an error
$O(\hskip 0.50003pt\delta t\hskip 0.50003pt^{2})$ with respect to the true
flow. This error actually depends on the parametrization of $\theta$.
So the IGO updates for different parametrizations coincide at first order in
$\hskip 0.50003pt\delta t\hskip 0.50003pt$, and may, in general, differ by
$O(\hskip 0.50003pt\delta t\hskip 0.50003pt^{2})$. For instance the difference
between the CMA-ES and xNES updates is indeed $O(\hskip 0.50003pt\delta
t\hskip 0.50003pt^{2})$, see Section 4.2.
For comparison, using the vanilla gradient results in a divergence of
$O(\hskip 0.50003pt\delta t\hskip 0.50003pt)$ at each step between different
parametrizations. So the divergence could be of the same magnitude as the
steps themselves.
In that sense, one can say that IGO algorithms are “more parametrization-
invariant” than other algorithms. This stems from their origin as a
discretization of the IGO flow.
The next proposition states that, for example, if one uses a family of
distributions on ${\mathbb{R}}^{d}$ which is invariant under affine
transformations, then our algorithm optimizes equally well a function and its
image under any affine transformation (up to an obvious change in the
initialization). This proposition generalizes the well-known corresponding
property of CMA-ES [HO01].
Here, as usual, the image of a probability distribution $P$ by a
transformation $\varphi$ is defined as the probability distribution
$P^{\prime}$ such that $P^{\prime}(Y)=P(\varphi^{-1}(Y))$ for any subset
$Y\subset X$. In the continuous domain, the density of the new distribution
$P^{\prime}$ is obtained by the usual change of variable formula involving the
Jacobian of $\varphi$.
###### Proposition 9 ($X$-invariance).
Let $\varphi:X\to X$ be a one-to-one transformation of the search space, and
assume that $\varphi$ globally preserves the family of measures $P_{\theta}$.
Let $\theta^{t}$ be the IGO flow trajectory for the optimization of function
$f$, initialized at $P_{\theta^{0}}$. Let $(\theta^{\prime})^{t}$ be the IGO
flow trajectory for optimization of $f\circ\varphi^{-1}$, initialized at the
image of $P_{\theta^{0}}$ by $\varphi$. Then $P_{(\theta^{\prime})^{t}}$ is
the image of $P_{\theta^{t}}$ by $\varphi$.
For the discretized algorithm with population size $N$ and step size $\hskip
0.50003pt\delta t\hskip 0.50003pt>0$, the same is true up to an error of
$O(\hskip 0.50003pt\delta t\hskip 0.50003pt^{2})$ per iteration. This error
disappears if the map $\varphi$ acts on $\Theta$ in an affine way.
The latter case of affine transforms is well exemplified by CMA-ES: here,
using the variance and mean as the parametrization of Gaussians, the new mean
and variance after an affine transform of the search space are an affine
function of the old mean and variance; specifically, for the affine
transformation $A:x\mapsto Ax+b$ we have
$(m,C)\mapsto(Am+b,ACA^{\mathrm{T}})$.
### 2.5 Speed of the IGO flow
###### Proposition 10.
The speed of the IGO flow, i.e. the norm of
$\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}$ in the Fisher metric, is at
most $\sqrt{\int_{0}^{1}w^{2}-(\int_{0}^{1}w)^{2}}$ where $w$ is the weighting
scheme.
This speed can be tested in practice in at least two ways. The first is just
to compute the Fisher norm of the increment $\theta^{t+\hskip 0.35002pt\delta
t\hskip 0.35002pt}-\theta^{t}$ using the Fisher matrix; for small $\hskip
0.50003pt\delta t\hskip 0.50003pt$ this is close to $\hskip 0.50003pt\delta
t\hskip 0.50003pt\|\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}\|$ with
$\|\cdot\|$ the Fisher metric. The second is as follows: since the Fisher
metric coincides with the Kullback–Leibler divergence up to a factor $1/2$, we
have $\mathrm{KL}\\!\left(P_{\theta^{t+\hskip 0.25002pt\delta t\hskip
0.25002pt}}\,||\,P_{\theta^{t}}\right)\approx\frac{1}{2}\hskip 0.50003pt\delta
t\hskip 0.50003pt^{2}\|\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}\|^{2}$ at
least for small $\hskip 0.50003pt\delta t\hskip 0.50003pt$. Since it is
relatively easy to estimate $\mathrm{KL}\\!\left(P_{\theta^{t+\hskip
0.25002pt\delta t\hskip 0.25002pt}}\,||\,P_{\theta^{t}}\right)$ by comparing
the new and old log-likelihoods of points in a Monte Carlo sample, one can
obtain an estimate of $\|\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}\|$.
###### Corollary 11.
Consider an IGO algorithm with weighting scheme $w$, step size $\hskip
0.50003pt\delta t\hskip 0.50003pt$ and sample size $N$. Then, for small
$\hskip 0.50003pt\delta t\hskip 0.50003pt$ and large $N$ we have
$\mathrm{KL}\\!\left(P_{\theta^{t+\hskip 0.25002pt\delta t\hskip
0.25002pt}}\,||\,P_{\theta^{t}}\right)\leqslant\frac{1}{2}\hskip
0.50003pt\delta t\hskip 0.50003pt^{2}\operatorname{Var}_{[0,1]}w+O(\hskip
0.50003pt\delta t\hskip 0.50003pt^{3})+O(1/\sqrt{N}).$
For instance, with $w(q)=\mathbbm{1}_{q\leqslant q_{0}}$ and neglecting the
error terms, an IGO algorithm introduces at most $\frac{1}{2}\hskip
0.50003pt\delta t\hskip 0.50003pt^{2}\,q_{0}(1-q_{0})$ bits of information (in
base $e$) per iteration into the probability distribution $P_{\theta}$.
Thus, the time discretization parameter $\hskip 0.50003pt\delta t\hskip
0.50003pt$ is not just an arbitrary variable: it has an intrinsic
interpretation related to a number of bits introduced at each step of the
algorithm. This kind of relationship suggests, more generally, to use the
Kullback–Leibler divergence as an external and objective way to measure
learning rates in those optimization algorithms which use probability
distributions.
The result above is only an upper bound. Maximal speed can be achieved only if
all “good” points point in the same direction. If the various good points in
the sample suggest moves in inconsistent directions, then the IGO update will
be much smaller. The latter may be a sign that the signal is noisy, or that
the family of distributions $P_{\theta}$ is not well suited to the problem at
hand and should be enriched.
As an example, using a family of Gaussian distributions with unkown mean and
fixed identity variance on ${\mathbb{R}}^{d}$, one checks that for the
optimization of a linear function on ${\mathbb{R}}^{d}$, with the weight
$w(u)=-\mathbbm{1}_{u>1/2}+\mathbbm{1}_{u<1/2}$, the IGO flow moves at
constant speed $1/\sqrt{2\pi}\approx 0.4$, whatever the dimension $d$. On a
rapidly varying sinusoidal function, the moving speed will be much slower
because there are “good” and “bad” points in all directions.
This may suggest ways to design the weighting scheme $w$ to achieve maximal
speed in some instances. Indeed, looking at the proof of the proposition,
which involves a Cauchy–Schwarz inequality, one can see that the maximal speed
is achieved only if there is a linear relationship between the weights
$W_{\theta}^{f}(x)$ and the gradient $\nabla_{\theta}\ln P_{\theta}(x)$. For
instance, for the optimization of a linear function on ${\mathbb{R}}^{d}$
using Gaussian measures of known variance, the maximal speed will be achieved
when the weighting scheme $w(u)$ is the inverse of the Gaussian cumulative
distribution function. (In particular, $w(u)$ tends to $+\infty$ when $u\to 0$
and to $-\infty$ when $u\to 1$.) This is in accordance with previously known
results: the expected value of the $i$-th order statistic of $N$ standard
Gaussian variates is the optimal $\widehat{w}_{i}$ value in evolution
strategies [Bey01, Arn06]. For $N\to\infty$, this order statistic converges to
the inverse Gaussian cumulative distribution function.
### 2.6 Noisy objective function
Suppose that the objective function $f$ is non-deterministic: each time we ask
for the value of $f$ at a point $x\in X$, we get a random result. In this
setting we may write the random value $f(x)$ as $f(x)=\tilde{f}(x,\omega)$
where $\omega$ is an unseen random parameter, and $\tilde{f}$ is a
deterministic function of $x$ and $\omega$. Without loss of generality, up to
a change of variables we can assume that $\omega$ is uniformly distributed in
$[0,1]$.
We can still use the IGO algorithm without modification in this context. One
might wonder which properties (consistency of sampling, etc.) still apply when
$f$ is not deterministic. Actually, IGO algorithms for noisy functions fit
very nicely into the IGO framework: the following proposition allows to
transfer any property of IGO to the case of noisy functions.
###### Proposition 12 (Noisy IGO).
Let $f$ be a random function of $x\in X$, namely, $f(x)=\tilde{f}(x,\omega)$
where $\omega$ is a random variable uniformly distributed in $[0,1]$, and
$\tilde{f}$ is a deterministic function of $x$ and $\omega$. Then the two
following algorithms coincide:
* •
The IGO algorithm (13), using a family of distributions $P_{\theta}$ on space
$X$, applied to the noisy function $f$, and where the samples are ranked
according to the random observed value of $f$ (here we assume that, for each
sample, the noise $\omega$ is independent from everything else);
* •
The IGO algorithm on space $X\times[0,1]$, using the family of distributions
$\tilde{P}_{\theta}=P_{\theta}\otimes U_{[0,1]}$, applied to the deterministic
function $\tilde{f}$. Here $U_{[0,1]}$ denotes the uniform law on $[0,1]$.
The (easy) proof is given in the Appendix.
This proposition states that noisy optimization is the same as ordinary
optimization using a family of distributions which cannot operate any
selection or convergence over the parameter $\omega$. More generally, any
component of the search space in which a distribution-based evolutionary
strategy cannot perform selection or specialization will effectively act as a
random noise on the objective function.
As a consequence of this result, all properties of IGO can be transferred to
the noisy case. Consider, for instance, consistency of sampling (Theorem 4).
The $N$-sample IGO update rule for the noisy case is identical to the non-
noisy case (14):
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip
0.50003pt\,I^{-1}(\theta^{t})\,\sum_{i=1}^{N}\widehat{w}_{i}\,\left.\frac{\partial\ln
P_{\theta}(x_{i})}{\partial\theta}\right|_{\theta=\theta^{t}}$
where each weight $\widehat{w}_{i}$ computed from (12) now incorporates noise
from the objective function because the rank of $x_{i}$ is computed on the
random function, or equivalently on the deterministic function $\tilde{f}$:
$\mathrm{rk}(x_{i})=\\#\\{j,\tilde{f}(x_{j},\omega_{j})<\tilde{f}(x_{i},\omega_{i})\\}$.
Consistency of sampling (Theorem 4) thus takes the following form: When
$N\to\infty$, the $N$-sample IGO update rule on the noisy function $f$
converges with probability $1$ to the update rule
$\displaystyle\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}$
$\displaystyle=\theta^{t}+\hskip 0.50003pt\delta t\hskip
0.50003pt\,\widetilde{\nabla}_{\\!\theta}\int_{0}^{1}\int
W_{\theta^{t}}^{\tilde{f}}(x,\omega)\,P_{\theta}({\mathrm{d}}x)\,{\mathrm{d}}\omega.$
$\displaystyle=\theta^{t}+\hskip 0.50003pt\delta t\hskip
0.50003pt\,\widetilde{\nabla}_{\\!\theta}\int\bar{W}_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x)$
(22)
where
$\bar{W}^{f}_{\theta}(x)=\mathbb{E}_{\omega}W_{\theta}^{\tilde{f}}(x,\omega)$.
This entails, in particular, that when $N\to\infty$, the noise disappears
asymptotically, as could be expected.
Consequently, the IGO flow in the noisy case should be defined by the $\hskip
0.50003pt\delta t\hskip 0.50003pt\to 0$ limit of the update (22) using
$\bar{W}$. Note that the quantiles $q^{\pm}(x)$ defined by (2) still make
sense in the noisy case, and are deterministic functions of $x$; thus
$W_{\theta}^{f}(x)$ can also be defined by (3) and is deterministic. However,
unless the weighting scheme $w(q)$ is affine, $\bar{W}_{\theta}^{f}(x)$ is
different from $W_{\theta}^{f}(x)$ in general. Thus, unless $w$ is affine the
flows defined by $W$ and $\bar{W}$ do not coincide in the noisy case. The flow
using $W$ would be the $N\to\infty$ limit of a slightly more complex algorithm
using several evaluations of $f$ for each sample $x_{i}$ in order to compute
noise-free ranks.
### 2.7 Implementation remarks
##### Influence of the weighting scheme $w$.
The weighting scheme $w$ directly affects the update rule (15).
A natural choice is $w(u)=\mathbbm{1}_{u\leqslant q}$. This, as we have
proved, results in an improvement of the $q$-quantile over the course of
optimization. Taking $q=1/2$ springs to mind; however, this is not selective
enough, and both theory and experiments confirm that for the Gaussian case
(CMA-ES), most efficient optimization requires $q<1/2$ (see Section 4.2). The
optimal $q$ is about $0.27$ if $N$ is not larger than the search space
dimension $d$ [Bey01] and even smaller otherwise.
Second, replacing $w$ with $w+c$ for some constant $c$ clearly has no
influence on the IGO continuous-time flow (5), since the gradient will cancel
out the constant. However, this is not the case for the update rule (15) with
a finite sample of size $N$.
Indeed, adding a constant $c$ to $w$ adds a quantity
$c\frac{1}{N}\sum\widetilde{\nabla}_{\theta}\ln P_{\theta}(x_{i})$ to the
update. Since we know that the $P_{\theta}$-expected value of
$\widetilde{\nabla}_{\theta}\ln P_{\theta}$ is $0$ (because
$\int(\widetilde{\nabla}_{\theta}\ln
P_{\theta})\,P_{\theta}=\int\widetilde{\nabla}P_{\theta}=\widetilde{\nabla}1=0$),
we have $\mathbb{E}\frac{1}{N}\widetilde{\nabla}_{\theta}\ln
P_{\theta}(x_{i})=0$. So adding a constant to $w$ does not change the expected
value of the update, but it may change e.g. its variance. The empirical
average of $\widetilde{\nabla}_{\theta}\ln P_{\theta}(x_{i})$ in the sample
will be $O(1/\sqrt{N})$. So translating the weights results in a
$O(1/\sqrt{N})$ change in the update. See also Section 4 in [SWSS09].
Determining an optimal value for $c$ to reduce the variance of the update is
difficult, though: the optimal value actually depends on possible correlations
between $\widetilde{\nabla}_{\theta}\ln P_{\theta}$ and the function $f$. The
only general result is that one should shift $w$ so that $0$ lies within its
range. Assuming independence, or dependence with enough symmetry, the optimal
shift is when the weights average to $0$.
##### Adaptive learning rate.
Comparing consecutive updates to evaluate a learning rate or step size is an
effective measure. For example, in back-propagation, the update sign has been
used to adapt the learning rate of each single weight in an artificial neural
network [SA90]. In CMA-ES, a step size is adapted depending on whether recent
steps tended to move in a consistent direction or to backtrack. This is
measured by considering the changes of the mean $m$ of the Gaussian
distribution. For a probability distribution $P_{\theta}$ on an arbitrary
search space, in general no notion of mean may be defined. However, it is
still possible to define “backtracking” in the evolution of $\theta$ as
follows.
Consider two successive updates $\delta\theta^{t}=\theta^{t}-\theta^{t-\hskip
0.35002pt\delta t\hskip 0.35002pt}$ and $\delta\theta^{t+\hskip
0.35002pt\delta t\hskip 0.35002pt}=\theta^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}-\theta^{t}$. Their scalar product in the Fisher metric
$I(\theta^{t})$ is
$\langle\delta\theta^{t},\delta\theta^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}\rangle=\sum_{ij}I_{ij}(\theta^{t})\,\delta\theta^{t}_{i}\,\delta\theta^{t+\hskip
0.35002pt\delta t\hskip 0.35002pt}_{j}.$
Dividing by the associated norms will yield the cosine $\cos\alpha$ of the
angle between $\delta\theta^{t}$ and $\delta\theta^{t+\hskip 0.35002pt\delta
t\hskip 0.35002pt}$.
If this cosine is positive, the learning rate $\hskip 0.50003pt\delta t\hskip
0.50003pt$ may be increased. If the cosine is negative, the learning rate
probably needs to be decreased. Various schemes for the change of $\hskip
0.50003pt\delta t\hskip 0.50003pt$ can be devised; for instance, inspired by
CMA-ES, one can multiply $\hskip 0.50003pt\delta t\hskip 0.50003pt$ by
$\exp(\beta(\cos\alpha)/2)$ or
$\exp(\beta(\mathbbm{1}_{\cos\alpha>0}-\mathbbm{1}_{\cos\alpha<0})/2)$, where
$\beta\approx\min(N/\dim\Theta,1/2)$.
As before, this scheme is constructed to be robust w.r.t. reparametrization of
$\theta$, thanks to the use of the Fisher metric. However, for large learning
rates $\hskip 0.50003pt\delta t\hskip 0.50003pt$, in practice the
parametrization might well become relevant.
A consistent direction of the updates does not necessarily mean that the
algorithm is performing well: for instance, when CEM/EMNA exhibits premature
convergence (see below), the parameters consistently move towards a zero
covariance matrix and the cosines above are positive.
##### Complexity.
The complexity of the IGO algorithm depends much on the computational cost
model. In optimization, it is fairly common to assume that the objective
function $f$ is very costly compared to any other calculations performed by
the algorithm. Then the cost of IGO in terms of number of $f$-calls is $N$ per
iteration, and the cost of using quantiles and computing the natural gradient
is negligible.
Setting the cost of $f$ aside, the complexity of the IGO algorithm depends
mainly on the computation of the (inverse) Fisher matrix. Assume an analytical
expression for this matrix is known. Then, with $p=\dim\Theta$ the number of
parameters, the cost of storage of the Fisher matrix is $O(p^{2})$ per
iteration, and its inversion typically costs $O(p^{3})$ per iteration.
However, depending on the situation and on possible algebraic simplifications,
strategies exist to reduce this cost (e.g. [LRMB07] in a learning context).
For instance, for CMA-ES the cost is $O(Np)$ [SHI09]. More generally,
parametrization by expectation parameters (see above), when available, may
reduce the cost to $O(p)$ as well.
If no analytical form of the Fisher matrix is known and Monte Carlo estimation
is required, then complexity depends on the particular situation at hand and
is related to the best sampling strategies available for a particular family
of distributions. For Boltzmann machines, for instance, a host of such
strategies are available. Still, in such a situation, IGO can be competitive
if the objective function $f$ is costly.
##### Recycling samples.
We might use samples not only from the last iteration to compute the ranks in
(12), see e.g. [SWSS09]. For $N=1$ this is indispensable. In order to preserve
sampling consistency (Theorem 4) the old samples need to be reweighted (using
the ratio of their new vs old likelihood, as in importance sampling).
##### Initialization.
As with other optimization algorithms, it is probably a good idea to
initialize in such a way as to cover a wide portion of the search space, i.e.
$\theta^{0}$ should be chosen so that $P_{\theta^{0}}$ has maximal diversity.
For IGO algorithms this is particularly relevant, since, as explained above,
the natural gradient provides minimal change of diversity (greedily at each
step) for a given change in the objective function.
## 3 IGO, maximum likelihood and the cross-entropy method
##### IGO as a smooth-time maximum likelihood estimate.
The IGO flow turns out to be the only way to maximize a _weighted_ log-
likelihood, where points of the current distribution are slightly reweighted
according to $f$-preferences.
This relies on the following interpretation of the natural gradient as a
weighted maximum likelihood update with infinitesimal learning rate. This
result singles out, in yet another way, the _natural_ gradient among all
possible gradients. The proof is given in the Appendix.
###### Theorem 13 (Natural gradient as ML with infinitesimal weights).
Let $\varepsilon>0$ and $\theta_{0}\in\Theta$. Let $W(x)$ be a function of $x$
and let $\theta$ be the solution of
$\theta=\operatorname*{arg\,max}_{\theta}\Bigg{\\{}(1-\varepsilon){\underbrace{\int\log
P_{\theta}(x)\,P_{\theta_{0}}({\mathrm{d}}x)}_{\text{$$maximal for
$\theta=\theta_{0}$$$}}}+\varepsilon\int\log
P_{\theta}(x)\,W(x)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}.$
Then, when $\varepsilon\to 0$, up to $O(\varepsilon^{2})$ we have
$\theta=\theta_{0}+\varepsilon\int\widetilde{\nabla}_{\\!\theta}\ln
P_{\theta}(x)\,\,W(x)\,P_{\theta_{0}}({\mathrm{d}}x).$
Likewise for discrete samples: with $x_{1},\ldots,x_{N}\in X$, let $\theta$ be
the solution of
$\theta=\operatorname*{arg\,max}\left\\{(1-\varepsilon)\int\log
P_{\theta}(x)\,P_{\theta_{0}}({\mathrm{d}}x)+\varepsilon\sum_{i}W(x_{i})\,\log
P_{\theta}(x_{i})\right\\}.$
Then when $\varepsilon\to 0$, up to $O(\varepsilon^{2})$ we have
$\theta=\theta_{0}+\varepsilon\sum_{i}W(x_{i})\,\,\widetilde{\nabla}_{\\!\theta}\ln
P_{\theta}(x_{i}).$
So if $W(x)=W_{\theta_{0}}^{f}(x)$ is the weight of the points according to
quantilized $f$-preferences, the weighted maximum log-likelihood necessarily
is the IGO flow (7) using the natural gradient—or the IGO update (14) when
using samples.
Thus the IGO flow is the unique flow that, continuously in time, slightly
changes the distribution to maximize the log-likelihood of points with good
values of $f$. Moreover IGO continuously updates the weight
$W_{\theta_{0}}^{f}(x)$ depending on $f$ and on the current distribution, so
that we keep optimizing.
This theorem suggests a way to approximate the IGO flow by enforcing this
interpretation for a given non-infinitesimal step size $\hskip 0.50003pt\delta
t\hskip 0.50003pt$, as follows.
###### Definition 14 (IGO-ML algorithm).
The _IGO-ML algorithm_ with step size $\hskip 0.50003pt\delta t\hskip
0.50003pt$ updates the value of the parameter $\theta^{t}$ according to
$\theta^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}=\operatorname*{arg\,max}_{\theta}\Bigg{\\{}{\textstyle(1-\hskip
0.50003pt\delta t\hskip 0.50003pt\sum\limits_{i}\widehat{w}_{i})}\int\log
P_{\theta}(x)\,P_{\theta^{t}}({\mathrm{d}}x)\,+\,\hskip 0.50003pt\delta
t\hskip 0.50003pt\sum_{i}\widehat{w}_{i}\log P_{\theta}(x_{i})\Bigg{\\}}$ (23)
where $x_{1},\ldots,x_{N}$ are sample points picked according to the
distribution $P_{\theta^{t}}$, and $\widehat{w}_{i}$ is the weight (12)
obtained from the ranked values of the objective function $f$.
As for the cross-entropy method below, this only makes algorithmic sense if
the argmax is tractable.
It turns out that IGO-ML is just the IGO algorithm in a particular
parametrization (see Theorem 15).
##### The cross-entropy method.
Taking $\hskip 0.50003pt\delta t\hskip 0.50003pt=1$ in (23) above corresponds
to a full maximum likelihood update, which is also related to the _cross-
entropy method_ (CEM). The cross-entropy method can be defined as follows
[dBKMR05] in an optimization setting. Like IGO, it depends on a family of
probability distributions $P_{\theta}$ parametrized by $\theta\in\Theta$, and
a number of samples $N$ at each iteration. Let also $N_{e}=\lceil qN\rceil$
($0<q<1$) be a number of _elite_ samples.
At each step, the cross-entropy method for optimization samples $N$ points
$x_{1},\ldots,x_{N}$ from the current distribution $P_{\theta^{t}}$. Let
$\widehat{w}_{i}$ be $1/N_{e}$ if $x_{i}$ belongs to the $N_{e}$ samples with
the best value of the objective function $f$, and $\widehat{w}_{i}=0$
otherwise. Then the _cross-entropy method_ or _maximum likelihoood_ update
(CEM/ML) for optimization is
$\theta^{t+1}=\operatorname*{arg\,max}_{\theta}\sum\widehat{w}_{i}\log
P_{\theta}(x_{i})$ (24)
(assuming the argmax is tractable).
A version with a smoother update depends on a step size parameter
$0<\alpha\leqslant 1$ and is given [dBKMR05] by
$\theta^{t+1}=(1-\alpha)\theta^{t}+\alpha\operatorname*{arg\,max}_{\theta}\sum\widehat{w}_{i}\log
P_{\theta}(x_{i}).$ (25)
The standard CEM/ML update corresponds to $\alpha=1$.
For $\alpha=1$ the standard cross-entropy method is independent of the
parametrization $\theta$, whereas for $\alpha<1$ this is not the case.
Note the difference between the IGO-ML algorithm (23) and the smoothed CEM
update (25) with step size $\alpha=\hskip 0.50003pt\delta t\hskip 0.50003pt$:
the smoothed CEM update performs a weighted average of the parameter value
_after_ taking the maximum likelihood estimate, whereas IGO-ML uses a weighted
average of current and previous likelihoods, _then_ takes a maximum likelihood
estimate. In general, these two rules can greatly differ, as they do for
Gaussian distributions (Section 4.2).
This interversion of averaging makes IGO-ML parametrization-independent
whereas the smoothed CEM update is not.
Yet, for exponential families of probability distributions, there exists one
particular parametrization $\theta$ in which the IGO algorithm and the
smoothed CEM update coincide. We now proceed to this construction.
##### IGO for expectation parameters and maximum likelihood.
The particular form of IGO for exponential families has an interesting
consequence if the parametrization chosen for the exponential family is the
set of _expectation parameters_. Let
$P_{\theta}(x)=\frac{1}{Z(\theta)}\exp\left(\sum\theta_{j}T_{j}(x)\right)\,H({\mathrm{d}}x)$
be an exponential family as above. The _expectation parameters_ are
$\bar{T}_{j}=\bar{T}_{j}(\theta)=\mathbb{E}_{P_{\theta}}T_{j}$, (denoted
$\eta_{j}$ in [AN00, (3.56)]). The notation $\bar{T}$ will denote the
collection $(\bar{T}_{j})$.
It is well-known that, in this parametrization, the maximum likelihood
estimate for a sample of points $x_{1},\ldots,x_{k}$ is just the empirical
average of the expectation parameters over that sample:
$\operatorname*{arg\,max}_{\bar{T}}\frac{1}{k}\sum_{i=1}^{k}\log
P_{\bar{T}}(x_{i})=\frac{1}{k}\sum_{i=1}^{k}T(x_{i}).$ (26)
In the discussion above, one main difference between IGO and smoothed CEM was
whether we took averages before or after taking the maximum log-likelihood
estimate. For the expectation parameters $\bar{T}_{i}$, we see that these
operations commute. (One can say that these expectation parameters “linearize
maximum likelihood estimates”.) A little work brings us to the
###### Theorem 15 (IGO, CEM and maximum likelihood).
Let
$P_{\theta}(x)=\frac{1}{Z(\theta)}\exp\left(\sum\theta_{j}T_{j}(x)\right)\,H({\mathrm{d}}x)$
be an exponential family of probability distributions, where the $T_{j}$ are
functions of $x$ and $H$ is some reference measure. Let us parametrize this
family by the expected values $\bar{T_{j}}=\mathbb{E}T_{j}$.
Let us assume the chosen weights $\widehat{w}_{i}$ sum to $1$. For a sample
$x_{1},\ldots,x_{N}$, let
$T_{j}^{\ast}=\sum_{i}\widehat{w}_{i}\,T_{j}(x_{i}).$
Then the IGO update (14) in this parametrization reads
$\bar{T}_{j}^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=(1-\hskip
0.50003pt\delta t\hskip 0.50003pt)\,\bar{T}_{j}^{t}+\hskip 0.50003pt\delta
t\hskip 0.50003pt\,T_{j}^{\ast}.$ (27)
Moreover these three algorithms coincide:
* •
The IGO-ML algorithm (23).
* •
The IGO algorithm written in the parametrization $\bar{T_{j}}$.
* •
The smoothed CEM algorithm (25) written in the parametrization $\bar{T_{j}}$,
with $\alpha=\hskip 0.50003pt\delta t\hskip 0.50003pt$.
###### Corollary 16.
The standard CEM/ML update (24) is the IGO algorithm in parametrization
$\bar{T}_{j}$ with $\hskip 0.50003pt\delta t\hskip 0.50003pt=1$.
Beware that the expectation parameters $\bar{T}_{j}$ are not always the most
obvious parameters [AN00, Section 3.5]. For example, for $1$-dimensional
Gaussian distributions, these expectation parameters are the mean $\mu$ and
the second moment $\mu^{2}+\sigma^{2}$. When expressed back in terms of mean
and variance, with the update (27) the new mean is $(1-\hskip 0.50003pt\delta
t\hskip 0.50003pt)\mu+\hskip 0.50003pt\delta t\hskip 0.50003pt\mu^{*}$, but
the new variance is $(1-\hskip 0.50003pt\delta t\hskip
0.50003pt)\sigma^{2}+\hskip 0.50003pt\delta t\hskip
0.50003pt(\sigma^{*})^{2}+\hskip 0.50003pt\delta t\hskip 0.50003pt(1-\hskip
0.50003pt\delta t\hskip 0.50003pt)(\mu^{*}-\mu)^{2}$.
On the other hand, when using smoothed CEM with mean and variance as
parameters, the new variance is $(1-\hskip 0.50003pt\delta t\hskip
0.50003pt)\sigma^{2}+\hskip 0.50003pt\delta t\hskip
0.50003pt(\sigma^{*})^{2}$, which can be significantly smaller for $\hskip
0.50003pt\delta t\hskip 0.50003pt\in(0,1)$. This proves, in passing, that the
smoothed CEM update in other parametrizations is generally _not_ an IGO
algorithm (because it can differ at first order in $\hskip 0.50003pt\delta
t\hskip 0.50003pt$).
The case of Gaussian distributions is further exemplified in Section 4.2
below: in particular, smoothed CEM in the $(\mu,\sigma)$ parametrization
exhibits premature reduction of variance, preventing good convergence.
For these reasons we think that the IGO-ML algorithm is the sensible way to
perform an interpolated ML estimate for $\hskip 0.50003pt\delta t\hskip
0.50003pt<1$, in a parametrization-independent way. In Section 6 we further
discuss IGO and CEM and sum up the differences and relative advantages.
Taking $\hskip 0.50003pt\delta t\hskip 0.50003pt=1$ is a bold approximation
choice: the “ideal” continuous-time IGO flow itself, after time $1$, does not
coincide with the maximum likelihood update of the best points in the sample.
Since the maximum likelihood algorithm is known to converge prematurely in
some instances (Section 4.2), using the parametrization by expectation
parameters with large $\hskip 0.50003pt\delta t\hskip 0.50003pt$ may not be
desirable.
The considerable simplification of the IGO update in these coordinates
reflects the duality of coordinates $\bar{T}_{i}$ and $\theta_{i}$. More
precisely, the natural gradient ascent w.r.t. the parameters $\bar{T}_{i}$ is
given by the vanilla gradient w.r.t. the parameters $\theta_{i}$:
$\widetilde{\nabla}_{\bar{T}_{i}}=\frac{\partial}{\partial\theta_{i}}$
(Proposition 22 in the Appendix).
## 4 CMA-ES, NES, EDAs and PBIL from the IGO framework
In this section we investigate the IGO algorithm for Bernoulli measures and
for multivariate normal distributions and show the correspondence to well-
known algorithms. In addition, we discuss the influence of the parametrization
of the distributions.
### 4.1 IGO algorithm for Bernoulli measures and PBIL
We consider on $X=\\{0,1\\}^{d}$ a family of Bernoulli measures
$P_{\theta}(x)=p_{\theta_{1}}(x_{1})\times\ldots\times p_{\theta_{d}}(x_{d})$
with $p_{\theta_{i}}(x_{i})=\theta_{i}^{x_{i}}(1-\theta_{i})^{1-x_{i}}$. As
this family is a product of probability measures $p_{\theta_{i}}(x_{i})$, the
different components of a random vector $y$ following $P_{\theta}$ are
independent and all off-diagonal terms of the Fisher information matrix (FIM)
are zero. Diagonal terms are given by $\frac{1}{\theta_{i}(1-\theta_{i})}$.
Therefore the inverse of the FIM is a diagonal matrix with diagonal entries
equal to $\theta_{i}(1-\theta_{i})$. In addition, the partial derivative of
$\ln P_{\theta}(x)$ w.r.t. $\theta_{i}$ can be computed in a straightforward
manner resulting in
$\frac{\partial\ln
P_{\theta}(x)}{\partial\theta_{i}}=\frac{x_{i}}{\theta_{i}}-\frac{1-x_{i}}{1-\theta_{i}}\enspace.$
Let $x_{1},\ldots,x_{N}$ be $N$ samples at step $t$ with distribution
$P_{\theta^{t}}$ and let $x_{1:N},\ldots,x_{N:N}$ be the ranked samples
according to $f$. The natural gradient update (15) with Bernoulli measures is
then given by
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}_{i}=\theta^{t}_{i}+\hskip
0.50003pt\delta t\hskip
0.50003pt\,\theta_{i}^{t}(1-\theta_{i}^{t})\sum_{j=1}^{N}w_{j}\left(\frac{[x_{j:N}]_{i}}{\theta_{i}^{t}}-\frac{1-[x_{j:N}]_{i}}{1-\theta_{i}^{t}}\right)$
(28)
where $w_{j}=w({(j-1/2)}/{N})/N$ and $[y]_{i}$ denotes the $i^{\text{th}}$
coordinate of $y\in X$. The previous equation simplifies to
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}_{i}=\theta^{t}_{i}+\hskip
0.50003pt\delta t\hskip
0.50003pt\,\sum_{j=1}^{N}w_{j}\left([x_{j:N}]_{i}-\theta_{i}^{t}\right)\enspace,$
or, denoting $\bar{w}$ the sum of the weights $\sum_{j=1}^{N}w_{j}$,
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}_{i}=(1-\bar{w}\hskip
0.50003pt\delta t\hskip 0.50003pt)\,\theta^{t}_{i}+\hskip 0.50003pt\delta
t\hskip 0.50003pt\,\sum_{j=1}^{N}w_{j}\,[x_{j:N}]_{i}\enspace.$
If we set the IGO weights as $w_{1}=1$, $w_{j}=0$ for $j=2,\ldots,N$, we
recover the PBIL/EGA algorithm with update rule towards the best solution only
(disregarding the mutation of the probability vector), with $\hskip
0.50003pt\delta t\hskip 0.50003pt={\rm LR}$ where ${\rm LR}$ is the so-called
learning rate of the algorithm [Bal94, Figure 4]. The PBIL update rule towards
the $\mu$ best solutions, proposed in [BC95, Figure 4]222Note that the
pseudocode for the algorithm in [BC95, Figure 4] is slightly erroneous since
it gives smaller weights to better individuals. The error can be fixed by
updating the probability in reversed order, looping from
NUMBER_OF_VECTORS_TO_UPDATE_FROM to 1. This was confirmed by S. Baluja in
personal communication. We consider here the corrected version of the
algorithm., can be recovered as well using
$\displaystyle\hskip 0.50003pt\delta t\hskip 0.50003pt$ $\displaystyle={\rm
LR}$ $\displaystyle w_{j}$ $\displaystyle=(1-{\rm LR})^{j-1},\text{ for
}j=1,\ldots,\mu$ $\displaystyle w_{j}$ $\displaystyle=0,\text{ for
}j=\mu+1,\ldots,N\enspace.$
Interestingly, the parameters $\theta_{i}$ are the expectation parameters
described in Section 3: indeed, the expectation of $x_{i}$ is $\theta_{i}$. So
the formulas above are particular cases of (27). Thus, by Theorem 15, PBIL is
both a smoothed CEM in these parameters and an IGO-ML algorithm.
Let us now consider another, so-called “logit” representation, given by the
logistic function $P(x_{i}=1)=\frac{1}{1+\exp(-{\tilde{\theta}}_{i})}$. This
${\tilde{\theta}}$ is the exponential parametrization of Section 2.3. We find
that
$\frac{\partial\ln
P_{{\tilde{\theta}}}(x)}{\partial{\tilde{\theta}}_{i}}=(x_{i}-1)+\frac{\exp(-{\tilde{\theta}}_{i})}{1+\exp(-{\tilde{\theta}}_{i})}=x_{i}-\mathbb{E}x_{i}$
(cf. (16)) and that the diagonal elements of the Fisher information matrix are
given by
$\exp(-{\tilde{\theta}}_{i})/(1+\exp(-{\tilde{\theta}}_{i}))^{2}=\operatorname{Var}x_{i}$
(compare with (17)). So the natural gradient update (15) with Bernoulli
measures now reads
${\tilde{\theta}}^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}_{i}={\tilde{\theta}}^{t}_{i}+\hskip 0.50003pt\delta t\hskip
0.50003pt(1+\exp({\tilde{\theta}}_{i}^{t}))\left(-\bar{w}+(1+\exp(-{\tilde{\theta}}_{i}^{t}))\sum_{j=1}^{N}w_{j}[x_{j:N}]_{i}\right)\enspace.$
To better compare the update with the previous representation, note that
$\theta_{i}=\frac{1}{1+\exp(-{\tilde{\theta}}_{i})}$ and thus we can rewrite
${\tilde{\theta}}^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt}_{i}={\tilde{\theta}}^{t}_{i}+\frac{\hskip 0.50003pt\delta t\hskip
0.50003pt}{\theta_{i}^{t}(1-\theta_{i}^{t})}\sum_{j=1}^{N}w_{j}\left([x_{j:N}]_{i}-\theta_{i}^{t}\right)\enspace.$
So the direction of the update is the same as before and is given by the
proportion of bits set to 0 or 1 in the sample, compared to its expected value
under the current distribution. The magnitude of the update is different since
the parameter ${\tilde{\theta}}$ ranges from $-\infty$ to $+\infty$ instead of
from $0$ to $1$. We did not find this algorithm in the literature.
These updates illustrate the influence of setting the sum of weights to $0$ or
not (Section 2.7). If, at some time, the first bit is set to 1 both for a
majority of good points and for a majority of bad points, then the original
PBIL will increase the probability of setting the first bit to $1$, which is
counterintuitive. If the weights $w_{i}$ are chosen to sum to $0$ this noise
effect disappears; otherwise, it disappears only on average.
### 4.2 Multivariate normal distributions (Gaussians)
Evolution strategies [Rec73, Sch95, BS02] are black-box optimization
algorithms for the _continuous_ search domain, $X\subseteq{\mathbb{R}}^{d}$
(for simplicity we assume $X={\mathbb{R}}^{d}$ in the following). They sample
new solutions from a multivariate normal distribution. In the context of
continuous black-box optimization, _Natural Evolution Strategies_ (NES)
introduced the idea of using a natural gradient update of the distribution
parameters [WSPS08, SWSS09, GSS+10]. Surprisingly, also the well-known
_Covariance Matrix Adaption Evolution Strategy_ , CMA-ES [HO96, HO01, HMK03,
HK04, JA06], turns out to conduct a natural gradient update of distribution
parameters [ANOK10, GSS+10].
Let $x\in{\mathbb{R}}^{d}$. As the most prominent example, we use mean vector
$m=\mathbb{E}x$ and covariance matrix
$C=\mathbb{E}(x-m)(x-m)^{\mathrm{T}}=\mathbb{E}(xx^{\mathrm{T}})-mm^{\mathrm{T}}$
to parametrize the distribution via $\theta=(m,C)$. The IGO update in (14) or
(15) depends on the chosen parametrization, but can now be entirely
reformulated without the (inverse) Fisher matrix, compare (18). The complexity
of the update is linear in the number of parameters (size of $\theta=(m,C)$,
where $(d^{2}-d)/2$ parameters are redundant). We discuss known algorithms
that implement variants of this update.
##### CMA-ES.
The CMA-ES implements the equations333The CMA-ES implements these equations
given the parameter setting $c_{1}=0$ and $c_{\sigma}=0$ (or
$d_{\sigma}=\infty$, see e.g. [Han09]) that disengages the effect of both so-
called evolution paths.
$\displaystyle m^{t+1}$
$\displaystyle=m^{t}+\eta_{\mathrm{m}}\sum_{i=1}^{N}\widehat{w}_{i}(x_{i}-m^{t})$
(29) $\displaystyle C^{t+1}$
$\displaystyle=C^{t}+\eta_{\mathrm{c}}\sum_{i=1}^{N}\widehat{w}_{i}((x_{i}-m^{t})(x_{i}-m^{t})^{T}-C^{t})$
(30)
where $\widehat{w}_{i}$ are the weights based on ranked $f$-values, see (12)
and (14).
When $\eta_{\mathrm{c}}=\eta_{\mathrm{m}}$, Equations (30) and (29) coincide
with the IGO update (14) expressed in the parametrization $(m,C)$ [ANOK10,
GSS+10]444 In these articles the result has been derived for
$\theta\leftarrow\theta+\eta\,\widetilde{\nabla}_{\\!\theta}\mathbb{E}_{{P_{\theta}}}f$,
see (9), leading to $f(x_{i})$ in place of $\widehat{w}_{i}$. No assumptions
on $f$ have been used besides that it does not depend on $\theta$. By
replacing $f$ with $W_{{\theta^{t}}}^{f}$, where ${\theta^{t}}$ is fixed, the
derivation holds equally well for
$\theta\leftarrow\theta+\eta\,\widetilde{\nabla}_{\\!\theta}\mathbb{E}_{{P_{\theta}}}W_{{\theta^{t}}}^{f}$..
Note, however, that the learning rates $\eta_{\mathrm{m}}$ and
$\eta_{\mathrm{c}}$ take essentially different values in CMA-ES, if
$N\ll\dim\Theta$555Specifically, let $\sum|\widehat{w}_{i}|=1$, then
$\eta_{\mathrm{m}}=1$ and $\eta_{\mathrm{c}}\approx 1\wedge
1/(d^{2}\sum\widehat{w}_{i}^{2})$. : this is in deviation from an IGO
algorithm. (Remark that the Fisher information matrix is block-diagonal in $m$
and $C$ [ANOK10], so that application of the different learning rates and of
the inverse Fisher matrix commute.)
##### Natural evolution strategies.
Natural evolution strategies (NES) [WSPS08, SWSS09] implement (29) as well,
but use a Cholesky decomposition of $C$ as parametrization for the update of
the variance parameters. The resulting update that replaces (30) is neither
particularly elegant nor numerically efficient. However, the most recent xNES
[GSS+10] chooses an “exponential” parametrization depending on the current
parameters. This leads to an elegant formulation where the additive update in
exponential parametrization becomes a multiplicative update for $C$ in (30).
With $C=AA^{\mathrm{T}}$, the matrix update reads
$A\leftarrow
A\times\exp\left(\frac{\eta_{\mathrm{c}}}{2}\sum_{i=1}^{N}\widehat{w}_{i}(A^{-1}(x_{i}-m)(A^{-1}(x_{i}-m))^{T}-\mathrm{I}_{d})\right)$
(31)
where $\mathrm{I}_{d}$ is the identity matrix. (From (31) we get $C\leftarrow
A\times\exp^{2}(\dots)\times A^{\mathrm{T}}$.) Remark that in the
representation
$\theta=(A^{-1}m,A^{-1}C{A^{\mathrm{T}}}^{-1})=(A^{-1}m,\mathrm{I}_{d})$, the
Fisher information matrix becomes diagonal.
The update has the advantage over (30) that even negative weights always lead
to feasible values. By default, $\eta_{\mathrm{m}}\not=\eta_{\mathrm{c}}$ in
xNES in the same circumstances as in CMA-ES (most parameter settings are
borrowed from CMA-ES), but contrary to CMA-ES the past evolution path is not
taken into account [GSS+10].
When $\eta_{\mathrm{c}}=\eta_{\mathrm{m}}$, xNES is consistent with the IGO
flow (6), and implements a slightly generalized IGO algorithm (14) using a
$\theta$-dependent parametrization.
##### Cross-entropy method and EMNA.
_Estimation of distribution algorithms_ (EDA) and the _cross-entropy method_
(CEM) [Rub99, RK04] estimate a new distribution from a censored sample.
Generally, the new parameter value can be written as
$\displaystyle\theta_{\mathrm{maxLL}}$
$\displaystyle=\arg\max_{\theta}\sum_{i=1}^{N}\widehat{w}_{i}\ln{P_{\theta}}(x_{i})$
(32)
$\displaystyle\longrightarrow_{N\to\infty}\arg\max_{\theta}\mathbb{E}_{P_{{\theta^{t}}}}W_{{\theta^{t}}}^{f}\ln{P_{\theta}}$
For positive weights, $\theta_{\mathrm{maxLL}}$ maximizes the weighted log-
likelihood of $x_{1}\dots x_{N}$. The argument under the $\arg\max$ in the RHS
of (32) is the negative cross-entropy between $P_{\theta}$ and the
distribution of censored (elitist) samples
$P_{{\theta^{t}}}W_{{\theta^{t}}}^{f}$ or of $N$ realizations of such samples.
The distribution $P_{\theta_{\mathrm{maxLL}}}$ has therefore minimal cross-
entropy and minimal KL-divergence to the distribution of the $\mu$ best
samples.666Let $P_{w}$ denote the distribution of the weighted samples:
$\mathrm{Pr}(x=x_{i})=\widehat{w}_{i}$ and $\sum_{i}\widehat{w}_{i}=1$. Then
the cross-entropy between $P_{w}$ and ${P_{\theta}}$ reads
$\sum_{i}P_{w}(x_{i})\ln 1/{P_{\theta}}(x_{i})$ and the KL-divergence reads
$\mathrm{KL}\\!\left(P_{w}\,||\,{P_{\theta}}\right)=\sum_{i}P_{w}(x_{i})\ln
1/{P_{\theta}}(x_{i})-\sum_{i}P_{w}(x_{i})\ln 1/P_{w}(x_{i})$. Minimization of
both terms in $\theta$ result in $\theta_{\mathrm{maxLL}}$. As said above,
(32) recovers the _cross-entropy method_ (CEM) [Rub99, RK04].
For Gaussian distributions, equation (32) can be explicitly written in the
form
$\displaystyle m^{t+1}$
$\displaystyle=m^{\ast}=\sum_{i=1}^{N}\widehat{w}_{i}x_{i}$ (33)
$\displaystyle C^{t+1}$
$\displaystyle=C^{\ast}=\sum_{i=1}^{N}\widehat{w}_{i}(x_{i}-m^{\ast})(x_{i}-m^{\ast})^{\mathrm{T}}$
(34)
the empirical mean and variance of the elite sample. The weights $\hat{w}_{i}$
are equal to $1/\mu$ for the $\mu$ best points and $0$ otherwise.
Equations (33) and (34) also define the most fundamental continuous domain
EDA, the _estimation of multivariate normal algorithm_ (EMNAglobal, [LL02]).
It might be interesting to notice that (33) and (34) only differ from (29) and
(30) in that the new mean $m^{t+1}$ is used in the covariance matrix update.
Let us compare IGO-ML (23), CMA (29)–(30), and smoothed CEM (25) in the
parametrization with mean and covariance matrix. For learning rate $\hskip
0.50003pt\delta t\hskip 0.50003pt=1$, IGO-ML and smoothed CEM/EMNA realize
$\theta_{\mathrm{maxLL}}$ from (32)–(34). For $\hskip 0.50003pt\delta t\hskip
0.50003pt<1$ these algorithms all update the distribution mean in the same
way; the update of mean and covariance matrix is computed to be
$\displaystyle m^{t+1}$ $\displaystyle=(1-\hskip 0.50003pt\delta t\hskip
0.50003pt)\,m^{t}+\hskip 0.50003pt\delta t\hskip 0.50003pt\,m^{*}$ (35)
$\displaystyle C^{t+1}$ $\displaystyle=(1-\hskip 0.50003pt\delta t\hskip
0.50003pt)\,C^{t}+\hskip 0.50003pt\delta t\hskip 0.50003pt\,C^{\ast}+\hskip
0.50003pt\delta t\hskip 0.50003pt(1-\hskip 0.50003pt\delta t\hskip
0.50003pt)^{j}\,(m^{*}-m^{t})(m^{*}-m^{t})^{\mathrm{T}},$
for different values of $j$, where $m^{\ast}$ and $C^{\ast}$ are the mean and
covariance matrix computed over the elite sample (with weights
$\widehat{w}_{i}$) as above. For CMA we have $j=0$, for IGO-ML we have $j=1$,
and for smoothed CEM we have $j=\infty$ (the rightmost term is absent). The
case $j=2$ corresponds to an update that uses $m^{t+1}$ instead of $m^{t}$ in
(30) (compare [Han06b]). For $0<\hskip 0.50003pt\delta t\hskip 0.50003pt<1$,
the larger $j$, the smaller $C^{t+1}$.
The rightmost term of (35) resembles the so-called rank-one update in CMA.
For $\hskip 0.50003pt\delta t\hskip 0.50003pt\to 0$, the update is independent
of $j$ at first order in $\hskip 0.50003pt\delta t\hskip 0.50003pt$ if
$j<\infty$: this reflects compatibility with the IGO flow of CMA and of IGO-
ML, but not of smoothed CEM.
##### Critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$.
Let us assume that the weights $\widehat{w}_{i}$ are non-negative, sum to one,
and $\mu<N$ weights take a value of $1/\mu$, so that the selection quantile is
$q=\mu/N$.
There is a critical value of $\hskip 0.50003pt\delta t\hskip 0.50003pt$
depending on this quantile $q$, such that above the critical $\hskip
0.50003pt\delta t\hskip 0.50003pt$ the algorithms given by IGO-ML and smoothed
CEM are prone to premature convergence. Indeed, let $f$ be a linear function
on ${\mathbb{R}}^{d}$, and consider the variance in the direction of the
gradient of $f$. Assuming further $N\to\infty$ and $q\leq 1/2$, then the
variance $C^{\ast}$ of the elite sample is smaller than the current variance
$C^{t}$, by a constant factor. Depending on the precise update for $C^{t+1}$,
if $\hskip 0.50003pt\delta t\hskip 0.50003pt$ is too large, the variance
$C^{t+1}$ is going to be smaller than $C^{t}$ by a constant factor as well.
This implies that the algorithm is going to stall. (On the other hand, the
continuous-time IGO flow corresponding to $\hskip 0.50003pt\delta t\hskip
0.50003pt\to 0$ does not stall, see Section 4.3.)
We now study the critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$ under
which the algorithm does not stall. For IGO-ML, ($j=1$ in (35), or
equivalently for the smoothed CEM in the expectation parameters
$(m,C+mm^{\mathrm{T}})$, see Section 3), the variance increases if and only if
$\hskip 0.50003pt\delta t\hskip 0.50003pt$ is smaller than the critical value
$\hskip 0.50003pt\delta t\hskip
0.50003pt_{\text{crit}}=qb\sqrt{2\pi}e^{b^{2}/2}{}$ where $b$ is the
percentile function of $q$, i.e. $b$ is such that
$q=\int_{b}^{\infty}e^{-x^{2}/2}/\sqrt{2\pi}{}$. This value $\hskip
0.50003pt\delta t\hskip 0.50003pt_{\text{crit}}$ is plotted as solid line in
Fig. 1. For $j=2$, $\hskip 0.50003pt\delta t\hskip 0.50003pt_{\text{crit}}$ is
smaller, related to the above by $\hskip 0.50003pt\delta t\hskip
0.50003pt_{\text{crit}}\leftarrow\sqrt{1+\hskip 0.50003pt\delta t\hskip
0.50003pt_{\text{crit}}}-1$ and plotted as dashed line in Fig. 1. For CEM
($j=\infty$), the critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$ is zero.
For CMA-ES ($j=0$), the critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$ is
infinite for $q<1/2$. When the selection quantile $q$ is above $1/2$, for all
algorithms the critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$ becomes
zero.
Figure 1: Critical $\hskip 0.50003pt\delta t\hskip 0.50003pt$ versus
truncation quantile $q$. Above the critical $\hskip 0.50003pt\delta t\hskip
0.50003pt$, the variance decreases systematically when optimizing a linear
function. For CMA-ES/NES, the critical $\hskip 0.50003pt\delta t\hskip
0.50003pt$ for $q<0.5$ is infinite.
We conclude that, despite the principled approach of ascending the natural
gradient, the choice of the weighting function $w$, the choice of $\hskip
0.50003pt\delta t\hskip 0.50003pt$, and possible choices in the update for
$\hskip 0.50003pt\delta t\hskip 0.50003pt>0$, need to be taken with great care
in relation to the choice of parametrization.
### 4.3 Computing the IGO flow for some simple examples
In this section we take a closer look at the IGO flow solutions of (6) for
some simple examples of fitness functions, for which it is possible to obtain
exact information about these IGO trajectories.
We start with the discrete search space $X=\\{0,1\\}^{d}$ and linear functions
(to be minimized) defined as $f(x)=c-\sum_{i=1}^{d}\alpha_{i}x_{i}$ with
$\alpha_{i}>0$. Note that the onemax function to be maximized $f_{\rm
onemax}(x)=\sum_{i=1}^{d}x_{i}$ is covered by setting $\alpha_{i}=1$. The
differential equation (6) for the Bernoulli measures
$P_{\theta}(x)=p_{\theta_{1}}(x_{1})\ldots p_{\theta_{d}}(x_{d})$ defined on
$X$ can be derived taking the limit of the IGO-PBIL update given in (28):
$\frac{{\mathrm{d}}\theta_{i}^{t}}{{\mathrm{d}}t}=\int
W_{\theta^{t}}^{f}(x)(x_{i}-\theta_{i}^{t})P_{\theta^{t}}(dx)=:g_{i}(\theta^{t})\enspace.$
(36)
Though finding the analytical solution of the differential equation (36) for
any initial condition seems a bit intricate we show that the equation admits
one critical stable point, $(1,\ldots,1)$, and one critical unstable point
$(0,\ldots,0)$. In addition we prove that the trajectory decreases along $f$
in the sense $\frac{{\mathrm{d}}f(\theta^{t})}{{\mathrm{d}}t}\leqslant 0$. To
do so we establish the following result:
###### Lemma 17.
On $f(x)=c-\sum_{i=1}^{d}\alpha_{i}x_{i}$ the solution of (36) satisfies
$\sum_{i=1}^{d}\alpha_{i}\frac{{\mathrm{d}}\theta_{i}}{{\mathrm{d}}t}\geqslant
0$; moreover $\sum\alpha_{i}\frac{{\mathrm{d}}\theta_{i}}{{\mathrm{d}}t}=0$ if
and only if $\theta=(0,\ldots,0)$ or $\theta=(1,\ldots,1)$.
###### Proof.
We compute $\sum_{i=1}^{d}\alpha_{i}g_{i}(\theta^{t})$ and find that
$\displaystyle\sum_{i=1}^{d}\alpha_{i}\frac{{\mathrm{d}}\theta_{i}^{t}}{{\mathrm{d}}t}$
$\displaystyle=\int
W_{\theta^{t}}^{f}(x)\left(\sum_{i=1}^{d}\alpha_{i}x_{i}-\sum_{i=1}^{d}\alpha_{i}\theta_{i}^{t}\right)P_{\theta^{t}}({\mathrm{d}}x)$
$\displaystyle=\int
W_{\theta^{t}}^{f}(x)(f(\theta^{t})-f(x))P_{\theta^{t}}({\mathrm{d}}x)$
$\displaystyle=\mathbb{E}[W_{\theta^{t}}^{f}(x)]\,\mathbb{E}[f(x)]-\mathbb{E}[W_{\theta^{t}}^{f}(x)f(x)]$
In addition, $-W_{\theta^{t}}^{f}(x)=-w(\Pr(f(x^{\prime})<f(x)))$ is a
nondecreasing bounded function in the variable $f(x)$ such that
$-W_{\theta^{t}}^{f}(x)$ and $f(x)$ are positively correlated (see [Tho00,
Chapter 1] for a proof of this result), i.e.
$\mathbb{E}[-W_{\theta^{t}}^{f}(x)f(x)]\geqslant\mathbb{E}[-W_{\theta^{t}}^{f}(x)]\mathbb{E}[f(x)]$
with equality if and only if $\theta^{t}=(0,\ldots,0)$ or
$\theta^{t}=(1,\ldots,1)$. Thus
$\sum_{i=1}^{d}\alpha_{i}\frac{{\mathrm{d}}\theta_{i}^{t}}{{\mathrm{d}}t}\geqslant
0$. ∎
The previous result implies that the positive definite function
$V(\theta)=\sum_{i=1}^{d}\alpha_{i}-\sum_{i=1}^{d}\alpha_{i}\theta_{i}$ in
$(1,\ldots,1)$ satisfies $V^{*}(\theta)=\nabla V(\theta)\cdot
g(\theta)\leqslant 0$ (such a function is called a Lyapunov function).
Consequently $(1,\ldots,1)$ is stable. Similarly
$V(\theta)=\sum_{i=1}^{d}\alpha_{i}\theta_{i}$ is a Lyapunov function for
$(0,\ldots,0)$ such that $\nabla V(\theta)\cdot g(\theta)\geqslant 0$.
Consequently $(0,\ldots,0)$ is unstable [AO08].
We now consider on ${\mathbb{R}}^{d}$ the family of multivariate normal
distributions $P_{\theta}=\mathcal{N}(m,\sigma^{2}I_{d})$ with covariance
matrix equal to $\sigma^{2}I_{d}$. The parameter $\theta$ thus has $d+1$
components $\theta=(m,\sigma)\in{\mathbb{R}}^{d}\times{\mathbb{R}}$. The
natural gradient update using this family was derived in [GSS+10]; from it we
can derive the IGO differential equation which reads:
$\displaystyle\frac{{\mathrm{d}}m^{t}}{{\mathrm{d}}t}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}W_{\theta^{t}}^{f}(x)(x-m^{t})p_{\mathcal{N}(m^{t},(\sigma^{t})^{2}I_{d})}(x)dx$
(37) $\displaystyle\frac{{\mathrm{d}}\tilde{\sigma}^{t}}{{\mathrm{d}}t}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\frac{1}{2d}\left\\{\sum_{i=1}^{d}\left(\frac{x_{i}-m^{t}_{i}}{\sigma^{t}}\right)^{2}-1\right\\}W_{\theta^{t}}^{f}(x)p_{\mathcal{N}(m^{t},(\sigma^{t})^{2}I_{d})}(x)dx$
(38)
where $\sigma^{t}$ and $\tilde{\sigma}^{t}$ are linked via
$\sigma^{t}=\exp(\tilde{\sigma}^{t})$ or $\tilde{\sigma}^{t}=\ln(\sigma^{t})$.
Denoting $\mathcal{N}$ a random vector following a centered multivariate
normal distribution with covariance matrix identity we write equivalently the
gradient flow as
$\displaystyle\frac{{\mathrm{d}}m^{t}}{{\mathrm{d}}t}$
$\displaystyle=\sigma^{t}\mathbb{E}\left[W_{\theta^{t}}^{f}(m^{t}+\sigma^{t}\mathcal{N})\mathcal{N}\right]$
(39) $\displaystyle\frac{{\mathrm{d}}\tilde{\sigma}^{t}}{{\mathrm{d}}t}$
$\displaystyle=\mathbb{E}\left[\frac{1}{2}\left(\frac{\|\mathcal{N}\|^{2}}{d}-1\right)W_{\theta^{t}}^{f}(m^{t}+\sigma^{t}\mathcal{N})\right]\enspace.$
(40)
Let us analyze the solution of the previous system on linear functions.
Without loss of generality (because of invariance) we can consider the linear
function $f(x)=x_{1}$. We have
$W_{\theta^{t}}^{f}(x)=w(\Pr(m^{t}_{1}+\sigma^{t}Z_{1}<x_{1}))$
where $Z_{1}$ follows a standard normal distribution and thus
$\displaystyle W_{\theta^{t}}^{f}(m^{t}+\sigma^{t}\mathcal{N})$
$\displaystyle=w(\Pr_{Z_{1}\sim\mathcal{N}(0,1)}(Z_{1}<\mathcal{N}_{1}))$ (41)
$\displaystyle=w(\mathcal{F}(\mathcal{N}_{1}))$ (42)
with $\mathcal{F}$ the cumulative distribution of a standard normal
distribution. The differential equation thus simplifies into
$\displaystyle\frac{{\mathrm{d}}m^{t}}{{\mathrm{d}}t}$
$\displaystyle=\sigma^{t}\left(\begin{array}[]{c}\mathbb{E}\left[w(\mathcal{F}(\mathcal{N}_{1}))\mathcal{N}_{1}\right]\\\
0\\\ \vdots\\\ 0\end{array}\right)$ (47)
$\displaystyle\frac{{\mathrm{d}}\tilde{\sigma}^{t}}{{\mathrm{d}}t}$
$\displaystyle=\frac{1}{2d}\mathbb{E}\left[(|\mathcal{N}_{1}|^{2}-1)w(\mathcal{F}(\mathcal{N}_{1}))\right]\enspace.$
(48)
Consider, for example, the weight function associated with truncation
selection, i.e. $w(q)=1_{q\leqslant q_{0}}$ where $q_{0}\in]0,1]$—also called
intermediate recombination. We find that
$\displaystyle\frac{{\mathrm{d}}m^{t}_{1}}{{\mathrm{d}}t}$
$\displaystyle=\sigma^{t}\mathbb{E}[\mathcal{N}_{1}1_{\\{\mathcal{N}_{1}\leqslant\mathcal{F}^{-1}(q_{0})\\}}]=:\sigma^{t}\beta$
(49) $\displaystyle\frac{{\mathrm{d}}\tilde{\sigma}^{t}}{{\mathrm{d}}t}$
$\displaystyle=\frac{1}{2d}\left(\int_{0}^{q_{0}}\mathcal{F}^{-1}(u)^{2}du-
q_{0}\right)=:\alpha\enspace.$ (50)
The solution of the IGO flow for the linear function $f(x)=x_{1}$ is thus
given by
$\displaystyle m^{t}_{1}$
$\displaystyle=m^{t}_{0}+\frac{\sigma^{0}\beta}{\alpha}\exp(\alpha t)$ (51)
$\displaystyle\sigma^{t}$ $\displaystyle=\sigma^{0}\exp(\alpha t)\enspace.$
(52)
The coefficient $\beta$ is strictly negative for any $q_{0}<1$. The
coefficient $\alpha$ is strictly positive if and only if $q_{0}<1/2$ which
corresponds to selecting less than half of the sampled points in an ES. In
this case the step-size $\sigma^{t}$ grows exponentially fast to infinity and
the mean vectors moves along the gradient direction towards minus $\infty$ at
the same rate. If more than half of the points are selected, $q_{0}\geqslant
1/2$, the step-size will decrease to zero exponentially fast and the mean
vector will get stuck (compare also [Han06a]).
## 5 Multimodal optimization using restricted Boltzmann machines
We now illustrate experimentally the influence of the natural gradient, versus
vanilla gradient, on diversity over the course of optimization. We consider a
very simple situation of a fitness function with two distant optima and test
whether different algorithms are able to reach both optima simultaneously or
only converge to one of them. This provides a practical test of Proposition 1
stating that the natural gradient minimizes loss of diversity.
The IGO method allows to build a natural search algorithm from an arbitrary
probability distribution on an arbitrary search space. In particular, by
choosing families of probability distributions that are richer than Gaussian
or Bernoulli, one may hope to be able to optimize functions with complex
shapes. Here we study how this might help optimize multimodal functions.
Both Gaussian distributions on ${\mathbb{R}}^{d}$ and Bernoulli distributions
on $\\{0,1\\}^{d}$ are unimodal. So at any given time, a search algorithm
using such distributions concentrates around a given point in the search
space, looking around that point (with some variance). It is an often-sought-
after feature for an optimization algorithm to handle multiple hypotheses
simultaneously.
In this section we apply our method to a multimodal distribution on
$\\{0,1\\}^{d}$: restricted Boltzmann machines (RBMs). Depending on the
activation state of the _latent variables_ , values for various blocks of bits
can be switched on or off, hence multimodality. So the optimization algorithm
derived from these distributions will, hopefully, explore several distant
zones of the search space at any given time. A related model (Boltzmann
machines) was used in [Ber02] and was found to perform better than PBIL on
some functions.
Our study of a bimodal RBM distribution for the optimization of a bimodal
function confirms that the natural gradient does indeed behave in a more
natural way than the vanilla gradient: when initialized properly, the natural
gradient is able to maintain diversity by fully using the RBM distribution to
learn the two modes, while the vanilla gradient only converges to one of the
two modes.
Although these experiments support using a natural gradient approach, they
also establish that complications can arise for estimating the inverse Fisher
matrix in the case of complex distributions such as RBMs: estimation errors
may lead to a singular or unreliable estimation of the Fisher matrix,
especially when the distribution becomes singular. Further research is needed
for a better understanding of this issue.
The experiments reported here, and the fitness function used, are extremely
simple from the optimization viewpoint: both algorithms using the natural and
vanilla gradient find an optimum in only a few steps. The emphasis here is on
the specific influence of replacing the vanilla gradient with the natural
gradient, and the resulting influence on diversity and multimodality, in a
simple situation.
### 5.1 IGO for restricted Boltzmann machines
##### Restricted Boltzmann machines.
Figure 2: The RBM architecture with the observed ($\mathbf{x}$) and latent
($\mathbf{h}$) variables. In our experiments, a single hidden unit was used.
A restricted Boltzmann machine (RBM) [Smo86, AHS85] is a kind of probability
distribution belonging to the family of undirected graphical models (also
known as a Markov random fields). A set of observed variables
$\mathbf{x}\in\\{0,1\\}^{n_{x}}$ are given a probability using their joint
distribution with unobserved latent variables $\mathbf{h}\in\\{0,1\\}^{n_{h}}$
[Gha04]. The latent variables are then marginalized over. See Figure 2 for the
graph structure of a RBM.
The probability associated with an observation $\mathbf{x}$ and latent
variable $\mathbf{h}$ is given by
$P_{\theta}(\mathbf{x},\mathbf{h})=\frac{e^{-E(\mathbf{x},\mathbf{h})}}{\sum_{\mathbf{x^{\prime}},\mathbf{h^{\prime}}}e^{-E(\mathbf{x^{\prime}},\mathbf{h^{\prime}})}}\,,\qquad
P_{\theta}(\mathbf{x})=\sum_{\mathbf{h}}P_{\theta}(\mathbf{x},\mathbf{h}).$
(53)
where $E(\mathbf{x},\mathbf{h})$ is the so-called energy function and
$\sum_{\mathbf{x^{\prime}},\mathbf{h^{\prime}}}e^{-E(\mathbf{x^{\prime}},\mathbf{h^{\prime}})}$
is the partition function denoted $Z$ in Section 2.3. The energy $E$ is
defined by
$E(\mathbf{x},\mathbf{h})=-\sum_{i}a_{i}x_{i}-\sum_{j}b_{j}h_{j}-\sum_{i,j}w_{ij}x_{i}h_{j}\;.$
(54)
The distribution is fully parametrized by the bias on the observed variables
$\mathbf{a}$, the bias on the latent variables $\mathbf{b}$ and the weights
$\mathbf{W}$ which account for pairwise interactions between observed and
latent variables: $\theta=(\mathbf{a},\mathbf{b},\mathbf{W})$.
RBM distributions are a special case of exponential family distributions with
latent variables (see (21) in Section 2.3). The RBM IGO equations stem from
Equations (16), (17) and (18) by identifying the statistics $T(x)$ with
$x_{i}$, $h_{j}$ or $x_{i}h_{j}$.
For these distributions, the gradient of the log-likelihood is well-known
[Hin02]. Although it is often considered intractable in the context of machine
learning where a lot of variables are required, it becomes tractable for
smaller RBMs. The gradient of the log-likelihood consists of the difference of
two expectations (compare (16)):
$\displaystyle\frac{\partial\ln P_{\theta}(x)}{\partial w_{ij}}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{P_{\theta}}[x_{i}h_{j}|x]-\mathbb{E}_{P_{\theta}}[x_{i}h_{j}]$
(55)
The Fisher information matrix is given by (20)
$I_{w_{ab}w_{cd}}(\theta)=\mathbb{E}[x_{a}h_{b}x_{c}h_{d}]-\mathbb{E}[x_{a}h_{b}]\mathbb{E}[x_{c}h_{d}]\\\
$ (56)
where $I_{w_{ab}w_{cd}}$ denotes the entry of the Fisher matrix corresponding
to the components $w_{ab}$ and $w_{cd}$ of the parameter $\theta$. These
equations are understood to encompass the biases $\mathbf{a}$ and $\mathbf{b}$
by noticing that the bias can be replaced in the model by adding two variables
$x_{k}$ and $h_{k}$ always equal to one.
Finally, the IGO update rule is taken from (14):
$\theta^{t+\delta t}=\theta^{t}+\delta
t\>I^{-1}(\theta^{t})\sum_{k=1}^{N}\left.\widehat{w}_{k}\frac{\partial\ln
P_{\theta}(\mathbf{x}_{k})}{\partial\theta}\right|_{\theta=\theta^{t}}$ (57)
##### Implementation.
In this experimental study, the IGO algorithm for RBMs is directly implemented
from Equation (57). At each optimization step, the algorithm consists in (1)
finding a reliable estimate of the Fisher matrix (see Eq. 56) which is then
inverted using the QR-Algorithm if it is invertible; (2) computing an estimate
of the vanilla gradient which is then weighted according to $W_{\theta}^{f}$;
and (3) updating the parameters, taking into account the gradient step size
$\delta t$. In order to estimate both the Fisher matrix and the gradient,
samples must be drawn from the model ${P_{\theta}}$. This is done using Gibbs
sampling (see for instance [Hin02]).
##### Fisher matrix imprecision.
The imprecision incurred by the limited sampling size may sometimes lead to a
singular estimation of the Fisher matrix (see p. 1.2 for a lower bound on the
number of samples needed). Although having a singular Fisher estimation
happens rarely in normal conditions, it occurs with certainty when the
probabilities become too concentrated over a few points. This situation arises
naturally when the algorithm is allowed to continue the optimization after the
optimum has been reached. For this reason, in our experiments, we stop the
optimization as soon as both optima have been sampled, thus preventing
${P_{\theta}}$ from becoming too concentrated.
In a variant of the same problem, the Fisher estimation can become close to
singular while still being numerically invertible. This situation leads to
unreliable estimates and should therefore be avoided. To evaluate the
reliability of the inverse Fisher matrix estimate, we use a cross-validation
method: (1) making two estimates $\hat{F}_{1}$ and $\hat{F}_{2}$ of the Fisher
matrix on two distinct sets of points generated from $P_{\theta}$, and (2)
making sure that the eigenvalues of $\hat{F}_{1}\times\hat{F}_{2}^{-1}$ are
close to $1$. In practice, at all steps we check that the average of the
eigenvalues of $\hat{F}_{1}\times\hat{F}_{2}^{-1}$ is between $1/2$ and $2$.
If at some point during the optimization the Fisher estimation becomes
singular or unreliable according to this criterion, the corresponding run is
stopped and considered to have failed.
##### When optimizing on a larger space helps.
A restricted Boltzmann machine defines naturally a distribution $P_{\theta}$
on both visible and hidden units $(\mathbf{x},\mathbf{h})$, whereas the
function to optimize depends only on the visible units $\mathbf{x}$. Thus we
are faced with a choice. A first possibility is to decide that the objective
function $f(\mathbf{x})$ is really a function of $(\mathbf{x},\mathbf{h})$
where $\mathbf{h}$ is a dummy variable; then we can use the IGO algorithm to
optimize over $(\mathbf{x},\mathbf{h})$ using the distributions
$P_{\theta}(\mathbf{x},\mathbf{h})$. A second possibility is to marginalize
$P_{\theta}(\mathbf{x},\mathbf{h})$ over the hidden units $\mathbf{h}$ as in
(53), to define the distribution $P_{\theta}(\mathbf{x})$; then we can use the
IGO algorithm to optimize over $\mathbf{x}$ using $P_{\theta}(\mathbf{x})$.
These two approaches yield slightly different algorithms. Both were tested and
found to be viable. However the first approach is numerically more stable and
requires less samples to estimate the Fisher matrix. Indeed, if
$I_{1}(\theta)$ is the Fisher matrix at $\theta$ in the first approach and
$I_{2}(\theta)$ in the second approach, we always have $I_{1}(\theta)\geqslant
I_{2}(\theta)$ (in the sense of positive-definite matrices). This is because
probability distributions on the pair $(\mathbf{x},\mathbf{h})$ carry more
information than their projections on $\mathbf{x}$ only, and so computed
Kullback–Leibler distances will be larger.
In particular, there are (isolated) values of $\theta$ for which the Fisher
matrix $I_{2}$ is not invertible whereas $I_{1}$ is. For this reason, we
selected the first approach.
### 5.2 Experimental results
In our experiments, we look at the optimization of the two-min function
defined below with a bimodal RBM: an RBM with only one latent variable. Such
an RBM is bimodal because it has two possible configurations of the latent
variable: $\mathbf{h}=0$ or $\mathbf{h}=1$, and given $\mathbf{h}$, the
observed variables are independent and distributed according to two Bernoulli
distributions.
Set a parameter $\mathbf{y}\in\\{0,1\\}^{d}$. The two-min function is defined
as follows:
$f_{\mathbf{y}}(\mathbf{x})=\min\left(\sum_{i}\left|x_{i}-y_{i}\right|,\sum_{i}\left|(1-x_{i})-y_{i})\right|\right)$
(58)
This function of $\mathbf{x}$ has two optima: one at $\mathbf{y}$, the other
at its binary complement $\mathbf{\bar{y}}$.
For the quantile rewriting of $f$ (Section 1.2), we chose the function $w$ to
be $w(q)=\mathbbm{1}_{q\leqslant 1/2}$ so that points which are below the
median are given the weight $1$, whereas other points are given the weight
$0$. Also, in accordance with (3), if several points have the same fitness
value, their weight $W_{\theta}^{f}$ is set to the average of $w$ over all
those points.
For initialization of the RBMs, the weights $\mathbf{W}$ are sampled from a
normal distribution centered around zero and of standard deviation
$1/\sqrt{n_{x}\times n_{h}}$, where $n_{x}$ is the number of observed
variables (dimension $d$ of the problem) and $n_{h}$ is the number of latent
variables ($n_{h}=1$ in our case), so that initially the energies $E$ are not
too large. Then the bias parameters are initialized as
$b_{j}\leftarrow-\sum_{i}\frac{w_{ij}}{2}$ and
$a_{i}\leftarrow-\sum_{j}\frac{w_{ij}}{2}+\mathcal{N}(\frac{0.01}{n_{x}^{2}})$
so that each variable (observed or latent) has a probability of activation
close to $1/2$.
In the following experiments, we show the results of IGO optimization and
vanilla gradient optimization for the two-min function in dimension $40$, for
various values of the step size $\delta t$. For each $\delta t$, we present
the median of the quantity of interest over $100$ runs. Error bars indicate
the 16th percentile and the 84th percentile (this is the same as
mean$\pm$stddev for a Gaussian variable, but is invariant by
$f$-reparametrization). For each run, the parameter $\mathbf{y}$ of the two-
max function is sampled randomly in order to ensure that the presented results
are not dependent on a particular choice of optima.
The number of sample points used for estimating the Fisher matrix is set to
$10,000$: large enough (by far) to ensure the stability of the estimates. The
same points are used for estimating the integral of (14), therefore there are
$10,000$ calls to the fitness function at each gradient step. These rather
comfortable settings allow for a good illustration of the theoretical
properties of the $N=\infty$ IGO flow limit.
The numbers of runs that fail after the occurrence of a singular matrix or an
unreliable estimate amount for less than $10\%$ for $\delta t\leqslant 2$ (as
little as $3\%$ for the smallest learning rate), but can increase up to $30\%$
for higher learning rates.
#### 5.2.1 Convergence
We first check that both vanilla and natural gradient are able to converge to
an optimum.
Figures 3 and 4 show the fitness of the best sampled point for the IGO
algorithm and for the vanilla gradient at each step. Predictably, both
algorithms are able to optimize the two-min function in a few steps.
The two-min function is extremely simple from an optimization viewpoint; thus,
convergence speed is not the main focus here, all the more since we use a
large number of $f$-calls at each step.
Figure 3: Fitness of sampled points during IGO optimization. Figure 4: Fitness
of sampled points during vanilla gradient optimization.
Note that the values of the parameter $\hskip 0.50003pt\delta t\hskip
0.50003pt$ for the two gradients used are not directly comparable from a
theoretical viewpoint (they correspond to parametrizations of different
trajectories in $\Theta$-space, and identifying vanilla $\hskip
0.50003pt\delta t\hskip 0.50003pt$ with natural $\hskip 0.50003pt\delta
t\hskip 0.50003pt$ is meaningless). We selected larger values of $\hskip
0.50003pt\delta t\hskip 0.50003pt$ for the vanilla gradient, in order to
obtain roughly comparable convergence speeds in practice.
#### 5.2.2 Diversity
As we have seen, the two-min function is equally well optimized by the IGO and
vanilla gradient optimization. However, the methods fare very differently when
we look at the distance from the sample points to _both_ optima. From (58),
the fitness gives the distance of sample points to the closest optimum. We now
study how close sample points come to the _other_ , opposite optimum. The
distance of sample points to the second optimum is shown in Figure 5 for IGO,
and in Figure 6 for the vanilla gradient.
Figure 5: Distance to the second optimum during IGO optimization. Figure 6:
Distance to second optimum during vanilla gradient optimization.
Figure 5 shows that IGO also reaches the second optimum most of the time, and
is often able to find it only a few steps after the first. This property of
IGO is of course dependent on the initialization of the RBM with enough
diversity. When initialized properly so that each variable (observed and
latent) has a probability $1/2$ of being equal to $1$, the initial RBM
distribution has maximal diversity over the search space and is at equal
distance from the two optima of the function. From this starting position, the
natural gradient is then able to increase the likelihood of the two optima at
the same time.
By stark contrast, the vanilla gradient is not able to go towards both optima
at the same time as shown in Fig. 6. In fact, the vanilla gradient only
converges to one optimum at the expense of the other. For all values of
$\delta t$, the distance to the second optimum increases gradually and
approaches the maximum possible distance.
As mentioned earlier, each state of the latent variable $\mathbf{h}$
corresponds to a mode of the distribution. In Figures 7 and 8, we look at the
average value of $\mathbf{h}$ for each gradient step. An average value close
to $1/2$ means that the distribution samples from both modes: $\mathbf{h}=0$
or $\mathbf{h}=1$ with a comparable probability. Conversely, average values
close to $0$ or $1$ indicate that the distribution gives most probability to
one mode at the expense of the other. In Figure 7, we can see that with IGO,
the average value of $\mathbf{h}$ is close to $1/2$ during the whole
optimization procedure for most runs: the distribution is initialized with two
modes and stays bimodal777In Fig. 7, we use the following adverse setting:
runs are interrupted once they reach both optima, therefore the statistics are
taken only over those runs which have _not yet_ converged and reached both
optima, which results in higher variation around $1/2$. The plot has been
stopped when less than half the runs remain. The error bars are relative only
to the remaining runs.. As for the vanilla gradient, statistics for
$\mathbf{h}$ are depicted in Figure 8 and we can see that they converge to
$1$: one of the two modes of the distribution has been lost during
optimization.
Figure 7: Average value of $\mathbf{h}$ during IGO optimization. Figure 8:
Average value of $\mathbf{h}$ during vanilla gradient optimization.
##### Hidden breach of symmetry by the vanilla gradient.
The experiments reveal a curious phenomenon: the vanilla gradient loses
multimodality by always setting the hidden variable $h$ to $1$, not to $0$.
(We detected no obvious asymmetry on the visible units $x$, though.)
Of course, exchanging the values $0$ and $1$ for the hidden variables in a
restricted Boltzmann machine still gives a distribution of another Boltzmann
machine. More precisely, changing $h_{j}$ into $1-h_{j}$ is equivalent to
resetting $a_{i}\leftarrow a_{i}+w_{ij}$, $b_{j}\leftarrow-b_{j}$, and
$w_{ij}\leftarrow-w_{ij}$. IGO and the natural gradient are impervious to such
a change by Proposition 9.
The vanilla gradient implicitly relies on the Euclidean norm on parameter
space, as explained in Section 1.1. For this norm, the distance between the
RBM distributions $(a_{i},b_{j},w_{ij})$ and
$(a^{\prime}_{i},b^{\prime}_{j},w^{\prime}_{ij})$ is simply
$\sum_{i}\left|\mskip
1.0mua_{i}-a^{\prime}_{i}\right|^{2}+\sum_{j}\left|\mskip
1.0mub_{j}-b^{\prime}_{j}\right|^{2}+\sum_{ij}\left|\mskip
1.0muw_{ij}-w^{\prime}_{ij}\right|^{2}$. However, the change of variables
$a_{i}\leftarrow a_{i}+w_{ij},\;{b_{j}\leftarrow-b_{j}},\;w_{ij}\leftarrow-
w_{ij}$ does _not_ preserve this Euclidean metric. Thus, exchanging $0$ and
$1$ for the hidden variables will result in two different vanilla gradient
ascents. The observed asymmetry on $h$ is a consequence of this implicit
asymmetry.
The same asymmetry exists for the visible variables $x_{i}$; but this does not
prevent convergence to an optimum in our situation, since any gradient descent
eventually reaches some local optimum.
Of course it is possible to cook up parametrizations for which the vanilla
gradient will be more symmetric: for instance, using $-1/1$ instead of $0/1$
for the variables, or defining the energy by
$E(\mathbf{x},\mathbf{h})=-{\textstyle\sum}_{i}A_{i}(x_{i}-\tfrac{1}{2})-{\textstyle\sum}_{j}B_{j}(h_{j}-\tfrac{1}{2})-{\textstyle\sum}_{i,j}W_{ij}(x_{i}-\tfrac{1}{2})(h_{j}-\tfrac{1}{2})$
(59)
with “bias-free” parameters $A_{i},B_{j},W_{ij}$ related to the usual
parametrization by
$w_{ij}=W_{ij},\;a_{i}=A_{i}-\frac{1}{2}\sum_{j}w_{ij}\;b_{j}=B_{j}-\frac{1}{2}\sum_{i}w_{ij}$.
The vanilla gradient might perform better in these parametrizations.
However, we chose a naive approach: we used a family of probability
distributions found in the literature, with the parametrization found in the
literature. We then use the vanilla gradient and the natural gradient on these
distributions. This directly illustrates the specific influence of the chosen
gradient (the two implementations only differ by the inclusion of the Fisher
matrix). It is remarkable, we think, that the natural gradient is able to
recover symmetry where there was none.
### 5.3 Convergence to the continuous-time limit
In the previous figures, it looks like changing the parameter $\hskip
0.50003pt\delta t\hskip 0.50003pt$ only results in a time speedup of the
plots.
Update rules of the type $\theta\leftarrow\theta+\hskip 0.50003pt\delta
t\hskip 0.50003pt\,\nabla_{\\!\theta}g$ (for either gradient) are Euler
approximations of the continuous-time ordinary differential equation
$\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}=\nabla_{\\!\theta}g$, with each
iteration corresponding to an increment $\hskip 0.50003pt\delta t\hskip
0.50003pt$ of the time $t$. Thus, it is expected that for small enough $\hskip
0.50003pt\delta t\hskip 0.50003pt$, the algorithm after $k$ steps approximates
the IGO flow or vanilla gradient flow at time $t=k.\hskip 0.50003pt\delta
t\hskip 0.50003pt$.
Figures 9 and 10 illustrate this convergence: we show the fitness w.r.t to
$\hskip 0.50003pt\delta t\hskip 0.50003pt$ times the number of gradient steps.
An asymptotic trajectory seems to emerge when $\hskip 0.50003pt\delta t\hskip
0.50003pt$ decreases. For the natural gradient, it can be interpreted as the
fitness of samples of the continuous-time IGO flow.
Thus, for this kind of optimization algorithms, it makes theoretical sense to
plot the results according to the “intrinsic time” of the underlying
continuous-time object, to illustrate properties that do not depend on the
setting of the parameter $\hskip 0.50003pt\delta t\hskip 0.50003pt$. (Still,
the raw number of steps is more directly related to algorithmic cost.)
Figure 9: Fitness of sampled points w.r.t. “intrinsic time” during IGO
optimization. Figure 10: Fitness of sampled points w.r.t. “intrinsic time”
during vanilla gradient optimization.
## 6 Further discussion
##### A single framework for optimization on arbitrary spaces.
A strength of the IGO viewpoint is to automatically provide optimization
algorithms using any family of probability distributions on any given space,
discrete or continuous. This has been illustrated with restricted Boltzmann
machines. IGO algorithms also feature good invariance properties and make a
least number of arbitrary choices.
In particular, IGO unifies several well-known optimization algorithms into a
single framework. For instance, to the best of our knowledge, PBIL has never
been described as a natural gradient ascent in the literature888Thanks to
Jonathan Shapiro for an early argument confirming this property (personal
communication)..
For Gaussian measures, algorithms of the same form (14) had been developed
previously [HO01, WSPS08] and their close relationship with a natural gradient
ascent had been recognized [ANOK10, GSS+10].
The wide applicability of natural gradient approaches seems not to be widely
known in the optimization community, though see [MMS08].
##### About quantiles.
The IGO flow, to the best of our knowledge, has not been defined before. The
introduction of the quantile-rewriting (3) of the objective function provides
the first rigorous derivation of quantile- or rank-based natural optimization
from a gradient ascent in $\theta$-space.
Indeed, NES and CMA-ES have been claimed to maximize
$-\mathbb{E}_{P_{\theta}}f$ via natural gradient ascent [WSPS08, ANOK10].
However, we have proved that when the number of samples is large and the step
size is small, the NES and CMA-ES updates converge to the IGO flow, not to the
similar flow with the gradient of $\mathbb{E}_{P_{\theta}}f$ (Theorem 4). So
we find that in reality these algorithms maximize
$\mathbb{E}_{{P_{\theta}}}W_{{\theta^{t}}}^{f}$, where $W_{{\theta^{t}}}^{f}$
is a decreasing transformation of the $f$-quantiles under the current sample
distribution.
Also in practice, maximizing $-\mathbb{E}_{P_{\theta}}f$ is a rather unstable
procedure and has been discouraged, see for example [Whi89].
##### About choice of $P_{\theta}$: learning a model of good points.
The choice of the family of probability distributions $P_{\theta}$ plays a
double role.
First, it is analogous to a _mutation operator_ as seen in evolutionary
algorithms: indeed, $P_{\theta}$ encodes possible moves according to which new
sample points are explored.
Second, optimization algorithms using distributions can be interpreted as
learning a probabilistic model of where the points with good values lie in the
search space. With this point of view, $P_{\theta}$ describes _richness of
this model_ : for instance, restricted Boltzmann machines with $h$ hidden
units can describe distributions with up to $2^{h}$ modes, whereas the
Bernoulli distribution used in PBIL is unimodal. This influences, for
instance, the ability to explore several valleys and optimize multimodal
functions in a single run.
##### Natural gradient and parametrization invariance.
Central to IGO is the use of natural gradient, which follows from
$\theta$-invariance and makes sense on any search space, discrete or
continuous.
While the IGO flow is exactly $\theta$-invariant, for any practical
implementation of an IGO algorithm, a parametrization choice has to be made.
Still, since all IGO algorithms approximate the IGO flow, two parametrizations
of IGO will differ less than two parametrizations of another algorithm (such
as the vanilla gradient or the smoothed CEM method)—at least if the learning
rate $\hskip 0.50003pt\delta t\hskip 0.50003pt$ is not too large.
On the other hand, natural evolution strategies have never strived for
$\theta$-invariance: the chosen parametrization (Cholesky, exponential) has
been deemed a relevant feature. In the IGO framework, the chosen
parametrization becomes more relevant as the step size $\hskip 0.50003pt\delta
t\hskip 0.50003pt$ increases.
##### IGO, maximum likelihood and cross-entropy.
The cross-entropy method (CEM) [dBKMR05] can be used to produce optimization
algorithms given a family of probability distributions on an arbitrary space,
by performing a jump to a maximum likelihood estimate of the parameters.
We have seen (Corollary 16) that the standard CEM is an IGO algorithm in a
particular parametrization, with a learning rate $\hskip 0.50003pt\delta
t\hskip 0.50003pt$ equal to $1$. However, it is well-known, both theoretically
and experimentally [BLS07, Han06b, WAS04], that standard CEM loses diversity
too fast in many situations. The usual solution [dBKMR05] is to reduce the
learning rate (smoothed CEM, (25)), but this breaks the reparametrization
invariance.
On the other hand, the IGO flow can be seen as a _maximum likelihood update
with infinitesimal learning rate_ (Theorem 13). This interpretation allows to
define a particular IGO algorithm, the IGO-ML (Definition 14): it performs a
maximum likelihood update with an arbitrary learning rate, and keeps the
reparametrization invariance. It coincides with CEM when the learning rate is
set to $1$, but it differs from smoothed CEM by the exchange of the order of
argmax and averaging (compare (23) and (25)). We argue that this new algorithm
may be a better way to reduce the learning rate and achieve smoothing in CEM.
Standard CEM can lose diversity, yet is a particular case of an IGO algorithm:
this illustrates the fact that reasonable values of the learning rate $\hskip
0.50003pt\delta t\hskip 0.50003pt$ depend on the parametrization. We have
studied this phenomenon in detail for various Gaussian IGO algorithms (Section
4.2).
Why would a smaller learning rate perform better than a large one in an
optimization setting? It might seem more efficient to jump directly to the
maximum likelihood estimate of currently known good points, instead of
performing a slow gradient ascent towards this maximum.
However, optimization faces a “moving target”, contrary to a learning setting
in which the example distribution is often stationary. Currently known good
points are likely not to indicate the _position_ at which the optimum lies,
but, rather, the _direction_ in which the optimum is to be found. After an
update, the next elite sample points are going to be located somewhere new. So
the goal is certainly not to settle down around these currently known points,
as a maximum likelihood update does: by design, CEM only tries to reflect
status-quo (even for $N=\infty$), whereas IGO tries to move somewhere. When
the target moves over time, a progressive gradient ascent is more reasonable
than an immediate jump to a temporary optimum, and realizes a kind of time
smoothing.
This phenomenon is most clear when the number of sample points is small. Then,
a full maximum likelihood update risks losing a lot of diversity; it may even
produce a degenerate distribution if the number of sample points is smaller
than the number of parameters of the distribution. On the other hand, for
smaller $\hskip 0.50003pt\delta t\hskip 0.50003pt$, the IGO algorithms do, by
design, try to maintain diversity by moving as little as possible from the
current distribution $P_{\theta}$ in Kullback–Leibler divergence. A full ML
update disregards the current distribution and tries to move as close as
possible to the elite sample in Kullback–Leibler divergence [dBKMR05], thus
realizing maximal diversity loss. This makes sense in a non-iterated scenario
but is unsuited for optimization.
##### Diversity and multiple optima.
The IGO framework emphasizes the relation of natural gradient and diversity:
we argued that IGO provides minimal diversity change for a given objective
function increment. In particular, provided the initial diversity is large,
diversity is kept at a maximum. This theoretical relationship has been
confirmed experimentally for restricted Boltzmann machines.
On the other hand, using the vanilla gradient does not lead to a balanced
distribution between the two optima in our experiments. Using the vanilla
gradient introduces hidden arbitrary choices between those points (more
exactly between moves in $\Theta$-space). This results in loss of diversity,
and might also be detrimental at later stages in the optimization. This may
reflect the fact that the Euclidean metric on the space of parameters,
implicitly used in the vanilla gradient, becomes less and less meaningful for
gradient descent on complex distributions.
IGO and the natural gradient are certainly relevant to the well-known problem
of exploration-exploitation balance: as we have seen, arguably the natural
gradient realizes the best increment of the objective function with the least
possible change of diversity in the population.
More generally, IGO tries to learn a model of where the good points are,
representing all good points seen so far rather than focusing only on some
good points; this is typical of machine learning, one of the contexts for
which the natural gradient was studied. The conceptual relationship of IGO and
IGO-like optimization algorithms with machine learning is still to be explored
and exploited.
## Summary and conclusion
We sum up:
* •
The information-geometric optimization (IGO) framework derives from invariance
principles and allows to build optimization algorithms from any family of
distributions on any search space. In some instances (Gaussian distributions
on ${\mathbb{R}}^{d}$ or Bernoulli distributions on $\\{0,1\\}^{d}$) it
recovers versions of known algorithms (CMA-ES or PBIL); in other instances
(restricted Boltzmann machine distributions) it produces new, hopefully
efficient optimization algorithms.
* •
The use of a quantile-based, time-dependent transform of the objective
function (Equation (3)) provides a rigorous derivation of rank-based update
rules currently used in optimization algorithms. Theorem 4 uniquely identifies
the infinite-population limit of these update rules.
* •
The IGO flow is singled out by its equivalent description as an infinitesimal
weighted maximal log-likelihood update (Theorem 13). In a particular
parametrization and with a step size of $1$, it recovers the cross-entropy
method (Corollary 16).
* •
Theoretical arguments suggest that the IGO flow minimizes the change of
diversity in the course of optimization. In particular, starting with high
diversity and using multimodal distributions may allow simultaneous
exploration of multiple optima of the objective function. Preliminary
experiments with restricted Boltzmann machines confirm this effect in a simple
situation.
Thus, the IGO framework is an attempt to provide sound theoretical foundations
to optimization algorithms based on probability distributions. In particular,
this viewpoint helps to bridge the gap between continuous and discrete
optimization.
The invariance properties, which reduce the number of arbitrary choices,
together with the relationship between natural gradient and diversity, may
contribute to a theoretical explanation of the good practical performance of
those currently used algorithms, such as CMA-ES, which can be interpreted as
instantiations of IGO.
We hope that invariance properties will acquire in computer science the
importance they have in mathematics, where intrinsic thinking is the first
step for abstract linear algebra or differential geometry, and in modern
physics, where the notions of invariance w.r.t. the coordinate system and so-
called gauge invariance play central roles.
## Acknowledgements
The authors would like to thank Michèle Sebag for the acronym and for helpful
comments. Y. O. would like to thank Cédric Villani and Bruno Sévennec for
helpful discussions on the Fisher metric. A. A. and N. H. would like to
acknowledge the Dagstuhl Seminar No 10361 on the Theory of Evolutionary
Computation (http://www.dagstuhl.de/10361) for inspiring their work on natural
gradients and beyond. This work was partially supported by the
ANR-2010-COSI-002 grant (SIMINOLE) of the French National Research Agency.
## Appendix: Proofs
### Proof of Theorem 4 (Convergence of empirical means and quantiles)
Let us give a more precise statement including the necessary regularity
conditions.
###### Proposition 18.
Let $\theta\in\Theta$. Assume that the derivative $\frac{\partial\ln
P_{\theta}(x)}{\partial\theta}$ exists for $P_{\theta}$-almost all $x\in X$
and that $\mathbb{E}_{P_{\theta}}\left|\mskip 1.0mu\frac{\partial\ln
P_{\theta}(x)}{\partial\theta}\right|^{2}<+\infty$. Assume that the function
$w$ is non-decreasing and bounded.
Let $(x_{i})_{i\in{\mathbb{N}}}$ be a sequence of independent samples of
$P_{\theta}$. Then with probability $1$, as $N\to\infty$ we have
$\frac{1}{N}\sum_{i=1}^{N}\widehat{W}^{f}(x_{i})\frac{\partial\ln
P_{\theta}(x_{i})}{\partial\theta}\to\int W_{\theta}^{f}(x)\,\frac{\partial\ln
P_{\theta}(x)}{\partial\theta}\,P_{\theta}({\mathrm{d}}x)$
where
$\widehat{W}^{f}(x_{i})=w\left(\frac{\mathrm{rk}_{N}(x_{i})+1/2}{N}\right)$
with $\mathrm{rk}_{N}(x_{i})=\\#\\{1\leqslant j\leqslant
N,f(x_{j})<f(x_{i})\\}$. (When there are $f$-ties in the sample,
$W^{f}(x_{i})$ is defined as the average of $w((r+1/2)/N)$ over the possible
rankings $r$ of $x_{i}$.)
###### Proof.
Let $g:X\to{\mathbb{R}}$ be any function with
$\mathbb{E}_{P_{\theta}}g^{2}<\infty$. We will show that
$\frac{1}{N}\sum\widehat{W}^{f}(x_{i})g(x_{i})\to\int
W_{\theta}^{f}(x)g(x)\,P_{\theta}({\mathrm{d}}x)$. Applying this with $g$
equal to the components of $\frac{\partial\ln P_{\theta}(x)}{\partial\theta}$
will yield the result.
Let us decompose
$\displaystyle\frac{1}{N}\sum\widehat{W}^{f}(x_{i})g(x_{i})=\frac{1}{N}\sum
W_{\theta}^{f}(x_{i})g(x_{i})+\frac{1}{N}\sum(\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i}))g(x_{i}).$
Each summand in the first term involves only one sample $x_{i}$ (contrary to
$\widehat{W}^{f}(x_{i})$ which depends on the whole sample). So by the strong
law of large numbers, almost surely $\frac{1}{N}\sum
W_{\theta}^{f}(x_{i})g(x_{i})$ converges to $\int
W_{\theta}^{f}(x)g(x)\,P_{\theta}({\mathrm{d}}x)$. So we have to show that the
second term converges to $0$ almost surely.
By the Cauchy–Schwarz inequality, we have
$\left|\mskip
1.0mu\frac{1}{N}\sum(\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i}))g(x_{i})\right|^{2}\leqslant\left(\frac{1}{N}\sum(\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i}))^{2}\right)\left(\frac{1}{N}\sum
g(x_{i})^{2}\right)$
By the strong law of large numbers, the second term $\frac{1}{N}\sum
g(x_{i})^{2}$ converges to $E_{P_{\theta}}g^{2}$ almost surely. So we have to
prove that the first term
$\frac{1}{N}\sum(\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i}))^{2}$ converges
to $0$ almost surely.
Since $w$ is bounded by assumption, we can write
$\displaystyle(\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i}))^{2}$
$\displaystyle\leqslant 2B\left|\mskip
1.0mu\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i})\right|$
$\displaystyle=2B\left|\mskip
1.0mu\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i})\right|_{+}+2B\left|\mskip
1.0mu\widehat{W}^{f}(x_{i})-W_{\theta}^{f}(x_{i})\right|_{-}$
where $B$ is the bound on $\left|\mskip 1.0muw\right|$. We will bound each of
these terms.
Let us abbreviate $q_{i}^{-}=\Pr_{x^{\prime}\sim
P_{\theta}}(f(x^{\prime})<f(x_{i}))$, $q_{i}^{+}=\Pr_{x^{\prime}\sim
P_{\theta}}(f(x^{\prime})\leqslant f(x_{i}))$,
$r_{i}^{-}=\\#\\{j\\!\leqslant\\!N,\,f(x_{j})<f(x_{i})\\}$,
$r_{i}^{+}=\\#\\{j\\!\leqslant\\!N,\,f(x_{j})\leqslant f(x_{i})\\}$.
By definition of $\widehat{W}^{f}$ we have
$\widehat{W}^{f}(x_{i})=\frac{1}{r_{i}^{+}-r_{i}^{-}}\sum_{k=r_{i}^{-}}^{r_{i}^{+}-1}w((k+1/2)/N)$
and moreover $W_{\theta}^{f}(x_{i})=w(q_{i}^{-})$ if $q_{i}^{-}=q_{i}^{+}$ or
$W_{\theta}^{f}(x_{i})=\frac{1}{q_{i}^{+}-q_{i}^{-}}\int_{q_{i}^{-}}^{q_{i}^{+}}w$
otherwise.
The Glivenko–Cantelli theorem states that $\sup_{i}\left|\mskip
1.0muq_{i}^{+}-r_{i}^{+}/N\right|$ tends to $0$ almost surely, and likewise
for $\sup_{i}\left|\mskip 1.0muq_{i}^{-}-r_{i}^{-}/N\right|$. So let $N$ be
large enough so that these errors are bounded by $\varepsilon$.
Since $w$ is non-increasing, we have $w(q_{i}^{-})\leqslant
w(r_{i}^{-}/N-\varepsilon)$. In the case $q_{i}^{-}\neq q_{i}^{+}$, we
decompose the interval $[q_{i}^{-};q_{i}^{+}]$ into $(r_{i}^{+}-r_{i}^{-})$
subintervals. The average of $w$ over each such subinterval is compared to a
term in the sum defining $w^{N}(x_{i})$: since $w$ is non-increasing, the
average of $w$ over the $k^{\text{th}}$ subinterval is at most
$w((r_{i}^{-}+k)/N-\varepsilon)$. So we get
$W_{\theta}^{f}(x_{i})\leqslant\frac{1}{r_{i}^{+}-r_{i}^{-}}\sum_{k=r_{i}^{-}}^{r_{i}^{+}-1}w(k/N-\varepsilon)$
so that
$W_{\theta}^{f}(x_{i})-\widehat{W}^{f}(x_{i})\leqslant\frac{1}{r_{i}^{+}-r_{i}^{-}}\sum_{k=r_{i}^{-}}^{r_{i}^{+}-1}(w(k/N-\varepsilon)-w((k+1/2)/N)).$
Let us sum over $i$, remembering that there are $(r_{i}^{+}-r_{i}^{-})$ values
of $j$ for which $f(x_{j})=f(x_{i})$. Taking the positive part, we get
$\frac{1}{N}\sum_{i=1}^{N}\left|\mskip
1.0muW_{\theta}^{f}(x_{i})-\widehat{W}^{f}(x_{i})\right|_{+}\leqslant\frac{1}{N}\sum_{k=0}^{N-1}(w(k/N-\varepsilon)-w((k+1/2)/N)).$
Since $w$ is non-increasing we have
$\frac{1}{N}\sum_{k=0}^{N-1}w(k/N-\varepsilon)\leqslant\int_{-\varepsilon-1/N}^{1-\varepsilon-1/N}w$
and
$\frac{1}{N}\sum_{k=0}^{N-1}w((k+1/2)/N)\geqslant\int_{1/2N}^{1+1/2N}w$
(we implicitly extend the range of $w$ so that $w(q)=w(0)$ for $q<0$). So we
have
$\frac{1}{N}\sum_{i=1}^{N}\left|\mskip
1.0muW_{\theta}^{f}(x_{i})-\widehat{W}^{f}(x_{i})\right|_{+}\leqslant\int_{-\varepsilon-1/N}^{1/2N}w-\int_{1-\varepsilon-1/N}^{1+1/2N}w\leqslant(2\varepsilon+3/N)B$
where $B$ is the bound on $\left|\mskip 1.0muw\right|$.
Reasoning symmetrically with $w(k/N+\varepsilon)$ and the inequalities
reversed, we get a similar bound for $\frac{1}{N}\sum\left|\mskip
1.0muW_{\theta}^{f}(x_{i})-\widehat{W}^{f}(x_{i})\right|_{-}$. This ends the
proof. ∎
### Proof of Proposition 5 (Quantile improvement)
Let us use the weight $w(u)=\mathbbm{1}_{u\leqslant q}$. Let $m$ be the value
of the $q$-quantile of $f$ under $P_{\theta^{t}}$. We want to show that the
value of the $q$-quantile of $f$ under $P_{\theta^{t+\hskip 0.25002pt\delta
t\hskip 0.25002pt}}$ is less than $m$, unless the gradient vanishes and the
IGO flow is stationary.
Let $p_{-}=\Pr_{x\sim P_{\theta^{t}}}(f(x)<m)$, $p_{m}=\Pr_{x\sim
P_{\theta^{t}}}(f(x)=m)$ and $p_{+}=\Pr_{x\sim P_{\theta^{t}}}(f(x)>m)$. By
definition of the quantile value we have $p_{-}+p_{m}\geqslant q$ and
$p_{+}+p_{m}\geqslant 1-q$. Let us assume that we are in the more complicated
case $p_{m}\neq 0$ (for the case $p_{m}=0$, simply remove the corresponding
terms).
We have $W_{\theta_{t}}^{f}(x)=1$ if $f(x)<m$, $W_{\theta_{t}}^{f}(x)=0$ if
$f(x)>m$ and
$W_{\theta_{t}}^{f}(x)=\frac{1}{p_{m}}\int_{p_{-}}^{p_{-}+p_{m}}w(u){\mathrm{d}}u=\frac{q-p_{-}}{p_{m}}$
if $f(x)=m$.
Using the same notation as above, let $g_{t}(\theta)=\int
W_{\theta^{t}}^{f}(x)\,P_{\theta}({\mathrm{d}}x)$. Decomposing this integral
on the three sets $f(x)<m$, $f(x)=m$ and $f(x)>m$, we get that
$g_{t}(\theta)=\Pr_{x\sim P_{\theta}}(f(x)<m)+\Pr_{x\sim
P_{\theta}}(f(x)=m)\frac{q-p_{-}}{p_{m}}$. In particular,
$g_{t}(\theta^{t})=q$.
Since we follow a gradient ascent of $g_{t}$, for $\hskip 0.50003pt\delta
t\hskip 0.50003pt$ small enough we have $g_{t}(\theta^{t+\hskip
0.35002pt\delta t\hskip 0.35002pt})>g_{t}(\theta^{t})$ unless the gradient
vanishes. If the gradient vanishes we have $\theta^{t+\hskip 0.35002pt\delta
t\hskip 0.35002pt}=\theta^{t}$ and the quantiles are the same. Otherwise we
get $g_{t}(\theta^{t+\hskip 0.35002pt\delta t\hskip
0.35002pt})>g_{t}(\theta^{t})=q$.
Since $\frac{q-p_{-}}{p_{m}}\leqslant\frac{(p_{-}+p_{m})-p_{-}}{p_{m}}=1$, we
have $g_{t}(\theta)\leqslant\Pr_{x\sim P_{\theta}}(f(x)<m)+\Pr_{x\sim
P_{\theta}}(f(x)=m)=\Pr_{x\sim P_{\theta}}(f(x)\leqslant m)$.
So $\Pr_{x\sim P_{\theta^{t+\hskip 0.25002pt\delta t\hskip
0.25002pt}}}(f(x)\leqslant m)\geqslant g_{t}(\theta^{t+\hskip 0.35002pt\delta
t\hskip 0.35002pt})>q$. This implies that, by definition, the $q$-quantile
value of $P_{\theta^{t+\hskip 0.25002pt\delta t\hskip 0.25002pt}}$ is smaller
than $m$.
### Proof of Proposition 10 (Speed of the IGO flow)
###### Lemma 19.
Let $X$ be a centered $L^{2}$ random variable with values in
${\mathbb{R}}^{d}$ and let $A$ be a real-valued $L^{2}$ random variable. Then
$\|\mathbb{E}(AX)\|\leqslant\sqrt{\lambda\operatorname{Var}A}$
where $\lambda$ is the largest eigenvalue of the covariance matrix of $X$
expressed in an orthonormal basis.
###### Proof of the lemma.
Let $v$ be any vector in ${\mathbb{R}}^{d}$; its norm satisfies
$\|v\|=\sup_{w,\,\|w\|\leqslant 1}(v\cdot w)$
and in particular
$\displaystyle\|\mathbb{E}(AX)\|$ $\displaystyle=\sup_{w,\,\|w\|\leqslant
1}(w\cdot\mathbb{E}(AX))$ $\displaystyle=\sup_{w,\,\|w\|\leqslant
1}\mathbb{E}(A\,(w\cdot X))$ $\displaystyle=\sup_{w,\,\|w\|\leqslant
1}\mathbb{E}((A-\mathbb{E}A)\,(w\cdot X))\quad\text{since $(w\cdot X)$ is
centered}$ $\displaystyle\leqslant\sup_{w,\,\|w\|\leqslant
1}\sqrt{\operatorname{Var}A}\,\sqrt{\mathbb{E}((w\cdot X)^{2})}$
by the Cauchy–Schwarz inequality and using the fact that $A$ is centered.
Now, in an orthonormal basis we have
$(w\cdot X)=\sum_{i}w_{i}X_{i}$
so that
$\displaystyle\mathbb{E}((w\cdot X)^{2})$
$\displaystyle=\mathbb{E}\left(({\textstyle\sum}_{i}w_{i}X_{i})({\textstyle\sum}_{j}w_{j}X_{j})\right)$
$\displaystyle={\textstyle\sum}_{i}{\textstyle\sum}_{j}\mathbb{E}(w_{i}X_{i}w_{j}X_{j})$
$\displaystyle={\textstyle\sum}_{i}{\textstyle\sum}_{j}w_{i}w_{j}\mathbb{E}(X_{i}X_{j})$
$\displaystyle={\textstyle\sum}_{i}{\textstyle\sum}_{j}w_{i}w_{j}C_{ij}$
with $C_{ij}$ the covariance matrix of $X$. The latter expression is the
scalar product $(w\cdot Cw)$. Since $C$ is a symmetric positive-semidefinite
matrix, $(w\cdot Cw)$ is at most $\lambda\|w\|^{2}$ with $\lambda$ the largest
eigenvalue of $C$. ∎
For the IGO flow we have
$\frac{{\mathrm{d}}\theta^{t}}{{\mathrm{d}}t}=\mathbb{E}_{x\sim
P_{\theta}}W_{\theta}^{f}(x)\tilde{\nabla}_{\theta}\ln P_{\theta}(x)$.
So applying the lemma, we get that the norm of
$\frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}$ is at most
$\sqrt{\lambda\operatorname{Var}_{x\sim P_{\theta}}W_{\theta}^{f}(x)}$ where
$\lambda$ is the largest eivengalue of the covariance matrix of
$\tilde{\nabla}_{\theta}\ln P_{\theta}(x)$ (expressed in a coordinate system
where the Fisher matrix at the current point $\theta$ is the identity).
By construction of the quantiles, we have $\operatorname{Var}_{x\sim
P_{\theta}}W_{\theta}^{f}(x)\leqslant\operatorname{Var}_{[0,1]}w$ (with
equality unless there are ties). Indeed, for a given $x$, let $\mathcal{U}$ be
a uniform random variable in $[0,1]$ independent from $x$ and define the
random variable $Q=q^{-}(x)+(q^{+}(x)-q^{-}(x))\mathcal{U}$. Then $Q$ is
uniformly distributed between the upper and lower quantiles $q^{+}(x)$ and
$q^{-}(x)$ and thus we can rewrite $W_{\theta}^{f}(x)$ as
$\mathbb{E}(w(Q)|x)$. By the Jensen inequality we have
$\operatorname{Var}W_{\theta}^{f}(x)=\operatorname{Var}\mathbb{E}(w(Q)|x)\leqslant\operatorname{Var}w(Q)$.
In addition when $x$ is taken under $P_{\theta}$, $Q$ is uniformly distributed
in $[0,1]$ and thus $\operatorname{Var}w(Q)=\operatorname{Var}_{[0,1]}w$, i.e.
$\operatorname{Var}_{x\sim
P_{\theta}}W_{\theta}^{f}(x)\leqslant\operatorname{Var}_{[0,1]}w$.
Besides, consider the tangent space in $\Theta$-space at point $\theta^{t}$,
and let us choose an orthonormal basis in this tangent space for the Fisher
metric. Then, in this basis we have $\tilde{\nabla}_{i}\ln
P_{\theta}(x)=\partial_{i}\ln P_{\theta}(x)$. So the covariance matrix of
$\tilde{\nabla}\ln P_{\theta}(x)$ is $\mathbb{E}_{x\sim
P_{\theta}}(\partial_{i}\ln P_{\theta}(x)\partial_{j}\ln P_{\theta}(x))$,
which is equal to the Fisher matrix by definition. So this covariance matrix
is the identity, whose largest eigenvalue is $1$.
### Proof of Proposition 12 (Noisy IGO)
On the one hand, let $P_{\theta}$ be a family of distributions on $X$. The IGO
algorithm (13) applied to a random function $f(x)=\tilde{f}(x,\omega)$ where
$\omega$ is a random variable uniformly distributed in $[0,1]$ reads
$\theta^{t+\hskip 0.35002pt\delta t\hskip 0.35002pt}=\theta^{t}+\hskip
0.50003pt\delta t\hskip
0.50003pt\sum_{i=1}^{N}\hat{w_{i}}\widetilde{\nabla}_{\theta}\ln
P_{\theta}(x_{i})$ (60)
where $x_{i}\sim P_{\theta}$ and $\hat{w_{i}}$ is according to (12) where
ranking is applied to the values $\tilde{f}(x_{i},\omega_{i})$, with
$\omega_{i}$ uniform variables in $[0,1]$ independent from $x_{i}$ and from
each other.
On the other hand, for the IGO algorithm using $P_{\theta}\otimes U_{[0,1]}$
and applied to the deterministic function $\tilde{f}$, $\hat{w_{i}}$ is
computed using the ranking according to the $\tilde{f}$ values of the sampled
points $\tilde{x}_{i}=(x_{i},\omega_{i})$, and thus coincides with the one in
(60).
Besides,
$\widetilde{\nabla}_{\theta}\ln P_{\theta\otimes
U_{[0,1]}}(\tilde{x}_{i})=\widetilde{\nabla}_{\theta}\ln
P_{\theta}(x_{i})+\underbrace{\widetilde{\nabla}_{\theta}\ln
U_{[0,1]}(\omega_{i})}_{=0}$
and thus the IGO algorithm update on space $X\times[0,1]$, using the family of
distributions $\tilde{P}_{\theta}=P_{\theta}\otimes U_{[0,1]}$, applied to the
deterministic function $\tilde{f}$, coincides with (60).
### Proof of Theorem 13 (Natural gradient as ML with infinitesimal weights)
We begin with a calculus lemma (proof omitted).
###### Lemma 20.
Let $f$ be real-valued function on a finite-dimensional vector space $E$
equipped with a definite positive quadratic form $\|\cdot\|^{2}$. Assume $f$
is smooth and has at most quadratic growth at infinity. Then, for any $x\in
E$, we have
$\nabla f(x)=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\operatorname*{arg\,max}_{h}\left\\{f(x+h)-\frac{1}{2\varepsilon}\|h\|^{2}\right\\}$
where $\nabla$ is the gradient associated with the norm $\|\cdot\|$.
Equivalently,
$\operatorname*{arg\,max}_{y}\left\\{f(y)-\frac{1}{2\varepsilon}\|y-x\|^{2}\right\\}=x+\varepsilon\nabla
f(x)+O(\varepsilon^{2})$
when $\varepsilon\to 0$.
We are now ready to prove Theorem 13. Let $W$ be a function of $x$, and fix
some $\theta_{0}$ in $\Theta$.
We need some regularity assumptions: we assume that the parameter space
$\Theta$ is non-degenerate (no two points $\theta\in\Theta$ define the same
probability distribution) and proper (the map $P_{\theta}\mapsto\theta$ is
continuous). We also assume that the map $\theta\mapsto P_{\theta}$ is smooth
enough, so that $\int\log P_{\theta}(x)\,W(x)\,P_{\theta_{0}}({\mathrm{d}}x)$
is a smooth function of $\theta$. (These are restrictions on
$\theta$-regularity: this does not mean that $W$ has to be continuous as a
function of $x$.)
The two statements of Theorem 13 using a sum and an integral have similar
proofs, so we only include the first. For $\varepsilon>0$, let $\theta$ be the
solution of
$\theta=\operatorname*{arg\,max}\Bigg{\\{}(1-\varepsilon)\int\log
P_{\theta}(x)\,P_{\theta_{0}}({\mathrm{d}}x)+\varepsilon\int\log
P_{\theta}(x)\,W(x)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}.$
Then we have
$\displaystyle\theta$
$\displaystyle=\operatorname*{arg\,max}\Bigg{\\{}\int\log
P_{\theta}(x)\,P_{\theta_{0}}({\mathrm{d}}x)+\varepsilon\int\log
P_{\theta}(x)\,(W(x)-1)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}$
$\displaystyle=\operatorname*{arg\,max}\Bigg{\\{}\int\log
P_{\theta}(x)\,P_{\theta_{0}}({\mathrm{d}}x)-\int\log
P_{\theta_{0}}(x)\,P_{\theta_{0}}({\mathrm{d}}x)+\varepsilon\int\log
P_{\theta}(x)\,(W(x)-1)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}$ (because the
added term does not depend on $\theta$)
$\displaystyle=\operatorname*{arg\,max}\Bigg{\\{}-\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)+\varepsilon\int\log
P_{\theta}(x)\,(W(x)-1)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}\ $
$\displaystyle=\operatorname*{arg\,max}\Bigg{\\{}-\frac{1}{\varepsilon}\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)+\int\log
P_{\theta}(x)\,(W(x)-1)\,P_{\theta_{0}}({\mathrm{d}}x)\Bigg{\\}}$
When $\varepsilon\to 0$, the first term exceeds the second one if
$\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)$ is too large
(because $W$ is bounded), and so
$\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)$ tends to $0$. So
we can assume that $\theta$ is close to $\theta_{0}$.
When $\theta=\theta_{0}+\delta\theta$ is close to $\theta_{0}$, we have
$\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)=\frac{1}{2}\sum
I_{ij}(\theta_{0})\,\delta\theta_{i}\,\delta\theta_{j}+O(\delta\theta^{3})$
with $I_{ij}(\theta_{0})$ the Fisher matrix at $\theta_{0}$. (This actually
holds both for $\mathrm{KL}\\!\left(P_{\theta_{0}}\,||\,P_{\theta}\right)$ and
$\mathrm{KL}\\!\left(P_{\theta}\,||\,P_{\theta_{0}}\right)$.)
Thus, we can apply the lemma above using the Fisher metric $\sum
I_{ij}(\theta_{0})\,\delta\theta_{i}\,\delta\theta_{j}$, and working on a
small neighborhood of $\theta_{0}$ in $\theta$-space (which can be identified
with ${\mathbb{R}}^{\dim\Theta}$). The lemma states that the argmax above is
attained at
$\theta=\theta_{0}+\varepsilon\widetilde{\nabla}_{\theta}\int\log
P_{\theta}(x)\,(W(x)-1)\,P_{\theta_{0}}({\mathrm{d}}x)$
up to $O(\varepsilon^{2})$, with $\widetilde{\nabla}$ the natural gradient.
Finally, the gradient cancels the constant $-1$ because
$\int(\widetilde{\nabla}\log P_{\theta})\,P_{\theta_{0}}=0$ at
$\theta=\theta_{0}$. This proves Theorem 13.
### Proof of Theorem 15 (IGO, CEM and IGO-ML)
Let $P_{\theta}$ be a family of probability distributions of the form
$P_{\theta}(x)=\frac{1}{Z(\theta)}\exp\left(\sum\theta_{i}T_{i}(x)\right)\,H({\mathrm{d}}x)$
where $T_{1},\ldots,T_{k}$ is a finite family of functions on $X$ and
$H({\mathrm{d}}x)$ is some reference measure on $X$. We assume that the family
of functions $(T_{i})_{i}$ together with the constant function $T_{0}(x)=1$,
are linearly independent. This prevents redundant parametrizations where two
values of $\theta$ describe the same distribution; this also ensures that the
Fisher matrix $\operatorname{Cov}(T_{i},T_{j})$ is invertible.
The IGO update (14) in the parametrization $\bar{T}_{i}$ is a sum of terms of
the form
$\widetilde{\nabla}_{\bar{T}_{i}}\ln P(x).$
So we will compute the natural gradient $\widetilde{\nabla}_{\bar{T}_{i}}$ in
those coordinates. We first need some general results about the Fisher metric
for exponential families.
The next proposition gives the expression of the Fisher scalar product between
two tangent vectors $\delta P$ and $\delta^{\prime}P$ of a statistical
manifold of exponential distributions. It is one way to express the duality
between the coordinates $\theta_{i}$ and $\bar{T}_{i}$ (compare [AN00, (3.30)
and Section 3.5]).
###### Proposition 21.
Let $\delta\theta_{i}$ and $\delta^{\prime}\theta_{i}$ be two small variations
of the parameters $\theta_{i}$. Let $\delta P(x)$ and $\delta^{\prime}P(x)$ be
the resulting variations of the probability distribution $P$, and
$\delta\bar{T}_{i}$ and $\delta^{\prime}\bar{T}_{i}$ the resulting variations
of $\bar{T}_{i}$. Then the scalar product, in Fisher information metric,
between the tangent vectors $\delta P$ and $\delta^{\prime}P$, is
$\langle\delta
P,\delta^{\prime}P\rangle=\sum_{i}\delta\theta_{i}\,\delta^{\prime}\bar{T}_{i}=\sum_{i}\delta^{\prime}\theta_{i}\,\delta\bar{T}_{i}.$
###### Proof.
By definition of the Fisher metric:
$\displaystyle\langle\delta P,\delta^{\prime}P\rangle$
$\displaystyle=\sum_{i,j}I_{ij}\,\delta\theta_{i}\,\delta^{\prime}\theta_{j}$
$\displaystyle=\sum_{i,j}\delta\theta_{i}\,\delta^{\prime}\theta_{j}\int_{x}\frac{\partial\ln
P(x)}{\partial\theta_{i}}\,\frac{\partial\ln P(x)}{\partial\theta_{j}}\,P(x)$
$\displaystyle=\int_{x}\sum_{i}\frac{\partial\ln
P(x)}{\partial\theta_{i}}\delta\theta_{i}\,\sum_{j}\frac{\partial\ln
P(x)}{\partial\theta_{j}}\delta^{\prime}\theta_{j}\,P(x)$
$\displaystyle=\int_{x}\sum_{i}\frac{\partial\ln
P(x)}{\partial\theta_{i}}\,\delta\theta_{i}\;\delta^{\prime}(\ln P(x))\,P(x)$
$\displaystyle=\int_{x}\sum_{i}\frac{\partial\ln
P(x)}{\partial\theta_{i}}\,\delta\theta_{i}\;\delta^{\prime}P(x)$
$\displaystyle=\int_{x}\sum_{i}(T_{i}(x)-\bar{T}_{i})\delta\theta_{i}\;\delta^{\prime}P(x)\qquad\text{by
\eqref{eq:gradexp}}$
$\displaystyle=\sum_{i}\delta\theta_{i}\left(\int_{x}T_{i}(x)\,\delta^{\prime}P(x)\right)-\sum_{i}\delta\theta_{i}\bar{T}_{i}\int_{x}\delta^{\prime}P(x)$
$\displaystyle=\sum_{i}\delta\theta_{i}\,\delta^{\prime}\bar{T}_{i}$
because $\int_{x}\delta^{\prime}P(x)=0$ since the total mass of $P$ is $1$,
and $\int_{x}T_{i}(x)\,\delta^{\prime}P(x)=\delta^{\prime}\bar{T}_{i}$ by
definition of $\bar{T}_{i}$. ∎
###### Proposition 22.
Let $f$ be a function on the statistical manifold of an exponential family as
above. Then the components of the natural gradient w.r.t. the expectation
parameters are given by the vanilla gradient w.r.t. the natural parameters:
$\widetilde{\nabla}_{\bar{T}_{i}}f=\frac{\partial f}{\partial\theta_{i}}$
and conversely
$\widetilde{\nabla}_{\theta_{i}}f=\frac{\partial f}{\partial\bar{T}_{i}}.$
(Beware this does _not_ mean that the gradient ascent in any of those
parametrizations is the vanilla gradient ascent.)
We could not find a reference for this result, though we think it known.
###### Proof.
By definition, the natural gradient $\widetilde{\nabla}f$ of a function $f$ is
the unique tangent vector $\delta P$ such that that, for any other tangent
vector $\delta^{\prime}P$, we have
$\delta^{\prime}f=\langle\delta P,\delta^{\prime}P\rangle$
with $\langle\cdot,\cdot\rangle$ the scalar product associated with the Fisher
metric. We want to compute this natural gradient in coordinates $\bar{T}_{i}$,
so we are interested in the variations $\delta\bar{T}_{i}$ associated with
$\delta P$.
By Proposition 21, the scalar product above is
$\langle\delta
P,\delta^{\prime}P\rangle=\sum\delta\bar{T}_{i}\,\delta^{\prime}\theta_{i}$
where $\delta\bar{T}_{i}$ is the variation of $\bar{T}_{i}$ associated with
$\delta P$, and $\delta^{\prime}\theta_{i}$ the variation of $\theta_{i}$
associated with $\delta^{\prime}P$.
On the other hand we have $\delta^{\prime}f=\sum_{i}\frac{\partial
f}{\partial\theta_{i}}\,\delta^{\prime}\theta_{i}$. So we must have
$\sum_{i}\delta\bar{T}_{i}\,\delta^{\prime}\theta_{i}=\sum_{i}\frac{\partial
f}{\partial\theta_{i}}\,\delta^{\prime}\theta_{i}$
for any $\delta^{\prime}P$, which leads to
$\delta\bar{T}_{i}=\frac{\partial f}{\partial\theta_{i}}$
as needed. The converse relation is proved _mutatis mutandis_. ∎
Back to the proof of Theorem 15. We can now compute the desired terms:
$\widetilde{\nabla}_{\bar{T}_{i}}\ln P(x)=\frac{\partial\ln
P(x)}{\partial\theta_{i}}=T_{i}(x)-\bar{T}_{i}$
by (16). This proves the first statement (27) in Theorem 15 about the form of
the IGO update in these parameters.
The other statements follow easily from this together with the additional fact
(26) that, for any set of weights $a_{i}$ with $\sum a_{i}=1$, the value
$T^{\ast}=\sum_{i}a(i)T(x_{i})$ is the maximum likelihood estimate of
$\sum_{i}a(i)\log P(x_{i})$.
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|
arxiv-papers
| 2011-06-19T06:18:07 |
2024-09-04T02:49:19.798137
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Yann Ollivier (LRI, INRIA Saclay - Ile de France), Ludovic Arnold\n (LRI, INRIA Saclay - Ile de France), Anne Auger (INRIA Saclay - Ile de\n France), Nikolaus Hansen (LRI, INRIA Saclay - Ile de France, MSR - INRIA)",
"submitter": "Yann Ollivier",
"url": "https://arxiv.org/abs/1106.3708"
}
|
1106.3712
|
# Gap symmetry and structure of Fe-based superconductors
P.J. Hirschfeld Department of Physics, University of Florida, Gainesville,
Florida 32611, USA pjh@ufl.edu M.M. Korshunov Department of Physics,
University of Florida, Gainesville, Florida 32611, USA L.V. Kirensky
Institute of Physics, Siberian Branch of Russian Academy of Sciences, 660036
Krasnoyarsk, Russia korshunov@phys.ufl.edu I.I. Mazin Code 6393, Naval
Research Laboratory, Washington, D.C. 20375, USA mazin@nrl.navy.mil
###### Abstract
The recently discovered Fe pnictide and chalcogenide superconductors display
low temperature properties suggesting superconducting gap structures which
appear to vary substantially from family to family, and even within family as
a function of doping or pressure. We propose that this apparent
nonuniversality can actually be understood by considering the predictions of
spin fluctuation theory and accounting for the peculiar electronic structure
of these systems, coupled with the likely “sign-changing $s$-wave” ($s_{\pm}$)
symmetry. We review theoretical aspects, materials properties, and
experimental evidence relevant to this suggestion, and discuss which further
measurements would be useful to settle these issues.
###### type:
Review Article
###### Contents
1. 1 Introduction.
1. 1.1 Aim and scope of this article.
2. 1.2 Fe-based superconductors.
1. 1.2.1 Comparison with cuprates.
2. 1.2.2 Comparison with MgB2.
3. 1.2.3 Conceptual importance.
4. 1.2.4 Gap symmetry and structure.
2. 2 Electronic structure.
1. 2.1 First principles.
1. 2.1.1 General properties of FeBS electronic structure.
2. 2.1.2 Limitations of DFT calculations.
2. 2.2 Minimal band models.
3. 3 Theoretical background.
1. 3.1 Spin fluctuation pairing.
2. 3.2 Alternative approaches.
3. 3.3 Multiband BCS theory.
4. 3.4 Disorder in multiband superconductors.
1. 3.4.1 Intra- vs. interband scattering.
2. 3.4.2 Effect on $T_{c}$.
3. 3.4.3 Anisotropic states.
4. 3.4.4 Single impurity problem.
5. 3.4.5 Magnetic impurities.
6. 3.4.6 Orbital effects.
5. 3.5 Dimensionality.
4. 4 Gap symmetry.
1. 4.1 Triplet or singlet?
2. 4.2 Chiral or not?
3. 4.3 $d$ or $s$?
5. 5 Gap structure.
1. 5.1 Does the gap in FeBS change sign?
1. 5.1.1 Spin-resonance peak.
2. 5.1.2 Josephson junctions.
3. 5.1.3 Quasiparticle interference.
4. 5.1.4 Coexistence of magnetism and superconductivity.
2. 5.2 Evidence for low-energy subgap excitations.
1. 5.2.1 Penetration depth.
2. 5.2.2 Specific heat.
3. 5.2.3 Thermal conductivity.
4. 5.2.4 The ARPES “paradox”.
5. 5.2.5 NMR $1/T_{1}$.
6. 5.2.6 Electronic Raman scattering.
3. 5.3 Alkali-intercalated iron selenide
4. 5.4 Differences among materials: summary.
6. 6 Conclusions.
* Satisfactoriness has to be measured by a multitude of standards, of which some, for aught we know, may fail in any given case; and what is more satisfactory than any alternative in sight, may to the end be a sum of pluses and minuses, concerning which we can only trust that by ulterior corrections and improvements a maximum of the one and a minimum of the other may some day be approached.
William James, Meaning of Truth
## 1 Introduction.
### 1.1 Aim and scope of this article.
The iron arsenide superconductor LaFeAsO with critical temperature 26K was
discovered in 2008 by Hideo Hosono and collaborators [1]. Within two months,
materials based on substitution of La with other rare earths had been
synthesized, raising the critical temperature of Fe-based superconductors
(FeBSs) to 55K. This rapid sequence of discoveries captured the attention of
the high-temperature superconductivity community. The following three years
saw the discovery of several related families of materials, the rapid
calculation of their electronic structure within density functional theory
(DFT), and the development of microscopic models for superconductivity largely
based on these DFT calculations. The existence of a second class of high-
temperature superconductors is generally agreed to be important not only for
the possible existence of materials with even higher $T_{c}$’s within the same
class of Fe-based materials, but because the comparison with the cuprates can
allow one to potentially understand the essential ingredients of high-
temperature superconductivity. Because of the extremely rapidly advancing
nature of the field, the perils of writing a review are obvious. Several
authors have nevertheless recently attempted to summarize the status of
research in this area, and we have benefited greatly from the existence of
these works [2, 3, 4, 5, 6, 7, 8, 9, 10].
We intend in this smaller scope review to focus on one particular question
among the several fascinating issues surrounding the Fe-based superconductors,
namely the symmetry and structure of the superconducting gap. In the study of
cuprate superconductors, the $d_{x^{2}-y^{2}}$ symmetry of the gap, with $\cos
k_{x}-\cos k_{y}$ structure, was empirically established soon after high
quality samples were prepared, by penetration depth, ARPES, NMR and phase
sensitive Josephson tunneling experiments. After three years of intensive
research on the Fe-based superconductors, no similar consensus on any
universal gap structure has been reached, and there is strong evidence that
small differences in electronic structure can lead to strong diversity in
superconducting gap structures, including gaps with nodes in some and full
gaps in other materials. The actual symmetry class of most of the materials
may be the same, of generalized $A_{1g}$ ($s$-wave) type, probably involving a
sign change of the order parameter between Fermi surface sheets in most
materials. In addition, there have been recent suggestions that some related
materials, furthest from the nearly compensated semimetal band structure of
the originally discovered compounds, may have $d$-wave symmetry. Understanding
both the symmetry character of the superconducting ground states and the
detailed structure should provide clues to the microscopic pairing mechanism
in the pnictides and thereby a deeper understanding of the phenomenon of high-
temperature superconductivity.
A complete review of the literature of even the more focussed problem of gap
structure we consider here is beyond the scope of our paper. We have attempted
to select those works we find most relevant (or at least sufficiently
representative) to the questions we believe to be important:
* •
What do experiments tell us about the pairing symmetry and gap structure of
the Fe-based materials, and what systematic trends can one identify?
* •
How do changes in electronic structure drive changes in gap symmetry and
structure?
* •
What physical effect drives pairing in Fe-based materials? Is more than one
important, at least in some materials?
* •
What role does disorder play, and how is it related to changes in carrier
densities, electronic structure, and pairing interactions?
If any works in this category have been omitted or slighted, we beg the
authors will attribute it to ignorance or haste rather than malice!
### 1.2 Fe-based superconductors.
#### 1.2.1 Comparison with cuprates.
High-$T_{c}$ cuprates are known for their high critical temperature,
unconventional superconducting state, and unusual normal state properties. The
Fe-based superconductors (FeBS), with $T_{c}$ up to 55K in SmFeAsO1-xFx stand
in second place after cuprates and 15K above MgB2. When superconductivity in
the FeBS was discovered, the question immediately arose: how similar are they
to cuprates? Let us compare some of their properties.
Both cuprates and FeBS have 2-dimensional lattices of $3d$ transition metal
ions as the building blocks. In both cases orthorhombic distortions can be
present at small doping. The main structural difference in these planes is
that the $2p$-ligands lie very nearly in the plane with the Cu in cuprates,
while in FeBS As, P, Se, or Te lie in nearly tetrahedral positions above and
below the Fe plane. The in-plane subset of Cu $d$-orbitals $e_{g}$ are both
present near the Fermi level, and of these, the planar $d_{x^{2}-y^{2}}$ is
quite dominant (see, however, [11]), allowing in principle the reduction of
the multiband electronic structure to a low-energy effective one-band model.
In FeBS, on the other hand, out-of-plane As hybridize well with the $t_{2g}$
Fe $d$-orbitals and all three of them have weight at the Fermi surface. In
addition (as opposed to the cuprates), there is substantial overlap between
the $d$-orbitals. The minimal model is then essentially multiband and that
makes FeBS in this respect more similar, e.g. to ruthenates than to cuprates.
At first glance, the phase diagrams of cuprates and many FeBS are similar. In
both cases the undoped compounds exhibit antiferromagnetism, which vanishes
with doping; superconductivity appears at some nonzero doping and then
disappears, such that $T_{c}$ forms a “dome”. While in cuprates the long range
ordered Néel phase vanishes before superconductivity appears, in FeBS the
competition between these orders can take several forms. In LaFeAsO, for
example, there appears to be a first order transition between the magnetic and
superconducting states at a critical doping value, whereas in the 122 systems
the superconducting phase coexists with magnetism over a finite range and then
persists to higher doping. It is tempting to conclude that the two classes of
superconducting materials show generally very similar behavior, but there are
profound differences as well. The first striking difference is that the
undoped cuprates are Mott insulators, but FeBS are metals. This suggests that
the Mott-Hubbard physics of a half-filled Hubbard model is not a good starting
point for pnictides, although some authors have pursued strong-coupling
approaches. It does not of course exclude effects of correlations in FeBS, but
they may be moderate or small. In any case, DFT-based approaches describe the
observed Fermi surface and band structure reasonably well for the whole phase
diagram, contrary to the situation in cuprates, especially in undoped and
underdoped regimes.
The second important difference pertains to normal state properties.
Underdoped cuprates manifest pseudogap behavior in both one-particle and two-
particle charge and spin observables, as well as a variety of competing
orders. At least for hole-doped cuprates, a strange metal phase near optimal
doping is characterized by linear-$T$ resistivity over a wide range of
temperatures. In FeBS, different temperature power laws for the resistivity,
including linear $T$-dependence of the resistivity for some materials, have
been observed near optimal doping and interpreted as being due to multiband
physics and interband scattering [12]. The FeBS do not manifest a robust
pseudogap behavior in a wide variety of observable properties.
The mechanism of doping deserves additional discussion. Doping in cuprates is
accomplished by replacing one of the spacer ions with another one with
different valence or adding extra out-of-plane oxygen, e.g. La2-xSrxCuO2,
Nd2-xCexCuO2, and YBa2Cu3O6+δ. The additional electron or hole is then assumed
to dope the plane in an itinerant state. In FeBS, the nature of doping is not
completely understood: similar phase diagrams are obtained by replacing the
spacer ion as in LaFeAsO1-xFx and Sr1-xKxFe2As2, or by in-plane substitution
of Fe with Co or Ni as in Ba(Fe1-xCox)2As2 and Ba(Fe1-xNix)2As2, or by
replacing Ba with K, Ba1-xKxFe2As2. Whether these heterovalent substitutions
dope the FeAs or FeP plane as in the cuprates was not initially clear [13],
but now it is well established that they affect the Fermi surface consistent
with the formal electron count doping [14, 15]. Another mechanism to vary
electronic and magnetic properties is via the possibility of isovalent doping
with phosphorous in BaFe2(As1-xPx)2 or ruthenium in BaFe2(As1-xRux)2.
‘Dopants’ can act as potential scatterers and change the electronic structure
because of differences in ionic sizes or simply by diluting the magnetic ions
with nonmagnetic ones. But crudely the phase diagrams of all FeBS are quite
similar, challenging workers in the field to seek a systematic structural
observable which correlates with the variation of $T_{c}$. Among several
proposals, the height of the pnictogen or chalcogenide above the Fe plane has
frequently been noted as playing some role in the overall doping dependence
[16, 17, 18].
It is well established that the superconducting state in the cuprates is
universally $d$-wave. By contrast, we review evidence below that the gap
symmetry and/or structure of the FeBS can be quite different from material to
material. Nevertheless, it seems quite possible that the ultimate source of
the pairing interaction in both systems is fundamentally similar, although
essential details such as pairing symmetry and the gap structure in the FeBS
depend on the FS geometry, orbital character, and degree of correlations.
#### 1.2.2 Comparison with MgB2.
MgB2 was the first example of multigap superconductivity (or at least, the
first one recognized as such). There is little doubt that this property is
shared by FeBS, as discussed below; therefore it is instructive to see which
multiband features have been discovered in MgB2 and what similarities can be
found in FeBS.
The thermodynamic properties of MgB2 show very characteristic behavior which
can easily be understood within multiband BCS theory (see Section 3.3)
assuming a weak coupling between bands. At the critical temperature the larger
gap is clearly visible in thermodynamics, and at a lower temperature, roughly
corresponding to the critical temperature of the weaker band alone, the second
gap becomes manifest in thermodynamic properties. The second gap, while
formally appearing at the same temperature as the first gap, remains very
small until a much lower temperature. Such considerations of course lead one
to examine also the opposite situation where the bands are strongly coupled
(as in essentially all theories of superconductivity in FeBS). In this case
both gaps gradually diminish as T is raised, but one or both may show non-BCS
behavior, and the thermodynamic properties cannot be accurately described by
one gap; the sum of two gaps, on the other hand, can provide a realistic
description. This is indeed the case in many FeBS, as probed by specific heat,
penetration depth, NMR relaxation rate etc. (see Section 5). On the other
hand, the picture is additionally clouded, as compared to MgB2, because of
presumably larger gap anisotropy and pair-breaking effects of impurity
scattering, as discussed below.
Another manifestation of multiband superconductivity is found in the thermal
conductivity. The reduced thermal conductivity $\kappa/T$ is, generally
speaking, zero at $T=0$, if the Fermi surface is fully gapped in the
superconducting state (although pair breaking effects due to magnetic
impurities may, in principle, create mobile quasiparticle states with zero
energy [19]). In MgB2, as well as in many FeBS, this is the case. Upon
applying magnetic field, Abrikosov vortices form in the system. As soon as
these vortices begin to overlap, the thermal conductivity starts growing. This
happens at field on the order of $H_{c2}/3$. Now, if there are two gaps in the
system, one substantially smaller than the other, one may think that the
vortex overlap will start at much smaller fields. Indeed, the distance between
the vortices is proportional to $H^{-1/2}$, while their size is defined by the
coherence length and thus inversely proportional to $\Delta$. So, the critical
field where the “weaker” band will be smaller than that for the “stronger”
band by a factor of, roughly. $(\Delta_{1}/\Delta_{2})^{2}$, which is, for
MgB2, about 10. So, the argument goes [20], one cannot observe the flat low-
field part of $\kappa(H)/T$, and experimentally the dependence looks linear at
the smallest accessible $H$. Of course, one must see flattening at $H\lesssim
H_{c2}/30$, but so far nobody has observed this. We only emphasize here that
many FeBS studied by this technique show a linear increase of
$\kappa/T|_{T\rightarrow 0}$ with $H$ at small $H$, which, in a traditional
multiband interpretation, suggest a considerable disparity between the largest
and the smallest gaps, or possibly strong gap anisotropy.
Another interesting lesson that one can derive from the MgB2 studies is
negative. One of the reasons why a number of theorists were initially
reluctant to accept the two-gap scenario for this material is the fact that
nonmagnetic impurities, in the Abrikosov-Gor’kov theory, should suppress
$T_{c}$ linearly with fairly large slope until the gaps are averaged. This
effect has not been observed in MgB2. While there have been reasonable
explanations of why particular impurities may have little effect on $T_{c}$
[21], in retrospect it is clear that the impurity effect is weaker than that
expected from the theory in many different cases.
Finally, it is worth looking back at the normal state of MgB2. Detailed
quantum oscillation studies [22] prove unambiguously that the two band systems
($\pi$ and $\sigma$) are shifted with respect to each other by up to 100 meV.
Similarly, in FeBS quantum oscillations shows that the hole bands and the
electron bands are shifted with respect to each other by up to 70 meV, so that
the hole and the electron Fermi surface become smaller relative to DFT
predictions. This holds both in magnetic [23] and nonmagnetic [24] cases. It
has been ascribed to correlation effects [25], but the comparison with MgB2
demonstrates that these effects beyond LDA are, if anything, less severe than
in MgB${}_{2},$ which is not generally considered to be a correlated metal.
#### 1.2.3 Conceptual importance.
While the FeBS may not signify a particular advance in terms of practical
applications—their $T_{c}$ is only 15K higher than that of MgB2, and, just as
the cuprates, they are expensive to make and difficult to work on—their
conceptual value is hard to overestimate. Indeed, fullerides and MgB2 clearly
belong to a different class than the cuprates, being in certain respects
exotic, but still phonon-driven superconductors. Not surprisingly, there had
been a growing feeling among physicists that phonon superconductivity will
probably never grow past 50-60K, while true high-temperature superconductivity
is a strong-correlation phenomenon limited to the unique family of layered
cuprates. It had been justly pointed out that the CuO2 layers have many unique
properties, largely coming from the fact that Cu is the last $3d$ transition
metal and as such is by far the most strongly correlated of all, yet its
simple one-orbital electronic structure provided for a simple and large Fermi
surface when doped. One can point to many aspects in which cuprates are
unique, and many people did.
What the discovery of the FeBS brought onto the table was the understanding
that however unique cuprates may be, these features are not prerequisites for
non-phonon, high-temperature superconductivity. And, if that is true, there
are likely many other crystallochemical families to be discovered, some of
which may have higher critical temperature or be better suited for
applications than cuprates and FeBS.
In a twisted way, we are lucky that FeBS and cuprate are so different in so
many aspects. This makes it more reasonable to look for those few
commonalities which exist and to assume, even without profound theoretical
insight, that these commonalities are important for high $T_{c}$. Some of
these obviously include proximity to magnetism and quantum criticality, or
substantial anisotropy of the Fermi surface (quasi-2D) and it is has already
been argued by many that one should look for a combination of these factors to
search for novel superconductors [26].
#### 1.2.4 Gap symmetry and structure.
*
Figure 1: Cartoon of order parameters under discussion in the Fe-pnictide
superconductors represented in the 2-dimensional, 1-Fe Brillouin zone (see
Section 2). Different colors stands for different signs of the gap.
The group theoretical classification of gap structures in unconventional
superconductors is somewhat arcane and has been amply reviewed elsewhere [27].
Here we present the simplest notions relevant to the discussion of symmetry
and structure of the order parameters under discussion in the Fe-based
superconductors at present. In the absence of spin-orbit coupling, the total
spin of the Cooper pair is well-defined and can be either $S=1$ or $S=0$.
Experimental data appear to rule out spin triplet states (see Section 4), so
we focus on the spin singlet case. We focus first on simple tetragonal point
group symmetry. In a 3D tetragonal system, group theory allows only for five
one-dimensional irreducible representations: $A_{1g}$ (“$s$-wave”), $B_{1g}$
(“$d$-wave” [$x^{2}-y^{2}$]), $B_{2g}$ (“$d$-wave” [$xy$]), $A_{2g}$
(“$g$-wave” [$xy(x^{2}-y^{2})$]), and $E_{g}$ (“$d$-wave” [$xz,yz$]) according
to how the order parameter transforms under rotations by 90∘ and other
operations of the tetragonal group. In figure 1 we have illustrated two of
these symmetries, namely $s$-wave and $d_{x^{2}-y^{2}}$-wave. Note that the
$s_{++}$ state and $s_{\pm}$ states represented all have the same symmetry,
i.e. neither changes sign if the crystal axes are rotated by 90∘. By contrast,
the $d$-wave state changes sign under a 90∘ rotation. Note further that the
mere existence of the single hole and single electron pocket shown lead to new
ambiguities in the sign structure of the various states. In addition to a
global change of sign, which is equivalent to a gauge transformation, one can
have individual rotations on single pockets and still preserve symmetry; for
example, if one rotates the gap on the hole pocket for the $d$-wave case in
figure 1 by 90∘ but keeps the electron pocket signs fixed, it still represents
a $B_{1g}$ state. Note also that $B_{2g}$ states, while not shown in the
figure, are also possible by symmetry and would have nodes on the electron
pockets. Further, more complicated, gap functions with differing relative
phases on the different pockets become possible when more pockets are present,
and/or when 3D effects are included (See Section 3.5).
These symmetry properties are distinct from gap structure, a term we use to
designate the $k$-dependent variation of an order parameter within a given
symmetry class. Gaps with the same symmetry may have very different
structures, as also illustrated in the figure, where three different types of
$s$-wave states are shown. The isotropic, fully gapped $s_{++}$ and $s_{\pm}$
states differ only by a relative phase of $\pi$ in the latter case between the
hole and electron pockets. On the other hand, in the nodal $s$ case, the gap
is shown vanishing at certain points on the electron pockets. This particular
case shows a case we will sometimes refer to as “nodal $s_{\pm}$”, in that the
sign on the hole pockets is still opposite the average sign on the electron
pockets. Nodes of this type are sometimes described as “accidental”, since
their existence is not dictated by symmetry, but rather by the details of the
pair interaction. As such, they can be removed continuously, resulting in
either an $s_{++}$ or an $s_{\pm}$ state.
## 2 Electronic structure.
### 2.1 First principles.
#### 2.1.1 General properties of FeBS electronic structure.
The basic crystallographic element of the FeBS is the FeAs (where instead of
As one can also have P, Se or Te) plane with an ${a}\times{a}$ square plane of
Fe ions, and two $\tilde{a}\times\tilde{a}$ square planes of As above and
below (where $\tilde{a}={a}\sqrt{2}).$ The minimal unit cell of the entire
FeAs plane is, therefore, also $\tilde{a}\times\tilde{a}$ and includes two
formula units. In some, but not all cases the low-energy part of the
electronic structure can be “unfolded” into a Brillouin zone (BZ) which is
twice as large, corresponding to the ${a}\times{a}$ unit cell, so that the
real band structure can be recovered by folding the 2D Brillouin zone in such
a way that the “unfolded” $\mathrm{X}=(\pi/a,0)$ and $\mathrm{Y}=(0,\pi/a)$
points fold on top of each other, forming the
$\tilde{\mathrm{M}}=(\pi/\tilde{a},\pi/\tilde{a})$ point in the small
Brillouin zone. Here and throughout the article we always use the untilded
notations in the one-Fe unit cell and the “unfolded” Brillouin zone, and the
tilded notations in the crystallographic unit cells and the corresponding
Brillouin zone.
Despite the large variety of crystal structure and chemical compositions, all
FeBS share the same gross features of the electronic structure. These can be
listed as follows:
1\. In the nonmagnetic state, the band structure is of semimetallic nature,
with two or more hole band crossing the Fermi level near the $\Gamma$ point
and two electron bands crossing the Fermi level near the $\tilde{M}$ point.
2\. Two hole bands, universally present in all superconducting compositions,
are formed by the $xz$ and $yz$ derived Fe band, which are degenerate (without
the spin-orbit effects) at the $\Gamma$ point, but split apart (and acquire
some $z^{2}$ character) in a relatively uniform manner away from it. The two
electron bands take their origin from the downfolding effect described above,
and are formed mostly by the ${x}{z}$ and ${y}{z}$ orbitals, respectively,
plus the ${x}{y}$ orbital.
3\. As a result, there are always at least two hole Fermi surfaces and two
electron Fermi surfaces, which are well separated in the reciprocal space.
Moreover, their respective centers are removed from each other exactly by
$\bi{Q}=\widetilde{(\pi/\tilde{a},\pi/\tilde{a})}=(\pi/a,0)$. In general, the
Fermi surfaces have sufficiently different shapes so that one cannot speak of
a good nesting here, only of a quasi-nesting.
On the other hand, many aspects of the electronic structure vary from material
to material. For instance, in some materials another hole band appears, which
may be either of $z^{2}$ character, in which case it is substantially 3D, or
${x}{y},$ which is even more 2D than the $xz/yz$ bands.
Different FeBS may have different degree of charge doping, of either sign,
ranging from 0.5 h to 0.15 e per Fe. It appears though that in all this
compositional range, the general structure of the FS almost always survives.
That is, while either the electron or the hole FS shrinks, they never entirely
disappear in the superconducting range of dopings, even though the nesting
conditions may have drastically deteriorated. It worth remembering that in
strongly anisotropic systems the size of a FS has little correlation with its
density of states at the Fermi level. In addition, recently two FeBS systems
have been discussed that are superconducting but which may lie outside this
“typical” range; in KFe2As2 the Fermi surface pockets near the $\mathrm{X}$
point nearly disappear, while the hole pockets around $\Gamma$ are greatly
expanded, and KFe2Se2 may (or may not; see Section 5.3) be completely lacking
the hole pockets.
By contrast, the $k_{z}$ dispersion can vary substantially from material to
material. This depends mostly on two factors: the thickness of the “filler”
layer between the FeAs or FeSe layer, and on crystallographic symmetry.
Obviously, materials with the 1111 structure are more anisotropic, in fact
nearly 2D, than those with the same $P4/nmm$ symmetry but no filler species,
that is, with the 11 structure. Less obviously, materials with a body center
symmetry, such as 122 ($I4/mmm$), have an additional reason for a 3D
character: the downfolding procedure in that case projects the electronic
states near the $\left({\pi/a},{0},{0}\right)$ point onto the states near
$\left(0,{\pi/a},{\pi/c}\right)$. Crossings occur at general $k$-point, and
therefore hybridization between these states is not forbidden, but depends on
$k_{z}$. As a result these materials tend to be even more 3D than the 11
compounds, despite having a filling monolayer in the former. We explore the
consequences of $k_{z}$ dispersion for superconductivity in Section 3.5.
#### 2.1.2 Limitations of DFT calculations.
First principles calculations were very important in the beginning of the FeBS
era, and they have informed the emerging understanding of the physics of the
FeBS much more than in the case of cuprate superconductors, although due to
somewhat stronger correlations and complexity of materials they have not yet
proven as definitively useful as in MgB2. In FeBS, they successfully predicted
the right topology of the FS [28], as well as the correct magnetic ordering in
the normal state [29]. The most successful proposal regarding the pairing
symmetry so far has been made based on band structure calculations [29]. These
facts have been instrumental boosting the reputation of the band theory in the
superconducting context, but one should remember that while DFT is not a snake
oil, it is not a panacea either, and nor are any of its generalizations such
as DFT+DMFT etc. Let us list below the most important shortcomings and
limitations of the DFT calculations as regards FeBS.
1\. The DFT is, by construction, a mean-field theory (but not a low-energy
theory, as is sometimes incorrectly asserted). It is a more sophisticated mean
field theory than many, for it includes in the energy functional (and thus in
the mean field potential), all correlation effects and integrates in all
fluctuations. On the other hand, the actual implementations of the DFT, such
as the local density approximation or the generalized gradient approximation,
by construction only include those correlations and those fluctuations that
are present in the reference system, the uniform electron gas, at densities
comparable with the electron densities in real solids. Remember that the
uniform electron gas at such densities is very far from magnetism, and even
farther from the electron localization (Wigner crystallization). The
corresponding physics is, therefore, largely missing when DFT is used in a
“local” approximation. This belongs to two major classes: (i) on-site Coulomb
fluctuations, also called Hubbard correlations, which are included in a very
limited way on the level of the Stoner magnetic interaction (reflecting the
first Hund’s rule), and (ii) quantum critical fluctuations; examples of such
are long range ferromagnetic fluctuations in nearly ferromagnetic metals [30].
The hallmark of the former is underestimation of the tendency to magnetism in
a DFT calculation, of the latter - overestimation.
In cuprates, the DFT calculations suffer from the former problem, in FeBS
mostly from the latter. From the density functional point of view, it is
rather curious that despite the fact that the calculated magnetic moments are
large, as opposed to such known cases of near-quantum-critical materials ZrZn2
or Fe3Al, yet the effect of such long-range fluctuations appears to be strong.
The explanation is that magnetic moments in this system are quite soft (in the
calculations they can change from nearly zero to more than a Bohr magneton
depending on the magnetic pattern), and on top of the transverse spin
fluctuations typical for strong antiferromagnets, there are longitudinal
fluctuations characteristic of itinerant magnets and quite efficient in
reducing the ordered magnetic moment. Moreover, there is a possibility that
other, so-called “nematic” fluctuations, may play an additional role in
reducing the ordered moment.
This does not mean that the first DFT problem, underestimation of on-site
Coulomb correlations, is nonexistent in these materials. It is relatively
mild, and secondary compared to the other deficiency, yet it exists and it
manifests itself, for instance, in the bandwidth renormalization. From the
point of view of this Section, the important corollary of the above is that
superconductivity in FeBS develops not on the background of a non-magnetic
state, but of a paramagnetic state which still has fluctuating local
magnetization of Fe ions. Therefore, the calculated bands and the Fermi
surfaces are true only as long as averaging over these fluctuating quasi-local
magnetic moments is equivalent to dropping the spin-dependent part of the
crystal potential entirely. Experimental evidence so far has been
inconclusive. De Haas - van Alphen experiments generally agree well with the
DFT calculation, up to some uniform shift of different bands with respect to
each other and overall mass renormalization. ARPES derived Fermi surfaces,
while conforming with the general topology, predicted by DFT, differ in
details substantially. It is fair to say that that the DFT bands are a
reasonable, but not exceedingly good approximation of the actual band
structure, even after accounting for the bandwidth renormalization and the
band shifts. They appear to be renormalized by a factor up to three, and may
be even larger for some systems, and the renormalization appears to be
stronger as the system approaches the AFM quantum critical point [24]. Also,
additional repulsive interaction between the holes and the electrons seems to
be operative, pushing the (mostly occupied) hole bands down, and (mostly
empty) electron bands up. The last effect is responsible for shrinking all
Fermi surfaces compared to DFT, but this effect is weak (but is, again,
stronger near the quantum critical point) [25].
The band renormalization comes from both the on-site and long-range
fluctuations. Existing DMFT calculations, while qualitatively agreeing among
themselves, disagree on the exact share of the total mass renormalization
provided by the on-site vs. long range fluctuations. Indeed, all groups find
that the effect comes predominantly from Hund’s $J$, and not Hubbard’s $U$,
and that the 11 family is substantially more correlated than other families.
At the same time, the Rutgers group has obtained mass renormalizations closely
matching the experiment, leaving basically no room for the long-range
fluctuations, while the other DMFT groups’ results suggest that both effects
provide comparable contributions to the total renormalization in pnictides
[31, 32, 33].
### 2.2 Minimal band models.
On the basis of the DFT band structure one can make a simplified model which
then can be studied by sophisticated theoretical methods like a Green’s
function formalism. There is always a trade off between complexity of a model
and physical effects captured by it. Here we discuss several popular models
with increasing levels of complexity.
*
Figure 2: (a) FeAs lattice indicating As above and below the Fe plane. Dashed
green and solid blue squares indicate 1- and 2-Fe unit cells, respectively.
(b) Schematic 2-dimensional Fermi surface in the 1-Fe Brillouin zone whose
boundaries are indicated by a green dashed square. Arrow indicates folding
wave vector $\bi{Q}_{F}$. (c) Fermi sheets in the folded Brillouin zone whose
boundaries are now shown by a solid blue square.
According to DFT, FeBS are essentially multiband systems and a minimal model
must include both hole and electron bands. The first complication comes from
the As, which forms square lattice planes between the lattice sites of, but
also above and below, the square lattice of Fe. This alternating pattern of As
makes the correct real space unit cell twice the one-Fe unit cell. The
corresponding 2-Fe BZ is twice as small as the 1-Fe one and called the “folded
BZ”, see figure 2. For the simplest case of single-layer FeBS the folding wave
vector is 2-dimensional and equal to $\bi{Q}_{F}=(\pi,\pi)$. Most experimental
results and DFT band structures are reported in the folded BZ since
crystallographically it is the correct one. However, some experiments
sensitive to the Fe positions, like neutron scattering on Fe moments, may have
more meaning in the 1-Fe (“unfolded”) zone. Theoretically, the virtue of using
this zone is its simplicity, since the number of bands is smaller by a factor
of two. From the point of view of symmetry, one might think of the As height
as a perturbation: if $z_{As}=0$, the electronic structure is correctly
reproduced in the unfolded BZ, and when $z_{As}$ increases, the procedure of
unfolding becomes less and less justified from the structural point of view.
Despite the fact that the As displacements from the Fe plane are not small,
use of the Fe-only band structure in the 1-Fe zone is frequently a good
approximation (see, however, Section 5.3) since DFT calculations predict the
band structure near the Fermi level to be mostly due to Fe $d$-bands, while
the As $p$-bands are about 2eV below [28].
The simplest model accounting for distinct electron and hole Fermi surfaces
would be a model in the 1-Fe zone with parabolic dispersions [34, 35],
$H=\sum_{\bi{k},\sigma,i=\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}}\varepsilon^{i}_{\bi{k}}c_{i\bi{k}\sigma}^{\dagger}c_{i\bi{k}\sigma}.$
(1)
Here, $c_{i\bi{k}\sigma}$ is the annihilation operator for an electron with
momentum $\bi{k}$, spin $\sigma$, and band index $i$,
$\varepsilon^{\alpha_{1,2}}_{\bi{k}}=-\frac{\bi{k}^{2}}{2m_{1,2}}+\mu$,
$\varepsilon^{\beta_{1}}_{\bi{k}}=\frac{(k_{x}-\pi/a)^{2}}{2m_{x}}+\frac{k_{y}^{2}}{2m_{y}}-\mu$,
and
$\varepsilon^{\beta_{2}}_{\bi{k}}=\frac{k_{x}^{2}}{2m_{y}}+\frac{(k_{y}-\pi/a)^{2}}{2m_{x}}-\mu$
are the dispersions of hole $\alpha_{i}$ and electron $\beta_{i}$ bands.
The electron pockets have to be distinct since they are located in the
different points of the Brillouin zone, but since the two hole pockets around
$\Gamma$ point are close in size, some physics can be captured by
approximating them as one band. After the folding procedure, this model
produces a Fermi surface topology which agrees with the results of DFT
calculations, see figure 2. This is useful for qualitative analysis of the
physics near the Fermi level, like magnetic susceptibility and formation of
the SDW state. The simplest extensions are the tight-binding models which can
be formulated in either the folded [36] or the unfolded BZ [37]. These can
reproduce correctly the Fermi surface and Fermi velocities, but neglect the
orbital character of the bands. The orbital character of electrons in
different bands is important e.g. for a correct analysis of scattering in
particle-particle and particle-hole channels. Furthermore, local Coulomb
interactions like Hubbard $U$ and Hund’s exchange $J$ are momentum-independent
only in the orbital representation.
According to DFT analysis, the band structure near the Fermi level consists
mainly of Fe $d$-orbitals, since the out-of-plane As $p$ orbitals hybridize
most effectively with Fe orbitals with both out-of-plane and in-plane
components. These conditions are satisfied for $d_{xz,yz}$ orbitals, so their
contribution at the Fermi level is dominant. The second largest contribution
comes from the $d_{xy}$ orbitals. The other two $d$ orbitals,
$d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$, also contribute at low energies, but
their weight at the Fermi surface is minimal except in some materials near the
top of the Brillouin zone.
A possible minimal model for the FeBS is then one which includes two orbitals,
$d_{xz}$ and $d_{yz}$ [38]. This has the virtue of being simple while having
mostly correct orbital character along the Fermi surface – $(0,\pi)$ and
$(\pi,0)$ pockets have dominant $d_{xz}$ and $d_{yz}$ contributions,
respectively, and hole pockets with a mixture of these two orbitals. This
model has several significant disadvantages [39, 40], however. The first one
is that the Fermi velocities are incorrect, leading to incorrect tendencies
towards superconductivity and SDW formation. Second, it is missing small
patches of $d_{xy}$ character at the tips of the electron pockets, which can
be important for node formation [41] and transport [42]. Finally, a serious
flaw is the position of the larger hole pocket which is located at $(\pi,\pi)$
point of the 1-Fe zone. After the folding, this Fermi surface sheet is
centered at the $\Gamma$ point and resembles DFT results for the Fermi
surface. But the fact that the two bands forming hole sheets are not
degenerate at the $(0,0)$ point of the unfolded zone contradicts the symmetry
of the DFT wave functions. This problem can be adjusted by adding a $d_{xy}$
orbital to the model [43], but this three-orbital model has other pathologies
and fails to reproduce the peak at the nesting wave vector $\bi{Q}$ [44],
which is established both experimentally and theoretically. Although the
origin of the problem is not obvious, it is related to the matrix elements of
transformation from the orbital to the band basis.
The next step in the direction of more realistic models is to include four or
all five Fe $t_{2g}$ orbitals. Models of this type [40, 45] work well in
reproducing the DFT Fermi surface and band structure, and they are free from
the disadvantages described above. The kinetic energy in [40] is then given by
the Hamiltonian
$H_{0}=\sum_{\bi{k}\sigma}\sum_{\ell\ell^{\prime}}\left(\xi_{\ell\ell^{\prime}}(\bi{k})+\epsilon_{\ell}\delta_{\ell\ell^{\prime}}\right)d_{\ell\bi{k}\sigma}^{\dagger}d_{\ell^{\prime}\bi{k}\sigma},$
(2)
where $d_{\ell\bi{k}\sigma}^{\dagger}$ creates a particle with momentum
$\bi{k}$ and spin $\sigma$ in the orbital $\ell$,
$\xi_{\ell\ell^{\prime}}(\bi{k})$ are the hoppings, and $\epsilon_{\ell}$ are
the single-site level energies. This model with the parameters obtained by
Wannier fits to $k_{z}=0$ cuts of the 1111 band structure of Cao et al [46]
gives rise to the Fermi surfaces shown in figure 3 displayed in the unfolded
BZ [40, 47]. The undoped material has completely filled $d^{6}$ orbitals
corresponding to electron number $n=6$. Note for the hole doped case $n=5.95$
shown in figure 3 there is an extra hole FS $\gamma$ around the $(\pi,\pi)$
point. An important role is played by the orbital matrix elements
$a^{\ell}_{\nu}(\bi{k})=\langle\ell|\nu\bi{k}\rangle$ which relate the orbital
and band states. The dominant ($>50\%$) orbital weights
$|a^{\ell}_{\nu}(\bi{k})|^{2}$ on the Fermi surfaces are illustrated in figure
3 by the colors indicated.
*
Figure 3: Fermi sheets of the five-band model in the unfolded BZ for $n=6.03$
(top) and $n=5.95$ (bottom) with colors indicating majority orbital character
(red=$d_{xz}$, green=$d_{yz}$, blue=$d_{xy}$). Note the $\gamma$ Fermi surface
sheet is a hole pocket which appears for $\sim 1\%$ hole doping [47].
The orbital matrix elements and the small patches of the $d_{xy}$ contribution
to the electron sheets play an important role in the formation of nodes in the
superconducting order parameter as will be discussed in Section 3.1. Natural
generalizations of the multiorbital models are 1) to include dispersion along
the $k_{z}$ direction; and 2) to include the proper effects of the 122 body
centered cubic symmetry. These effects appear to have important consequences
for the pairing in Ba1-xKxFe2As2, where effects of three-dimensionality are
more important than in 1111 systems [48, 49].
## 3 Theoretical background.
### 3.1 Spin fluctuation pairing.
It has become commonplace when a new class of superconductors is discovered to
discuss electronic pairing mechanisms as soon as there is some evidence that
the electron phonon mechanism is not strong enough to produce observed
critical temperatures; this was the case in both the cuprates and Fe-based
superconductors. Among many candidates for electronic pairing, Berk-Schrieffer
[50] type spin fluctuation theories are popular because they are relatively
simple to express and give some qualitatively correct results in the well-
known cases of 3He and the cuprates. The interesting history of the
development of this theory in the one-band case has been reviewed by Scalapino
[51]. It is important to keep in mind that this type of description cannot be
regarded as the complete answer even in superfluid 3He, where the true pairing
interaction contains a significant density fluctuation component, while in the
cuprates it is controversial whether the full pairing interaction can be
described by a simple boson exchange theory at all. Nevertheless spin
fluctuation theories can explain the symmetry of the order parameter in both
systems quite well, in part because other interaction channels are projected
out in the ground state. For example, in the cuprates, the $d$-wave nature of
the pair wave function follows from the strongly peaked spin susceptibility at
$(\pi,\pi)$, characteristic of repulsive local interactions between electrons
hopping on a square lattice. In the Fe-based superconductors, the early
realization that the Fermi surface consisted of small, nearly nested electron
and hole pockets led to the analogous anticipation of a strongly peaked
susceptibility near $\bi{Q}=(\pi,0)$, and a corresponding pairing instability
with sign change between electron and hole sheets. Below we illustrate the
basic equations leading to the canonical $d$-wave case within the one-band
Berk-Schrieffer approach, and sketch the generalization to the
multiorbital/multiband case. Many authors have obtained similar results with a
variety of related methods (see Section 3.1, “similar approaches” below).
It is important to emphasize that while spin fluctuation theories come in many
varieties and flavors, they share more commonalities than differences. Indeed,
as will be discussed later in this Section, in the singlet channel spin
fluctuations exchange always leads to a repulsive interaction, and therefore
can only realize sign-changing superconducting states. If this interaction is
sufficiently strong at some particular momentum it will necessarily result in
superconductivity. In case of a single Fermi surface this superconductivity
will necessarily be nodal, usually of a $d$-wave symmetry. Examples of this
situation are high-$T_{c}$ cuprates and, possibly, overdoped KFe2As2 FeBS. On
the other hand, in a multiband system there may be a possibility to avoid
nodes, while still preserving a sign-changing structure. Examples of this are:
$d$-wave superconductivity that can develop in a cubic system with Fermi
surface pockets around the $\mathrm{X}$ points and in a hexagonal system with
pockets around the $\mathrm{M}$ points [52], $d$-wave superconductivity in a
tetragonal system with FS pockets near $\mathrm{X}$/$\mathrm{Y}$ points [45],
$s_{\pm}$ superconductivity proposed for bilayer cuprates (where the bonding
and the antibonding band have opposite signs of the order parameter) [53, 54],
and the electron-hole $s_{\pm}$ superconductivity that can develop in
semimetals [55].
While all these options had been discussed theoretically many years ago, all
of them have been revisited in connection to FeBS. The last option is now the
leading contender for the majority of pnictides and selenides, while $d$-wave
superconductivity has been proposed for KFe2As2 that is on the verge of losing
its semimetallic character (see Section 5.4), a version of the nodeless
$d$-wave state first discussed in [45] has been proposed for Se-based 122
materials (see Section 5.3), and the “bonding-antibonding” nodeless $s_{\pm}$
state, analogous to that discussed in [53, 54] has been also proposed for
these selenides. What is important, however, is that as long as the spin
fluctuations are strong and nonuniform, some superconducting state will
unavoidably form, and the details of the electronic structure and of the
pairing interaction will decide which particular symmetry will form, often in
close competition with other symmetries.
##### Historical: ferromagnetic spin fluctuations.
The original proposal of superconducting pairing arising from magnetic
interactions was put forward by Emery [56] and by Berk and Schrieffer [50],
who were interested primarily in transition metal elements and nearly
ferromagnetic metals. Such systems are considered to be close to a
ferromagnetic ordering transition in the Stoner sense, i.e. their
susceptibility may be approximated by $\chi=\chi_{0}/(1-U\chi_{0})$, where $U$
is a local Hubbard-like Coulomb matrix element assumed to be large since
$U\chi_{0}\simeq 1$ ($\chi_{0}$ is the susceptibility in the absence of
interactions). Physically this means a spin up electron traveling through the
medium polarizes the spins around it ferromagnetically, lowering the system’s
energy as illustrated in figure 4.
*
Figure 4: Heuristic snapshot of pairing of two test spins by ferromagnetic
spin fluctuations.
The excitations being “exchanged” in such a picture are not well- defined
collective modes of the system such a phonons or magnons, but rather
”paramagnons”, defined by the existence of a peak-like structure in the the
imaginary (absorptive) part of the small q susceptibility. Diagramatically,
only the ring-type diagrams shown in figure 5 contribute to the equal-spin
channel $\Gamma_{\uparrow\uparrow}$, whereas both ring and ladder-type
diagrams contribute to the effective pairing vertex in the opposite-spin
channel $\Gamma_{\uparrow\downarrow}$. These series may be summed in the usual
way to give:
$\displaystyle\Gamma_{\uparrow\uparrow}$ $\displaystyle=$
$\displaystyle{-U^{2}\chi_{0}(\bi{k}^{\prime}-\bi{k})\over
1-U^{2}{\chi_{0}}^{2}(k^{\prime}-k)},$ (3)
$\displaystyle\Gamma_{\uparrow\downarrow}$ $\displaystyle=$
$\displaystyle{U\over
1-U^{2}{\chi_{0}}^{2}(\bi{k}^{\prime}-\bi{k})}+{U^{2}\chi_{0}(\bi{k}^{\prime}+\bi{k})\over
1-U^{2}{\chi_{0}}^{2}(\bi{k}^{\prime}+\bi{k})}$ (4) $\displaystyle=$
$\displaystyle U^{2}\left({3\over 2}\chi^{s}-{1\over 2}\chi^{c}\right)+U,$ (5)
where we have defined $\chi^{s}\equiv\chi_{0}/(1-U\chi_{0})$ and
$\chi^{c}=\chi_{0}/(1+U\chi_{0})$, and in the last step we have changed
$-\bi{k}$ to $\bi{k}$ in the second term of $\Gamma_{\uparrow\downarrow}$
because we assume we work in the even parity (singlet pairing) channel. The
total pairing vertex in the triplet (singlet) channel is $\Gamma_{t}={1\over
2}\Gamma_{\uparrow\uparrow}$ ($\Gamma_{s}={1\over
2}(2\Gamma_{\uparrow\downarrow}-\Gamma_{\uparrow\uparrow})$). In the original
paramagnon theory, $\chi_{0}(\bi{q})$ is the noninteracting susceptibility of
the (continuum) Fermi gas, i.e. the Lindhard function. This function at small
frequency has a maximum at q=0, meaning correlations are indeed ferromagnetic.
Thus due to the negative sign in the equation for $\Gamma_{\uparrow\uparrow}$
(note $\chi_{0}>0$ and $U\chi_{0}<1$ to prevent a magnetic instability),
pairing is attractive in the triplet channel and singlet superconductivity is
suppressed, which explains why Pd, for example, does not superconduct [50].
*
Figure 5: Effective pairing interaction between (a) equal spins and (b)
opposite spins. Solid lines are electron $G$’s, dashed lines Hubbard $U$’s,
i.e. interactions between electrons of opposite spin only.
##### Antiferromagnetic spin fluctuations.
In the context of heavy fermion systems it was realized [57, 58] that strong
antiferromagnetic spin fluctuations in either the weak or strong coupling
limit lead naturally to spin singlet, $d$-wave pairing. The weak coupling
argument has been elegantly reviewed by Scalapino [59]. Suppose the
susceptibility is strongly peaked near some wave vector $\bi{Q}$. The form of
the singlet interaction is
$\displaystyle\Gamma_{s}(\bi{k},\bi{k}^{\prime})=\frac{3}{2}U^{2}\frac{\chi_{0}(\bi{q})}{1-U\chi_{0}(\bi{q})}$
(6)
if we neglect terms which are small near the RPA instability
$U\chi_{0}(\bi{q})\rightarrow 0$ [60]. This now implies that
$\Gamma_{s}(\bi{q})$ is also peaked at this wavevector, but is always
repulsive. Nevertheless, if one examines the BCS gap equation for this
interaction
$\Delta{k}=-{\sum_{\bi{k}^{\prime}}}^{\prime}\Gamma_{s}(\bi{k},\bi{k}^{\prime}){\Delta{k}^{\prime}\over
2E{k}^{\prime}}{\rm tanh}{E{k}^{\prime}\over 2T},$ (7)
one sees immediately that an isotropic state cannot be a solution, but that if
the state changes sign,
$\displaystyle\Delta{k}=-\Delta_{\bi{k}+\bi{Q}},$ (8)
a solution will be allowed.
In the cuprates, $\chi$ is peaked at $\bi{Q}\simeq(\pi,\pi)$, and the two
possible states of this type which involve pairing on nearest neighbor bonds
only are
$\displaystyle\Delta{k}^{d,s}$ $\displaystyle=$ $\displaystyle\Delta_{0}(\cos
k_{x}a\mp\cos k_{y}a).$ (9)
Which state will be stabilized then depends on the Fermi surface in question.
So we need to use the fact that the states close to the Fermi surface are the
most important in Equation (7), and examine the pairing kernel for these
momenta. For example, for a $(\pi/a,0)\rightarrow(0,\pi/a)$ scattering,
$\Delta{k}^{s}$ satisfies Equation (8) by being zero, whereas $\Delta{k}^{d}$
is nonzero and changes sign, contributing to the condensation energy. It
should therefore not be surprising that the end result of a complete numerical
evaluation of Equation (7) over a cuprate Fermi surface gives $d$-wave
pairing.
An alternative way to approach the question of how a purely repulsive
interaction allows for pair formation is to examine the interaction Fourier
transformed back to real space, where it shows regions (in the cuprates with
$\bi{Q}=(\pi,\pi)$, on nearest neighbor sites), where the pair potential
becomes attractive [59] if the interaction is sufficiently nonuniform in
momentum space.
##### Effect of 2 bands.
The discussion in Section 2 described the unusual, fully compensated Fermi
surface of the parent Fe-pnictide materials. In the Fe-based superconductors,
the proximity of the Fe $d$-states to the Fermi level has led many authors to
consider a Hamiltonian which takes a kinetic energy $H_{0}$ consisting of the
bands derived from the 5 Fe $d$-orbitals found in DFT, approximated within
some tight-binding or other scheme. The 2D Fermi surface in the 1-Fe zone thus
obtained is shown in figure 3. Like DFT, this model Fermi surface is
characterized by small concentric hole pockets around the $\Gamma$ point and
slightly elliptical electron pockets around the $\mathrm{M}$ points. Mazin and
collaborators [29] pointed out that modeling these pockets in the simplest
possible way, allowing for one hole and one electron pocket, led to a very
simple and elegant generalization of the “standard” argument for $d$-wave
pairing in the cuprates (a similar result was reached in a strong-coupling
approach by Seo et al [61]). In a weak-coupling approach, the near-nesting of
the hole and electron pockets suggested the existence of a peak in the spin
susceptibility at $\bi{Q}=(\pi,0)$ in the 1-Fe zone. The gap equation (7) then
admits a solution with the property (8) if there is a sign change of
$\Delta_{k}$ between electron and hole pockets. In the simplest version of
this theory, the anisotropy on each electron sheet is neglected, with the
argument that the pockets are small. This leads to the so-called isotropic
“$s_{\pm}$” state (figure 1) which has become the leading candidate for the
discussion of many of the superconducting properties of these materials. Note
that such a state has the full symmetry of the crystal lattice and is
therefore formally an $A_{1g}$ or “$s$-wave” state, but with fundamentally
different gap structure which leads to many nontrivial superconducting
properties.
##### Spin fluctuation pairing in multiorbital systems.
More realistic analyses of pairing in these systems by electronic interactions
soon followed. Many authors began with a Hamiltonian consisting of a kinetic
energy $H_{0}$ for the effective Fe bands as described above, plus an
interaction $H_{int}$ containing all possible two-body on-site interactions
between electrons in Fe orbitals as a good starting point for a microscopic
description of this system,
$\displaystyle H$ $\displaystyle=$ $\displaystyle
H_{0}+\bar{U}\sum_{i,\ell}n_{i\ell\uparrow}n_{i\ell\downarrow}+\bar{U}^{\prime}\sum_{i,\ell^{\prime}<\ell}n_{i\ell}n_{i\ell^{\prime}}$
$\displaystyle+\bar{J}\sum_{i,\ell^{\prime}<\ell}\sum_{\sigma,\sigma^{\prime}}c_{i\ell\sigma}^{\dagger}c_{i\ell^{\prime}\sigma^{\prime}}^{\dagger}c_{i\ell\sigma^{\prime}}c_{i\ell^{\prime}\sigma}$
$\displaystyle+\bar{J}^{\prime}\sum_{i,\ell^{\prime}\neq\ell}c_{i\ell\uparrow}^{\dagger}c_{i\ell\downarrow}^{\dagger}c_{i\ell^{\prime}\downarrow}c_{i\ell^{\prime}\uparrow}$
where $n_{i\ell}=n_{i,\ell\uparrow}+n_{i\ell\downarrow}$. The Coulomb
parameters ${\bar{U}}$, ${\bar{U}}^{\prime}$, ${\bar{J}}$, and
${\bar{J}}^{\prime}$ are in the notation of Kuroki et al [45], and are related
to those used by Graser et al [40] by $\bar{U}=U$, $\bar{J}=J/2$,
$\bar{U}^{\prime}=V+J/4$, and $\bar{J}^{\prime}=J^{\prime}$. The
noninteracting $H_{0}$ is given by a tight-binding model spanned by the 5 Fe
$d$-orbitals, Equation (2), which give rise to the Fermi surfaces shown in
figure 3.
In Equation (3.1), we have distinguished the intra- and inter-orbital Coulomb
repulsion, as well as the Hund’s rule exchange $\bar{J}$ and “pair hopping”
term $\bar{J}^{\prime}$ for generality, but if they are generated from a
single two-body term with spin rotational invariance they are related by
$\bar{U}^{\prime}=\bar{U}-2\bar{J}$ and $\bar{J}^{\prime}=\bar{J}$. In a real
crystal, such a local symmetry will not always hold.
The generalization of the simple 1-band Berk-Schrieffer spin fluctuation
theory to the multiorbital case was discussed by many authors [62, 63]. The
effective pair scattering vertex $\Gamma(\bi{k},\bi{k}^{\prime})$ between
bands $i$ and $j$ in the singlet channel is
$\displaystyle{\Gamma}_{ij}(\bi{k},\bi{k}^{\prime})$ $\displaystyle=$
$\displaystyle\mathrm{Re}\left[\sum_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}a_{\nu_{i}}^{\ell_{2},*}(\bi{k})a_{\nu_{i}}^{\ell_{3},*}(-\bi{k})\right.$
$\displaystyle\left.\times{\Gamma}_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}(\bi{k},\bi{k}^{\prime},\omega=0)a_{\nu_{j}}^{\ell_{1}}(\bi{k}^{\prime})a_{\nu_{j}}^{\ell_{4}}(-\bi{k}^{\prime})\right]$
where the momenta $\bi{k}$ and $\bi{k}^{\prime}$ are confined to the various
Fermi surface sheets with $\bi{k}\in C_{i}$ and $\bi{k}^{\prime}\in C_{j}$.
The orbital vertex functions $\Gamma_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}$
describe the particle-particle scattering of electrons in orbitals
$\ell_{1},\ell_{4}$ into $\ell_{2},\ell_{3}$ (see figure 6) and in the
fluctuation exchange formulation [64, 63] are given by
$\displaystyle{\Gamma}_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}(\bi{k},\bi{k}^{\prime},\omega)=\left[\frac{3}{2}\bar{U}^{s}\chi_{1}^{\rm
RPA}(\bi{k}-\bi{k}^{\prime},\omega)\bar{U}^{s}+\right.\,~{}~{}~{}~{}~{}~{}\,$
$\displaystyle\,~{}~{}~{}~{}~{}\left.\frac{1}{2}\bar{U}^{s}-\frac{1}{2}\bar{U}^{c}\chi_{0}^{\rm
RPA}(\bi{k}-\bi{k}^{\prime},\omega)\bar{U}^{c}+\frac{1}{2}\bar{U}^{c}\right]_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}},$
(12)
where each of the quantities $\bar{U}^{s}$, $\bar{U}^{c}$, $\chi_{1}$, etc
represent matrices in orbital space which depend on the interaction
parameters. This is the multiorbital generalization of Eq. (5). Here
$\chi_{1}^{\rm RPA}$ describes the spin-fluctuation contribution and
$\chi_{0}^{\rm RPA}$ the orbital (charge)-fluctuation contribution, determined
by Dyson-type equations as
$(\chi_{0}^{RPA})_{st}^{pq}=\chi_{st}^{pq}-(\chi_{0}^{RPA})_{uv}^{pq}(U^{c})_{wz}^{uv}\chi_{st}^{wz}$
(13)
and
$(\chi_{1}^{RPA})_{st}^{pq}=\chi_{st}^{pq}+(\chi_{1}^{RPA})_{uv}^{pq}(U^{s})_{wz}^{uv}\chi_{st}^{wz},$
(14)
where repeated indices are summed over. Here $\chi_{st}^{pq}$ is a generalized
multiorbital susceptibility (see [47]).
*
Figure 6: Top: pairing vertex $\Gamma_{\ell_{1},\ell_{2},\ell_{3},\ell_{4}}$
defined in terms of orbital states $\ell_{i}$ of incoming and outgoing
electrons. Bottom: representative examples of classes of orbital vertices
referred to in the text: intra-, inter- and mixed orbital vertices. From [47].
##### Results of microscopic theory.
The simplest goal of the microscopic approach is to calculate the critical
temperature $T_{c}$ via the linearized gap equation and determine the symmetry
of the pairing instability there. If one writes the superconducting order
parameter $\Delta(\bi{k})$ as $\Delta g(\bi{k})$, with $g(\bi{k})$ a
dimensionless function describing the momentum dependence on the Fermi
surface, then $g(\bi{k})$ is given as the stationary solution of the
dimensionless pairing strength functional [40]
$\lambda[g(\bi{k})]=-\frac{\sum_{ij}\oint_{C_{i}}\frac{dk_{\parallel}}{v_{F}(\bi{k})}\oint_{C_{j}}\frac{dk_{\parallel}^{\prime}}{v_{F}(\bi{k}^{\prime})}g(\bi{k}){\Gamma}_{ij}(\bi{k},\bi{k}^{\prime})g(\bi{k}^{\prime})}{(2\pi)^{2}\sum_{i}\oint_{C_{i}}\frac{dk_{\parallel}}{v_{F}(\bi{k})}[g(\bi{k})]^{2}}$
(15)
with the largest eigenvalue $\lambda$, which provides a dimensionless measure
of the pairing strength. Here $\bi{k}$ and $\bi{k}^{\prime}$ are restricted to
the various Fermi surfaces $\bi{k}\in C_{i}$ and $\bi{k}^{\prime}\in C_{j}$
and $v_{F,\nu}(\bi{k})=|\nabla{k}E_{\nu}(\bi{k})|$ is the Fermi velocity on a
given Fermi sheet.
In figure 7, we plot the leading dimensionless gap function $g(k)$ derived
from the RPA theory around the electron $\beta_{1}$ sheet for two different
values of the doping, for spin-rotationally invariant parameters $U=1.3$ and
$J=0.2$. The gap on the hole sheets is seen to be essentially isotropic, while
on the electron sheets the average of the gap is of opposite sign compared to
the hole sheets, and is highly anisotropic, with nodes in the case of electron
doping. One would like to understand the origin of the anisotropy and its
doping dependence.
*
Figure 7: Top: false color plots of dimensionless gap function $g(k)$ on
various Fermi surface sheets for electron doped $n=6.03$ (left) and hole doped
$n=5.95$ (right). Bottom: detail of $g(k)$ on $\beta_{1}$ pocket for $U=1.3$
and $n=5.95,\bar{J}=0.2$ (red squares) and $n=6.01,\bar{J}=0.2$ (blue
circles). Here the angle $\phi$ is measured from the $k_{x}$-axis. From Kemper
et al [47].
##### Physical origins of anisotropy of pair state and node formation.
An unusual element which emerges from the spin fluctuation pairing analysis
based on (3.1) is that the orbital structure of the Fermi surface can have a
significant impact on the anisotropy of the pair state. The intra-orbital
scattering of $d_{xz}$ and $d_{yz}$ pairs between the $\alpha$ and $\beta$
Fermi surfaces by $(\pi,0)$ and $(0,\pi)$ spin fluctuations (figure 6) leads
naturally to a gap which changes sign between the $d_{xz}/d_{yz}$ parts of the
$\alpha$ Fermi surfaces and the $d_{yz}$ and $d_{xz}$ parts of the $\beta_{1}$
and $\beta_{2}$ electron pockets, just as in the early proposal of Mazin et al
[29], see figure 3. However, as discussed by Maier et al [41], Kuroki et al
[16], and Kemper et al [47], scattering between the $\beta_{1}$ and
$\beta_{2}$ Fermi surfaces frustrates the formation of an isotropic $s_{\pm}$
gap there. Furthermore, this anisotropy can also be driven by the effect of
the intraband Coulomb interaction. Finally, inter-orbital pair scattering can
also occur, depending upon $\bar{U}^{\prime}$ and $\bar{J}^{\prime}$. The
contributions to the total pairing interaction arising from mixed and
interorbital processes may be shown explicitly to be subdominant to the
intraorbital processes but important for nodal formation [47].
On the other hand, the $\beta_{1}-\beta_{2}$ $d_{xy}$ orbital frustration is
weaker or does not occur if an additional hole pocket $\gamma$ of $d_{xy}$
character is present (see figure 3); these scattering processes are at
($\pi,0$) in the unfolded zone and therefore support isotropic $s_{\pm}$
pairing (see figure 7). Kuroki et al [16] took the important step of relating
the crystal structure, electronic structure, and pairing, by noting that the
As height above the Fe plane in the 1111 family was a crucial variable
controlling the appearance of the $\gamma$ pocket and thus driving the
isotropy of the $s_{\pm}$ state. It is important to note that the transition
between nodal and nodeless $A_{1g}$ gap structures, investigated by a number
of authors [16, 65, 47, 66, 67, 68], does not involve any symmetry change, and
relies only on the details of the pairing interaction.
##### Similar approaches.
In this discussion we have presented “spin fluctuation theory” in terms of an
RPA approximation to the pair scattering vertices $\Gamma$, which also
includes subleading charge/orbital contributions via Equation (3.1). It is
important to note that other approaches have obtained very similar results for
the Hamiltonian (3.1). The most closely related technique is the FLEX
(fluctuation exchange) approximation [64], which is a conserving approximation
to the Luttinger-Ward functional and the self-energy. Several authors have
applied this approach to the pairing problem [69, 70, 71, 72, 73], employing a
five-band model for FeBS based on Wannier fits to DFT results. Qualitatively,
results are similar for the leading pairing instabilities, including the
doping dependence [72] (see figure 8), although node formation or strong
angular anisotropy have not been observed in FLEX so far. In addition, FLEX
has certain well-known peculiarities which need to be handled carefully. In
particular, the real part of the self-energy, which shifts the band structure
in a momentum-dependent way, includes some of the correlations already
included in DFT; thus the use of a kinetic energy $H_{0}$ fit to DFT plus the
self-consistent FLEX self energy tends to overcount these correlations. This
problem is traditionally handled by subtracting the real part of the static
self-energy [73].
*
Figure 8: Phase diagram in FLEX treatment of SDW and SC instabilities (from
[72]).
A further popular approach is the functional renormalization group [74] (fRG),
which has the advantage that it is capable of studying various instabilities
of the system on an equal footing, which is of particular relevance for
studying the competition between SDW and superconductivity in these systems in
realistic models. Numerical RG equations are derived by dividing the Brillouin
zone into $N$ patches, and then summing at each RG step over the five one-loop
Feynman diagrams to compute the renormalized 4-point vertex function. This
technique was pioneered in connection with FeBS by Fang et al [75],
generalizing numerical work on 1-band systems [76], and has been extended and
applied to various FeBS [66, 67, 68, 77, 78]. These works are the intellectual
offspring of earlier analytical (logarithmic) RG calculations in two-band
models [79, 80] which were later generalized to include low-order angular
harmonics [65] to describe gap anisotropy. Recently this approach has been
extended and compared directly to the RPA results, such that the doping
evolution of the fundamental band interactions could be obtained [81].
It is remarkable that rather dissimilar techniques, none of which are
controlled in the usual perturbative sense, give such similar results. To
illustrate this, in figure 9 we plot results for a FeBS Fermi surface for
LaFePO or LaFeAsO including only the two inner hole pockets, obtained by fRG
and RPA techniques. Although the scale of the interactions are quite
different, the ratio $J/U$ is $\ll 1$ in both cases. Note that the order
parameter on both hole pockets is quite isotropic, but that on the electron
sheets nodes appear, but consistent with the overall average $s_{\pm}$
character of the state.
*
Figure 9: Comparison of fRG and RPA results for the gap function $g(\theta)$
vs. $\theta$, with $\theta$ parametrizing each Fermi pocket, for 4-Fermi
pocket model of LaFePO/ LaFeAsO. Left: FRG results from [67] with
$\bar{U}=3.5$, $\bar{U}^{\prime}=2.0$, $\bar{J}=\bar{J}^{\prime}=0.7$. Right:
RPA results from [41] with $\bar{U}=1.67$, $\bar{U}^{\prime}=1.46$,
$\bar{J}=\bar{J}^{\prime}=0.10$. Bands $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$
and $\beta_{2}$ correspond to inner and outer hole pockets, and two electron
pockets, respectively. Energy units for parameters are in eV, units on
vertical axes of both figures are arbitrary.
Finally, we comment on so-called “strong-coupling” approaches to pairing,
based on the $J_{1}-J_{2}$ multiple competing exchange model of the magnetism
in these systems [82, 83, 84]. Note that this term frequently appears in
reference to the large size of the Hubbard interaction relative to the
bandwidth, which in the 1-band case at half-filling allows for a description
in terms of the spin degrees of freedom as the charge degrees of freedom
become localized. The spin dynamics can become more localized in situations
other than the canonical Mott-Hubbard case, however, and are in fact well
described by DFT, which for the FeBS gives a typical energy scale for the
magnetic interactions of order 100 meV or larger. Thus the moments are large
and quite localized. This is not a contradiction; even in genuinely itinerant
systems (elemental 3D metals are a good example) magnetic interactions are
essentially local, decaying with distance as a power law.
This magnetic model leads to a competition between the Neel and stripe-
collinear orders, also present in the (itinerant) DFT calculation,
corresponding to the same ground state magnetic pattern and to a similar
structure of spin fluctuations in the reciprocal state [a maximum near
$(\pi,0)$]. In [61], a $t-J_{1}-J_{2}$ model with two bands was studied, and
the exchange terms were decoupled in mean field in the pairing channel. In
this procedure, the nearest neighbor exchange $J_{1}$ induces competing $\cos
k_{x}+\cos k_{y}$ and $\cos k_{x}-\cos k_{y}$ ($s$\- and
$d_{x^{2}-y^{2}}$-wave) pairing harmonics, while the next nearest neighbor
exchange leads to $\cos k_{x}\cos k_{y}$ and $\sin k_{x}\sin k_{y}$ ($s$\- and
$d_{xy}$-wave). In the region of the general phase diagram with $J_{2}\gtrsim
J_{1}$, $\cos k_{x}\cos k_{y}$ was the leading instability for the 2-band
Fermi surface, leading to a nodeless $s_{\pm}$ state, and a similar ground
state was also found for a 5-orbital model [85].
Mean field theories of the strong coupling type (see also models where $U$ and
$J_{1,2}$-type terms are treated as independent before the mean field step
[86, 43]) show an intriguing set of circumstantial agreement with the
predictions of itinerant weak-coupling models, despite the lack of logical
continuity between the two types of models. Yet it is not that surprising.
Indeed, the pairing symmetry in any spin-fluctuation model is mainly defined
by matching the structure of the spin fluctuations and the FS geometry. Since
both are rather similar in the two approaches, not surprisingly, the main
results agree. In fact, Wang et al have shown that in the case with 5 Fermi
surface pockets, the low energy spin and charge excitations in the fRG
treatment of the 5-orbital model (3.1) overlap very well with those of the
$t-J_{1}-J_{2}$ model. At the Hamiltonian level, however, there is no way to
derive these particular strong coupling models (with unrenormalized kinetic
term) from the general model with on-site interactions.
An advantage of such strong coupling approaches is that they capture the local
character and large amplitude of Fe magnetic moments, in agreement with the
DFT calculations. They also have some attractive conceptual simplicity. On the
other hand, they have several uncontrollable shortcomings, which make proper
application of this approach tricky.
First and foremost, such theories artificially separate the itinerant
electrons and the local moments, as if the latter were coming from a separate
atomic species. Yet, the moments are formed by exactly the same electrons that
form the band structure, which also mediate the magnetic interaction mapped
onto the $J_{1}-J_{2}$ Heisenberg Hamiltonian. In some papers an attempt to
account for this fact is made by adding spin susceptibility of itinerant
electrons to the aforementioned Heisenberg Hamiltonian, which essentially
amounts to a double counting.
Second, actual band structure calculations [87, 88, 89] cannot be mapped onto
a Heisenberg Hamiltonian of any range. They can be mapped onto a Heisenberg
Hamiltonian with biquadratic exchange, or possibly to a more complicated
Hamiltonian (such as ring exchange), but not onto a pure Heisenberg model.
Third, “strong-coupling” models essentially fix the shape of the spin-
fluctuation induced interaction; therefore, the resulting solution fixes the
structure of the gap nodes in momentum space, so that the amplitude,
anisotropy and possible nodes on actual FSs depends only on the proximity of
these FSs to the imaginary nodal lines. This result is not corroborated by the
weak coupling calculations and is likely unphysical.
The partial agreement between fRG and the $t-J_{1}-J_{2}$ results will
therefore remain a curiosity until a more concrete understanding why the
similarity of the low energy sectors of the two theories is observed.
##### Pairing in multiorbital systems from DFT perspective.
RPA calculations, as well as other approaches discussed above, use the same
model Hamiltonian (3.1), but resort to different approximate methods to solve
the problem of superconductivity emerging from this Hamiltonian. Comparing
results of such different approaches one can get an idea of how accurate these
solutions are. Yet this Hamiltonian itself is a rather uncontrollable
approximation, and one can legitimately ask the question, to what extent the
simplifications introduced when constructing this Hamiltonian are justified.
Indeed, some of the qualitative results discussed above, such as anisotropy
(and possible nodes) of the order parameter, are intimately related to the
details of the model that may or may not be sufficiently universal.
Specifically, there are two aspects of the model that appear to be
qualitatively significant. First, the leading spin-dependent term in this
Hamiltonian, $\bar{U}\sum_{i,\ell}n_{i\ell\uparrow}n_{i\ell\downarrow},$ is
orbital diagonal (there are two other spin-dependent terms, the Hund’s term,
proportional to $\bar{J},$ and pair hopping term proportional to
$\bar{J}^{\prime},$ but these are smaller). That makes the pairing interaction
rather sensitive to the orbital composition, with the parts of the Fermi
surface whose orbital content is not matched by that of the rest of the Fermi
surface effectively decoupled from the rest. This is true in RPA, FLEX, fRG or
any other method based on the same Hamiltonian. On the other hand, there is
growing belief [90] that physics of these materials is controlled by the Hund
rule’s coupling rather than by the direct Coulomb repulsion, as implied for
instance in the original expressions (3), (3.1). The second aspect of the
model is that it does not include the retardation effects, at least on the RPA
level. The large Coulomb repulsion that in real life is logarithmically
reduced in the calculations needs to be avoided, which can be done, for an on-
site local repulsion, by tuning the order parameter in such a way that it
integrates to zero. These two problems, characteristic for this model, suggest
that the tendency towards gap anisotropy is probably somewhat overestimated in
this approach.
It is instructive to compare the effect of direct Coulomb repulsion in
conventional superconductors, such as V, to another $3d$ transition metal with
the interaction parameters in the charge channel not much different from Fe
based superconductors. There the Coulomb repulsion, obviously strong, is
renormalized as
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle UN_{0}$ (16)
$\displaystyle\mu^{*}$ $\displaystyle=$
$\displaystyle\mu/[1+\mu\ln\frac{E_{C}}{E_{b}}],$ (17)
where $N_{0}$ is the density of states at the Fermi level, $E_{C}$ is a
characteristic electronic energy (it may be the total band width, or the
plasmon energy, or a combination thereof), and $E_{b}$ is the energy of the
pairing bosons. Given that $\mu\ln\frac{E_{C}}{E_{b}}\gg 1,$ $\mu^{*}\approx
1/\ln\frac{E_{C}}{E_{b}}\sim 0.1-0.15$ for typical conventional
superconductors. For most Fe superconductors $T_{c}$ is only a factor of two
or three larger than for the best transition metal superconductors (niobum’s
$T_{c}$ is 10 K, that of some binary alloys reaches 23 K), and the full d-band
widths are comparable, so if the Coulomb repulsion were uniform one could
apply similar reasoning and conclude that for these systems $\mu^{*}\sim
0.15-0.2$. If, on the other hand, the unrenormalized Coulomb repulsion is
different in the interband and intraband channels, the renormalization
equation should be written differently, namely
$\mu_{ij}^{*}=\mu_{ij}-\sum_{n}\mu_{in}\ln\frac{E_{C}}{E_{b}}\mu_{nj}^{*}$
(18)
where $i$, $j$, and $n$ are band indices. The solution in the limit
$|\mu_{i\neq j}-\mu_{ii}|\ln\frac{E_{C}}{E_{b}}\gg 1$ (which may or may not
fold for FeBS) reads: $\mu_{ii}\approx 1/\ln\frac{E_{C}}{E_{b}}$, $\mu_{i\neq
j}\approx const/\ln^{2}\frac{E_{C}}{E_{b}}$. Thus the renormalization in a
multiband system has two effects: (1) it strongly reduces the effect of
Coulomb repulsion in general and (2) it suppresses interband repulsion
compared with intraband. The latter effect has important implication – effect
of ‘Coulomb avoidance’. Indeed, if the Coulomb repulsion does not depend on
the wave vector, the condition for it to cancel out of the equations on
$T_{c}$ is that the order parameter, when averaged over the entire FS and all
its sheets, integrates to zero. If the Coulomb repulsion is only present in
the intraband channel, than to avoid its effect on $T_{c}$ entirely, the order
parameter must integrate to zero in each band separately.
It should be kept in mind that the preceding argument implicitly relies upon
an analogy with the Eliashberg theory, assuming that all relevant interactions
can be separated into groups: a pairing interaction (boson exchange) that is
restricted to sufficiently low energies (recall that the spin resonance in
FeBS occurs at the energies around 30-40 meV), and is also subject to the
Migdal theorem, and a direct Coulomb repulsion that exists at all energies,
and gets renormalized after the ladder diagrams are summed. Most existing
approaches explicitly or implicitly make this very assumption, with the
notable exception of the renormalization group techniques. However, in reality
for electronic (as opposed to a phonon) superconductivity all interactions
emerge from electrons themselves, and such separation is not always possible.
Nor is it always possible to limit the approximation to the ladder diagrams.
In more complex cases, formally involving parquet diagrams, the solution may
depend on whether the Coulomb interaction is first screened and then
renormalized or first renormalized and then screened, etc [91]. On the other
hand, in the renormalization group approach interactions are not separated
into these channels, and in principle all retardation effects are supposed to
be included. While it is very difficult to make a one-to-one correspondence
between such approaches and RPA (and similar) calculations, is quite obvious
that they are based on substantially different physical assumptions. Why,
nevertheless, results obtained in RPA and in functional renormalization group
approaches are quantitatively similar, is unclear and needs to be understood.
Given these uncertainties involved in RPA calculations described in the
previous sections, as well as other similar approaches, it is interesting to
look at the problem at hand not from the Hubbard model point of view, but from
the opposite, density functional one. Indeed many believe that, as opposed to
the high $T_{c}$ cuprates, DFT is a reasonable starting point for these
materials, and in some sense it is more consistent to use DFT for the full
susceptibility as long as we use the DFT band structure for the noninteracting
one. For instance, in classical semiconductors, such as Si, DFT (even the
exact DFT) underestimates the fundamental gap, and thus yields an incorrect
noninteracting susceptibility, yet the full susceptibility calculated entirely
within DFT is by definition exact [92]. Note that within DFT there are only
two electronic interactions: the charge interaction is defined as
$\delta^{2}E_{tot}/\delta n(\bi{r})\delta n(\bi{r}^{\prime})$, which includes
the Coulomb (Hartree) and the exchange-correlation interactions, and the spin
interaction is defined as
$I_{xc}(\bi{r},\bi{r}^{\prime})=\delta^{2}E_{tot}/\delta
n_{\uparrow}(\bi{r})\delta n_{\downarrow}(\bi{r}^{\prime})$. Therefore, RPA is
exact in DFT. Obviously, LDA and GGA approximations to DFT are not exact, yet
one may think that the full spin susceptibility calculated, say, within LDA-
DFT is not a bad approximation. The only caveat is that in a quantum critical
material LDA deviates from the exact DFT in one systematic way: the reduction
of the Hund rule coupling due to long range spin fluctuations is not included.
Emprically, it can be accounted for by scaling $I_{xc}$ down by 15-20%. After
that, one can use Equations (3), (4) using
$I_{xc}(\bi{r},\bi{r}^{\prime})=I_{xc}(\bi{r})\delta(\bi{r}-\bi{r}^{\prime})$
instead of $U.$
One can work in the orbital basis again, just as one does in the Hubbard
model, except that the spin-dependent term now reads
$\sum_{i,\ell,\ell^{\prime}}I_{\ell\ell^{\prime}}n_{i\ell\uparrow}n_{i\ell^{\prime}\downarrow}.$
It is easy to show that $I_{\ell\neq\ell^{\prime}}=I_{\ell\ell}/3,$ so some
orbital dependence remains. Whether it will be enough to provide for nodal
lines, is unclear. The charge channel should also be revisited in this case.
The Coulomb repulsion term should be added in form of a matrix in band
indices, $\mu_{ij}^{\ast}$, as defined in Equation (18). The diagonal elements
of this matrix should be taken as 0.15-0.20, and nondiagonal between 0 and the
diagonal ones, depending on how different are the matrix elements of $U$ in
the band representation.
Although the course of action outlined above is straightforward, so far it has
not been tried yet. In principle, such calculations could be very useful for
understanding the stability of the qualitative results, such as nodal
structure, with respect to principal approximation, because the Hubbard model
and the DFT in many aspects represent two opposite approximations.
### 3.2 Alternative approaches.
Historically, many other pairing agents have been proposed proposed as
mediators for unconventional (non-phonon) superconductivity [93]. Arguably,
historically the first suggestions were those of Little [94] and Ginzburg
[95], who proposed, respectively, quasi-1D metal chains and quasi 2D metal
planes embedded in highly polarizable media. This was dubbed “excitonic
superconductivity”, after the simple physical picture of Cooper pairs living
in the metallic subsystem, and the intermediate bosons being excitons
localized in the surrounding nonmetalic media. In the context of FeBS, this
proposition was recently brought back into limelight by Sawatzky et al, who
pointed out that As and Se are large ions and thus have large polarizability
[13, 96]. This model is still being discussed; the arguments usually brought
up against it include the fact that Fe-As hybridization is not small, as the
model requires, that ion-ion interaction should also be subject to screening
by polarizable As ions, but no anomalous phonon softening is observed, and,
finally, that superconductivity seems to be always adjacent in the phase
diagram to antiferromagnetism.
Other proposals rely on long wavelength electron charge fluctuations known as
plasmons, particularly on acoustic plasmons. Theories of this type were
attempted for cuprates, for MgB2, and, most recently, for intercalated
graphites. So far this model, however, has not had any confirmed success. One
shortcoming that plagues the papers advocating this mechanism is that they
hardly ever address the lattice stability, yet any sort of overscreening (and
essentially any attractive interactions in the charge channel can be
considered as a sophisticated overscreening) tends to overscreen phonons as
well and render them unstable. This is of course a problem for other
“excitonic” mechanisms as well; one can achieve pairing and keep phonons
stable only through invoking vertex corrections that appear in electron-
electron but not in electron-ion vertices.
*
Figure 10: (a) Phase diagram for pairing in Hubbard-Holstein model [97].
$g(0)$ is the bare electron-electron interaction due to electron-phonon
coupling. (b) Interorbital susceptibility mixing $xz$ and $xy$ orbitals; (c)
Intraorbital susceptibility for $xz$ orbitals.
Of all alternatives to the spin fluctuation model, the one that has received
most attention in the context of FeBS is the orbital fluctuations model of
Kontani and co-workers [98, 97, 99]. The possible importance of orbital
fluctuations was pointed out by several authors early on due to the
possibility of orbital ordering in the Fe $d$ states at the orthorhombic
transition [100, 84, 101, 102]. The stripe magnetic order will drive an
orbital ordering to some degree even in itinerant models. Fluctuating order of
this type, taken in isolation, can in principle lead to an attractive
mechanism for pairing, although it is hard to disentangle such orbital
fluctuations from spin fluctuations of the same symmetry.
It is useful to note that orbital fluctuations of this type are present in the
standard fluctuation exchange approach (3.1), and of course driven by the
interorbital Coulomb matrix elements $\bar{U}^{\prime}$, $\bar{J}^{\prime}$.
It is normally assumed (and verified by ab initio calculations [103]) that
$\bar{U}>\bar{U}^{\prime}$, but one can ask, taking $\bar{U}^{\prime}$ as an
independent parameter, what happens to the effective electron pairing vertex?
One may show that for sufficiently large $\bar{U}^{\prime}$,$\bar{J}^{\prime}$
the instability in the charge/orbital channel dominates the spin channel
contribution to $\Gamma_{ij}(\bi{k},\bi{k}^{\prime})$, Equation (3.1), even
for purely electronic interactions. In the former channel the interorbital
pair vertex becomes peaked at $(0,0)$ (compare figure 10), such that the
leading instability occurs in the $A_{1g}$ channel but without sign change,
i.e. a $s_{++}$ state.
These strong orbital fluctuations are unphysical in the required limit
$\bar{U}^{\prime}>\bar{U},$ $\bar{J}^{\prime}>\bar{J}$, relying on electronic
interactions alone. However, as pointed out by Kontani and Onari [97], certain
in plane Fe phonons can in principle “bootstrap” the interorbital processes
such that they dominate the spin fluctuation part of the interaction (a
different phonon was considered by Yanagi [104]). The electron-phonon coupling
to these phonons is included in an RPA-type calculation [97], and it is found
indeed that effective interorbital couplings $\tilde{U}^{\prime}$,
$\tilde{J}^{\prime}$, renormalized by the effective electron-electron
interaction due to phonons $g(0)$, enhance $s_{++}$ pairing. The phase diagram
of this model for fixed $\tilde{U}^{\prime}/\tilde{U}$ is shown in figure 10.
Note that orbitals 2 and 4 in the Kontani-Onari scheme are $xz$ and $xy$.
* (a) (b) (c)
Figure 11: Diagrams for the electron-phonon vertex renormalization. Straight
red lines are electron propagators, wavy green line is the phonon propagator,
and the curly magenta line is a high-energy electronic excitation (orbital
fluctuation).
While the concept of orbital fluctuation pairing is quite clear, there are
several open questions concerning this idea. Probably the most intriguing one
is that how model calculations of Kontani and Onari should be reconciled with
the density functional calculations. Indeed, electron-phonon coupling in the
linear harmonic adiabatic approximation is included in the standard linear
response calculations of the phonon frequencies and coupling strength. These
calculations find only very moderate coupling strength for representative FeBS
[105], and include diagrams of the type shown in figure 11(a), where the
vertex is computed adiabatically, as a derivative of the electronic Green’s
function with respect to ionic displacement. An enhancement found by Kotani
and Onari can only come from vertex corrections. However, vertex corrections
of the type shown in figure 11(b) are excluded by virtue of the Migdal
theorem, and vertex corrections of the type shown in figure 11(c), where the
springy line is a high-energy electronic excitation (like an orbital
fluctuation) are included in the adiabatic density functional calculations
(any orbital repopulation due to a static ionic displacement is accounted
for). Thus, the slow (compared to the electronic scale) orbital fluctuations
are excluded by the Migdal theorem, and fast ones (compared to the phonon
scale) are already included in the DFT calculations. This appears to leave
rather little room for strong renormalization of the electron-phonon coupling.
Similarly, one can ask how such a huge enhancement will affect the
corresponding phonon self-energy. Indeed pretty much the same vertex
corrections enter the equation on the phonon frequency and one would expect
that if the density functional theory fails so miserably in calculating the
electron-phonon coupling for a particular mode, it should also drastically
overestimate the frequency of this mode compared to the experiment. Yet this
is definitely not the case [106].
Thus, the question to which extent this interesting orbital-fluctuation model
is operative in real compounds remains open. It is possible that, due to
varying interactions strengths, orbital fluctuations dominate in some of the
FeBS and spin fluctuation in the others. The authors of these works believe
that orbital fluctuations, and thus $s_{++}$ pairing, are dominant in most of
these systems, pointing to weak impurity scattering [107], the broad neutron
scattering resonance observed in most FeBS [98], and the natural explanation
[99] of the Lee plot (maximum of $T_{c}$ within family at tetrahedral Fe-As
angle [108]) as evidence in favor of this scenario. We postpone the question
of whether or not $s_{++}$ pairing has in fact been observed to Section 5.1.
### 3.3 Multiband BCS theory.
The famous BCS formula is derived in the assumption that the pairing amplitude
(superconducting gap, order parameter) is the same at all points on the Fermi
surface. The variational character of the BCS theory makes one think that
giving the system an additional variational freedom of varying the order
parameter over the Fermi surface should always lead to a higher transition
temperature. For a case of two bands with uniform order parameters in each of
them this problem was solved first in 1959 by Matthias, Suhl, and Walker [109]
and by Moskalenko [110]. It can be easily generalized onto a general
$\bi{k}$-dependent order parameter. In the weak coupling limit it reads
$\Delta(\bi{k})=\int\Lambda(\bi{k},\bi{k}^{\prime})\Delta(\bi{k}^{\prime})F[\Delta(\bi{k}^{\prime}),T]d\bi{k}^{\prime},$
(19)
where summation over $\bi{k}$ implies also summation over all bands crossing
the Fermi level. A strong coupling generalization in the spirit of Eliashberg
theory is straightforward. Here the matrix $\Lambda$ characterizes the pairing
interaction, and
$F=\int_{0}^{\omega_{B}}dE\tanh(\frac{\sqrt{E^{2}+\Delta^{2}}}{2T})/\sqrt{E^{2}+\Delta^{2}}$
. The intermediate boson frequency sets the cut-off frequency. Assuming that
the order parameter $\Delta$ varies little within each sheet of the Fermi
surface, while differing between the different sheets, Equation (19) is
reduced to the original expression of [109, 110]:
$\Delta_{i}=\sum_{j}\Lambda_{ij}\Delta_{j}F(\Delta_{j},T),$ (20)
where $i,j$ are the band indices and $\Lambda$ is an asymmetric matrix related
to the symmetric matrix of the pairing interaction $V$,
$\Lambda_{ij}=V_{ij}N_{j}$, where $N_{i}$ is the contribution of the $i$-th
band to the total DOS. It can be shown that in the BCS weak coupling limit the
critical temperature is given by the standard BCS relation,
$kT_{c}=\hbar\omega_{D}\exp(-1/\lambda_{eff}),$ where $\lambda_{eff}$ is the
largest eigenvalue of the matrix $\Lambda.$ The ratios of the individual order
parameters are given by the corresponding eigenvector. Note that although the
matrix $\Lambda$ is not symmetric, its eigenvalues (but not eigenvectors!) are
the same as those of the symmetric matrix $\sqrt{N}V\sqrt{N}$.
Furthermore it is evident from Equation 20 that unless all $V_{ij}$ are the
same, the temperature dependence of individual gaps does not follow the
canonical BCS behavior. For instance, in a two-band superconductor where the
intraband coupling dominates, the smaller gap opens initially at a very small
value, and only at a temperature corresponding to its own superconducting
transition (not induced by the larger gap) it starts to grow. This effect is
gradually suppressed as the interband coupling approaches the geometrical
average of the intraband couplings, but as the interband coupling starts to
dominate the gaps again show non-BCS temperature dependence. In this limit,
however, it is the larger gap that deviates more from the BCS behavior.
It was realized in 1972 [55] that Equations (19) and (20) may have solutions
even when all elements of the interaction matrices $\Lambda$ are negative,
i.e., repulsive. The simplest example is an off-diagonal repulsion:
$V_{11}=V_{22}=0$, $V_{12}=V_{21}=-V<0$. In this case the solution reads:
$\lambda_{eff}=\sqrt{\Lambda_{12}\Lambda_{21}}=|V_{12}|\sqrt{N_{1}N_{2}},$
$\Delta_{1}(T_{c})/\Delta_{2}(T_{c})=-\sqrt{N_{2}/N_{1}}$. At lower
temperatures the gap ratio becomes somewhat closer to 1.
It is important to bear in mind that actual FeBS materials have more than two
bands — rather four or five. These may all have different gap magnitudes, and
possibly angular dependences, as discussed in Section 5.
### 3.4 Disorder in multiband superconductors.
*
Figure 12: Schematic representation of two Fermi surface pockets with
superconducting gaps indicated with their signs. Top: interband scattering by
impurities mixes $\Delta_{1}$ and $\Delta_{2}$. Bottom: intraband scattering
mixes states on each pocket.
Experiments on impurity substitution [111, 112, 113] and proton irradiation
[114] have given the impression that $T_{c}$ is suppressed generally more
slowly than the maximum rate obtained for pure interband scattering in the
symmetric model, which is identical to the AG universal $T_{c}$ suppression
curve. It is worth advising the reader to interpret $T_{c}$ suppression
experimental results with caution, for several reasons. First, in some cases
not all the nominal concentration of impurity substitutes in the crystal.
Second, “slow” and “fast” $T_{c}$ suppression cannot be determined by plotting
$T_{c}$ vs. impurity concentration, but only vs. a scattering rate directly
comparable to a theoretical scattering rate (see below), which is generally
difficult to determine. The alternative is to plot $T_{c}$ vs. residual
resistivity change $\Delta\rho$, but a) this is only possible if the $\rho(T)$
curve shifts rigidly with disorder, and b) if comparisons with theory include
a proper treatment of the transport rather than the quasiparticle lifetime
[115]. Finally, the effect of a chemical substitution in a Fe-based
superconductor is quite clearly not describable solely in terms of a potential
scatterer, but the impurity may dope the system or cause other electronic
structure changes which influence the pairing interaction.
Given the many uncertainties present in the basic modeling of a single
impurity, as well as the multiband nature of the Fe-based materials, it is
reasonable to assume that systematic disorder experiments may not play a
decisive role in identifying order parameter symmetry as they did, e.g. in the
cuprates. Nevertheless, one can perhaps draw useful qualitative conclusions
about the effects of impurities on the various types of superconducting states
under discussion by attempting to study models of pairbreaking by impurities
which are generalizations of the conventional Abrikosov-Gor’kov (AG) approach
[116].
#### 3.4.1 Intra- vs. interband scattering.
In conventional 2-band superconductors with two different isotropic gaps,
nonmagnetic impurities can either scatter quasiparticles between bands or
within the same band. Interband processes (see figure 12) will average the
gaps and can thus lead to some initial $T_{c}$ suppression, after which
$T_{c}$ will saturate until localization effects become important. Interband
scattering is much more profound in a sign-changing 2-band system [117, 118,
119], where nonmagnetic impurities with interband component of the scattering
potential are pairbreaking even if the gaps and densities of states are equal
on both bands (symmetric model). In such a situation, $T_{c}$ will eventually
be suppressed to zero at a finite critical concentration as in the theory of
scattering by magnetic impurities in a 1-band $s$-wave system [116]. In the
context of the FeBS, these general considerations were pointed out early on by
several groups [29, 120, 121, 122, 123]. A given type of chemical impurity in
a given host will be characterized crudely by an effective interband potential
$u$ and an intraband potential $v$, and results for various quantities in the
superconducting state will depend crucially on the size and relative strengths
of these two quantities.
Most calculations are performed in the framework of the $T$-matrix
approximation to calculate the average impurity self-energy for pointlike
scatterers $\hat{\Sigma}^{imp}(\omega_{n})$,
$\hat{\Sigma}^{imp}=n_{imp}\hat{\mathbf{U}}+\hat{\mathbf{U}}\hat{{G}}(\omega_{n})\hat{\Sigma}^{imp}({\mathrm{i}}\omega_{n}),$
(21)
where $\hat{\mathbf{U}}=\mathbf{U}\otimes\hat{\tau}_{3}$, $n_{imp}$ is the
impurity concentration, and the $\tau_{i}$ are Pauli matrices in particle-hole
space. Here ${\mathbf{U}}$ is a matrix in band space, frequently taken for
simplicity to be represented by constant intra- and inter-band potentials $v$
and $u$, respectively, such that
$(\mathbf{U})_{\alpha\beta}=(v-u)\delta_{\alpha\beta}+u$. This completes the
specification of the equations which determine the Green’s functions
$\hat{G}(\bi{k},\omega)^{-1}=\hat{G^{0}}(\bi{k},\omega)^{-1}-\hat{\Sigma}^{imp}(\bi{k},\omega),$
(22)
where $\hat{G}^{0}$ is the Green’s function for the pure system. Note that the
self-consistent $T$-matrix approximation includes only diagrams corresponding
to multiple scattering from a single impurity, and is well-known to have some
pathologies in two dimensions [124]. In the context of impurities in an
$s_{\pm}$ state, it has been claimed to produce inaccurate results in the
statistics of subgap states [19]. Nevertheless, for qualitative purposes–and
we will argue below that one cannot go beyond a qualitative analysis here
anyway–it seems quite adequate.
#### 3.4.2 Effect on $T_{c}$.
*
Figure 13: Critical temperature for various $\tilde{\sigma}$ and $\eta$ as a
function of (a) the impurity scattering rate $\Gamma_{1}$ and (b) the
effective interband scattering rate $\tilde{\Gamma}_{12}$. The parameters are:
$N_{2}/N_{1}=2$, coupling constants are
for $\langle\lambda\rangle>0$: $\lambda_{11}=3$, $\lambda_{12}=-0.2$,
$\lambda_{21}=-0.1$, $\lambda_{22}=0.5$,
for $\langle\lambda\rangle=0$: $\lambda_{11}=2$, $\lambda_{12}=-2$,
$\lambda_{21}=-1$, $\lambda_{22}=1$,
for $\langle\lambda\rangle<0$: $\lambda_{11}=1$, $\lambda_{12}=-2$,
$\lambda_{21}=-1$, $\lambda_{22}=1$. From [125].
Properties in the presence of disorder can depend sensitively on the coupling
constants $\lambda_{ij}$ of the two-band superconductor (see Section 3.3) as
well, since these enter the BCS gap equations including impurities. In general
even the two-band problem can seem to be quite dependent on many parameters
which are difficult to determine as a practical matter. Recently a
simplification was pointed out in [125] whereby the suppression of $T_{c}$ can
be expressed solely in terms of a universal pairbreaking parameter
$\tilde{\Gamma}_{12}=\Gamma_{1(2)}\frac{(1-\tilde{\sigma})}{\tilde{\sigma}(1-\tilde{\sigma})\eta\frac{(N_{1}+N_{2})^{2}}{N_{1}N_{2}}+(\tilde{\sigma}\eta-1)^{2}},$
(23)
where $\tilde{\sigma}=(\pi^{2}N_{1}N_{2}u^{2})/(1+\pi^{2}N_{1}N_{2}u^{2})$ and
$\Gamma_{1(2)}=n_{imp}\pi N_{2(1)}u^{2}(1-\tilde{\sigma})$ are cross-section
and normal state scattering rate parameters, respectively. The parameter
$\eta=v/u$ is the ratio of intra-band to inter-band scattering. In the weak
scattering (Born) limit, $\tilde{\sigma}\to 0$, while for $\tilde{\sigma}\to
1$ the unitary limit (strong scattering) is reached. Note the strange result
that $\tilde{\Gamma}_{12}\rightarrow 0$ in the unitary limit, i.e. nonmagnetic
impurities do not affect $T_{c}$ in an $s_{\pm}$ state [119]. While this may
be an artifact of the 2-band model used, it is an indication that a rather
robust set of parameters may produce significantly weaker effects of $T_{c}$
suppression than expected.
Expressed in terms of (23), all $T_{c}$ suppression curves collapse onto one
of three “universal” curves, depending on whether the average pairing strength
parameter $\langle\lambda\rangle$ is positive, negative, or zero, as shown in
figure 13. It is clear that in the case where intraband scattering dominates,
even in the “standard” $s_{\pm}$ scenario with $\langle\lambda\rangle<0$, the
$T_{c}$ suppression will be much slower than the AG result; thus experimental
results need not be taken as evidence against the $s_{\pm}$ state. In the
interesting and relatively unexplored $\langle\lambda\rangle>0$ case, a
transition below $T_{c}$ from $s_{\pm}$ to $s_{++}$ with increasing disorder
is possible [125].
#### 3.4.3 Anisotropic states.
*
Figure 14: (Color online) Normalized spectral gap $\Omega_{G}(\phi)/T_{c0}$
vs. angle $\phi$ on the Fermi surface for an extended $s$-wave state
$\Delta(\phi)=\Delta_{0}(1+r\cos 2\phi)$, with $\Delta_{0}/T_{c0}=1$, $r=1.3$
and Born limit scattering rate $\Gamma/T_{c0}=0$ (dashed), 0.3 (dotted), 1.0
(solid), and 3.1 (dashed-dotted). From [126].
There is substantial evidence from low-energy thermodynamics and transport
experiments (Section 5) that low-energy quasiparticle excitations are present
in many Fe-based materials, indicating either very small minimum gaps or true
nodes of the order parameter on one or more Fermi surface sheets. The effect
of nonmagnetic impurities on $s$-wave states of this type also depends on the
character of the scattering, in particular whether or not inter- or intraband
processes dominate. If intraband scattering processes are considered by
themselves, they simply average the angular structure of the order parameter
on each Fermi surface sheet, as in the conventional $s$-wave case [127].
$T_{c}$ will fall initially and then saturate. If gap nodes are present, they
will be lifted by the averaging process at a critical value of disorder
(figure 14), and give rise to a crossover at the lowest temperature from power
laws in $T$ to exponential behavior [126] with increasing disorder. Thus if
intraband scattering dominates, such nodal or near-nodal systems will display,
in the clean limit, the low-energy excitations characteristics of nodes, while
dirty systems will be gapped with reduced $T_{c}$.
In the early literature on disorder in Fe-based system it was frequently
assumed, to the contrary,that the order parameter was isotropic $s_{\pm}$, and
that interband scattering dominates. In this case, under special circumstances
(see Section 3.4.4), bound states at the Fermi level may be induced and mimic
the effect of nodes in some experiments [120, 121]. In such a situation, the
opposite behavior with disorder is to be expected: clean systems will display
exponential $T$-dependence and dirty systems the power laws expected from
impurity-induced residual densities of states.
#### 3.4.4 Single impurity problem.
In the most general and presumably realistic situation, anisotropic multiband
order parameters are present with both intra- and interband scattering.
Intraband scattering probably dominates in most situations (see below) and for
intermediate to strong impurity potentials, interband scattering effects are
largely irrelevant. To understand why this is the case, we consider the single
impurity problem in a symmetric $s_{\pm}$ state. Equation (21) for
$\hat{\Sigma}^{imp}$ is essentially identical as the $T$-matrix for a single
impurity, whose poles at $\Omega_{0}$ indicate the existence of impurity bound
states [128]. In general, energies nested near the gap edge correspond to weak
pairbreaking, while energies near the Fermi level correspond to strong
pairbreaking. A plot of the single impurity resonance energy position in a
symmetric $s_{\pm}$ state is given in figure 15, and shows that in order to
influence the states near the Fermi level a very specific fine tuning of
interband and intraband scattering is required. For the symmetric model, this
corresponds to $\eta=u/v=1$ in the intermediate to strong potential range, but
this criterion will be different in the asymmetric model $N_{1}\neq N_{2}$,
$\Delta_{1}\neq\Delta_{2}$. It thus seems a priori unlikely that the impurity
band in an isotropic $s_{\pm}$ state explanation for experiments indicating
low-lying quasiparticle states is correct. For further discussion, see Section
5.
*
Figure 15: Energy of single impurity bound state $\Omega/\Delta_{0}$ in
symmetric $s_{\pm}$ superconductor with gaps $\pm\Delta_{0}$ as a function of
interband ($u$) and intraband ($v$) scattering (A.F. Kemper, private
communication).
The single impurity problem has been considered in more detail in a 2-band
model in [123, 129, 130, 131, 132, 133, 134, 135] and in a 5-band model in
[136]. These calculations show a much richer structure of the local density of
states around a single impurity than is found, e.g. in the one band $d$-wave
problem [128], as might be expected, and a complicated dependence on
interaction and impurity parameters. It seems unlikely, given this complexity,
that the combination of STM imaging of impurity states [128] and theories of
this type will be able to provide definitive information on order parameter
symmetry or structure in these systems.
One approach to reducing the number of parameters and making theories of this
type more predictive has been to try to calculate impurity potentials from
first principles methods. For example, Kemper et al [137] calculated the
nonmagnetic and magnetic impurity potentials for a Co substituting for an Fe
in BaFe2As2 within density functional theory. Nakamura et al [138] later
performed similar calculations for several impurity types. In principle, such
calculations can provide important input into phenomenological treatments of
disorder by specifying $u$ and $v$ in band space, but band-resolved results of
this type have not yet been given. Kemper et al found in their calculations
the nonmagnetic potential was significantly larger than the magnetic one, and
that the interband scattering was perhaps a factor of three smaller than
intraband. In orbital space, results from [137] and [138] disagree
substantively for the Co potential, but it is not clear whether this arises
from the fact that the former calculations were performed in the spin-
polarized phase, or due to a different treatments of the nonlocal LDA
potential [138].
#### 3.4.5 Magnetic impurities.
A qualitative rule of thumb when considering the effect of magnetic and
nonmagnetic impurities on multiband, anisotropic superconductors is as
follows: When a nonmagnetic impurity scatters a pair from one point on the FS
into another point, such that the order parameter does not change sign,
scattering is not pair breaking; if the order parameter flips its sign, it is
pair breaking. For a magnetic impurity, the opposite is true: scattering with
an order parameter sign change is not pairbreaking, otherwise it is. However,
quantifying this rule of thumb may be complicated and results are sometimes
counterintuitive.
In terms of concrete calculations for magnetic impurities in $s_{\pm}$ states,
Golubov and Mazin [118] showed that for the symmetric model considered above,
$T_{c}$ is suppressed by magnetic and nonmagnetic impurities at the same rate
in the disorder averaged theory. The analogous single impurity problem was
treated by Akbari et al [139] within an Anderson model approach to a rare
earth impurity in such a system in the $s_{\pm}$ state. In both situations
only quantitative differences in the scattering from magnetic impurities
relative to the usual $s$-wave case were found. In special symmetric
situations, interband magnetic scattering in an $s_{\pm}$ state can give rise
to arbitrarily weak pairbreaking, whereas intraband scattering is strongly
pairbreaking as expected from AG theory [132].
#### 3.4.6 Orbital effects.
In Section 3.4.4 above it was argued that fine tuning of intra-and interband
pairing amplitudes was required in order to create substantial pairbreaking in
an $s_{\pm}$ state, e.g. create bound states near the Fermi level. In a more
realistic approach, however, these parameters ( $u$ and $v$) should not be
considered arbitrary, For example, if one starts from a local atomic picture
where an impurity is assumed to modify the orbital occupation energies of the
Fe-derived $d$ states, and then transforms to a band basis, it is easy to see
that intraband scattering terms of the same order as interband ones are
automatically generated by the unitary transformation from orbitals to bands.
This was the basis of the argument made in [107] that interband scattering
generically leads to $u\simeq v$ and therefore to large $T_{c}$ suppression in
the symmetric $s_{\pm}$ model; this was taken as evidence against $s_{\pm}$
pairing. However as seen in Section 3.4.4, a fine tuning is required to
produce significant pairbreaking, so we consider it generically unlikely that
for a given set of four or five Fermi surface pockets, with differing density
of states and order parameter magnitudes, that the condition for maximal
(Abrikosov-Gor’kov like) $T_{c}$ suppression will be achieved accidentally.
This point of view is also borne out by the argument in [125] and exhibited in
figure 13.
### 3.5 Dimensionality.
In this section we will concentrate on the qualitative effects due to 3D
dispersion of the electronic bands. It should be kept in mind that
dimensionality also plays a role in magnetism, in particular, it may affect
the extent of the magnetic part of the phase diagram and magnetic-orthorhombic
splitting [140], and also it can manifest itself through anisotropy of elastic
properties, phonon dispersion and electron-phonon coupling (which does not
seem to be the case in FeBS). Again, we will not discuss these effects, but
only the effects of dimensionality on the band structure, and, via the latter,
on superconductivity.
Such effects can be, roughly speaking, divided into three groups. First, there
is a generic issue of the correspondence between the number of carrier and
their density of states. Let us compare conventional superconductors, MgB2 and
B-doped diamond. Both exhibit hole-doped covalent bonds, and electron-phonon
matrix elements are practically identical. Note that it is rather difficult to
dope such bonds, so the number of carriers is small in both cases. Yet in the
quasi-2D $\sigma$-bands of MgB2 this small amount creates a sizeable DOS
(recall that in ideal 2D parabolic bands DOS does not depend on the carrier
concentration at all), and a critical temperature of 40 K, while in Boron-
doped diamond the DOS remains small, according to the small number of holes,
and so does $T_{c}$.
The second effect is the geometry of the Fermi surface. As we know, in the
spin-fluctuation model the structure of the order parameter is defined by the
interplay between the $q$-dependence of the spin fluctuations and the shape of
the Fermi surface. This is an interesting possibility that has been explored
mostly by that part of the community which traces the spin fluctuations to
nearest- and second nearest neighbor superexchange (see Section 3.1). In that
case the nodal lines are fixed in reciprocal space at $k_{x}=\pm\pi/2$, and at
$k_{y}=\pm\pi/2$. If at some particular $k_{z}$ a Fermi surface expands as to
cross these lines, actual gap nodes develop.
Finally, the last group of effects is related not so much to possible changes
with $k_{z}$ of the Fermi surface shape but to the orbital composition of the
states forming the Fermi surface. These may come from two sources. First, near
the $\Gamma$ point, besides the ubiquitous $xz/yz$ band, occasionally other
bands may cross the Fermi level, including the $z^{2}$ band that is very
dispersive in the direction. This band hybridizes with the $xz/yz$ band
everywhere except the high-symmetry planes, leading to Fermi surface pockets
that rapidly change their character as $k_{z}$ changes (see figure 16).
*
Figure 16: $d_{z^{2}}$ orbital character on the Fermi surface of 5% electron
doped Ba-122 according to DFT. The two outer hole Fermi surfaces are clipped
at $\pm\pi/2c$ to show the parts carrying most of the $d_{z^{2}}$ character.
Another, more subtle source of 3D effect related to the orbital character of
the bands comes from the fact that the electron bands, as discussed in Section
2, are never pure $xz/yz$, but always have an admixture of the $xy$ symmetry,
in the outer barrel. This fact was first noted and explained by Lee and Wen
[39], and elaborated in a review paper by Andersen and Boeri [141]. The
relevant physics also controls the warping and the twisting of the electron
FSs.
*
Figure 17: Band dispersion in the unfolded BZ corresponding to a 1-Fe unit
cell. Major orbital contributions are labeled.
To understand this, we will start as usual with the unfolded band structure
(figure 17), corresponding to a single Fe unit cell (figure 16(b)). The
unfolded Fermi surface geometry of the electron pockets is, essentially,
defined by their ellipticity and its variation with $k_{z}$. The ellipticity
at a given $k_{z}$ in the unfolded zone is determined by the relative position
of the $xy$ and $xz/yz$ levels of Fe, and the relative dispersion of the bands
derived from them. Indeed, the point on the Fermi surface located between
$\Gamma$ and $\mathrm{X}$ has a purely $xy$ character, while that between
$\mathrm{X}$ and $\mathrm{M}$ a pure $yz$ character. At the $\mathrm{X}$ point
the $xy$ state is slightly below the $yz$ state, but has a stronger
dispersion, therefore depending on the system parameters and the Fermi level
the corresponding point of the Fermi surface may be more removed from
$\mathrm{X}$, or less. In the 1111 compounds, for instance, the dispersion of
the $xy$ band is not high enough to reverse the natural trend, so the Fermi
surface remains elongated in the $\Gamma\mathrm{M}$ (1,0) direction.
Both $xy$ and $xz/yz$ orbitals point away from the Fe-Fe bond (as opposed to
the $x^{2}-y^{2}$) orbital, therefore their hopping mainly proceeds via As
(Se) $p-$orbitals. The $xy$ states near the $\mathrm{X}$ point mainly hop
through the $p_{z}$ orbital (see [141] for more detailed discussions), and
$xz$ ($yz$) via $p_{y}$ ($p_{x}$) orbitals (not because of the orbitals’
shape, but because of their phases at $\mathrm{Y}$[$\mathrm{X}$]). If there is
a considerable interlayer hopping between the $p$ orbitals, whether direct (11
family) or assisted (122 family), the ellipticity becomes $k_{z}$-dependent.
For instance, in FeSe there is noticeable overlap between the Se $p_{z}$
orbitals, so that they form a dispersive band with the maximum at $k_{z}=0$
and the minimum at $k_{z}=\pi/c$. Obviously, hybridization is stronger when
the $p_{z}$ states are higher, therefore the Fermi surface ellipticity is
practically absent in the $k_{z}$=0 plane, while it is rather strong in the
$k_{z}=\pi/c$ plane, which leads to formation of the characteristic “bellies”
in the Fermi surface of FeSe. On the other hand, $p_{x,y}$ orbitals in FeSe
hardly overlap in the neighboring layers, so the $xz$ and $yz$ bands have very
little $k_{z}$ dispersion, so that the inner barrels of the electronic pockets
in this compound are practically 2D.
In 122, the interlayer hopping proceeds mainly via the Ba (K) sites, and thus
the $k_{z}$ dispersion is comparable, but opposite in sign (for instance, at
the $\Gamma$ point the hopping amplitudes from Ba $s$ to As $p_{z}$ orbitals
above and below have opposite signs, while those for the As $p_{x,y}$ orbitals
have the same signs) for the $xy$ and $xz/yz$ bands. Note this difference
between the 122 and 1111 materials is related simply to the structural
difference between “in-phase” and “out of phase” FeAs layers in the unit cell.
As a result, when going from the $k_{z}=0$ plane to the $k_{z}=\pi/c$ plane
the longer axis of the Fermi pocket shrinks, and the shorter expands. In
BaFe2As2 the average ellipticity is very small, while the As-Ba overlap is
large, so that the actual ellipticity changes sign when going from $k_{z}=0$
to $k_{z}=\pi/c$. On the other hand, in Se based 122 systems the Se-K hopping
is quite small, so ellipticity is small for all $k_{z}$.
Importantly, the symmetry operation that folds down the single-Fe Brillouin
zone when the unit cell is doubled according to the As (Se) site symmetry is
different in the 11 and 1111 structures, as compared to the 122 structure. In
the former case, the operation in question is the translation by
$({\pi},{\pi},0)$,without any shift in the $k_{z}$ direction, in the latter by
$({\pi},{\pi},{\pi})$. Thus the folded Fermi surface in 11 and in 1111 has
full fourfold symmetry, while that in the 122 has such symmetry only for one
particular $k_{z}$, namely $k_{z}=\pi/2c$. Furthermore, in 122 the folded
bands are not degenerate along $\tilde{M}\tilde{X}$ as they were in 11/1111.
Finally, there is a considerable (at least on the scale of the superconducting
gap) hybridization when the folded bands cross (except for $k_{z}=0$). As a
result, although the band structure calculations for 122 materials actually
produce two detached (except for one plane) cylinders for the electronic FSs,
one cannot “unfold” these two cylinders as if one of them was folded down into
the other. The actual folded FSs intersect, yet we do not observe these
intersections because of hybridization induced by the As (Se) potential.
All these effects of 3-dimensionality of the electronic structure manifest
themselves in the 3D gap structure, tending to complicate the simple 2D
theoretical pictures. We comment below on some of the most significant ways in
which this occurs.
## 4 Gap symmetry.
The first question to be asked regarding the pairing state in a novel
superconductor is, what is the symmetry of the order parameter? Of course, the
symmetry itself does not fully describe the structure of the gap. For
instance, even a full-symmetry ($s$-wave, or, synonymously, $A_{1g},$
symmetry) order parameter may have “accidental”, that is, not required by
symmetry, nodes, as long as these nodes transform into each other by all point
group operations. Yet, establishing the right symmetry is arguably the most
important step towards uncovering the full gap structure.
### 4.1 Triplet or singlet?
In materials with an inversion center, so-called centrosymmetric, any Cooper
pair can be characterized by its parity, in the sense that the spatial part of
its wave function (the order parameter) can either change sign or remain the
same under the inversion operation. Electrons being fermions, the full wave
function is always antisymmetric. Since inversion corresponds to swapping the
pair components, if they have total spin $S=1$ (triplet), the spatial part of
the wave function should be odd, and if they have $S=0$ (singlet) it should be
even.
A triplet Cooper pair with $S_{z}=\pm 1$ can screen an external magnetic
field, just as individual electrons can. The spin-orbit interaction can
prevent this for some directions, but not for others. On the other hand, a
singlet pair has no net spin and does not contribute to magnetic
susceptibility as $T\rightarrow 0$. Thus for singlet superconductivity one can
expect the uniform spin susceptibility to diminish below $T_{c}$. The easiest
and the most accurate way to probe the latter is via the Knight shift. This
experiment has been performed on several FeBS including Ba(Fe1-xCox)2As2
[142], LaFeAsO1-xFx [143], PrFeAsO0.89F0.11 [144], Ba1-xKxFe2As2 [145, 146],
LiFeAs [147, 148], and BaFe2(As0.67P0.33)2 [149], and it was found that the
Knight shift decreases in all crystallographic directions. This effectively
excluded triplet symmetries such as $p$-wave or $f$-wave.
### 4.2 Chiral or not?
Another way to classify the superconducting state is according to whether it
breaks time-reversal symmetry or not, and according to whether it is chiral
(finite expectation value of the magnetic moment operator in the ground state)
or not. A singlet chiral superconducting state is allowed in tetragonal
symmetry [27], and is transformed under the symmetry operations as
$xz+{\mathrm{i}}yz$ (of course the complex conjugate state,
$xz-{\mathrm{i}}yz$, is also allowed). There are several ways to detect
$T$-breaking and chirality, but the simplest is probably the $\mu SR$
spectroscopy. This technique is sensitive to small local magnetic fields, as
long as they are static. In a chiral superconductor, as long as the balance
between the two conjugate states is broken, which should be happening near
crystallographic defects, spontaneous orbital current appear, and should be
visible by the $\mu$SR technique. Such an effect was observed, for instance,
in Sr2RuO4 [150], but not in FeBS [151]. In principle, a chiral state can also
be generated by mixing two 1D representations, e.g. a so-called
$d+id^{\prime}$ (note a $s+id$ state, also possible in principle, is
$T$-breaking but is not chiral). However, this requires separate transition
temperatures corresponding to the two representations; since this has not been
observed, we do not discuss this possibility further.
### 4.3 $d$ or $s$?
Having excluded triplet ($p$ and $f$) symmetries, as well as the chiral state,
we are left with the following singlet pairing possibilities (in a 3D system
with tetragonal symmetry): $s$-wave [$A_{1g}$]; $d(xy)$ [$B_{2g}]$;
$d(x^{2}-y^{2})$ [$B_{1g}$]; $g$-wave $(xy(x^{2}-y^{2}))$ [$A_{2g}]$; and
$d(xz\pm yz)$ [$E_{g}$]. It is important to note that, while $d$-wave does not
necessarily imply the existence of gap nodes, in combination with a quasi-2D
Fermi surface centered around the $\Gamma Z$ line such nodes are unavoidable,
either vertical for the $A$ and $B$ symmetries, or horizontal, for the $E$
symmetry. As will be discussed later in this review, the surface probes, such
as ARPES and tunneling show full gaps with no nodes, and, at least for some
compounds bulk probes show exponential low-temperature behavior.
There are also direct experiments that provide evidence against $d$-wave. The
$d$-wave representations, $B_{1g}$ and $B_{2g}$, should not exhibit any
Josephson current when weakly coupled to a known $s$-wave superconductor, by
symmetry, if the current flow precisely along the $z$ axis. However, such
current was observed in the 122 single crystals [152]. This observation is
difficult to explain away by deviation from the correct geometry, because the
observed current was strong and showed a well-defined Fraunhofer diffraction.
Another piece of evidence comes from the so-called anomalous Meissner, or
Wohlleben effect. This effect was predicted in the beginning of the cuprates
era [153] and since then has been routinely observed in $d$-wave
superconductors. In a nutshell, this effect appears in polycrystalline samples
with random orientation of grains. For any $d$-wave superconductor one expects
roughly 50% of weak links to have a zero phase shift, and 50% a $\pi$ phase
shift. One can show that in this case the response to a weak external magnetic
field is paramagnetic, i.e., opposite to the standard diamagnetic
superconducting response. This effect has been searched for in FeBS [154], but
not found.
These separate pieces of evidence strongly suggest that the pairing symmetry
is $s$, and not $d$. However, we want to stress that direct testing similar to
that performed in cuprates, namely a single-crystal experiment with a 90∘
Josephson junction forming a closed loop, is still missing, and it is highly
desirable for experimentalists to perform this ultimate test. In addition, it
should be borne in mind that no law of nature forbids different FeBS materials
from having different order parameter symmetries, although our previous
experience with other novel superconductors tends to argue against this.
Indeed, there are several proposals that, while most FeBS are $s$-wave, those
with unusual Fermi surfaces with only one type of pocket can be $d$ wave, see
Section 5.3.
## 5 Gap structure.
### 5.1 Does the gap in FeBS change sign?
Even though there is convincing evidence that the point symmetry of the order
parameter is for most, if not for all compounds, $s$-wave, it does not tell us
much about the actual structure of the order parameter and the excitation gap.
As opposed to the $d$-wave case, where nodes are mandated on the hole pockets
by symmetry, in an extended $s$-wave scheme they may appear on either type of
pocket if higher harmonics in the angular expansion of the order parameter are
sufficiently large. As discussed in Section 3.1, there are microscopic reasons
why this may be the case. Moreover, since nodeless $s_{\pm}$, nodeless
$s_{++}$, and an extended $s$ with accidental nodes all belong to the same
symmetry class, the difference between them is only quantitative (but
important). In this regard, several experiments appear relevant.
#### 5.1.1 Spin-resonance peak.
One obvious effect that was mentioned even in the very first paper proposing
the $s_{\pm}$ scenario [29] and later elaborated in detail [36, 155, 156], is
the neutron spin resonance. Neutron scattering is a powerful tool to measure
the dynamical spin susceptibility $\chi_{s}(\bi{q},\omega)$. For the local
interactions (Hubbard and Hund’s exchange, see Equation (3.1)), $\chi_{s}$ can
be obtained in the RPA from the bare electron-hole bubble
$\chi_{0}(\bi{q},\omega)$ by summing up a series of ladder diagrams to give
$\displaystyle\chi_{s}(\bi{q},\omega)=\left[I-U_{s}\chi_{0}(\bi{q},\omega)\right]^{-1}\chi_{0}(\bi{q},\omega),$
(24)
where $I$ is a unit matrix in orbital space and all other quantities are
matrices as well.
The fact that $\chi_{0}(\bi{q},\omega)$ describes particle-hole excitations
has interesting consequences in the case of an unconventional superconducting
state. Excitations are gapped below approximately $2\Delta_{0}$; (at $T=0$)
only above this threshold does $\mathrm{Im}\chi_{0}(\bi{q},\omega)$ become
non-zero. The term arising from the anomalous Green functions is proportional
to
$\sum{k}\left[1-\frac{\Delta{k}\Delta_{\bi{k}+\bi{q}}}{E_{\bi{k}}E_{\bi{k}+\bi{q}}}\right]...$
(25)
where $...$ represents the kernel of the BCS susceptibility (see e.g. [157]).
At the Fermi level,
$E_{\bi{k}}\equiv\sqrt{\varepsilon{k}^{2}+\Delta{k}^{2}}=|\Delta{k}|$. If
$\Delta{k}$ and $\Delta_{\bi{k}+\bi{q}}$ have the same sign, the coherence
factor in square brackets in (25) vanishes, leading to a smooth increase of
the magnetic response with frequency above the $T=0$ threshold of
$\Omega_{c}=\min\left(|\Delta{k}|+|\Delta_{\bi{k}+\bi{q}}|\right)$. In case of
unconventional superconductors [158], when for a given $\bi{q}$,
$\mathrm{sgn}\Delta{k}\neq\mathrm{sgn}\Delta_{\bi{k}+\bi{q}}$, the coherence
factor is non-zero and the imaginary part of $\chi_{0}$ possesses a
discontinuous jump at $\Omega_{c}$. Due to the Kramers-Kronig relations, the
real part exhibits a logarithmic singularity. For a range of interaction
values entering the matrix $U_{s}$, $\mathrm{Im}\chi_{0}=0$ and non-zero
$\mathrm{Re}\chi_{0}$ result in the divergence of
$\mathrm{Im}\chi_{s}(\bi{q},{\mathrm{i}}\omega_{m})$ according to Equation
(24). Such an enhancement of the spin susceptibility is called a “spin
resonance”. The corresponding peak appears at a frequency below $\Omega_{c}$
with the exact position $\Omega_{res}$ pushed below $\Omega_{c}$ by an amount
which scales with the strength of $U_{s}$.
*
Figure 18: (a) Calculated $\mathrm{Im}\chi(\bi{q}=\bi{Q},\omega)$ in the
normal state and for the $d_{x^{2}-y^{2}}$ and $s_{\pm}$ pairing symmetries
[36]. In the latter case, the resonance is clearly seen around
$\omega=2\Delta_{0}$. (b) Cartoon of the order parameter symmetries and a wave
vector $\bi{Q}$ connecting different Fermi sheets. (c)
$\mathrm{Im}\chi(\bi{q},\omega)$ for the $s_{\pm}$ state as a function of
frequency $\omega$ and momentum along the $(1,1)$ direction [36]. (d)
Experimental neutron data showing appearance of the spin resonance in
BaFe1.85Co0.15As2 below $T_{c}=25$K [159].
Scattering between nearly nested hole and electron Fermi surfaces in FeBS
produce a peak in the normal state magnetic susceptibility at or near
$\bi{q}=\bi{Q}=(\pi,0)$. For the uniform $s$-wave gap,
$\mathrm{sgn}\Delta{k}=\mathrm{sgn}\Delta_{\bi{k}+\bi{Q}}$ and there is no
resonance peak. For the $s_{\pm}$ order parameter, $\bi{Q}$ connects Fermi
sheets with mostly different signs of the gaps, see figure 18(b). This fulfils
the resonance condition for the interband susceptibility, and a well defined
spin resonance peak is formed (compare normal and $s_{\pm}$ superconductor’s
response in figure 18(a)). Moreover, the intraband bare susceptibilities are
small at this wave vector due to the direct gap, i.e. no states at the Fermi
level can be connected by intraband scattering with wave vector $\bi{Q}$.
Therefore, a single pole will occur for all components of the RPA spin
susceptibility at $\Omega_{res}\leq\Omega_{c}$ and a spin exciton forms [36,
155].
In the case of the $d_{x^{2}-y^{2}}$ superconducting gap under discussion in
FeBS, the situation is more complicated. $\bi{Q}$ connects states rather close
to the nodes of the order parameter on the hole sheets (see figure 18(b)) and
the overall gap in $\mathrm{Im}\chi_{0}$ determined by $\Omega_{c}$ is
significantly reduced. Still, the resonance condition can be fulfilled due to
the fact that for some $\bi{k}$’s $\Delta{k}=-\Delta_{\bi{k}+\bi{Q}}$.
However, because of the smallness of $\Omega_{c}$ the discontinuous jump in
$\mathrm{Im}\chi_{0}$ is vanishingly small or zero. Thus the total RPA
susceptibility shows a moderate enhancement with respect to the normal state
value, as seen in figure 18(a). The same holds for $d_{xy}$\- and
$d_{x^{2}-y^{2}}+{\mathrm{i}}d_{xy}$-wave symmetries [36] and a triplet
$p$-wave [155]. Of course, for states besides $s_{\pm}$ a resonance can occur
at a different wave vector $\bi{Q}$ connecting surfaces where the gap changes
sign; in the $d$-wave case a resonance has been predicted for the wave vector
$\bi{q}\approx(\pi,\pi)$ connecting the two electron pockets [156].
Thus, the resonance peak at $(\pi,0)$ is pronounced only for the sign-changing
$s$-wave order parameter like $s_{\pm}$. Such a distinct behavior for the
$s$\- and $d$-wave gaps can be clearly resolved via the inelastic neutron
scattering experiments and therefore it is a direct probe for the gap symmetry
in FeBS [36, 155, 156]. This situation is similar to the high-$T_{c}$ cuprates
and heavy fermion superconductors where a bound state (spin resonance) with a
high intensity also forms below $T_{c}$ [160, 161, 162].
The existence of the spin resonance in FeBS was first calculated theoretically
[36, 155] and subsequently discovered experimentally. No pronounced features
were observed in the earliest work [163], presumably due to sample quality
issues. This was followed, however, by many reports of well-defined spin
resonances near $(\pi,0)$ in 1111, 122, and 11 families of FeBS [159, 164,
165, 166, 167, 168, 175, 176]. Although the ratios of $2\Delta_{0}/T_{c}$ vary
from material to material [177], the gross features of the spin excitations
are similar: they are gapped below $\Omega_{c}$ at $T<T_{c}$, and there is an
enhancement at $\Omega_{res}$, which vanishes at temperatures above $T_{c}$.
Results for BaFe1.85Co0.15As2 shown in figure 18(d) are representative in many
respects. For this material, $\Omega_{res}/2\Delta_{0}=(0.79\pm 0.15)$ [159]
which is close to 0.64, which has been claimed to be a universal value for
cuprates, heavy-fermion superconductors, and FeBS [177, 178] (note there is no
compelling theoretical reason for this to be the case). While the resonance in
FeBS and cuprates are similar in many aspects, there are some differences. For
instance, the temperature evolution of $\Omega_{res}$ in Ba(Fe1-xCox)2As2 is
BCS-like without a signature of the pseudogap [159]. Also, because in cuprates
the AFM wave vector $\bi{Q}_{AFM}$ connects different parts of the same Fermi
surface, for $\bi{q}<\bi{Q}_{AFM}$ the gap becomes smaller than at
$\bi{Q}_{AFM}$ since we are closer to the $d$-wave nodes; the resonance
therefore shows downward dispersion. On the other hand, in FeBS with the
$s_{\pm}$ gap symmetry the resonance disperses upwards, see figure 18(c).
There are still a few puzzles connected with the spin resonance. In figure
18(c), we show the total RPA spin susceptibility as a function of both
momentum and frequency for almost perfectly nested Fermi surfaces. Note that
the $s_{\pm}$ gap changes only slightly on the hole and electron Fermi sheets
and can be considered nearly as a constant. Therefore, one always finds
$\Delta{k}=-\Delta_{\bi{k}+\bi{q}}$ as long as the wave vector $\bi{q}<\bi{Q}$
connects the states on the distant Fermi surfaces. The nesting condition is
very sensitive to the variation of $\bi{q}$ away from $\bi{Q}$ and already at
$\bi{q}\approx 0.995\bi{Q}$, the $\mathrm{Re}\chi_{0}(\bi{q},\Omega_{r})$ is
much smaller than its value at $\bi{Q}$. As a result the resonance peak is
confined to the nesting wave vector and does not disperse very much as occurs
in high-Tc cuprates. Therefore, one expects that when a system doped away from
the perfectly nested case, the spin resonance should become incommensurate
with $\bi{q}\neq\bi{Q}$. This is however not the case in the 1111 and Co-doped
122 families where it stays at $\bi{Q}$ independently of doping (within
experimental accuracy) [165, 166, 159, 167]. On the other hand,
incommensurability was found in the Fe(Se,Te) [168], and recently in the
K-doped 122 system as well [169]. Note that in the latter case the K-doping
was far from the optimal doping studied in [164], which may explain why the
incommensurability was easier to detect. However, there is currently no clear
understanding of why the resonance appears to be commensurate in some cases
and not others.
Another puzzle is connected with the anisotropy in the spin space observed
with polarized neutrons in the non-magnetic phase of Ba(Fe1-xNix)2As2 [170].
It was found that $\mathrm{Im}\chi_{+-}$ and $2\mathrm{Im}\chi_{zz}$ are
different, displaying different resonance frequencies and intensities. This
contradicts spin-rotational invariance (SRI) condition
$\left<S_{+}S_{-}\right>=2\left<S_{z}S_{z}\right>$ which must be obeyed in the
paramagnetic system. The relation $\mathrm{Im}\chi_{+-}>2\mathrm{Im}\chi_{zz}$
was confirmed by measurements of the NMR spin-lattice relaxation rate in the
perpendicular magnetic fields [145, 171]. One possible solution to the puzzle
could be the presence of the spin-orbit interaction, which can break the SRI
as it does in Sr2RuO4 [172].
Recently, it was suggested that the theoretically predicted peak for the
isotropic $s_{\pm}$ state is is too sharp and too strong compared to the
maximum observed in the experiment. Onari et al [98] proposed an alternative
explanation for the spin resonance that does not involve a sign change of the
order parameter. They noted that if there is a collapse of the scattering rate
below the pairbreaking edge, the redistribution of the spectral weight upon
entering the superconducting state can lead to the enhancement of the spin
response below $T_{c}$ as compared to the normal state. This effect does not
represent a true spin resonance in the sense that there is no divergence in
$\mathrm{Im}\chi$, but depending on the parameters one can gain significant
enhancement, and the observed resonance is indeed generally broader than
predicted in theories of the neutron response in a clean $s_{\pm}$ state.
On the other hand, the similar spin resonance in cuprates, albeit somewhat
sharper than that in FeBS, is also rather broad, and in this case there there
is little doubt that the scattering involves a sign-changing gap. Broadening
of a spin excitation can of course arise from many sources, the most obvious
one in this case being significant anisotropy of the $s_{\pm}$ gap. Another
problem with the explanation of [98] is that it may require a special form of
scattering in the normal state, $\mathrm{Im}\Sigma(\bi{q},\omega)=A(\pi
T+\omega)$, and in addition one needs to fine-tune the parameter $A$. The
exact effect of various assumptions regarding the scattering has in fact been
the subject of some debate [173, 174]. The fact that extremely similar
features of the spin excitations are observed in all families of FeBS would
seem to argue against the possibility of an isotropic $s_{++}$-wave gap.
#### 5.1.2 Josephson junctions.
While in cuprates the existence of the neutron resonance mode was a strong
argument in favor of $d$-wave pairing, really instrumental in establishing
that beyond reasonable doubt were Josephson-based experiments. A direct probe
of the symmetry of the order parameter was performed by creating a current
loop that included two Josephson contacts, one at the $a$ face of a crystal,
and the other at the $b$ face. It is easy to show that if the phase difference
across both contacts is $0$, the allowed values of magnetic flux through the
loop, expressed in flux quanta, are integer, while if one of the two contacts
has a phase shift of $\pi$, they are half-integer. The two cases are easily
distinguishable in the experiment.
The problem with applying this technique to FeBS is that in the $s_{\pm}$ case
there is no direction where symmetry would be dictating the phase of the order
parameter. The $x$ and the $y$ directions are indistinguishable by symmetry.
On may think about a Josephson loop with the contacts in the $ab$ plane, but
along inequivalent directions (for instance, $[10]$ and $[11]$), in the hope
that the numbers that define the Josephson current (orbital composition of the
wave function, relative gap sizes, etc) will conspire in such a way that the
current along one direction will be dominated by holes and the other by
electrons [179, 180]. Unfortunately, such contacts are not only difficult to
make but also there is no guarantee that the numerics will work out right;
existing theoretical estimates are on the borderline. Some other designs have
been proposed, but they either incorrectly (and too favorably) estimate the
condition of the $\pi$ contact formation, or are even less practical.
*
Figure 19: Sandwich design suggested in [179].
More promising is another suggestion [179]. This one utilizes a so-called
sandwich design (figure 19), wherein an epitaxial film of a hole-doped FeBS is
grown on top of an electron-doped one (or vice versa). If the parallel
momentum is conserved at the interface (which is why epitaxial growth is
necessary), the phase coherence is established among the holes in both slabs,
and correspondingly among electrons. A point contact to the hole-doped slab
will be dominated by the hole current, because of the prevalence of this type
of carriers, and the one to the electron-doped slab by the electron current.
If these contacts are now connected in a loop, the desired phase difference is
achieved.
So far no such (or similar) experiment has been performed. However, there is
an experiment that presents indirect evidence that Josephson loops with a
$\pi$ phase shift can be formed in these materials [181]. In this experiment,
rather than carefully preparing two Josephson contacts that are dominated by
hole and electron currents, respectively, one measures a very large number of
randomly formed contact pairs, in hope that some of them will accidentally
fulfil the condition that the two contacts have the required phase shift. In
general, a point contact to an electron-doped sample (on which the experiment
[181] was performed) will be dominated by electron current. Nevertheless,
there is a possibility that some of the randomly formed contacts will have a
sufficiently thick tunneling barrier, in which case the current may be
dominated by the states near the zone center, i.e. the hole states. Thus, one
expects in such an experiment to see a small, but not negligible fraction of
the formed loops to exhibit the $\pi$ phase difference. This is exactly what
was reported in the experiment [181].
#### 5.1.3 Quasiparticle interference.
Another important experiment providing information on gap structure is the so-
called quasiparticle interference scattering (QPI). The idea of this method is
simple: any sort of impurity or defect in a metal is screened by the
conducting electrons. This leads to the well known Friedel oscillations of the
charge and spin density around the imperfection. In real space, interference
among such oscillations stemming from random impurities is currently
unresolvable in these systems, but the Fourier transform of the measured
electron density will reflect the structure of the charge susceptibility in
reciprocal space. A natural way to map the electron density near the Fermi
level is by scanning tunneling spectroscopy. The theory of this effect in a
$d$-wave superconductor was proposed by Wang and Lee [182] for a single
impurity, and subsequently established for a finite density of impurities
[183, 184].
This technique can be used to gain information on the phases of the
superconducting order parameter. Indeed, let us begin with a simple BCS theory
with a uniform gap. Let us further assume that we have tuned our tunneling
bias to a voltage slightly above the gap value. The quasiparticle density of
states is enhanced at this voltage, showing a coherence peak as the voltage
approaches the gap value. As with any generalized susceptibility, there are
coherence factors involved [185, 186]. It turns out that for scattering from
magnetic impurities, or from order parameter suppressions, including vortices,
the coherence factors are proportional to
$(u_{k}u_{k^{\prime}}+v_{k}v_{k^{\prime}})$, that is, they are constructive
when $\Delta_{k}\Delta_{k^{\prime}}>0$, and destructive otherwise. Just as
with impurity pair breaking effects, discussed in Section 3.4, the situation
is reversed when the impurity is nonmagnetic, and the coherence factor is
proportional to $(u_{k}u_{k^{\prime}}-v_{k}v_{k^{\prime}})$. Thus, magnetic
impurities and vortices emphasize processes that scatter pairs without
flipping the sign of the order parameter, and the nonmagnetic defects
emphasize the sign-changing processes. We do not have a tool to change
dynamically the impurity concentration, but we can introduce vortices by
applying an external magnetic field, and then the QPI features associated with
the same-sign scattering will be enhanced in comparison with those due to the
sign-flip scattering [187].
In the Fe-based superconductors, theoretical predictions for the dispersion of
the QPI $\bi{q}$-peaks have been made for models with electron and hole
pockets [188] in the presence of $s_{\pm}$ superconducting order [77, 68], in
the SDW state with no superconducting order [189] and in the coexistence phase
[190]. All give differing signatures depending on the evolution of the
contours of constant quasiparticle energy in the various reconstructed Fermi
surfaces. The QPI signatures depend strongly on the sign change of the gap,
but also on details of the Fermi surface. A problem that prevents QPI from
being as useful a tool as it was in cuprates, is that in a $d$-wave
superconductor the tunneling current at low biases is dominated by “hot spots”
on the underlying Fermi surface where the superconducting gap is exactly equal
to the bias voltage [186]. In an isotropic nodeless superconductor there are
no “hot spots” and the entire theoretical picture is therefore blurred
compared to QPI in cuprates.
Experimentally, QPI measurements on Ca-122 lightly doped with Co [191] in the
magnetic phase revealed strong breaking of tetragonal symmetry, qualitatively
consistent with the observed SDW [189] and with DFT calculations with the
observed stripelike magnetic order [192]. It was also pointed out that
scattering in this system may be affected by anisotropic impurity states
around the Co sites imaged in the experiment, enhancing or modifying the
background anisotropy caused by the stripelike magnetism.
In the superconducting state of an Fe(Se,Te) superconductor near optimal
doping, a QPI experiment in a varying magnetic field was performed by Hanaguri
et al [193]. They found three features, one associated with the hole-electron
scattering (the smallest momentum), and two associated with two different
electron-electron scattering options. The last two features grow with respect
to the first one, which led Hanaguri et al to conclude that the holes and
electrons have opposite signs of the order parameter, as dictated by the
$s_{\pm}$ model.
It should be noted [194] that the last two wave vectors coincide with the two
smallest reciprocal lattice vectors, so QPI features at these vectors (if any)
will coincide with Bragg peaks. Hanaguri et al [195] argued that the
corresponding features can be decomposed into sharp peaks reflecting the Bragg
reflection, and broader paddings than must arise from QPI. The problem however
remains that both the sharp peaks and the paddings show similar dependence on
magnetic field (although the Bragg peaks should be insensitive to the vortex
concentration), which makes one suspect the real effect of magnetic peaks is
suppression of the small-moment feature, rather than enhancing the other two.
Thus the issue of whether Hanaguri et al have really observed QPI features and
can make conclusions regarding the order parameter is open.
Finally, there is another intriguing aspect of the QPI spectroscopy. If the
order parameter has nodes or deep minima, and there is a good experimental
reason to believe that this is the optimally doped Fe(Se,Te) [231], the same
mechanism that creates hot spots, dominating the QPI picture in cuprates, will
kick in as soon as the bias voltage is larger than the gap minimum. After
that, the spectrum should be dominated by these hot spots, creating, similarly
to the cuprates, a complicated patter of multiple very sharp spots, dispersing
with the bias. No trace of this effect as been observed.
#### 5.1.4 Coexistence of magnetism and superconductivity.
This discussion will not be complete without mentioning an experiment that
Mother Nature has performed for us, namely that in the phase diagram of the
Co-doped BaFe2As2 there is a well established range with microscopic
coexistence of weak antiferromagnetism and superconductivity [196]. Moreover,
the “backbending” of the SDW instability line in the coexistence phase,
observed in Co-doped Ba-122, indicates that the magnetism and
superconductivity are carried by the same electrons so that the two
instabilities compete for the same carriers. It can be shown [197, 198] that
in this case an $s_{\pm}$ superconductivity can easily coexist with an SDW
state, but $s_{++}$ can only for a very narrow range of parameter. Thus,
$s_{\pm}$ appears to be a much more natural state given that the coexistence
appears in a large part of the phase diagram, and probably exists also in
other FeBS, although for the other materials direct microscopic probes (e.g.,
NMR) are still missing.
This is a quantitative argument. One can also make another, slightly more
subtle, qualitative argument. It was noted already some time ago [199] that if
conventional ($s_{++}$ in our language) superconductivity develops on the
background of a spin density wave, the order parameter develops nodes. These
appear everywhere where new band crossings occur due to SDW symmetry lowering
(see [200] for a more detailed discussion). On the other hand, when we
introduce such an SDW in an otherwise nodeless $s_{\pm}$ superconductor, it
can be shown that no nodes develop even if in the new, downfolded Brillouin
zone, the “$s_{+}$” and the “$s_{-}$” bands cross [201].
The importance of this theorem can be appreciated if we remember that the in-
plane thermal conductivity in this very part of the phase diagram show the
clear absence of any vertical nodal lines (there are indication of possible
horizontal or so-called “$c$-axis” nodal lines, but the above mentioned SDW-
induced BZ band folding creates full vertical lines of new band crossings,
where the hole and the electron Fermi surfaces now overlap). Thus, if the
order parameter has the same sign on all FS pockets, in the coexistence region
vertical nodal lines must appear, and they must show up in the in-plane
thermal conductivity. If that does not happen, it leaves only one possibility:
an $s_{\pm}$ order parameter.
### 5.2 Evidence for low-energy subgap excitations.
In the few years of experimental studies on Fe-pnictide superconductors, the
hope that one might quickly identify a universal form of the superconducting
order parameter was confounded by an unexpectedly wide diversity of
experimental results, with some consistent with fully gapped behavior, and
others providing evidence for very low energy excitations consistent with gap
nodes. Early discussion focussed on the possibility that variations could be
explained exclusively by the effect of disorder (see Section 3.4), so that
some varying results on different samples of the same material could be
explained in this way. It may still be true that in some situations disorder
plays a key role and needs to be understood. However, in the past year or so
different experimental probes, particularly penetration depth experiments and
thermal conductivity, both bulk probes, are now providing a consistent picture
of the evolution of the low-energy quasiparticle density across the phase
diagram of the 122 materials, and to a lesser extent in other families as
well. This suggests a picture in which the gap structure is sensitively
related to the details of the Fermi surface as it evolves across the phase
diagram from hole- to electron-doped systems (see figure 20).
From the point of view of spin fluctuation theory outlined in Section 3, this
evolution is relatively easy to anticipate. As we move away from the parent
compound, the spin pairing interaction weakens. On the other hand, in the
ordered phase superconductivity is suppressed entirely by the competition with
magnetism for states near the Fermi level, until the SDW amplitude is
sufficiently weak, when $T_{c}$ can begin to grow. The optimal doping is thus
expected to be not far from the antiferromagnetic quantum critical point, as
it is indeed in reality. One can also understand on a similarly qualitative
level the tendency to node formation in the overdoped regime. Suppose that
local interaction parameters do not vary significantly with doping, as might
be expected if they are derived generally from Fe atomic orbitals (and
possibly ligand polarization effects, see [13]). Generally speaking, the
highest pairing strengths are predicted for systems that have taken full
advantage of the available condensation energy, i.e. optimally doped systems
should be maximally isotropic. Note that this may be a gap close to the
idealized isotropic $s_{\pm}$ state, or one with considerably more anisotropy,
depending on details of the band structure and the interactions themselves.
Optimal doping is then determined by a compromise between “nesting”, which is
generally maximal for the undoped parent compound, and the proximity of the
ordered SDW. As one overdopes the system, nesting deteriorates, and $T_{c}$
decreases. As discussed in Section 3, the dominant orbital interaction is
between $d_{xz}$ and $d_{yz}$ orbitals on the electron and hole pockets, while
the subdominant one which drives nodal behavior derives from the $d_{xy}$
interactions. The former interactions are weakened with overdoping while the
latter remain constant, leading to a relative enhancement of the frustrating
interactions and a tendency towards nodes which grows with overdoping.
Finally, in the common 1111 and 122 systems, there is a further effect which
drives an overall asymmetry of the $T$ vs. doping phase diagram. This is the
existence of the additional $d_{xy}$ hole pocket which appears with sufficient
hole doping. As explained in Section 3, this implies that hole doped systems
should be generally more isotropic than electron-doped ones. The above
“standard scenario” should now be tested against experiment.
*
Figure 20: Schematic phase diagram of Fe-based superconductors vs. doping,
with order parameter expected from 2D spin fluctuation theory plotted in one
quadrant of Brillouin zone as false color on Fermi surface [red=+, blue=-].
#### 5.2.1 Penetration depth.
Magnetic penetration depth measurements, summarized in figure 21, are bulk
probes of quasiparticle excitations which can provide evidence for nodal
structures or small gaps. In general, fits to theory over the entire
temperature range are difficult particularly for multiband systems, and depend
sensitively on details, so information obtained at low temperatures is simpler
to relate directly to gap structure. The relation of different gap nodal
structures to power laws in temperature $\Delta\lambda\sim T^{n}$ was pointed
out by Gross et al [202]. In a fully gapped system, at low temperatures
relative to the smallest full gap in the system an exponentially activated
behavior is expected; if this gap is very small, however, the penetration
depth can typically be fit to a power law in $T$ over some intermediate
temperature range. Another factor complicating the interpretation is disorder;
at the lowest temperatures, impurity scattering can lead to a $T^{2}$
dependence [202] if a residual density of states at the Fermi level is induced
(see Section 3.4). Thus fits to low-temperature power laws at low but not
asymptotically low temperatures may not be completely straightforward to
interpret, but do provide evidence for low-lying quasiparticle excitations.
Only in the case that a true linear power law $\Delta\lambda\sim T$ is
observed may one make definitive statements about the existence of (line)
nodes.
In both the LaFePO system [203, 204] and in BaFe2As1-xPx [205], a linear-$T$
dependence of the low-$T$ penetration depth $\Delta\lambda(T)$ was reported.
By contrast, in Ba(Fe1-xCox)2As2 and Ba(Fe1-xNix)2As2, $\Delta\lambda$ was
initially reported to vary close to $T^{2}$ over most of the phase diagram
[206, 207]; these power laws are in contrast to the activated temperature
dependencies expected for an isotropic gap. While $T^{2}$ is the power law one
naively expects for a (dirty) line nodal state, one may also show that in an
isotropic ($s^{\pm}$) superconductor, disorder can create subgap states [118]
under certain conditions, depending on the ratio of inter- to intraband
impurity scattering [126] (see Section 3.4). . If these states are at the
Fermi level, a fully gapped $s^{\pm}$ state will also lead to
$\Delta\lambda\sim T^{2}$. Fits of the same or very similar data on these
systems Ba(Fe1-xCox)2As2 and Ba(Fe1-xNix)2As2 near optimal doping are also
possible for an isotropic multigap model [208], and at optimal doping the
$T^{2}$ fit is rather poor, suggesting a small true gap. In this context, it
is worth noting that a number of multigap fits—to penetration depth, specific
heat, and other observables—in the literature violate BCS theory by taking
arbitrary ratios of the gaps $\Delta_{i}/T_{c}$ as fit parameters. Kogan et al
[209] have warned that unphysical results can be obtained by this procedure
even if the gaps are truly isotropic, since the various gaps are coupled
through the BCS gap equation. At the moment, substantial experience has been
accumulated by researchers from various groups that indicates that full
solution of a multiband Eliashberg equations in realistic cases always yields
at least one gap that is larger than the isotropic gap with the same $T_{c}$,
and one smaller [209].
Finally, there are some systems where a large full gap has been reported. For
example, in optimally doped Ba1-xKxFe2As2, a minimum gap of $1.3k_{B}T_{c}$
was extracted in [210, 211], and similar behavior was found for LiFeAs [212].
Early reports of exponential behavior in 1111 systems with any rare earth
except La were probably “contaminated” by the magnetic susceptibility of the
rare earth ion [213], and LaFeAsO1-xFx itself has been reported to have a
power law $T$ dependence close to $n=2$ [214].
*
Figure 21: Superfluid density temperature variation of Fe-pnictide
superconductors. (a) $(\lambda(T_{min})^{2}/\lambda(T)^{2}$ vs. $T$(K) for
optimally doped Ba1-xKxFe2As2 [210]; (b) $T-$dependent change in penetration
depth $\Delta\lambda(T)$(nm) vs. $(T/T_{c})^{2}$ for Ba(Fe1-xNix)2As2 [207];
(c) $\Delta\lambda(T)/\lambda(0)$ for optimally doped BaFe2(As1-xPx)2 [205];
(d) $\Delta\lambda(T)$(Å) for LaFePO [204].
Experimental attempts have been made to correlate disorder to the low-$T$
penetration depth to see if conclusions could be drawn regarding the structure
of the underlying order parameter. Hashimoto et al [210] reported not only the
sample which fit well to an exponential $T$-dependence, but a second sample,
considered dirtier due to its smaller $T_{c}$, which exhibited a $T^{2}$
behavior. This was interpreted as pairbreaking caused by interband scattering
in an $s_{\pm}$ state. A similar model was employed to study changes in
low-$T$ power laws with disorder, explicitly calculating the low-energy
density of states induced by interband impurity scattering and its effect on
the penetration depth and $T_{c}$ simultaneously [215]. While these studies
are suggestive, they cannot be regarded as conclusive regarding either the
structure of the order parameter or the nature of the disorder scattering, due
to the uncertainties regarding the difficulty of determining impurity model
parameters, see Section 3.4.
#### 5.2.2 Specific heat.
Specific heat was pioneered [216] as a tool for investigating the gap
structure on YBCO. Since the density of states of an unconventional
superconductor with line nodes varies as $N(\omega)\sim\omega$, the
temperature dependence of the Sommerfeld coefficient
$\gamma=\lim_{T\rightarrow 0}C/T$ in a clean superconductor with lines of
nodes varies as $T$ as $T\rightarrow 0$. This is a difficult measurement since
disorder generally gives rise to a residual density of states $N(0)$ which
induces a linear-$T$ term in $\gamma(T)$ below some disorder scale. It is
therefore sometimes more useful to examine the field dependence, which is also
quite sensitive to low-energy excitations.
In a clean nodal system, the theory of Volovik [218] predicts
$\gamma\sim\sqrt{H}$ in a clean superconductor with lines of nodes (disorder
changes this behavior slightly, to $\gamma\sim H\log H$ [219] for a disordered
superconductor with lines of nodes). These power laws can be derived very
easily from the Doppler shift of the low-energy nodal quasiparticles in the
superflow field of the vortex lattice. For a fully gapped superconductor,
$\gamma$ should vary linearly with H at low fields due to the localized
Caroli-de Gennes-Matricon states in the vortex cores. This probe provides a
first indication of the variability of thermodynamic properties in the Fe-
pnictide materials: in Ba1-xKxFe2As2 $\gamma\sim H$ [220] (implies fully
gapped superconductivity), while $\gamma\sim H^{1/2}$ in LaFeAsO1-xFx [221]
(implies nodal superconductivity). Recently Gofryk et al [222] performed
measurements on Ba(Fe1-xCox)2As2 across the electron doping range, and
reported a nonmonotonic dependence of the density of excitations with doping.
At optimal doping, a very weak field dependence consistent with a small gap or
weak nodes was reported, with quasiparticle contributions increasing on either
side of optimal doping (figure 22). Note that the underdoped sample is in the
SDW-SC coexistence phase. In the isovalent analog system BaFe2(As1-xPx)2, an
early report of linear-$H$ behavior which appeared inconsistent with
penetration depth measurements [223] reporting linear $T$ behavior as
discussed above [205], as well as evidence for nodes from NMR [149] and
thermal conductivity measurements [205], was recently superseded by a
measurement reporting a small Volovik term at low fields crossing over to
linear behavior at higher fields [224]. This work underlined the importance of
determining on which sheets the nodes occur, since sheets with smaller mass
and longer relaxation times will dominate transport, while larger mass alone
will determine specific heat. Theoretically, the field crossover of $\gamma$
was examined in a semiclassical multigap $s_{\pm}$ framework [240] whose
validity is questionable since it neglects the contribution of the core
states, which must contribute significantly in a fully gapped superconductor.
Fits to the $H^{1/2}\rightarrow H$ behavior were however obtained within a
multiband Eilenberger approach [224] assuming a highly anisotropic state on
one band.
*
Figure 22: Magnetic field dependence of the low-temperature specific heat of
Ba(Fe1-xCox)2As2 (circles, $x=0.045$; triangles, $x=0.105$; squares,
$x=0.08$). Empty and full symbols represent unannealed and annealed data, The
dotted, solid, and dashed lines described the field dependencies of the low-
temperature specific heat according to clean $s$-, anisotropic $s$-, and clean
$d$-wave. From [225].
If the system has gap nodes or deep minima, the semiclassical theory of
specific heat of an unconventional superconductor predicts [226] that the
measured specific heat should oscillate as a function of magnetic field
direction relative to the crystal axes. At the lowest temperature and fields,
it is generally expected that the minima in the specific heat will correspond
to fields pointing in nodal directions. It should be noted, however, that this
result depends sensitively on the phase space available for quasiparticle
scattering, and need not be universal for arbitrary superconducting states.
The roles of minima and maxima in the specific heat can also reverse as a
function of temperature or field, as is found even in the well-known $d$-wave
case [227, 228]. Early on in the discussion of the symmetry of the Fe-based
superconducting order, it was noted that phase sensitive experiments of the
type which led ultimately to the definitive determination of $d$-wave pairing
in the cuprates [229] would be difficult in the new systems, both due to
sample preparation difficulties and because of the complexity of interpreting
Josephson-based experiments in multiband systems. As an alternative, it was
proposed that specific heat oscillations might provide important information
as to the $k$-space structure of the order parameter [230]. This experiment
was first perfomed on the Fe(Te,Se) system by Zeng et al [231] (figure 23).
The positions of the specific heat minima along the $\Gamma-\mathrm{M}$ axis
are consistent [230, 232, 233] with an anisotropic gap with minima at these
angles, as predicted by spin fluctuation theories (see Section 3).
*
Figure 23: (a) Angle dependence of specific heat coefficient $C/T$ for
FeSe0.45Te0.55 at $H=9$T; (b) possible positions of gap minima consistent with
measurements from [231].
#### 5.2.3 Thermal conductivity.
The experimental probe of the bulk order parameter that has so far been
performed at the lowest temperatures is thermal conductivity. In a system of
normal conducting electrons, the thermal conductivity varies linearly with
$T$. While in principle thermal currents are carried by phonons as well, at
low $T$ the contribution to the thermal conductivity from phonons
$\kappa_{ph}$ typically varies as $T^{3}$ or nearby power, depending on the
phonon mean free path; thus at sufficiently low $T$, any linear-$T$ term in
$\kappa$ may be attributed to electronic excitations. In a superconductor,
this term provides information on both the superconducting gap structure and
the role of disorder. The technique has been applied extensively over many
years to cuprates and unconventional superconductors [234]. In the presence of
a magnetic field which creates a vortex state, quasiparticles are Doppler
shifted as in the case of the specific heat, not only in the density of states
but also in the lifetime. In a clean nodal superconductor, a field dependence
of $\Delta\kappa\sim H\log H$ is expected [235].
In principle, then, thermal conductivity should be one of the best probes of
low-energy quasiparticle excitations arising from gap structure. However, even
taken alone the thermal conductivity data on the various Fe-based systems
present a complex picture. In the 122 systems, the $ab$-plane thermal
conductivity data for both electron- and hole-doping exhibit zero or extremely
small linear-$T$ term in zero magnetic field, reflecting the apparent absence
of any nodes in the superconducting gap. The field dependence, however, is
significantly stronger than that expected for a large-gap superconductor [236,
237, 238] (see, however, a discussion in Section 1.2.2), particularly away
from optimal doping (see, e.g. figure 24(a)). Mishra et al [239] then proposed
that such results could most naturally be explained in terms of a gap with A1g
symmetry with no nodes but deep minima on the electron sheets. Bang proposed
that such strong field dependence could also be explained phenomenologically
by an isotropic “$s$-wave” state with very small gap on one Fermi surface
sheet [240] (on the other hand, such an explanation appears to be ruled out by
$c$-axis thermal conductivity, see below). To provide a scale to interpret
statements about the size of the low-$T$ thermal conductivity in the
superconducting state, we remind the reader that in a 2D nodal superconductor
$\frac{\kappa_{ab}}{T}|_{T\rightarrow 0}\simeq
aN_{\mathrm{nodes}}{k_{B}^{2}m^{*}\over\hbar
d}\left[\frac{v_{F,ab}^{2}}{k_{F}v_{\Delta,ab}}\right]_{\mathrm{node}},$ (26)
where $a$ is a dimensionless constant which depends on the nodal phase space,
$d$ is the distance between planes, $N_{\mathrm{nodes}}$ is the number of
distinct nodal surfaces, assumed equivalent, and $m^{*}$ is the effective mass
for motion of quasiparticles in the $ab$-plane. The conclusion of all
experiments on near-optimally doped K- doped or Co-doped Ba-122 systems was
that the measured linear term was much less than expected from this
expression, i.e. $a\ll 1$.
*
Figure 24: (a) Field dependence of in-plane $\kappa_{a}/T$ in Ba(Fe1-xCox)2As2
from [241]. (b) Same for both normalized $\kappa_{a}$ and $\kappa_{c}$ in
Ba1-xKxFe2As2, [242]. (c) Normalized $a$ and $c$ thermal conductivities as
functions of doping, [241].
Recently, Reid et al [241] showed that in the same samples where the ab-plane
thermal conductivity was very small (consistent with zero within experimental
error), a significant $c$-axis linear-$T$ thermal conductivity was reported
(note that “significant” here refers to values normalized to normal state
values determined by the Wiedemann-Franz law; absolute values are still of the
same order as quoted error bars for the in-plane conductivity). This is
paradoxical at first glance, because if order parameter nodes exist, they
should influence transport properties in both directions, the only difference
being a weight factor of $v_{F,i}^{2}$ for $i=ab,c$, as seen, e.g. in Equation
(26). The only possible interpretation [241] is that the nodes are located on
flared portions of the Fermi surface where the $c$-axis velocities are very
high. Mishra et al [243] confirmed this picture and pointed out that that the
result $a\ll 1$ above in the ab-plane implies that the phase space for the
nodes producing the $c$-axis signal must be very small. Their result is
consistent with “weak nodes”, which exist over a small portion of the Fermi
surface rather than running the length of the Fermi cylinders. These could be
the “V-shaped” or “loop” nodes found in [48, 49] on the hole pockets near the
Z point, or small loop nodes on electron pockets as suggested in [244]. In
figure 24(c) the data of Reid et al are plotted as a function of doping at
$H=0$ and $H=H_{c2}/4$. It is seen that the normalized thermal conductivity,
reflecting the existence of low-energy quasiparticles, increases dramatically
away from optimal doping for currents along the $c$-axis in zero field. In
nonzero field, however, the response is quite isotropic. One is tempted to
conclude that only one Fermi surface sheet plays a role in thermal transport
at higher field, such that the thermal current is isotropic when normalized;
this is far from obvious, however, given that the field is aligned in the $c$
direction in both cases, so that the averaging over the inhomogeneous response
function is quite different for the two directions. These issues are discussed
in [243].
The existence of nodes on the flared portion of the Fermi surfaces may appear
an unlikely accident, but it is easy to see within the context of spin
fluctuation theory that changes in orbital character on the Fermi surface tend
to produce nodes because of the strong tendency of like orbitals to pair. As
discussed in Section 3.5, while the 122 systems look quite similar to the 1111
systems at $k_{z}=0$ (except for differences in ellipticity), at higher
$k_{z}$ other bands, particularly the $z^{2}$, mix strongly, such that at the
top of the Brillouin zone, where the hole pockets are most flared, there is a
strong admixture of several bands whose weight varies around the sheet. This
is most likely the origin of possible weak nodes in the hole band [48, 243].
The above discussion has applied exclusively to the Ba-122 system doped with K
or Co. The measurements on optimally doped BaFe2(As1-xPx)2 find a strong
linear-$T$ term, suggesting a strong nodal component [205], consistent with
other measurements on this material. Measurements in FeSe find a small linear
term [246], which the authors reported as consistent with nonzero gap, but
whose size is of order that found by Reid et al for the $ab$ plane values. We
are not aware of any thermal conductivity measurements on the 111 or 1111
families.
The oscillations of thermal conductivity in a rotating magnetic field provide
similar information to angle-dependent specific heat oscillations. They tend
to be easier to observe and somewhat less straightforward to interpret [245].
At this writing, this experiment has only been performed on P-doped Ba-122,
where significant oscillations are observed [247], and interpreted in terms of
loop nodes on the electron pockets. As mentioned above, both penetration depth
and thermal conductivity experiments have provided evidence for nontrivial 3D
nodal structures in the Ba(Fe1-xCox)2As2 and now the BaFe2(As1-xPx)2 systems.
A variety of such structures have been suggested, by microscopic theory [48,
49], and phenomenology [244, 247], which we summarize in figure 25.
*
Figure 25: (a) Nodal structure on $\alpha$ (hole) and $\beta$ (electron)
pockets in 3D spin fluctuation calculation for Ba(Fe1-xCox)2As2 by Graser et
al [48]; (b) Similar result for $\alpha$ sheet of BaFe2(As1-xPx)2 [49];
$\kappa_{a}$ and $\kappa_{c}$ in Ba1-xKxFe2As2, [242]; loop-like nodes around
point of maximum (A), rather than minimum (B) Fermi velocity on the outer
$\beta$ pocket in BaFe2(As1-xPx)2 deduced from angle-dependent thermal
conductivity [247], similar to that found in [244] from analysis of Raman
scattering on Ba(Fe1-xCox)2As2.
For the most part, we have not discussed experiments in the so-called
“coexistence phases” of the FeBS phase diagram where superconductivity and
magnetic order are simultaneously present. In part this is because the
microscopic homogeneity of these phases is not firmly established, and in part
because theoretical calculations to predict transport properties have not yet
been performed. We note that the spin fluctuation theory calculations for the
instability line of $T_{c}$ vs. doping in the absence of magnetic order do not
show a suppression of superconductivity in the underdoped regime (see, e.g.
figure 8), suggesting that the suppression is due to the competition with
magnetic order. Recent experimental results in this regime also imply that the
magnetic ordering plays an important role in gap structure in the coexistence
regime. Penetration depth measurements on Ba(Fe1-xCox)2As2 early on reported a
sharp rise in the coefficient of the $T^{2}$ term in $\Delta\lambda$ in the
underdoped regime [206]. More recently, thermal transport measurements [241]
reported a similar sharp increase in the normalized linear-$T$ $\kappa_{c}$
term (figure 24(c)). While these two results on underdoped Ba(Fe1-xCox)2As2
have not been understood completely within a single model (see however [243]),
they are strong indications that the number of quasiparticles increases in the
coexistence phase due to a strengthening of nodal behavior. On the hole-doped
side, the effect is even stronger and appears quite abruptly in
$\kappa_{ab}/T$ [242]. This effect will be important to understand
theoretically to complete the picture of the 122 materials, but should be
approached with caution because issues of inhomogeneity are not settled,
particularly on the K-doped side.
#### 5.2.4 The ARPES “paradox”.
Low-energy excitations whose existence is implied by the above measurements
should be visible in angle-resolved photoemission spectroscopy (ARPES). In
fact, ARPES is arguably the most direct probe of the superconducting gap
structure. Yet at this writing no ARPES experiment [248, 249, 250, 251, 252,
253, 254, 255] has reported the existence of gap nodes or even significant gap
anisotropy, in dramatic contrast to the cuprate case, where ARPES was one of
the experiments providing definitive evidence for $d$-wave superconductivity
[256]. The disagreement between bulk probes and ARPES on Fe-based
superconductors is an important point which requires resolution if we are to
rely on both types of measurements to study superconductivity, as we have in
the past. There are several possibilities to explain the discrepancy:
* •
Surface electronic reconstruction One possibility is that the electronic
structure at the surface is different from that of the bulk. This was the
point of view advocated by Kemper et al [47], who calculated the bulk and
surface band structures of BaFe2As2 from density functional theory (DFT), and
discovered that the surface bands included an additional pocket of $xy$
orbital character at the Fermi level, and gave arguments to the effect that
such a pocket would stabilize an isotropic pair state.
* •
Surface depairing. Given the common assumption that the ground state of many
(but probably not all) Fe-based superconductors display anisotropic $A_{1g}$
order, it is easy to see that the anisotropic component of the gap (whether on
hole or electron sheet) will be destroyed by in-plane intraband scattering by
the rough surface, since it does not conserve the parallel (to the surface)
quasiparticle momentum. Thus, as one approaches the surface, the
superconducting gap associated with a pair of momentum $\bi{k},-\bi{k}$ should
become isotropic and the nodes lifted. The phenomenon is similar to that
observed for the same type of superconducting order in the presence of
intraband scattering by impurities [126]. Surface scattering and electronic
reconstruction should be smaller for non-polar surfaces which occur, e.g. in
the LiFeAs material. In this case, however, spin fluctuation theory [257],
thermal conductivity [258], penetration depth [212], ARPES [254] and STM [259]
are all in agreement that the gap is fully developed.
* •
Resolution issues. Gaps on the hole sheets around the $\Gamma$ point should be
imaged with relative ease by ARPES, and it is possible to imagine reconciling
the largely isotropic gaps found on these sheets with thermodynamic
measurements, since other evidence points primarily to nodes on the electron
sheets, as discussed above. It is noteworthy that the nominally highest
resolution experiments using laser sources find full gaps but otherwise
qualitatively different results than synchrotron-based ARPES [255]. Most
calculations predicting nodal effects are done in the 1-Fe zone; when folding
such states one gets gaps from 2 electron pockets centered at the $\mathrm{M}$
points. Given that the resolution around these points (not probed by laser
ARPES) is typically several meV, it seems possible that the anisotropy of the
gaps on the electron sheets might be missed if the spectral peaks from both
sheets were broadened into one with averaged–and hence isotropic–dispersion.
We note that although ARPES experiments to date are apparently providing
unreliable measures of the superconducting gap anisotropy, this does not
necessarily mean that they are inconsistent with bulk gap scales, as shown by
the comparison of two gaps extracted from ARPES and specific heat on LiFeAs
[260].
#### 5.2.5 NMR $1/T_{1}$.
In addition to the Knight shift, which allows one to distinguish between
singlet and triplet pairing (see Section 4.1), NMR can probe the spin-lattice
relaxation rate $1/T_{1}$ that corresponds to a the spin susceptibility
integrated over the Brillouin zone,
$\frac{1}{T_{1}T}\propto\lim_{\omega\rightarrow
0}\sum_{\bi{q}}\frac{\mathrm{Im}\chi(\bi{q},\omega)}{\omega}.$ (27)
As in the case with the spin resonance, Section 5.1.1, $1/T_{1}$ carries
information about the underlying gap symmetry and structure. For example, an
isotropic $s$-wave state is characterized by a Hebel-Slichter peak just below
$T_{c}$ and an exponential low-$T$ temperature dependence. It is well-known
that $d$-wave superconductors exhibit weak or absent peak and demonstrate
$T_{1}^{-1}\sim T^{3}$ behavior for $T\ll T_{c}$. In the case of FeBS, the
situation is somewhat more complicated. Typical data for some 1111 and 122
systems are shown in figure 26(a). Apparently, there is no peak below $T_{c}$
and the temperature dependence does not follow the same simple power or
exponential law in all systems. However, simple arguments can enable us to
understand the main features found in experiments.
*
Figure 26: Temperature dependence of $1/T_{1}$ in FeBS. (a) Experimental
results for two classes of materials, 1111 and 122, from [146]. (b) Log-log
plot summarizing experimental data from several groups [144, 143, 262],
theoretical curve for the $s_{\pm}$ superconductor with intermediate strength
of impurity scattering ($0\leq\sigma\leq 1$) and pairbreaking parameter
$\gamma_{interband}=0.4\Delta_{0}$, and $T^{2.5}$ curve to demonstrate a
power-law dependence (from [120]).
In case of a weakly coupled clean two-band superconductor below $T_{c}$,
assuming that the main contribution to $\mathrm{Im}\chi(\bi{q},\omega)$ comes
from interband interactions, we have
$\frac{1}{T_{1}T}\propto\sum_{\bi{k}\bi{k}^{\prime}}\left[1+\frac{\Delta_{\bi{k}}\Delta_{\bi{k}^{\prime}}}{E_{\bi{k}}E_{\bi{k}^{\prime}}}\right]\left(-\frac{\partial
f(E_{\bi{k}})}{\partial
E_{\bi{k}}}\right)\delta\left(E_{\bi{k}}-E_{\bi{k}^{\prime}}\right),$ (28)
where $\bi{k}$ and $\bi{k}^{\prime}$ lie on hole and electron Fermi sheets,
respectively, and $E_{\bi{k}}$ is the quasiparticle energy in the
superconducting state. This is a straightforward generalization of the
textbook expression [157]. As in the spin resonance case, the coherence factor
in square brackets gives rise to an important distinction between different
symmetries of the gap. In the NMR $T_{1}^{-1}$ case, we see that the internal
sign is different from that which occurs in Equation (25) for the neutron spin
resonance effect. Assuming first an isotropic $s_{++}$-wave gap with
$\Delta_{\bi{k}}=\Delta_{\bi{k}^{\prime}}=\Delta$, one finds
$\frac{1}{T_{1}}\propto\int\limits_{\Delta(T)}^{\infty}dE\frac{E^{2}+\Delta^{2}}{E^{2}-\Delta^{2}}~{}\mathrm{sech}^{2}\left(\frac{E}{2T}\right).$
(29)
The denominator gives rise to a peak just below $T_{c}$, which is the famous
Hebel-Slichter peak. As pointed out earlier in [29], it is suppressed for the
$s_{\pm}$ state. Indeed, if
$\Delta_{\bi{k}}=-\Delta_{\bi{k}^{\prime}}=\Delta$,
$\frac{1}{T_{1}}\propto\int\limits_{\Delta(T)}^{\infty}dE\frac{E^{2}-\Delta^{2}}{E^{2}-\Delta^{2}}~{}\mathrm{sech}^{2}\left(\frac{E}{2T}\right)=\int\limits_{\Delta(T)}^{\infty}dE~{}\mathrm{sech}^{2}\left(\frac{E}{2T}\right),$
(30)
which is just the Yoshida function, which decreases monotonically as
temperature is decreased below $T_{c}$. The same can be shown for a more
general $s_{\pm}$ case of $|\Delta_{\bi{k}}|\neq|\Delta_{\bi{k}^{\prime}}|$
[120].
It is well known that pair-breaking impurity scattering dramatically increases
the subgap density of states just below $T_{c}$, and even weak magnetic
scattering can broaden and eliminate the Hebel-Slichter peak in conventional
superconductors. In FeBS, the same effect is present due to the nonmagnetic
interband scattering [118]. Since the Hebel-Slichter peak is not present in
this scenario even in a clean sample, see Equation (30), the pair-breaking
effect is more subtle: it changes exponential behavior below $T_{c}$ to a more
power-law like one. If the impurity-induced bound state lies at the Fermi
level (Section 3.4), the relaxation rate acquires a low-temperature linear-$T$
Korringa-like term over a range of temperatures corresponding to the impurity
bandwidth [261].
Qualitative arguments suggest that neither pure Born nor pure unitary limits
with a simple isotropic $s_{\pm}$ state are well suited for explaining the
observed $1/T_{1}$ behavior: the former leads to an exponential behavior at
low temperatures in a relatively clean system, the latter to Korringa
behavior. Various early data on the 1111 systems appeared to be between these
two limits. On the other hand, early theory by Parker et al studying the
intermediate scattering regime seemed to be rather promising in this respect.
Figure 26(b) shows various experimental data for 1111 systems [144, 143, 262]
together with a calculation of $T_{1}^{-1}$ for the simple $s_{\pm}$ gap
[120]. We observe that the $s_{\pm}$ state result exhibits no coherence peak
and as opposed to the Born and unitary limits, intermediate-$\sigma$
scattering is capable of reproducing the experimental behavior [120, 121, 122,
123]. It is not clear that these results, taken alone, should be taken as
evidence for an isotropic $s_{\pm}$ state, since strong gap anisotropy is
probably present in some of these systems, and will also lead to a higher
density of quasiparticles contributing at intermediate temperatures. From the
data, one can say with certainty only that the K-doped Ba-122 system appears
to have a large full gap, while the 1111 systems show a much higher density of
low-energy excitations.
Regarding other systems, data obtained on BaFe2(As1-xPx)2 shows a linear-$T$
term in $T_{1}^{-1}$ for an optimally doped sample, crossing over to something
roughly approximating $T^{3}$ above $\sim 0.1T_{c}$ [149, 263], consistent
with reports of nodes in this material from other probes. Low-temperature data
on Co-doped Ba(Fe1-xCox)2As2 is not available at this writing. In
Ba0.68K0.32Fe2As2, $1/T_{1}$ shows an exponential decrease below $T\approx
0.45T_{c}$ consistent with a full $s_{\pm}$ gap [264]. Finally, consistent
with other measurements, NMR in the LiFeAs system also shows a full gap [148].
#### 5.2.6 Electronic Raman scattering.
Because the momentum and polarization of incoming and outgoing photons can be
controlled in a Raman scattering measurement, this technique is useful to
probe selectively different parts of the Fermi surface. The nonresonant
electronic Raman intensity can to a good approximation be represented as an
electron-hole bubble with Raman vertices
$\gamma_{n\bi{k}}=\varepsilon_{\alpha}^{i}(\partial\epsilon_{n\bi{k}}/\partial
k_{\alpha}\partial k_{\beta})\varepsilon_{\beta}^{f}$, where the
$\hat{\varepsilon}$’s are the incident and scattered (final) photon
polarizations, $\epsilon_{n\bi{k}}$ is the electronic dispersion and $n$ is
the band index. Muschler et al [265] measured Raman scattering on an optimally
doped sample of Ba(Fe1-xCox)2As2 and presented a simple approximation to these
vertices which suggested that the $A_{1g}$ polarization (symmetric
configuration of $\hat{\varepsilon}^{i,f}$) intensity is maximal near the BZ
center and thus probes the hole Fermi sheets; similarly, $B_{1g}$ probes the
electron pockets, and $B_{2g}$ is maximal near $(\pi/2,\pi/2)$ points where
there is no Fermi surface (we use the notation of the 1-Fe zone here, i.e.
$B_{1g}=\widetilde{B_{2g}}$). Within this interpretation, the large $B_{1g}$
peak observed corresponds to twice the maximum gap in the system, yielding a
value of $\Delta_{max}\simeq$70cm-1 on the electron sheets. Furthermore,
Muschler et al showed that in this polarization excitations were present down
to the lowest measurement frequency, indicating nodes or deep gap minima less
than their resolution of order 10 cm-1. Strong in-plane anisotropy of the
$B_{1g}$ peak was also reported across a wider range of dopings in [266].
The theory of electronic Raman scattering in the superconducting state has
been reviewed by Devereaux and Hackl [267]. An early discussion of the
intensities to be expected in a two-band isotropic $s_{\pm}$ state [268]
predicted a peak at $2\Delta_{0}$ and a resonance below
$2\Delta_{0}$—analogous to the neutron spin resonance (Section 5.1.1)—in the
$A_{1g}$ channel. No peak or resonance was observed later in Muschler et al.
It was then pointed out by Boyd et al [269] that Coulomb backflow effects in
the doped multiband system, which did not occur in [268], would strongly
suppress the $2\Delta_{0}$ peak. Boyd et al, however, did not consider vertex
corrections due to short-range interactions which are important for a
formation of the resonance below $2\Delta_{0}$ [268] and a subgap resonance
for $A_{1g}$ polarization may therefore still be possible. The experimental
situation in this channel is still controversial with the reported observation
of a weak peak in BaFe1.84Co0.16As2 [270].
Analysis of the $B_{1g}$ channel by Muschler et al [265] and Boyd et al [269]
in terms of highly anisotropic $s_{\pm}$ states provides internally consistent
evidence for order parameter nodes or deep minima on the electron pockets in
the electron doped 122 system. But it was also argued that the gap on the
electron pocket observed in the $B_{1g}$ channel can be strongly affected by
disorder [271], which not only broadens the $2\Delta_{0}$ peaks but can lift
the nodes, as apparently observed by Muschler et al upon doping.
More recently, a more detailed analysis of the correct Raman vertices for
these systems based on DFT was attempted by Mazin et al [244], who concluded
that the earlier approximation for the $B_{1g}$ vertex (which weighted the
entire electron pocket essentially equally) was too crude, and used the
Muschler et al data to argue that gap nodes or deep minima had to be present
in the form of loops on the electron barrels circling the $\Gamma-\mathrm{X}$
axis. It is interesting to note that this identification is consistent with
that of Yamashita et al [247] from angle-dependent magnetic field thermal
conductivity measurements on the BaFe2(As1-xPx)2 system, supporting the notion
that the gap minima in the Ba(Fe1-xCox)2As2 might deepen and evolve into nodes
in the BaFe2(As1-xPx)2 system.
### 5.3 Alkali-intercalated iron selenide
As this review was being finalized, a new intriguing FeBS material was
discovered, challenging both theory and experiment with its novel properties.
As of now, this is still work-in-progress, and the field remains very
controversial. Some would argue that the subject is not ripe for a review yet,
and indeed it is too early to pass any judgement on the superconducting
mechanism, superconducting symmetry, or even physical properties of this
system. Nevertheless, the authors of this review are of the opinion that it is
worth to review here the preliminary results, both on the experimental and
theoretical sides, as a matter of a status report, rather than an analytical
review along the lines of the previous Sections.
In November 2010 a new superconductor, believed at that time to have the
chemical formula of K0.8Fe2Se2, was reported, with a maximum critical
temperature of 33K. The formal electron count makes this compound electron-
doped at the level of 0.4 $e$ per Fe, the same as Ba(Fe0.6Co0.4)2, which is
far beyond the superconducting dome of the Co-doped system, and significantly
past the level of doping at which the hole pockets completely sink under the
Fermi level. The calculated band structure (figure 27(a)) shows no hole
pockets at all, but rather a large electron pocket at the corner of the
Brillouin zone ($\tilde{M}$) and a small electron pocket at the center
($\Gamma)$. Several ARPES measurements were reported within a few months
[272], all agreeing among themselves on the Fermi surface shown in figure
27(a). In addition, some reported a uniform nodeless gap around the large FS
pocket.
* (a)
(b) (c)
Figure 27: (a) Fermi surface of K0.8Fe2Se2 (from [273]). (b) Cartoon showing a
generic 3D Fermi surface for an AFe2Se2 material in the unfolded (one Fe/cell)
Brillouin zone. Different colors show the signs of the order parameter in a
nodeless $d$-wave state, allowed in the unfolded zone. The $\Gamma$ point is
in the center (no Fermi surface pockets around $\Gamma$), and the electron
pockets are around the $\mathrm{X}$, $\mathrm{Y}$ points. (c) Same as (b), but
assuming a finite ellipticity (still zero $k_{z}$ dispersion). Different
colors show the signs of the order parameter in a $d$-wave state. Wherever the
two colors meet, turning on hybridization due to the Se potential creates
nodes in the order parameter.
This, by itself, is rather inspiring. Indeed, in the absence of the central
hole pocket the basis for the $s_{\pm}$ pairing is essentially lost, and the
whole theory seems to be in need of revisiting. Such revisiting, which we will
discuss in more detail later, did come nearly immediately from numerous groups
[274, 275, 278, 279].
However, the next wave of experiments, probing the bulk of the samples, came
to different conclusions, not readily compatible with the results of
photoemission. It appears that the most accurate methods for determining the
exact composition in the bulk, such as neutron scattering, invariably yield
the so-called charge-balanced compositions, namely K2xFe2-xSe2, where each
extra electron brought in by alkaline intercalation is compensated by holes
introduced by Fe vacancies [280]. Obviously, if that is the case, and one
neglects all effects of vacancy ordering, this compositions brings us back to
the parent FeSe material, with equal hole and electron FSs, favorable for the
$s_{\pm}$ model and completely inconsistent with the ARPES data. One can, as
usual, ascribe the difference to surface effects, but that would require the
surface to be stoichiometric in Fe, with 20% Fe vacancies in the bulk, which
seems unlikely.
Figure 28: Measured and calculated band structure of
K${}_{0}.8$Fe${}_{1}.7$Se2. (a) ARPES data and DFT calculations from [281].
(b) A modification of the DFT bands according to the suggestion in [281]. (c)
Modified DFT bands alone. Note absence of a band gap between the electron
states at $\tilde{M}$ and hole states at $\Gamma$ points.
It was also suggested [281] that the ARPES-measured band structures differs
from DFT calculations in the sense that the hole bands are strongly shifted
down, and the electron bands up, so that the bottom of the latter is above the
top of the former. The authors of [281] claim that their actual composition is
K0.8Fe1.7Se2, corresponding to 0.1 $e$/Fe doping, close to optimal doping in
Ba(Fe,Co)2As2. However, because of the assumed band shifts, they argue, the
Fermi level only crosses the electron bands. This explanation, however, is
only valid for this particular composition, and it does not seem very
plausible that this composition is special in any other way. Indeed, figure
28(a) shows the data from [281], overlaid with their own band structure
calculations. The suggested band shifts are illustrated in figure 28(b). As
one can see, particularly in figure 28(c) (where only the bands modified
according to [281]’s suggestion are shown), if the composition is reduced to
no electron doping (K0.8Fe1.6Se2), the Fermi level will shift down and will
cross a completely different band, neither the familiar hole pockets near
$\Gamma$, not the familiar electron pockets near $\tilde{M}$.
The surprises do not end there. Particularly stable appears the composition
with $x=0.4,$ K0.8Fe1.6Se${}_{2},$ which can be also written as K2Fe4Se5,
suggesting a particular superstructure with a 5-fold unit cell. Indeed such a
superstructure was found [282], and corresponds to Fe vacancies forming a
$\sqrt{5}\times\sqrt{5}$ structure. The formula unit contains one vacancy and
four Fe ions, forming a square plaquette. Each plaquette is ordered
ferromagnetically, forming a rigid ferromagnetic cluster. First principles
calculations [283, 284] have independently arrived at the same picture. Inside
the plaquette the calculated Fe-Fe bonds are noticeably shorter than the
average Fe-Fe distance [283], also in agreement with the experiment [282].
Additionally, Yan et al [283] have shown that shrinking of Fe-Fe bonds is not
a magnetic effect, even though the ferromagnetic ordering benefits from such
shrinking, but a covalent effect existing apart from Fe spin polarization
[283]. Moreover, both experiment and theory suggest an extremely large ordered
magnetic moment on Fe, up to 3.3 $\mu_{B}$ in the experiment and even larger
in the calculations. Note that this corresponds to a plaquette with a
supermoment of at least 13$\mu_{B}$. Several groups claim, nevertheless, that
this ordered magnetic state coexists microscopically with superconductivity.
Several considerations are in order here. This vacancy-ordered structure
corresponds to a lattice parameter of the order of 6Å. The coherence length
has been measured [286] to be less than 60Å. One can estimate [273] that a net
misalignment of the moments of the order of 0.05 degree will result in a net
exchange field larger than $H_{c2}$. It is hard to see how one can avoid this
in real samples with strain, grain boundaries, etc.. Furthermore, calculations
unambiguously show that the ideally ordered stoichiometric K2Fe4Se5 is a band
insulator with a large gap [283, 284]. Experimental samples with this claimed
composition are metallic, but an insulating phase has been found (and
initially identified as a Mott insulator, although in view of the most recent
data (see, e.g., [285]) a band insulator appears more likely) nearby in the
phase diagram; the metallicity of the ordered stoichiometric composition, in
fact, may be an experimental artifact.
So now we face not only the fact that ARPES and bulk probes suggest different
valence states for Fe, we also seem to have high-temperature superconductivity
in a strongly magnetic insulator. An interesting, and possibly correct
explanation of this controversy was suggested recently in [287]. They suggest
that although superconductivity and magnetism occur in the same sample, and
each involves nearly 100% of carriers, they never occur simultaneously. In
short, the statement is that whenever vacancies are disordered, the sample is
superconducting, and whenever they order it becomes antiferromagnetic, but not
superconducting. The authors’ claim that the vacancies reversibly order upon
heating and disorder upon cooling sounds counterintuitive. On the other hand,
this picture was recently lent support by Li et al[289], who found, using STM,
that areas of a K-intercalated Fe selenide with ordered Fe vacancies were not
superconducting, but areas closer to stoichiometric KFe2Se2 were.
One can say that in this contradictory experimental situation any speculations
are out of place. Yet, it is interesting to discuss what can possibly happen
in the superoverdoped regime that ARPES suggests. Indeed this intriguing Fermi
surface topology was already discussed in one of the first theoretical papers,
by Kuroki et al [45] who pointed out that in the absence of the hole pockets
in the unfolded BZ this band structure is an ideal 2D representation of the
“Agterberg-Barzykin-Gor’kov” gapless $d$-wave superconductivity [52] [figure
27(b)]. Indeed, in this case the quasi-nesting between the hole and electron
pockets is supplanted by the quasi-nesting between the electron pockets,
resulting in a $d$-wave state with alternating signs of the order parameters,
while the symmetry-required nodal lines fall between the Fermi surfaces. After
the first ARPES experiments on the Se-based 122 systems appeared, several
theoretical groups have revisited this idea [274, 275].
However, the original Agterberg et al paper unambiguously identifies possible
locations of the zone-corner pockets allowing for a gapless $d$-wave
superconductor. These are: $(\pi,0)$ and equivalent points in a tetragonal
symmetry, and $(\pi,0,0)$ and equivalent in a cubic one. This is the case in
the unfolded BZ, but in the folded zone the pockets are at the
$(\pm\pi,\pm\pi)$ points, connected among themselves by reciprocal lattice
vector. The pockets in question are formed by shifting the unfolded FSs by
$(\pi,\pi,\pi)$, and overlapping the resulting pockets (figure 27). As
discussed in details in [201, 290], this unavoidably leads to node formation.
At the same time, not only ARPES data, but also various bulk probes [291, 292]
indicate that superconductivity in KxFeySe2 is nodeless. Since the nodes in
this case are driven by the hybridization of the two unfolded FSs, the phase
space for quasiparticle excitations associated with such nodes is less than
generic $d$ wave nodes if the hybridization gap is small (cf. [201]). Roughly,
the phase space affected is reduced compared to $d$ wave as $V/\Delta E$,
where $V$ is the matrix element of the symmetry-breaking Se potential and
$\Delta E\simeq v_{F}\delta k$, where $\delta k$ is the ellipticity of the
electron FS pocket. While in actual DFT calculations it appears that
$V\approx\Delta E$, and ARPES seems to show small ellipticity as well, one
cannot exclude at this stage a possibility that $V/\Delta E$ is considerably
smaller than one, in which case spectroscopic signatures of a $d$ state with
no nodes enforced by tetragonal symmetry but weak ones imposed by
hybridization of the two electron bands will be much harder to detect.
At the present writing, at least three qualitatively different proposals
exist. One is, as mentioned, a “quasi-nodeless” $d$-wave (nodes only induced
by hybridization) (figure 27). The second is, essentially, the same $s_{\pm}$
as in other materials, but with the “minus” pairs formed not on the Fermi
surface but on the hole band 50-100 meV below the Fermi level (the problem
here is that superconductivity appears only in higher orders in the coupling
constant) [275, 276]; such a state would look for most practical purposes as
regular same-sign s-wave, so it can be called “incipient $s_{\pm}$”. The third
state that allows for a sign-changing order parameter is an $s_{\pm}$ one, but
entirely different form the one discussed throughout this review. Rather, this
is an $s_{\pm}$ state that was discussed more than a decade ago in connection
with bilayer cuprates. Here one refers to the fact that in a bilayer system
every band is split in a bonding and an antibonding combination, and these may
have different signs of the order parameter. Since DFT calculations predicts
that the two electron-pockets in KxFeySe2 are strongly hybridized, over most
of the Fermi surface the calculations predict similar bonding-antibonding
splitting and a possibility of a strictly-nodeless sign-changing s-wave
superconductivity [290]. The three different types of states are summarized in
figure 29.
*
Figure 29: This cartoon shows three proposed pairing states for KxFe2-ySe2 in
the 2-Fe Brillouin zone. As suggested by the first principles calculations, a
finite gap between the inner and the outer Fermi surface sheets is introduced.
(a) $d$-wave state, including small parts of the Fermi surface where the gap
is small; (b) the “incipient” $s_{\pm}$ state, with hole bands in proximity of
the Fermi level, but not crossing it; (c) the “bonding-antibonding” $s_{\pm}$
state. Note that (a) and (c), but not (b), can give rise to a spin resonance
at $(\pi,\pi)$ (in the unfolded Brillouin zone).
One of the latest experimental developments relevant to the the order
parameter in KxFeySe2 is a recent inelastic neutron scattering measurement
[277]. In agreement with ARPES-measured band structure, these authors did not
find any peak around the $(\pi,0)$ wave vector, indicating the absence of the
conventional electron–hole nesting. In agreement with theoretical expectation
[274], there is not much scattering at exactly $(\pi,\pi)$, even though this
is the vector of nearly exact electron–electron nesting. The reason is that
the real part of the noninteracting spin-susceptibility is large when the
Fermi velocities of the initial and the final states are opposite, and the
real part controls the Stoner enhancement of the full susceptibility. Thus, a
peak in susceptibility is expected when the FSs displaced by the given
momentum just touch; if the radius of the electronic FSs in KxFeySe2 is
$k_{F}$, then a peak in the neutron scattering is expected near
$Q=(\pi/a,\pi/a)-(k_{F},k_{F})$. Actual calculations [274] show that due to
the somewhat squarish shape of the FS the peak appears to be asymmetric and
located at ($\pi,0.625\pi)$ (for $0.1e$ doping). Experimentally, a peak is
observed at ($\pi,\pi/2)$, not far from this predicted position, and found to
be resonantly enhanced below $T_{c}$. The latter fact indicates that this wave
vector connects two points on the FS, and these points have order parameters
of the opposite signs, consistent, in principle, with the “quasi-nodeless”
$d$-wave or with the bonding-antibonding $s_{\pm}$, but not with the
“incipient s±”.
A second look, however, reveals that this straightforward interpretation may
be too naive. Indeed, the FS suggested by ARPES has by far too small electron
pockets to provide any states removed from each other by ($\pi,\pi/2)$. One
either needs to assume that the Fe content is grossly underestimated in [277],
or that sizeable hole pockets are present (in which cases a feature at
$q\approx(\pi,0)$ would be expected). Combined with the complex
antiferromagnetic structure found in the same compound, and possibly (even
though unlikely) coexisting with superconductivity, one may think that this
antiferromagnetic structure, either present or incipient in the
superconducting state, may drastically change the physics of superconductivity
in these materials. In this regard, resolving the mystery of interplay and
possible coexistence between magnetism and superconductivity in these
selenides is the most burning issue at this writing.
### 5.4 Differences among materials: summary.
In the above discussion we have focussed primarily on the heterovalently doped
BaFe2As2 materials, on which most experiments have been performed to date, and
argued that thermodynamic properties tend to support fully gapped behavior
near optimal doping, and increasing anisotropy and eventual nodal behavior
away from it, consistent with expectations from a spin fluctuation picture.
The 122 family may indeed be representative of Fe-based superconductors, but
it is worth noting that it is among the most weakly correlated families, and
the most 3D as well. Therefore it is important to review some of the other
families, in the hope that new aspects of the physics may be gleaned from the
comparison. In Table 1, we have tried to group materials into three
categories, according to whether they display—in bulk thermodynamic properties
alone—a large gap (at least several meV), deep gap minima or weak nodes, or
clear nodal behavior. The middle category is clearly somewhat delicate, since
it contains materials which have in some cases been explored extensively,
whereas others have received less experimental attention. In addition, the
distinction between deep gap minima and weak nodes (situations where the gap
changes sign only over a very small part of the Fermi surface) is not a clear
one, since in A1g symmetry, a gap with deep minima can be adiabatically
transformed into a one with weak nodes, and vice versa, with small changes in
electronic structure, disorder, etc. Note we have deliberately omitted AFe2Se2
from the Table due to its unclear materials properties and incomplete
experimental information on the superconducting state at this writing.
Table 1: Gap structures in Fe-based materials deduced from thermodynamic and transport measurements. OD=overdoped, OP=optimally doped, UD=underdoped. Symbol * indicates possible evidence for “$c$-axis nodes”. Family | Full gap | Highly anisotropic | Strong nodal
---|---|---|---
1111 | PrFeAsO1-y[52K] [293] | LaFeAs(O,F)[26K] [214] | LaFePO[6K] [203, 204, 294]
| SmFeAs(O,F)[55K] [295] | NdFeAs(O,F) [214] |
122 | (Ba,K)Fe2As2[40K] [146, 236, 296, 242] | Ba(Fe,Co)2As2 [OD] [238, 241]* | KFe2As2 [4K] [211, 309]
| Ba(Fe,Co)2As2 [OP,23K] [238, 208] | Ba(Fe,Ni)2As2 [297]* | BaFe2(As,P)2[OP,31K] [205, 149]
| | Ba(Fe,Co)2As2 [UD] [241]* | (Ba,K)Fe2As2 [UD] [242]
111 | LiFeAs [18K] [298, 258] | | LiFeP [6K] [299]
11 | | Fe(Se,Te) [27K] [231, 246] |
In our discussion of other materials, we first focus on the isovalently doped
BaFe2(As1-xPx)2 system, which displays clear nodal behavior. This material is
interesting in several regards, including its remarkable “quantum critical”
normal state properties and its phase diagram vs P concentration remarkably
similar to Ba(Fe1-xCox)2As2 despite the isovalent nature of the substitution
of P for As [300]. It has the highest critical temperature (31K at optimal
doping) of the confirmed nodal superconductors. ($T_{c}$’s for LaFePO and
KFe2As2 are 6K and 4K, respectively). The fact that the three P-doped
superconductors BaFe2(As1-xPx)2, LaFePO, and LiFeP are nodal whereas their As-
doped counterparts Ba1-xKxFe2As2, Ba(Fe1-xCox)2As2, LaFeAsO, and LiFeAs are
either fully gapped or anisotropic is also striking. In the case of the
LaFeAsO/LaFePO comparison, the claim has been made by Kuroki et al [16] and
Wang et al [68] that the distinction arises from the lack of a third hole
pocket in the LaFePO band structure. This does not appear to provide an
explanation in the case of LiFeAs and LiFeP, where both materials have a third
hole ($\gamma$) pocket. On the other hand, the third pocket is important
because it allows $(\pi,0)$ processes to couple to the $d_{xy}$ states on the
electron pockets, as discussed in Section 3.1. Hashimoto et al [299] pointed
out that a preponderance of $d_{xy}$ orbital character on the $\alpha_{2}$
hole pocket of LiFeAs (negligible in the LiFeP case) may play the same role,
and induces a more isotropic state. Other possibilities to explain differences
between As- and P-based compounds include the effective local interactions on
Fe orbitals, which might be different due to the local polarizabilities of the
ligand As or P [13]. Such effects should be accounted for in DFT calculations;
however these do not show significant differences in static local interactions
$U$,$U^{\prime}$,$J$, and $J^{\prime}$ for As- and P-based systems [103].
As emphasized in Table 1, the 1111 materials based on As are both gapped and
possibly nodal; it was this discrepancy among measurements on these early
samples which led to the initial diversity of low-$T$ results and impression
of nonuniversality of the superconducting state in these materials. The
substitution on the rare earth site is known to affect the lattice constants,
in particular the $c$-axis constant, which shrinks as $T_{c}$ grows. The
concomitant changes in electronic structure have been ascribed primarily to
the pnictogen height by Kuroki et al [16], who parameterized them in an
attempt to explain the sequence of $T_{c}$’s in this family. It is clearly
significant that the more isotropic materials have the highest $T_{c}$’s, as
would generally be expected from the spin fluctuation theory argument, but
better crystals need to be made and other experiments performed to confirm the
assignments given here.
In the past year, a great deal of progress has been made in preparing high
quality single crystals of LiFeAs ($T_{c}$=17K). Since these crystals cleave
well at the nonpolar Li surface, one might expect that surfaces would be
excellent, and recent unpublished STM work with atomic resolution indeed shows
beautiful surfaces [259]. No surface states near the Fermi level are expected
on the basis of DFT calculations [301]. But the earliest ARPES experiment on
this system, by Borisenko et al [254], reported a Fermi surface very different
from the rather conventional set of hole and electron pockets predicted by
DFT, in particular less clear nesting of hole and electron pockets. More
recently, on the other hand, de Haas-van Alphen measurements [302] showed good
agreement with bulk DFT.
A second controversy relates to the spin excitations of the material. While
early NMR work reported a strongly temperature dependent Knight shift and
$1/T_{1}$ below $T_{c}$, consistent with $s$-wave pairing [304], this has been
challenged recently by Baek et al [305], who report a Knight shift in some
magnetic field directions with no $T$-dependence and claim consistency with
recent theoretical analysis proposing triplet pairing for this system [306].
While the existence of a spin resonance does not definitively exclude triplet
pairing [307], the LiFeAs Fermi surface seems much more likely to support a
resonance in an $s_{\pm}$ state with wave vector $(\pi,0)$ [308]. These
debates clearly need to be settled before it can be decided if LiFeAs fits
into the usual framework discussed above, with pairing driven by spin
fluctuations. What is fairly certain phenomenologically is that this system
has a full gap, as determined by various superconducting state measurements
[212, 254, 258, 259, 299].
A final system which deserves special comment is the strongly hole-doped
compound KFe2As2, with a $T_{c}$ of 4K, which may be considered to be the end
point of the Ba1-xKxFe2As2 series with Ba entirely replaced by K. The Fermi
surface calculated by DFT shows no electron pockets, and experiments have
reported large quasiparticle densities at low $T$ in the superconducting
state, consistent with gap nodes [211, 309, 310]. From a theoretical
standpoint, this material is particularly interesting because it allows one to
ask what the subleading spin-fluctuation pairing channel is when the pair
interaction between electrons and holes no longer drives superconductivity. An
fRG study by Thomale et al [311] found $d$-wave pairing, which was discussed
in terms of scattering between $\Gamma$-centered and $\mathrm{M}$-centered
pockets (1-Fe zone) yielding strong spin fluctuations at $\bi{q}=(\pi,\pi)$.
Maiti et al [81] discussed this system in a more general context, studying the
evolution of pairing within the RPA, and noted the decay of the $s_{\pm}$
channel relative to the $d$ channel as the system was doped in either
direction. They also found a leading $d$-wave pairing instability for KFe2As2.
However, a recent study [312] of the vortex lattices in KFe2As2 claimed the
results inconsistent with an in-plane anisotropy of the order parameter
(including a $d$-wave state), and invoking a horizontal nodal line instead.
Lacking more direct probes of the gap anisotropy, though, the controversy
about the possible $d$-wave pairing in KFe2As2 remains.
## 6 Conclusions.
In the past year, a confusing variety of experiments on the superconducting
state of FeBS indicating a much wider diversity of gap structure among
materials in the same family than expected have, with the help of theory, been
classified in such a way that most variations can be qualitatively understood.
The general hypothesis, which still requires further experimental and
theoretical underpinning, is that in a “typical” FeBS, a spin fluctuation
interaction between like orbitals on nearly nested hole and electron Fermi
surfaces leads to an $s$-wave state which changes sign between these sheets.
At optimal doping, the condensation energy obtained from such an interaction
is maximal and there is a true spectral gap; the extent to which the overall
gap is still somewhat anisotropic depends on details of the system, including
whether three or two hole pockets are present. Hole-doped systems are, within
this picture, generally expected to have higher $T_{c}$ and be more
anisotropic relative to their electron-doped counterparts. In both cases,
however, as one dopes the system away from optimal, the relative importance of
subdominant interactions, in particular the intraband Coulomb interaction and
pair scattering between the electron-like Fermi surface sheets, increases and
frustrates the isotropic $s_{\pm}$ interaction, leading to anisotropy and
eventually nodes. There is considerable experimental evidence to support this
picture in the As-based 122 family.
The $s$-wave ($A_{1g}$) nature of the superconducting allows for any
distortion of the gap structure consistent with invariance under the
operations of the tetragonal group; in particular nodes can be created or
lifted without any thermodynamic singularities (except weak ones at $T=0$).
Thus it is perhaps not a surprise that on the one hand the P-doped Ba-122
system exhibits a phase diagram which resembles strongly that of the Co-doped
Ba-122 system, yet the gap remains nodal across the doping range. We have
reviewed evidence in favor of deep gap minima on the electron pocket in the
Co-doped system, which can easily shift to create nodes. The microscopic
origin of this tendency is still unknown, but we have pointed to the striking
tendency of the P-doped materials to be more anisotropic than their FeAs
analogs, representing an obvious challenge to microscopic theory.
With the above explanation of the “nonuniversality” of the gap structure, the
obvious question is whether one can develop an intuitive understanding of what
physical principles are at work controlling the degree of anisotropy and, of
course, the size of $T_{c}$, without having to perform a full-blown
theoretical calculation for each material. One point emphasized early on by
Kuroki and co-workers was the importance of the third, $d_{xy}$-dominated hole
pocket, whose influence we noted already above. Their observation that this
feature of the electronic structure is controlled by the pnictogen height
worked well within the 1111 family, but appears not to apply directly in the
122 systems. This may be because of the stronger 3D character of these
systems. While few fully 3D theoretical calculations have been attempted thus
far due to technical limitations, it is clear that the strong $k_{z}$
dispersion of certain bands forces admixtures of new bands near the top of the
Brillouin zone in the 122 systems, and that this in turn can force strong gap
variations and possibly nodes.
One of the themes of this review is that new physics can be found with the
addition of more bands and more orbitals. The recipe for the highest $T_{c}$
is not clear, but the intuition gained from extensive spin fluctuation
calculations can be summarized as follows: at least in the general framework
of FeBS-type band structure, it is advantageous to maximize the number of
bands but minimize the number of orbitals present at the Fermi surface. If
variable orbital weight is present, it should be distributed such that large
momentum intraorbital pair scattering processes should be available to the
system.
One can also search for a general intuitive principle describing the effect of
disorder on gap structure. While disorder will always reduce $T_{c}$, in
complex multiband superconductors with gap anisotropy, we have discussed that
disorder can have several different effects on gap structure–even
diametrically opposite ones–depending on impurity scattering and band
interaction parameters. We are therefore somewhat pessimistic that systematic
disorder studies will in this case aid substantially in identifying order
parameter symmetry and structure.
We close by recalling the ancient remark that our knowledge is but a white
spot in a black vastness, and therefore the more we learn, the larger is the
circle that separates us from the unknown, and correspondingly larger is also
our awareness of the limits of our knowledge. In this case much hard work by
many experimentalists and theorists has gone into building up a plausible
“standard” story of the FeBS with “typical” electronic structure. The recent
discovery of the AFe2Se2 based materials has shown us that there is probably
further terra incognita for superconductivity yet to be explored even within
this particular set of chemical elements and structures. It is the grand
challenge for theory to guide the search through this terrain.
## Acknowledgments
We would like to thank R. Arita, A.V. Chubukov, T.P. Devereaux, O. Dolgov, I.
Eremin, R.M. Fernandes, S. Graser, H. Ikeda, A.F. Kemper, H. Kontani, K.
Kuroki, T.A. Maier, Y. Matsuda, V. Mishra, K.A. Moler, R. Prozorov, D.J.
Scalapino, T. Shibauchi, G.R. Stewart, I. Vekhter, W. Wang, Y. Wang, and H.-H.
Wen for useful discussions. Partial support was provided by DOE DE-
FG02-05ER46236 (PJH and MMFK) and NSF-DMR-1005625 (PJH). MMK acknowledges
support from RFBR (grant 09-02-00127), Presidium of RAS program “Quantum
physics of condensed matter” N5.7, FCP Scientific and Research-and-Educational
Personnel of Innovative Russia for 2009-2013 (GK P891 and GK 16.740.12.0731),
and President of Russia (grant MK-1683.2010.2). PJH and MMK are grateful for
the support of the Kavli Institute for Theoretical Physics and the Stanford
Institute for Materials & Energy Science during the writing of this work.
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|
arxiv-papers
| 2011-06-19T07:07:24 |
2024-09-04T02:49:19.822907
|
{
"license": "Public Domain",
"authors": "P. J. Hirschfeld, M. M. Korshunov, I. I. Mazin",
"submitter": "Maxim M. Korshunov",
"url": "https://arxiv.org/abs/1106.3712"
}
|
1106.3736
|
# Sweeping out sectional curvature
D. Panov and A. Petrunin is a Royal Society University Research Fellowwas
partially supported by NSF grant DMS 0905138
###### Abstract
We observe that the maximal open set of constant curvature $\kappa$ in a
Riemannian manifold of curvature $\geqslant\kappa$ or $\leqslant\kappa$ has a
convexity type property, which we call _two-convexity_. This statement is used
to prove a number of rigidity statements in comparison geometry.
## 1 Introduction
Denote by $\mathbb{M}^{m}[\kappa]$ the model $m$-space with curvature
$\kappa$; i.e., $\mathbb{M}^{m}[\kappa]$ is the simply connected
$m$-dimensional Riemannian manifold with constant curvature $\kappa$. We will
also use shortcuts $\mathbb{S}^{m}=\mathbb{M}^{m}[1]$ for the unit $m$-sphere,
and $\mathbb{E}^{m}=\mathbb{M}^{m}[0]$ for the Euclidean $m$-space.
In this paper we play with applications of the following lemma. The proof is
given in Section 3. This lemma was first discovered by Buyalo in the case of
nonpositive curvature; see [3, Lemma 5.8].
1.1. Buyalo’s lemma. Let $M$ be a complete Riemannian manifold with sectional
curvature either $\geqslant\kappa$ or $\leqslant\kappa$. Let $\Delta$ be a
tetrahedron in $\mathbb{M}^{3}[\kappa]$ and $\Lambda$ be a union of three out
of four faces of $\Delta$. Then any immersion $f\colon\Lambda\looparrowright
M$ which is isometric and geodesic on each face can be extended to an
isometric geodesic immersion $F\colon\Delta\looparrowright M$. Moreover, $F$
is uniquely determined by $f$.
Here is an immediate corollary:
1.2. Corollary. Let $g$ be a complete Riemannian metric on $\mathbb{R}^{3}$
with curvature $\geqslant 0$ (or $\leqslant 0$) such that all three coordinate
planes of $\mathbb{R}^{3}$ are flat geodesic hypersurfaces in
$(\mathbb{R}^{3},g)$. Then $(\mathbb{R}^{3},g)$ is isometric to Euclidean
space.
We would suggest that reader checks that the last statement does not follow
from the standard theorems; in particular the splitting theorems can not help
here directly.
Let us now introduce some terminology to state further applications.
* $\diamond$
A Riemannian manifold (possibly not complete) of constant curvature $\kappa$
will be called _$\kappa$ -flat_.
* $\diamond$
A $\kappa$-flat Riemannian manifold (possibly not complete) which satisfies
the conclusion of Buyalo’s Lemma will be called _two-convex_. This definition
is discussed in more details in Section 2.
* $\diamond$
Given a Riemannian manifold $M$, its maximal open subset of constant curvature
$\kappa$ will be called _$\kappa$ -flat domain of $M$_ and it will be denoted
as $\mathrm{Flat}^{\kappa}M$.
From Buyalo’s Lemma one easily gets the following; a formal proof is given in
Section 3.
1.3. Observation. Let $M$ be a complete Riemannian manifold either with
curvature $\geqslant\kappa$ or $\leqslant\kappa$. Then
$\mathrm{Flat}^{\kappa}M$ is two-convex.
Here is an application.
1.4. Theorem. Let $m\geqslant 3$ and $M$ be a complete connected
$m$-dimensional manifold with curvature $\geqslant 1$ or $\leqslant 1$ which
admits a totally geodesic immersion of the closed unit hemisphere
$\imath\colon\mathbb{S}^{2}_{+}\looparrowright M$ and an open neighborhood of
$\imath(\mathbb{S}^{2}_{+})$ in $M$ has constant curvature $1$. Then $M$ has
constant curvature $1$.
Remarks.
* $\diamond$
Note that diameter-sphere rigidity does not help here directly; in principle,
the diameter of $M$ might be $<\pi$.
* $\diamond$
Note that $\mathbb{C}\mathrm{P}^{2}$ equipped with the canonical metric is an
example of a space with curvature $\geqslant 1$ and $\leqslant 4$, which
admits totally geodesic immersions of 2-spheres of constant curvature $1$ and
$4$. I.e., the condition in Theorem 1 that the curvature is constant in a
neighborhood of $\imath(\mathbb{S}^{2}_{+})$ is necessary.
* $\diamond$
In the case of curvature $\geqslant 1$, Theorem 1 also holds in dimension $2$;
this is proved by Zalgaller in [14]; see Theorem Appendix: Zalgaller’s
rigidity. and the discussion around.
To prove the theorem, one needs to show that if a neighborhood $\Omega$ of
$\mathbb{S}^{2}_{+}$ in $\mathbb{S}^{m}$ admits an immersion in a two-convex
manifold $\Phi$ then $\Phi$ has to be complete. Then Observation 1 implies
that $\mathrm{Flat}^{1}M=M$; i.e., $M$ is a spherical space form. In other
words, any neighborhood $\Omega$ of $\imath(\mathbb{S}^{2}_{+})$ in
$\mathbb{S}^{m}$ is _exhaustive_ in the sense of the following definition.
1.5. Definition. Let $\Omega$ be a $\kappa$-flat manifold. Assume that any
connected two-convex manifold $\Phi$ that appears as the target of an open
isometric immersion $\Omega\looparrowright\Phi$ is complete, and at least one
such $\Phi$ exists. Then we say that $\Omega$ is _exhaustive_.
Using this definition, we can formulate the following generalization of
Theorem 1:
1.6. Theorem. Let $M$ be a complete connected Riemannian manifold with
curvature $\geqslant\kappa$ or $\leqslant\kappa$. Assume there is an open
isometric immersion $\Omega\looparrowright M$ from an exhaustive $\kappa$-flat
manifold $\Omega$. Then $M$ has constant curvature $\kappa$.
In order to apply this theorem one only has to find a source of exhaustive
manifolds. In Section 2, we introduce the notion of the two-hull of a
$\kappa$-flat simply connected manifold $\Omega$; in some sense this is the
minimal simply connected two-convex manifold which contains an _immersed copy_
of $\Omega$. It is easy to see that if the two-hull of a manifold $\Omega$ is
isometric to $\mathbb{M}^{m}[\kappa]$ then $\Omega$ is exhaustive. This
permits one to present a number of examples of exhaustive manifolds. This is
done in Section 4, here is a list of examples:
(Proposition 4.)
For $m\geqslant 3$, any non-empty open subset of $\mathbb{M}^{m}[\kappa]$ with
convex complement.
(Proposition 4.)
More generally: any open simply connected subset
$\Omega\subset\nobreak\mathbb{M}^{m}[\kappa]$ which satisfies the following
property. For any $p\in\mathbb{M}^{m}[\kappa]$ there is a 3-dimensional
subspace $W_{p}$ of $\mathbb{M}^{m}[\kappa]$ containing $p$ ($W_{p}$ is an
isometric copy of $\mathbb{M}^{3}[\kappa]$) such that
$W_{p}\cap\Omega\not=\varnothing$ and each connected component of
$W_{p}\backslash\Omega$ is a convex set.
In particular,
$\Omega=\left\\{\,\left.{(x_{1},x_{2},\dots,x_{m})\in\mathbb{E}^{m}}\vphantom{1+x_{1}^{2}+x_{2}^{2}>x_{3}^{2}+x_{4}^{2}+\dots+x_{m}^{2}}\,\right|\,{1+x_{1}^{2}+x_{2}^{2}>x_{3}^{2}+x_{4}^{2}+\dots+x_{m}^{2}}\,\right\\}$
is exhaustive.
(Proposition 4.)
Any open subset of $\mathbb{S}^{m}$ which contains the standard 2-dimensional
hemisphere. This type of manifolds is used in Theorem 1.
(This list can be continued.)
### Related results.
One outcome of Theorem 1 is a sufficient condition on the piece111the
complement of $\Omega$ of the model space $\mathbb{M}^{m}[\kappa]$, which _can
not_ be exchanged to another piece that has sectional curvature not smaller or
not bigger. This condition is nontrivial only for $m\geqslant 3$.
The similar conditions for scalar and Ricci curvature were studied. The case
of deformation with nondecreasing curvature turned out to be very different
from the one with nonincreasing curvature.
After rescaling one can only consider three cases $\kappa=-1,0$ or $1$.
Nondecreasing curvature. If $\kappa=0$, the case of nondecreasing scalar
curvature leads to so called _positive mass conjecture_ which is proved by
Schoen–Yau and Witten in [12] and [13]. This implies in particular that the
metric of Euclidean space can not be perturbed in a bounded region so that the
scalar curvature does not decrease.
An analogous statement holds for $\kappa=-1$; i.e., the metric of Lobachevsky
space can not be perturbed in a bounded region so that the scalar curvature
does not decrease. The later was proved by Min-Oo in [9].
The case $\kappa=1$ was considered in [10], where Min-Oo makes an attempt to
show that the standard metric on the $m$-sphere can not be perturbed inside of
hemisphere so that the scalar curvature does not decrease. But in [4],
Brendle, Marques and Neves find a counterexample. One can not perturb the
metric in a sufficiently small domain of sphere, but optimal bounds on such
domain seem to be not known.
On the other hand as it was shown by Hang and Wang in [6], one can not perturb
the metric of the standard sphere inside its hemisphere with nondecreasing
Ricci curvature.
The two-dimensional case of the above statements for $\kappa=0$ and $-1$
follows from Gauss–Bonnet formula and the case $\kappa=1$ was done by
Zalgaller (see the Appendix).
Nonincreasing curvature. In [7], Lohkamp proves that for all $m\geqslant 3$,
one can perturb the metric of $\mathbb{M}^{m}[\kappa]$ in any open region in
such a way that its Ricci curvature does not increase. Moreover, this can be
done without changing the topology and with arbitrary small change of the
geometry of the space.
In two-dimensional case, attaching a handle can be done in arbitrary small
region with decreasing its curvature. On the other hand, if we fix the
topology, for $\kappa=0$ and $-1$, Gauss–Bonnet formula prevents any change of
metric in bounded regions with nonincreasing curvature. For $\kappa=1$, even
if topology is fixed, the metric can be changed (by inserting a bubble) in
arbitrary small open subset so that the curvature in the region decreases.
However, it seems that for proper subsets of hemisphere, there is no
continuous deformation of this type.
## 2 Two-convexity and two-hull
2.1. Definition. Let $\Omega$ be a $m$-dimensional $\kappa$-flat manifold. We
say that $\Omega$ is _two-convex_ if the following condition holds: given a
tetrahedron222i.e. 3-simplex $\Delta$ in $\mathbb{M}^{3}[\kappa]$ with a
choice of a subset $\Lambda\subset\Delta$ formed by 3 out of 4 faces, any
immersion $f\colon\Lambda\looparrowright M$ which is isometric and geodesic on
each face of $\Lambda$ can be extended to an isometric geodesic immersion
$F\colon\Delta\looparrowright M$.
2.2. Definition. Let $\Omega$ be a simply connected $m$-dimensional
$\kappa$-flat manifold. A two-convex manifold $\Phi$ is called the _two-hull_
of $\Omega$ (briefly $\Phi=\Omega^{(2)}$) if there is an open immersion
$\varphi\colon\Omega\looparrowright\Phi$ such that for any open isometric
immersion $\psi\colon\Omega\looparrowright\nobreak\Psi$ into a two-convex
manifold $\Psi$ there is a isometric immersion
$\vartheta\colon\Phi\looparrowright\nobreak\Psi$ which makes the following
diagram commutative:
$\begindc{\commdiag}[10]\obj(3,5)[aa]{$\Omega$}\obj(0,0)[bb]{$\Phi$}\obj(6,0)[cc]{$\Psi$}\mor{aa}{bb}{$\varphi$}[-1,0]\mor{aa}{cc}{$\psi$}\mor{bb}{cc}{$\vartheta$}\enddc$
Further the immersion $\varphi\colon\Omega\looparrowright\Phi$ will be called
_two-hull immersion_.
Let us notice that even though for some manifolds $\Omega$ the two-hull
immersion $\varphi\colon\Omega\looparrowright\Phi$ is in fact an embedding, in
general one should not expect this.
Our next goal is to prove existence of the two-hull.
2.3. Proposition. For any simply connected $\kappa$-flat manifold $\Omega$,
its two-hull $\Phi$ is uniquely defined up to isometry.
Moreover,
1. i)
If $\varphi\colon\Omega\looparrowright\Phi$ and
$\varphi^{\prime}\colon\Omega\looparrowright\Phi^{\prime}$ are two-hull
immersions then there is an isometry $\vartheta:\Phi\to\Phi^{\prime}$ such
that $\varphi^{\prime}=\vartheta\circ\varphi$.
2. ii)
$\Phi$ is simply connected.
To prove the above proposition, we mimic the proof of existence of ordinary
convex hull as the intersection of all convex sets containing the given set.
Proof. Fix a simply connected $m$-dimensional $\kappa$-flat manifold $\Omega$.
Note that $\Omega$ admits an open isometric immersion
$\imath:\Omega\looparrowright\mathbb{M}^{m}[\kappa]$.
Let us construct a category $\mathcal{C}_{\Omega}$. The class of objects in
$\mathcal{C}_{\Omega}$ is formed by all open isometric immersion
$\psi\colon\Omega\looparrowright\Psi$ where target $\Psi$ is a two-convex
$\kappa$-flat manifold, and the morphisms are commutative triangles of
isometric immersions
$\begindc{\commdiag}[10]\obj(3,5)[aa]{$\Omega$}\obj(0,0)[bb]{$\Psi_{1}$}\obj(6,0)[cc]{$\Psi_{2}$}\mor{aa}{bb}{$\psi_{1}$}[-1,0]\mor{aa}{cc}{$\psi_{2}$}\mor{bb}{cc}{$\vartheta$}\enddc$
$None$
The category $\mathcal{C}_{\Omega}$ contains at least one object, the
immersion $\imath\colon\Omega\looparrowright\nobreak\mathbb{M}^{m}[\kappa]$
mentioned above (this is the terminal object of $\mathcal{C}_{\Omega}$). The
existence of the two-hull of $\Omega$ is equivalent to the existence of an
initial object in $\mathcal{C}_{\Omega}$.
A choice of point $x_{\psi}\in\Psi$ for each object
$\psi\colon\Omega\looparrowright\Psi$ in $\mathcal{C}_{\Omega}$ is called the
_inverse point system_ , if for any morphism as in $({*})$ we have
$x_{\psi_{2}}=\vartheta(x_{\psi_{1}})$. Note that for any point $p\in\Omega$,
the choice of points $x_{\psi}=\psi(p)\in\Psi$ forms an inverse point system.
Set $\Phi$ to be the set of all inverse point systems. Note that $\Phi$ comes
with natural maps $\varphi\colon\Omega\to\Phi$ and
$\vartheta_{\psi}\colon\Phi\to\Psi$ for any object
$\psi\colon\Omega\looparrowright\Psi$ in $\mathcal{C}_{\Omega}$ such that the
following diagram commutes.
$\begindc{\commdiag}[10]\obj(3,5)[aa]{\Omega}\obj(0,0)[bb]{\Phi}\obj(6,0)[cc]{\Psi}\mor{aa}{bb}{\varphi}[-1,0]\mor{aa}{cc}{\psi}\mor{bb}{cc}{\vartheta_{\psi}}\enddc$
Let us equip $\Phi$ with the weakest topology which makes all maps
$\vartheta_{\psi}$ continuous. Clearly, with this topology all
$\vartheta_{\psi}$ and $\varphi$ become immersions.
Let $\Phi^{\prime}$ be the maximal open set in $\Phi$ which is homeomorphic to
$m$-manifold. Note that $\Phi^{\prime}$ comes with a natural $\kappa$-flat
metric so that each $\vartheta_{\psi}$ is an open isometric immersion of
$\Phi^{\prime}$. It is easy to see that $\Phi^{\prime}$ is two-convex and it
contains $\varphi(\Omega)$. Therefore, the isometric immersion
$\varphi\colon\Omega\looparrowright\Phi^{\prime}$ is an object of
$\mathcal{C}_{\Omega}$. Hence there is an immersion
$\Phi\looparrowright\Phi^{\prime}$ which commutes with the natural embedding
$\Phi^{\prime}\hookrightarrow\Phi$; i.e., $\Phi=\Phi^{\prime}$. In other words
$\Phi$ is isometric to the two-hull of $\Omega$.
The last two statements of the proposition follow easily from above. In
particular, $\Phi$ coincides with its universal cover $\widetilde{\Phi}$
because $\varphi:\Omega\looparrowright\Phi$ lifts to
$\widetilde{\varphi}:\Omega\looparrowright\widetilde{\Phi}$
($\pi_{1}(\Omega)=0$), and there is a morphism from $\widetilde{\varphi}$ to
$\varphi$ in $\mathcal{C}_{\Omega}$, proving that $\widetilde{\varphi}$ and
$\varphi$ are isomorphic because $\varphi$ is initial in
$\mathcal{C}_{\Omega}$ by construction. ∎
## 3 Buyalo’s Lemma and the Observation
In this section we prove Buyalo’s Lemma 1 and Observation 1. The proof of the
following proposition is left to the reader.
3.1. Proposition. Let $X$ and $Y$ be (possibly noncomplete) Riemannian
manifolds and $\Gamma$ be an open set of unit speed geodesics in $X$, covering
all points of $X$. Then $f\colon X\to Y$ is an isometric geodesic immersion if
and only if for any $\gamma\in\Gamma$, the curve $f\circ\gamma$ is a unit
speed geodesic in $Y$.
Proof of Buyalo’s Lemma. Set $m=\mathop{\rm dim}\nolimits M$. Note that the
statement of Buyalo’s Lemma trivially holds if $m\leqslant 2$. Further we
assume $m\geqslant 3$.
By choosing an isometric geodesic embedding
$\Delta\hookrightarrow\mathbb{M}^{m}[\kappa]$, we can consider $\Delta$ as a
subset of $\mathbb{M}^{m}[\kappa]$. Let us denote by $\tilde{p}$ the common
vertex of the faces in $\Lambda$ and let $\tilde{x},\tilde{y},\tilde{z}$ be
the remaining vertexes of $\Delta$. Denote by $p,x,y,z$ the corresponding
points in $M$; i.e.
$\displaystyle p$ $\displaystyle=f(\tilde{p}),$ $\displaystyle x$
$\displaystyle=f(\tilde{x}),$ $\displaystyle y$ $\displaystyle=f(\tilde{y}),$
$\displaystyle z$ $\displaystyle=f(\tilde{z}).$
Set $R=2{\hskip 0.5pt\cdot\nobreak\hskip 0.5pt}\mathop{\rm
diam}\nolimits\Delta$. Assume first that the injectivity radius at any point
in $B_{R}(p)\subset M$ is at least $R$. In this case $f$ is distance
preserving on each face.
3.2. Claim. $f\colon\Lambda\to M$ is a distance preserving map.
Proof of the claim. On the geodesic $[px]$ consider two unit normal fields
that go in the directions of the images of the faces adjacent to $[px]$. Note
that both fields are parallel. Thus the angle between the images of the faces
in $\Lambda$ is constant along the common side. Taking the point on geodesic
$[px]$ close to $p$, one can see that angles between faces of $f(\Lambda)$ in
$M$ coincide with the corresponding angles in
$\Lambda\subset\mathbb{M}^{m}[\kappa]$.
Consider points
$\displaystyle\tilde{x}^{\prime}$ $\displaystyle\in[\tilde{p}\tilde{x}],$
$\displaystyle\tilde{y}^{\prime}$ $\displaystyle\in[\tilde{p}\tilde{y}],$
$\displaystyle\tilde{z}^{\prime}$ $\displaystyle\in[\tilde{p}\tilde{z}],$
$\displaystyle x^{\prime}$ $\displaystyle=f(\tilde{x}^{\prime}),$
$\displaystyle y^{\prime}$ $\displaystyle=f(\tilde{y}^{\prime}),$
$\displaystyle z^{\prime}$ $\displaystyle=f(\tilde{z}^{\prime}).$
From above, we have that corresponding angles in the triangles
$[x^{\prime}y^{\prime}z^{\prime}]$ and
$[\tilde{x}^{\prime}\tilde{y}^{\prime}\tilde{z}^{\prime}]$ are equal; i.e.,
the angles in triangle $[x^{\prime}y^{\prime}z^{\prime}]$ coincide with its
comparison angles.
Let $\tilde{v}$ and $\tilde{w}$ be arbitrary points on the sides of triangle
$[\tilde{x}^{\prime}\tilde{y}^{\prime}\tilde{z}^{\prime}]$ and
$v=f(\tilde{v})$ and $w=f(\tilde{w})$. In both cases (curvature
$\geqslant\kappa$ or $\leqslant\kappa$) the above angle equality implies that
$|v-w|_{M}=|\tilde{v}-\tilde{w}|_{\mathbb{M}^{m}[\kappa]}.$
(Here $|{*}-{*}|_{X}$ denotes distance function in a metric space $X$.)
Note that for any $\tilde{v},\tilde{w}\in\Lambda$ there is a triangle
$[\tilde{x}^{\prime}\tilde{y}^{\prime}\tilde{z}^{\prime}]$ as above which
contains $\tilde{v}$ and $\tilde{w}$ on its sides. Hence the claim follows. ∎
Note that there is a map $F\colon B_{R}(\tilde{p})\to B_{R}(p)$ satisfying the
following properties:
1. 1.
$F|_{\Lambda}=f$;
2. 2.
$F(\tilde{p})=p$, and the differential of $F$ at $p$ is an isometry
$\mathrm{T}_{\tilde{p}}\to\mathrm{T}_{p}$;
3. 3.
$F$ sends all unit speed geodesics through $\tilde{p}$ to unit speed geodesics
through $p$.
3.3. Claim. The restriction of any such $F$ to $\Delta$ satisfies Buyalo’s
Lemma;
This claim is proved separately in the two cases:
Proof of the claim in case of curvature $\geqslant\kappa$. By Toponogov
comparison theorem, the diffeomorphism $F\colon B_{R}(\tilde{p})\to B_{R}(p)$
is non-expanding. This fact together with Claim 3 imply that the restriction
of $F$ to $\Delta$ is distance preserving on any geodesic in $\Delta$ with
ends in $\Lambda$. Applying Proposition 3 we get that the restriction of $F$
to $\Delta$ is isometric and geodesic in the interior of $\Delta$ and hence
the same holds on whole $\Delta$.∎
Proof of the claim in case of curvature $\leqslant\kappa$. Set $\Upsilon$ to
be the set of all minimizing geodesics with ends in $f(\Lambda)$ and let
$\bar{\Upsilon}$ be the subset of $M$ covered by geodesics in $\Upsilon$. By
Toponogov comparison, the diffeomorphism $F\colon B_{R}(\tilde{p})\to
B_{R}(p)$ is distance nondecreasing, while its inverse $F^{-1}$ is a distance
non-increasing diffeomorphism. Since $f$ is distance preserving, it follows
that $F^{-1}$ is isometric on each of the geodesic in $\Upsilon$; moreover,
any minimizing geodesic between points in $\Lambda$ can be presented as
$F^{-1}\circ\gamma$ for some $\gamma\in\Upsilon$. It follows that
$F^{-1}(\bar{\Upsilon})=\Delta$, or equivalently $F(\Delta)=\bar{\Upsilon}$.
In particular, $F$ is distance preserving on each minimizing geodesic with
ends in $\Lambda$. Applying Proposition 3 the same way as above, we conclude
that the restriction of $F$ to $\Delta$ is distance preserving and geodesic.∎
The general case. To treat the general case, choose $\varepsilon>0$ so that
the injectivity radius at any point in $B_{R}(p)$ is at least $2{\hskip
0.5pt\cdot\nobreak\hskip 0.5pt}\varepsilon$. Note that one can cover the
interior of $\Delta$ by an infinite sequence of tetrahedra
$\Delta_{1},\Delta_{2},\dots$ with a choice of three faces $\Lambda_{i}$ in
each $\Delta_{i}$ such that $\mathop{\rm diam}\nolimits\Delta_{i}<\varepsilon$
and
$\Lambda_{n}\subset\Lambda\cup\left(\bigcup_{i<n}\Delta_{i}\right).$
for each $n$. Then it remains to apply the above argument sequentially to
$\Delta_{1},\Delta_{2},\dots$ and pass to the closure. ∎
Proof of Observation 1. Set $m=\mathop{\rm dim}\nolimits M$. Choose any point
$p\in\mathrm{Flat}^{\kappa}M$ and $\tilde{p}\in\mathbb{M}^{m}[\kappa]$. Choose
a map $F\colon\mathbb{M}^{m}[\kappa]\to\mathrm{Flat}^{\kappa}M$ such that
1. 1.
$F(\tilde{p})=p$, and the differential of $F$ at $p$ is an isometry
$\mathrm{T}_{\tilde{p}}\to\mathrm{T}_{p}$;
2. 2.
$F$ sends all unit speed geodesics through $\tilde{p}$ to unit speed geodesics
through $p$.
Let $\Omega_{p}\subset\mathbb{M}^{m}[\kappa]$ be the maximal open star-shaped
w.r.t. $\tilde{p}$ set such that the map $F$ induces an open isometric
immersion of $\Omega_{p}$. Let $\Psi_{p}$ be the set of all tetrahedra with
one vertex at $\tilde{p}$ and three adjacent faces in $\Omega_{p}$ and let
$\bar{\Psi}_{p}$ be the union of all tetrahedra in $\Psi_{p}$.
Clearly $\bar{\Psi}_{p}$ is open and $\bar{\Psi}_{p}\supset\Omega_{p}$.
According to Buyalo’s Lemma, the map $F$ is isometric on each geodesic lying
in a tetrahedron from $\Psi_{p}$. Applying Proposition 3, we get that $F$ is
an open isometric immersion $\bar{\Psi}_{p}\looparrowright M$. Thus,
$\bar{\Psi}_{p}=\Omega_{p}$ for any $p\in\mathrm{Flat}^{\kappa}M$, hence the
result. ∎
## 4 Exhaustive manifolds
Let $\Omega$ be a simply connected $\kappa$-flat manifold. Denote by
$\Omega^{(2)}$ the two-hull of $\Omega$ (see Definition 2). From the
definition of the two-hull, we have that if $\Omega^{(2)}$ is isometric to the
model space $\mathbb{M}^{m}[\kappa]$ then $\Omega$ is exhaustive (see
Definition 1).
In this section we use the above observation to construct examples of
exhaustive manifolds. The following two propositions follow directly from the
discussion above. (In other words, the proof is left to the reader.)
4.1. Proposition. Assume $m\geqslant 3$ and suppose
$\Omega\subset\mathbb{M}^{m}[\kappa]$ is an nonempty open set with convex
complement. Then $\Omega^{(2)}$ is isometric to $\mathbb{M}^{m}[\kappa]$. In
particular, $\Omega$ is exhaustive.
Here is a generalization of the above proposition:
4.2. Proposition. Suppose $m\geqslant 3$ and suppose
$\Omega\subset\mathbb{M}^{m}[\kappa]$ is an nonempty open set such that
through any point $p\in\mathbb{M}^{m}[\kappa]$ passes a $3$-dimensional
subspace $W_{p}$ (i.e., an isometric copy of $\mathbb{M}^{3}[\kappa]$) such
that each connected component of $W_{p}\backslash\Omega$ is a convex set.
Then $\Omega^{(2)}$ is isometric to $\mathbb{M}^{m}[\kappa]$. In particular,
$\Omega$ is exhaustive.
The proof of the following proposition requires some work.
4.3. Proposition. Assume $m\geqslant 3$ and suppose
$\Omega\subset\mathbb{S}^{m}\buildrel\mathrm{def}\over{=\\!\\!=}\mathbb{M}^{m}[1]$
admits a geodesic isometric immersion
$\mathbb{S}^{2}_{+}\hookrightarrow\Omega$. Then $\Omega^{(2)}$ is isometric to
$\mathbb{S}^{m}$.
Proof. Fix two embeddings
$\mathbb{S}^{2}_{+}\hookrightarrow\Omega\hookrightarrow\mathbb{S}^{m}$, denote
their composition by $\imath$. Note that for any point
$x\in\mathbb{S}^{m}\backslash\imath(\partial\mathbb{S}^{2}_{+})$ there is
unique embedding
$\imath_{x}\colon\mathbb{S}^{2}_{+}\hookrightarrow\mathbb{S}^{m}$ such that
$x\in\imath_{x}(\mathbb{S}^{2}_{+})$ and $\imath_{x}(z)=\imath(z)$ for any
$z\in\partial\mathbb{S}^{2}_{+}$. It is easy to see that one can choose a
tetrahedron $\Delta$ in $\mathbb{S}^{m}$, such that one face of $\Delta$
belongs to $\imath_{x}(\mathbb{S}^{2}_{+})$ and contains all points in the set
$\imath_{x}(\mathbb{S}^{2}_{+})\backslash\Omega$, while the rest of the faces
is arbitrary close to $\imath(\mathbb{S}^{2}_{+})$, in particular these faces
belong to $\Omega$.
Applying to $\Delta$ the definition of two-convexity, we get an isometric
geodesic immersion $F\colon\Delta\looparrowright\Omega^{(2)}$. It is easy to
see that the map $x\mapsto F(x)$ is independent on the choice of $\Delta$;
moreover, the obtained map $\mathbb{S}^{m}\to\Omega^{(2)}$ is an open
isometric immersion. Since $\Omega^{(2)}$ is simply connected (see Proposition
2) we have that $\Omega^{(2)}$ is isometric to $\mathbb{S}^{m}$. ∎
## 5 Comments and open problems
$\bm{k}$-convexity. The definition of two-convexity (2) can be generalized to
“$k$-convexity”; one has to change tetrahedron $\Delta$ to a
$(k+1)$-dimensional simplex and $\Lambda$ to the set formed by $k$ faces out
of $(k+1)$ in $\Delta$. In this case, $1$-convexity is equivalent to the usual
convexity of each connected component of $\Omega$.
In [5, Section $\tfrac{1}{2}$], Gromov introduced the following closely
related notion which we will call further as _Lefschetz- $k$-convexity_333We
state a slight variation of Gromov’s definition; in particular, we restrict
our consideration to open sets and change the meaning of $k$; in Gromov’s
notations Lefschetz-$k$-convexity in $\mathbb{E}^{m}$ is called
$(m-k)$-convexity..
5.1. Definition. An open set $\Omega$ in $\mathbb{E}^{m}$ is _Lefschetz-
$k$-convex_ if for any $k$-dimensional affine subspace $A$ the natural
homology homomorphism
$H_{k-1}(\Omega\cap A)\to H_{k-1}(\Omega)$ $None$
is injective.
This definition can be generalized to $\kappa$-flat manifolds, one only has to
replace $\Omega\cap A$ by $k$-dimensional manifolds $\Theta$ that admit
proper444An isometric immersion $\imath\colon\Theta\looparrowright\Omega$ of
Riemannian manifolds $\Theta$ and $\Omega$ is called _proper_ if for any point
$p\in\Omega$ there is $\varepsilon>0$ such that each connected component of
$\imath^{-1}(\bar{B}_{\varepsilon}(p))\subset\Theta$ is compact. isometric
geodesic immersion $\Theta\looparrowright\Omega$.
It is easy to show that Lefschetz-$k$-convexity in $\mathbb{E}^{m}$ implies
our $k$-convexity. We know that the converse holds in two trivial cases: $k=1$
and $m\leqslant k+1$, but in all other cases we do not know the answer to the
following question.
5.2. Open problem. Is it true that any $k$-convex open subset of
$\mathbb{E}^{m}$ is Lefschetz-$k$-convex?
Smooth approximation of two-convex sets. To get a feeling of definition of
$k$-convexity, it is useful to observe the following.
5.3. Proposition. If $\Omega$ is an open subset of $\mathbb{E}^{m}$ with
smooth boundary $\partial\Omega$, then it is $k$-convex if and only if the
hypersurface $\partial\Omega$ has at most $k-1$ negative principle curvatures
at any point.
It is well known that any convex set in $\mathbb{E}^{m}$ can be approximated
by a convex set with smooth boundary. It turns out that for $k$-convex sets
(as well as for Lefschetz-$k$-convex sets) this is no longer true.
One of the reasons comes from the fact that for $k$-convex sets with smooth
boundary the homeomorphism in $({*}{*})$ is injective for subspaces $A$ of
arbitrary dimension; the proof is an exercise in Morse theory, see [5, Section
$\tfrac{1}{2}$]. Thus, any $k$-convex set which does not satisfy this
condition can not be approximated.
To give an explicit example, let
$\Omega\subset\mathbb{E}^{4}=\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
be the complement to the union of the following two 3-dimensional halfspaces:
$(\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{\geqslant
0}\times\\{0\\})\cup(\\{0\\}\times\mathbb{R}_{\geqslant
0}\times\mathbb{R}\times\mathbb{R})$
Clearly $\Omega$ is 2-convex and simply connected, but
$H_{1}(A\cap\Omega)=\mathbb{Z}$ for
$A=\left\\{\,\left.{(x,y,z,t)\in\mathbb{E}^{4}}\vphantom{x+y+z+t=0}\,\right|\,{x+y+z+t=0}\,\right\\}.$
Two-hull in non simply connected case. The following example shows a problem
with extension of the two-hull construction to non simply connected case.
Consider an isometric action $\mathbb{Z}_{2}\curvearrowright\mathbb{S}^{3}$
with two fixed points; then take $\Omega$ to be the orbit space
$\mathbb{S}^{3}/\mathbb{Z}_{2}$ with singular orbits removed. Note that
$\Omega$ admits no open isometric immersion into a two-convex $1$-flat
manifold. Hence the two-hull of $\Omega$ can not be defined in the class of
manifolds. On the other hand it can be defined in the class of “Riemannian
megafolds”; these creatures were introduced in [11] and under a different name
in [8]; they look a lot like Riemannian manifolds, but fail to be topological
spaces. (In the above example, the two-hull of $\Omega$ is the Riemannian
orbifold $(\mathbb{S}^{3}:\mathbb{Z}_{2})$.)
More questions. Here is a possible generalization of Proposition 4:
5.4. Question. Is it true that the two-hull of any open simply connected set
$\Omega\subset\mathbb{S}^{m}$ which contains a closed geodesic is isometric to
$\mathbb{S}^{m}$?
The following question of D. Burago and B. Kleiner is open for long time. It
is not directly relevant to all above, but it was one of the initial
motivations for our work.
5.5. Question. Is it possible to construct a Riemannian metric $g$ on the
product of a torus and an open disc $T^{2}\times D^{2}$ such that the torus
$T^{2}\times\\{0\\}\hookrightarrow\nobreak T^{2}\times D^{2}$ is flat and the
curvature is strictly positive outside of $T^{2}\times\\{0\\}$?
An answer to this question might lead to a better understanding of manifolds
with _almost positive curvature_ (see [15]).
Let us yet mention two related questions from mathoverflow:
* $\diamond$
Question 55788 about two-convexity and Lefschetz property.
* $\diamond$
Question 50889 about possible generalization of Buyalo’s Lemma.
## Appendix: Zalgaller’s rigidity.
Here we briefly repeat the proof of a theorem from [14]. We do this since the
result which interests us (Theorem Appendix: Zalgaller’s rigidity.) was not
formulated as a separate statement; it appeared as an intermediate statement
in the proof.
A.1. Theorem. Let $A=a_{1}a_{2}\dots a_{n}$ and $B=b_{1}b_{2}\dots b_{n}$ be
two simple spherical polygons (not necessary convex) with equal corresponding
sides. Assume $A$ lies in an open hemisphere and $\angle a_{i}\geqslant\angle
b_{i}$ for each $i$. Then $A$ is congruent to $B$.
At first this result might look unrelated to the content of this article. But
the proof relies on the following 2-dimensional analog of Theorem 1. Recall
that _spherical polyhedron_ is a simplicial complex equipped with a metric
such that each simplex is isometric to a simplex in a standard sphere.
A.2. Theorem. Let $\Sigma$ be a spherical polyhedron which is homeomorphic to
$\mathbb{S}^{2}$ and has curvature $\geqslant 1$ in the sense of Alexandrov.
Assume that an open neighborhood of $\mathbb{S}^{2}_{+}$ in $\mathbb{S}^{2}$
admits a locally isometric immersion in $\Sigma$. Then $\Sigma$ is isometric
to the standard sphere.
To deduce Theorem Appendix: Zalgaller’s rigidity. from Theorem Appendix:
Zalgaller’s rigidity., Zalgaller cuts the polygon $A$ from the sphere and
glues instead polygon $B$. As a result he gets the spherical polyhedron
$\Sigma$ as in Theorem Appendix: Zalgaller’s rigidity.. (In fact, if we drop
the condition that $A$ lies in a hemisphere, we can obtain this way any
spherical polyhedral metric on $\mathbb{S}^{2}$ with curvature $\geqslant 1$.)
Theorem Appendix: Zalgaller’s rigidity. is proved by induction on the number
$n$ of singular points in $\Sigma$. The base case $n=1$ is trivial. To do the
induction step, choose two singular points $p,q\in\Sigma$, cut $\Sigma$ along
a geodesic $[pq]$ and patch the hole so that the obtained new polyhedron
$\Sigma^{\prime}$ has curvature $\geqslant 1$. The patch is obtained by gluing
two copies of a spherical triangle along two sides. For the right choice of
the triangle, the points $p$ and $q$ become regular in $\Sigma^{\prime}$ and
exactly one new singular point appears in the patch. This way, the case with
$n$ singular points is reduced to the case with $n-1$ singular points (if
$n>1$).
The patch construction above was introduced by Alexandrov in his famous proof
of convex embeddabilty of polyhedrons; the earliest reference we have found is
[2, VI, §7].
Applying polyhedral approximation, one can extend Theorem Appendix:
Zalgaller’s rigidity. to any surface with curvature $\geqslant 1$ in the sense
of Alexandrov; in particular, this shows that Theorem 1 holds in addition for
$m=2$ and curvature $\geqslant 1$.
## References
* [1] Alexander, S. B.; Berg, I. D.; Bishop, R. L. Geometric curvature bounds in Riemannian manifolds with boundary. Trans. Amer. Math. Soc. 339 (1993), no. 2, 703–716.
* [2] Aleksandrov, A. D. Intrinsic Geometry of Convex Surfaces (Russian) OGIZ, Moscow-Leningrad, (1948).
* [3] Buyalo, S. V. Volume and fundamental group of a manifold of nonpositive curvature. (Russian) Mat. Sb. (N.S.) 122(164) (1983), no. 2, 142–156; translated in Math. USSR-Sbornik 50 (1985), 137–150.
* [4] Brendle, S.; Marques, F. C.; Neves, A. Deformations of the hemisphere that increase scalar curvature. Invent. Math. 183, (2011).
* [5] Gromov, M. Sign and geometric meaning of curvature. Rend. Semin. Mat. Fis. Milano 61 (1991), 9–123.
* [6] Hang, F.; Wang, X. Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. J. Geom. Anal. 19, 628–642 (2009).
* [7] Lohkamp, J. Global and local curvatures. Riemannian geometry (Waterloo, ON, 1993), Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI, (1996).
* [8] Lott, J. On the long-time behavior of type-III Ricci flow solutions. Math. Ann. 339 (2007), 627–666.
* [9] Min-Oo, M. Scalar curvature rigidity of asymptotically hyperbolic spin manifolds. Math. Ann. 285 (1989), no. 4, 527–539.
* [10] Min-Oo, M. Scalar curvature rigidity of certain symmetric spaces, Geometry, topology, and dynamics (Montreal, 1995), 127–137, CRM Proc. Lecture Notes vol. 15, Amer. Math. Soc., Providence RI, 1998.
* [11] Petrunin, A.; Tuschmann, W. Diffeomorphism finiteness, positive pinching, and second homotopy. Geom. Funct. Anal. 9 (1999), no. 4, 736–774.
* [12] Schoen, R.; Yau, S-T. Proof of the positive mass theorem. II. Comm. Math. Phys. 79 (1981), no. 2, 231–260.
* [13] Witten, E. A new proof of the positive energy theorem. Comm. Math. Phys. 80 (1981), no. 3, 381–402.
* [14] Залгаллер, В. А. О деформациях многоугольника на сфере. УМН, 11:5(71) (1956), 177–178
* [15] Ziller, W. Examples of Riemannian manifolds with non-negative sectional curvature. Surveys in differential geometry. Vol. XI, 63–102, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007.
|
arxiv-papers
| 2011-06-19T12:11:02 |
2024-09-04T02:49:19.854016
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "D. Panov and A. Petrunin",
"submitter": "Anton Petrunin",
"url": "https://arxiv.org/abs/1106.3736"
}
|
1106.3757
|
August 27, 2024
Unstable particles in non-relativistic quantum
mechanics?
H. Hernandez-Coronado
XX Instituto Mexicano del Petróleo
Eje central Lázaro Cárdenas 152, 07730, México D.F., México.
hhernand@imp.mx X
Abstract: The Schrödinger equation is up-to-a-phase invariant under the
Galilei group. This phase leads to the Bargmann’s superselection rule, which
forbids the existence of the superposition of states with different masses and
implies that unstable particles cannot be described consistently in non-
relativistic quantum mechanics. In this paper we claim that Bargmann’s rule
neglects physical effects and that a proper description of non-relativistic
quantum mechanics requires to take into account this phase through the
Extended Galilei group and the definition of its action on spacetime
coordinates.
## 1 Introduction
The Schrödinger equation is not invariant under pure Galilean boosts, but up-
to-a-phase invariant. This implies that there is not a unitary representation
of the Galilei group in the Hilbert space of a non-relativistic particle, but
a projective representation111In general, for a projective representation, two
elements $g,g^{\prime}$ of the Galilei group are represented in the Hilbert
space by operators $U_{g}$ and $U_{g^{\prime}}$ such that
$U_{g}U_{g^{\prime}}=\omega(g,g^{\prime})U_{g\cdot g^{\prime}}$, with
$|\omega|=1$. [1, 2]. The Galilei projective representation can be understood
as a unitary representation of the Extended Galilei group, a central extension
of the Galilei group [1, 2] and it yields to the Bargmann’s superselection
rule, i.e., the impossibility of consistently describing superposition of
states with different mass in non-relativistic quantum mechanics [3].
However, the phase $\omega$ (see footnote) associated with the mentioned
projective representation describes physical effects, i.e., it is related to
relativistic time differences between the corresponding inertial observers to
order $1/c^{2}$. Moreover, as pointed out by Greenberger [4], there are
physical situations that the superselection rule neglects, e.g., binding
energy effects which “show up as inertial mass when one transforms to another
system [of reference]”. Also, Giulini has remarked that for the mass to define
a superselection rule, it has to have spectrum and then, it has to be
dynamical and the conjugate momentum of a canonical coordinate [5].
The situation is, as Anandan put it, that the symmetry group of non-
relativistic quantum mechanics (Extended Galilei group) is different to the
spacetime’s one (Galilei group) [6]. And the Bargmann’s rule arises from
assuming that the symmetry group describing the world in the low velocity
limit is the Galilei group and the phase associated with the Extended group is
unphysical.
In this paper we claim, echoing previous works [4, 7, 8, 9], that the phase
$\omega$ is physically meaningful, we provide as an example of which the
Sagnac effect and contrast it with the argument behind the Bargmann’s rule.
Thus, we propose that a consistent description of non-relativistic quantum
mechanics requires to realize that the symmetry of the physical world in the
non-relativistic limit is given by the extended Galilei group, and then that
we should define an action of it on the coordinates of spacetime consistent
with its representation in the corresponding Hilbert space.
## 2 Bargmann’s superselection rule
It is well known that the Schrödinger equation is invariant under Galilean
transformations up-to-a-phase. To see this, let us assume that the Schrödinger
equation in the presence of a scalar potential $V$ is valid in an inertial
reference frame $S$ with coordinates ($\mathbf{x},t$), i.e.,
$i\hbar\partial_{t}\psi(\mathbf{x},t)=\left(mc^{2}-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{x},t)\right)\psi(\mathbf{x},t),$
(1)
and then, let us define another inertial reference frame $S^{\prime}$ with
coordinates $(\mathbf{x}^{\prime},t^{\prime})$ related to the unprimed ones by
the following Galilean transformation,
$\mathbf{x^{\prime}}=\mathbf{x}-\mathbf{v}t$ and $t^{\prime}=t$, which imply
that $\partial_{t}^{\prime}=\partial_{t}+\mathbf{v}\cdot\nabla$ and
$\nabla^{\prime}=\nabla$, where $\mathbf{v}$ is a constant three dimensional
real vector. In the frame $S^{\prime}$, Eq. (1) can be written as:
$i\hbar\partial_{t}^{\prime}\psi^{\prime}(\mathbf{x^{\prime}},t^{\prime})=\left(mc^{2}-\frac{\hbar^{2}}{2m}\nabla^{\prime
2}+V^{\prime}(\mathbf{x^{\prime}},t^{\prime})\right)\psi^{\prime}(\mathbf{x^{\prime}},t^{\prime}),$
(2)
with
$\psi^{\prime}(\mathbf{x^{\prime}},t^{\prime})=e^{im(\mathbf{v}^{2}t/2-\mathbf{v}\cdot\mathbf{x})/\hbar}\psi(\mathbf{x},t),$
(3)
and $V^{\prime}(\mathbf{x^{\prime}},t^{\prime})=V(\mathbf{x},t)$. The relation
(3) is the standard transformation expression for the wavefunction under a
Galilean boost and it implies that the probability density is a Galilean
scalar when $m\in\mathbb{R}$ although the wavefunction is not. As it is also
well known, under a space translation by $\mathbf{a}\in\mathbb{R}^{3}$ the
wavefunction transforms as
$\psi_{\mathbf{a}}(\mathbf{x}+\mathbf{a},t)=\psi(\mathbf{x},t)$, where
$\psi_{\mathbf{a}}=e^{-i\mathbf{P}\cdot\mathbf{a}/\hbar}\psi$ with $P_{i}$
being the spacial translation generator along the $i$-th coordinate111Latin
indexes label spacial coordinates, i.e., $i,j=1,2,3$..
The relation (3) traduces in that the representation of the Galilei group in
the Hilbert space of a non-relativistic particle is not a true unitary
representation but a projective one. Projective representations are physically
relevant because pure states are represented by rays in the Hilbert space.
As a consequence of the above mentioned projective representation arises the
Bargmann’s superselection rule, i.e., the impossibility of describing
superposition of states with different mass in a consistent way. In order to
see this, we will reproduce here the original Bargmann’s argument [3]. Suppose
$\psi(\mathbf{x},t)=\psi_{m_{1}}(\mathbf{x},t)+\psi_{m_{2}}(\mathbf{x},t)$ is
a superposition of the states with two masses in an inertial reference frame
$S$. Now let us perform the following composed transformation: a translation
(by $\mathbf{a}$) and then a boost (by $\mathbf{v}$) followed by a translation
(by $-\mathbf{a}$) and finally a boost (by $-\mathbf{v}$) to return to the
original inertial system $S$ with coordinates $(\mathbf{x},t)$. In the light
of the transformation laws for the wavefunction under boosts (3) and spacial
translations, it is not difficult to see that under the previous composed
transformation the wavefunction $\psi(\mathbf{x},t)$ yields:
$\tilde{\psi}(\mathbf{x},t)=e^{im_{1}\mathbf{v}\cdot\mathbf{a}/\hbar}\psi_{m_{1}}(\mathbf{x},t)+e^{im_{2}\mathbf{v}\cdot\mathbf{a}/\hbar}\psi_{m_{2}}(\mathbf{x},t),$
(4)
which is equal to $\psi(\mathbf{x},t)$ (modulo a phase) only if the Bargmann’s
superselection rule is imposed, i.e., $m_{2}=m_{1}$.
### 2.1 Relativistic remnants
The phase appearing in expression (3) and giving rise to the Bargmann’s rule,
however, has a physical origin. To understand where it comes from it is
required to look into the relativistic case [7, 8, 9]. The Klein-Gordon
equation,
$\left(\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}-\nabla^{2}+\frac{m^{2}c^{2}}{\hbar^{2}}\right)\phi(x)=0,$ (5)
with the ansatz:
$\phi(x)=e^{-imc^{2}t/\hbar}\psi(\mathbf{x},t),$ (6)
reduces to:
$-\frac{\hbar^{2}}{2m}\nabla^{2}\psi(\mathbf{x},t)=i\hbar\partial_{t}\psi(\mathbf{x},t)-\frac{\hbar^{2}}{2mc^{2}}\partial_{t}^{2}\psi(\mathbf{x},t),$
(7)
and then by neglecting the last term in the r.h.s. of the above equation in
the limit $c\rightarrow\infty$, the Schrödinger equation is obtained. Now,
since $\phi$ is a scalar, it follows from (6) that the wavefunctions for two
Lorentz inertial observers are related by:
$\psi^{\prime}(\mathbf{x}^{\prime},t^{\prime})=e^{imc^{2}(t^{\prime}-t)/\hbar}\psi(\mathbf{x},t)=e^{im(\mathbf{v}^{2}t/2-\mathbf{v}\cdot\mathbf{x})/\hbar}\psi(\mathbf{x},t)+\mathcal{O}(1/c^{2}),$
such that in the non-relativistic limit, the previous expression reduces
precisely to relation (3). Thus, the phase in relation (3) can be identified
as some relativistic remnant: it is the time difference of order $1/c^{2}$
between the different inertial observers.
Consequently, by imposing the Bargmann’s superselection rule we may be
neglecting physical effects. We claim that this is indeed the case. In fact,
it has been suggested that the phase shift observed in the COW experiment [9]
as well as the Sagnac effect [6, 10] may lead to such a situation (for the
experimental measurement of such effects see [11, 12], respectively). We will
focus in the latter effect in the following subsection.
### 2.2 Sagnac effect
Consider a device (source/detector) mounted on the extreme of a disk of radius
$R$. Suppose that the device can emit two equal signals restricted to move
along circular paths whose center is the disk’s, with the same velocity and in
opposite directions. Now, the disk is set into rotation around its center with
angular velocity $\Omega$. We would like to know if there is a detectable
physical effect due to the rotation when they arrived back to the emission
point. We can describe the previous system from both, the non-relativistic and
relativistic frameworks and they produce different results [6, 13].
Non-relativistically, both signals arrive at the same time at the detector and
if the signals are Galilean invariant waves, their phases are equal such that
there is not interference. This result can be obtained by an observer in a
laboratory frame and also by a comoving observer (which can be thought of as
an instantaneous inertial Galilean observer).
Relativistically, on the other hand, the two signals do not arrive at the same
time, but with a difference $\Delta t=4\pi
R^{2}\Omega/\sqrt{c^{2}-\Omega^{2}R^{2}}$ (as measured by a comoving observer)
[13]. This time difference traduces in the phase difference $\Delta\phi=4\pi
R^{2}\Omega\omega/\sqrt{c^{2}-\Omega^{2}R^{2}}$, where $\omega$ is the signal
frequency. Again, this result can be obtained by an observer in a laboratory
frame and also by a comoving observer (which can be thought of as an
instantaneous inertial Lorentzian observer). By using the Einstein-Planck
relation $\hbar\omega=mc^{2}/\sqrt{1-v^{2}/c^{2}}$ and taking the non-
relativistic limit, the above phase difference reduces to $\Delta\phi_{N}=4\pi
R^{2}m\Omega/\hbar$.
Now, quantum mechanically, the signals’ phases transform according to the
projective representation (3) such that its difference corresponds to
$\Delta\phi_{NQM}=m(v+\Omega R)^{2}t/2\hbar-m(v-\Omega
R)^{2}t/2\hbar=\Delta\phi_{N},$
with $t=\pi/\Omega$.
Consequently, if a Sagnac-like experiment is performed with a superposition of
states with different masses $\psi=\psi_{m_{1}}+\psi_{m_{2}}$, we would expect
to obtain an interference term similar to the one obtained from the Bargmann’s
superselection rule (4) and we know that it makes sense physically, so then we
do not have to impose any superselection rule.
## 3 Extended Galilei group
The projective representation of the Galilei group we found is equivalent to a
true unitary representation of the extended Galilei group [1, 2, 6], which
consists of introducing another generator $M$ that commutes with all other
Galilei group generators and such that the corresponding algebra remains the
same except for the relation $[C_{i},P_{j}]=M\delta_{ij}$, where $C_{i}$ is
the Galilean boost generator along the $i$-th coordinate. In terms of this
group, the Bargmann composed transformation is represented by
$e^{-i\mathbf{v}\cdot\mathbf{C}/\hbar}e^{-i\mathbf{P}\cdot\mathbf{a}/\hbar}e^{i\mathbf{v}\cdot\mathbf{C}/\hbar}e^{i\mathbf{P}\cdot\mathbf{a}/\hbar}=e^{iM\mathbf{a}\cdot\mathbf{v}/\hbar},$
(8)
which is not the group identity (as was for the Galilei group).
In order to see that the previous result is physically meaningful, let us
compare with the relativistic case. In the Poincaré algebra the commutator
between the boost generators $\mathbf{K}$ and the spacial translations
$\mathbf{P}$ is given by $[K_{i},P_{j}]=H\delta_{ij}/c^{2}$, which reduces
precisely to the commutator between $C_{i}$ and $P_{j}$ to leading order in
the non-relativistic limit. Correspondingly, the relativistic version of the
transformation considered by Bargmann can be written, up to
$\mathcal{O}(1/c^{2})$, as:
$e^{-i\mathbf{v}\cdot\mathbf{K}/\hbar}e^{-i\mathbf{a}\cdot\mathbf{P}/\hbar}e^{i\mathbf{v}\cdot\mathbf{K}/\hbar}e^{i\mathbf{a}\cdot\mathbf{P}/\hbar}=e^{iH\mathbf{a}\cdot\mathbf{v}/\hbar
c^{2}}e^{i(\mathbf{v}\cdot{\mathbf{a}})(\mathbf{v}\cdot\mathbf{P})/2\hbar
c^{2}},$ (9)
and accordingly, under the above transformation, the coordinates of the event
$(\mathbf{x},t)$ are given by
$(\mathbf{x}^{\prime},t^{\prime})=(\mathbf{x}+(\mathbf{v}\cdot\mathbf{a})\mathbf{v}/2c^{2},t+\mathbf{v}\cdot\mathbf{a}/c^{2})+\mathcal{O}(1/c^{4})$.
Then the non-relativistic limit of expression (9) reduces precisely to
expression (8) and it produces a relativistic displacement in the coordinates
of spacetime. Therefore, the transformation considered in Bargmann’s argument
produces a non-relativistic phase when acting on the wavefunction of a non-
relativistic quantum particle which can be identified as the relativistic time
difference between the corresponding inertial observers.
The bottom line is that we should identify $M$ as the generator of time
translations of order $1/c^{2}$ (cf. r.h.s. of relation (8)). And then, the
proper description of superposition of states with different masses requires
to define an action of the extended Galilei group on the coordinates of
Newtoninan spacetime which is consistent with its action on the Hilbert space,
in particular, we need to define how $M$ transforms the coordinates of an
event such that under the transformation (8), we obtain (at least the time
coordinate in) the relativistic relation for
$(\mathbf{x}^{\prime},t^{\prime})$ up to order $\mathcal{O}(c^{0})$. As
claimed by Guilini [5], such description implies that the mass must be
dynamical and, accordingly, we have to introduce its conjugate coordinate (the
correction of $\mathcal{O}(1/c^{2})$ to relativistic time), and then, it may
be required to modify the notion of Newtonian spacetime. Such a discussion is
presented somewhere else [14].
## Acknowledgments
The author is indebted with Y. Bonder, C. Chryssomalakos, E. Okon and D.
Sudarsky for useful discussions. This work has been partially supported by the
Czech Ministry of Education, Youth and Sports within the project LC06002.
## References
* [1] J. A. de Azcárraga and J. M. Izquierdo, Lie groups, Lie algebras, cohomology groups and some applications in physics (Cambridge University Press, New York, 1995).
* [2] J. -M. Levy-Leblond, J. Math. Phys. 4 (1963) 776.
* [3] V. Bargmann, Ann. Math. 59 (1954), 1.
* [4] D. M. Greenberger, Phys. Rev. Lett. 87, (2001).
* [5] D. Giulini, Annals Phys., 249 (1996) 222-235.
* [6] J. Anandan, Phys. Rev. D 24, (1981) 338.
* [7] B. Mashhoon, “Quantum theory in accelerated frames of reference” in J. Ehlers and C. L$\ddot{\text{a}}$mmerzahl, Special Relativity, Lect. Notes Phys. 702 (Springer, Berlin Heidelberg 2006).
* [8] P. Holland and H. R. Brown, Am. J. Phys., 67 (1999) 204.
* [9] D. M. Greenberger, Rev. Mod. Phys. 55, (1983) 875.
* [10] D. Dieks, Found. Phys. Lett. 3, (1990) 347.
* [11] R. Colella, A.W. Overhauser and S. A. Werner, Phys. Rev. Lett. 34, (1975) 1472.
* [12] S. A. Werner, J. -L. Staudenmann, and R. Colella, Phys. Rev. Lett. 42, (1979) 1103-1106.
* [13] D. Dieks and G. Nienhuis, Am. J. Phys. 58, (1989) 650.
* [14] H. Hernandez-Coronado, in preparation.
|
arxiv-papers
| 2011-06-19T15:56:05 |
2024-09-04T02:49:19.861000
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Hernandez-Coronado",
"submitter": "Hector Hernandez-Coronado",
"url": "https://arxiv.org/abs/1106.3757"
}
|
1106.3787
|
# Calculating Ellipse Overlap Areas
gbhughes@calpoly.edu m.chraibi@fz-juelich.de
###### Abstract
We present a general algorithm for finding the overlap area between two
ellipses. The algorithm is based on finding a segment area (the area between
an ellipse and a secant line) given two points on the ellipse. The Gauss-Green
formula is used to determine the ellipse sector area between two points, and a
triangular area is added or subtracted to give the segment area. For two
ellipses, overlap area is calculated by adding the areas of appropriate
sectors and polygons. Intersection points for two general ellipses are found
using Ferrari’s quartic formula to solve the polynomial that results from
combining the two ellipse equations. All cases for the number of intersection
points (0, 1, 2, 3, 4) are handled. The algorithm is implemented in c-code,
and has been tested with a range of input ellipses. The code is efficient
enough for use in simulations that require many overlap area calculations.
###### keywords:
Ellipse Area, Ellipse Sector, Ellipse Segment, Ellipse Overlap, Algorithm,
Quartic Formula.
Gary B. Hughes
California Polytechnic State University
Statistics Department
San Luis Obispo, CA 93407-0405, USA
Mohcine Chraibi
Jülich Supercomputing Centre
Forschungszentrum Jülich GmbH
D-52425 Jülich, Germany
## 1 Introduction
Ellipses are useful in many applied scenarios, and in widely disparate fields.
In our research, which happens to be in two very different areas, we have
encountered a common need for efficiently calculating the overlap area between
two ellipses.
In one case, the design for a solar calibrator on-board an orbiting satellite
required an efficient algorithm for ellipse overlap area. Imaging systems
aboard satellites rely on semi-conductor detectors whose performance changes
over time due to many factors. To produce consistent data, some means of
calibrating the detectors is required; see, e.g., [1]. Some systems use the
sun as a light source for calibration. In a typical solar calibrator, incident
sunlight passes through an attenuator grating and impinges on a diffuser
plate, which is oriented obliquely to the attenuator grating. The attenuator
grating is a pattern of circular openings. When sunlight passes through the
circular openings, projections of the circles onto the oblique diffuser plate
become small ellipses. The projection of the large circular entrance aperture
on the oblique diffuser plate is also an ellipse. The total incident light on
the calibrator is proportional to the sum of all the areas of the smaller
ellipses that are contained within the larger entrance aperture ellipse.
However, as the calibration process proceeds, the satellite is moving through
its orbit, and the angle from the sun into the calibrator changes (~7∘ in 2
minutes). The attenuator grating ellipses thus move across the entrance
aperture, and some of the smaller ellipses pass in and out of the entrance
aperture ellipse during calibration. Movement of the small ellipses across the
aperture creates fluctuations in the total amount of incident sunlight
reaching the calibrator in the range of 0.3 to 0.5%. This jitter creates
errors in the calibration algorithms. In order to model the jitter, an
algorithm is required for determining the overlap area of two ellipses. Monte
Carlo integration had been used; however, the method is numerically intensive
because it converges very slowly, so it was not an attractive approach for
modeling the calibrator due to the large number of ellipses that must be
modeled.
In a more down-to-earth setting, populated places such as city streets or
building corridors can become quite congested while crowds of people are
moving about. Understanding the dynamics of pedestrian movement in these
scenarios can be beneficial in many ways. Pedestrian dynamics can provide
critical input to the design of buildings or city infrastructure, for example
by predicting the effects of specific crowd management strategies, or the
behavior of crowds utilizing emergency escape routes. Current research in
pedestrian dynamics is making steady progress toward realistic modeling of
local movement; see, e.g., [2]. The model presented in [2] is based on the
concept of elliptical volume exclusion for individual pedestrians. Each model
pedestrian is surrounded by an elliptical footprint area that the model uses
to anticipate obstacles and other pedestrians in or near the intended path.
The footprint area is influenced by an individuals’ velocity; for example, the
exclusion area in front of a fast-moving pedestrian is elongated when compared
to a slower-moving individual, since a pedestrian is generally thinking a few
steps ahead. As pedestrians travel through a confined space, their collective
exclusion areas become denser, and the areas will eventually begin to overlap.
A force-based model will produce a repulsive force between overlapping
exclusion areas, causing the pedestrians to slow down or change course when
the exclusion force becomes large. Implementing the force-based model with
elliptical exclusion areas in a simulation requires calculating the overlap
area between many different ellipses in the most general orientations. The
ellipse area overlap algorithm must also be efficient, so as not to bog down
the simulation.
Simulations for both the satellite solar calibrator and force-based pedestrian
dynamic model require efficient calculation of the overlap area between two
ellipses. In this paper, we provide an algorithm that has served well for both
applications. The core component of the overlap area algorithm is based on
determining the area of an ellipse segment, which is the area between a secant
line and the ellipse boundary. The segment algorithm forms the basis of an
application for calculating the overlap area between two general ellipses.
## 2 Ellipse area, sector area and segment area
### 2.1 Ellipse Area
Consider an ellipse that is centered at the origin, with its axes aligned to
the coordinate axes. If the semi-axis length along the x-axis is A, and the
semi-axis length along the y-axis is B, then the ellipse is defined by a locus
of points that satisfy the implicit polynomial equation
$\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1$ (1)
The same ellipse can be defined parametrically by:
$\left.\begin{array}[]{c}x=A\cdot\cos(t)\\\
y=B\cdot\sin(t)\end{array}\right\\}\ \ 0\leq t\leq 2\pi$ (2)
The area of such an ellipse can be found using the parameterized form with the
Gauss-Green formula:
$\begin{split}\text{Area}=&\frac{1}{2}\int^{B}_{A}{[x(t)\cdot
y^{\prime}(t)-y(t)\cdot x{{}^{\prime}}(t)]dt}\\\
=&\frac{1}{2}\int^{2\pi}_{0}{A\cdot\cos(t)\cdot
B\cdot\cos(t)-B\cdot\sin(t)\cdot(-A)\cdot\sin(t)]dt}\\\ =&\frac{A\cdot
B}{2}\int^{2\pi}_{0}{\cos^{2}(t)+\sin^{2}(t)]dt}=\frac{A\cdot
B}{2}\int^{2\pi}_{0}{dt}\\\ =&\pi\cdot A\cdot B\end{split}$ (3)
### 2.2 Ellipse Sector Areas
We define the ellipse sector between two points (${x}_{1}$, ${y}_{1}$) and
(${x}_{2}$, ${y}_{2}$) on the ellipse as the area that is swept out by a
vector from the origin to the ellipse, beginning at (${x}_{1}$, ${y}_{1}$), as
the vector travels along the ellipse in a counter-clockwise direction from
(${x}_{1}$, ${y}_{1}$) to (${x}_{2}$, ${y}_{2}$). An example is shown in Fig.
1. The Gauss-Green formula can also be used to determine the area of such an
ellipse sector.
$\begin{split}\text{Sector Area}=&\frac{A\cdot
B}{2}\int^{\theta_{2}}_{\theta_{1}}{dt}\\\
=&\frac{(\theta_{2}-\theta_{1})\cdot A\cdot B}{2}\end{split}$ (4)
Figure 1: The area of an ellipse sector between two points on the ellipse is
the area swept out by a vector from the origin to the first point as the
vector travels along the ellipse in a counter-clockwise direction to the
second point. The area of an ellipse sector can be determined with the Gauss-
Green formula, using the parametric angles $\theta_{1}$ and $\theta_{2}$.
The parametric angle $\theta$ that is formed between the $x$-axis and a point
($x$, $y$) on the ellipse is found from the ellipse parameterizations:
$\displaystyle x=$ $\displaystyle A\cdot\cos(\theta)\ \
\Longrightarrow\theta=\cos^{-1}(x/A)$ $\displaystyle y=$ $\displaystyle
B\cdot\sin(\theta)\ \ \Longrightarrow\theta=\sin^{-1}(y/B)\ $
For a circle ($A=B$ in the ellipse implicit polynomial form), the parametric
angle corresponds to the geometric (visual) angle that a line from the origin
to the point ($x$, $y$) makes with the $x$-axis. However, the same cannot be
said for an ellipse; that is, the geometric (visual) angle is not the same as
the parametric angle used in the area calculation. For example, consider the
ellipse in Fig. 1; the implicit polynomial form is
$\frac{x^{2}}{4^{2}}+\frac{y^{2}}{2^{2}}=1$ (5)
Suppose the point ($x_{1}$, $y_{1}$) is at
$\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$. The point is on the ellipse,
since
$\frac{{\left({4}/{\sqrt{5}}\right)}^{2}}{4^{2}}+\frac{{\left({4}/{\sqrt{5}}\right)}^{2}}{2^{2}}=\frac{{4^{2}}/{5}}{4^{2}}+\frac{{4^{2}}/{5}}{2^{2}}=\frac{1}{5}+\frac{4}{5}=1$
A line segment from the origin to $\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$
forms an angle with the $x$-axis of $\pi$/4 ($\approx$0.7485398). However, the
ellipse parametric angle to the same point is:
$\theta={{\cos}^{-1}\left(\frac{{4}/{\sqrt{5}}}{4}\right)\
}={{\cos}^{-1}\left(\frac{1}{\sqrt{5}}\right)\ }\approx 1.10715$
The same angle can also be found from the parametric equation for $y$:
$\theta={\sin^{-1}\left(\frac{4/\sqrt{5}}{2}\right)\
}={\sin^{-1}\left(\frac{2}{\sqrt{5}}\right)\ }\approx 1.10715$
The angle found by using the parametric equations does not match the geometric
angle to the point that defines the angle.
When determining the parametric angle for a given point ($x$, $y$) on the
ellipse, the angle must be chosen in the proper quadrant, based on the signs
of $x$ and $y$. For the ellipse in Fig. 1, suppose the point ($x_{2}$,
$y_{2}$) is at $\left(-3,-{\sqrt{7}}/{2}\right)$. The parametric angle that is
determined from the equation for $x$ is:
$\theta={\cos^{-1}\left(\frac{-3}{4}\right)\ }\approx 2.41886$
The parametric angle that is determined from the equation for $y$ is:
$\theta={{\sin}^{-1}\left(\frac{-{\sqrt{7}}/{2}}{2}\right)\
}={{\sin}^{-1}\left(\frac{-\sqrt{7}}{4}\right)\ }\approx-.722734$
The apparent discrepancy is resolved by recalling that inverse trigonometric
functions are usually implemented to return a ‘principal value’ that is within
a conventional range. The typical (principal-valued) $\theta=\arccos(x)$
function returns angles in the range 0 = $\theta$ = $\pi$, and the typical
(principal-valued) $\theta=\arcsin(x)$ function returns angles in the range
–$\pi/2=\theta=\pi/2$. When the principal-valued inverse trigonometric
functions return angles in the typical ranges, the ellipse parametric angles,
defined to be from the $x$-axis, with positive angles in the counter-clockwise
direction, can be found with the relations in Table 1.
Quadrant II ($x<0\;\text{and}\;y\geq 0$)
$\theta=\arccos(x/A)$
$=\pi-\arcsin(y/B)$ | Quadrant I ($x\geq 0\;\text{and}\;y\geq 0)$
$\theta=\arccos(x/A)$
$=\arcsin(y/B)$
---|---
Quadrant III ($x<0\;\text{and}\;y<0$)
$\theta=2\pi-\arccos(x/A)$
$=\pi-\arcsin(y/B)$ | Quadrant IV ($x\geq 0\;\;y<0$)
$\theta=2\pi-\arccos(x/A)$
$=2\pi+\arcsin(y/B)$
Table 1: Relations for finding the parametric angle that corresponds to a
given point (x, y) on the ellipse x2/A2 \+ y2/B2 = 1. The parametric angle is
formed between the positive x-axis and a line drawn from the origin to the
given point, with counterclockwise being positive. For the standard
(principal-valued) inverse trigonometric functions, the resulting angle will
be in the range 0 $\leq\theta<2\pi$ for any point on the ellipse.
The point at $\left(-3,-{\sqrt{7}}/{2}\right)$ on the ellipse of Fig. 1 is in
Quadrant III. Using the relations in Table 1, the parametric angle that is
determined from the equation for $x$ is:
$\theta=2\pi-\arccos(\frac{-3}{4})\approx 3.86433$
The parametric angle that is determined from the equation for $y$ is:
$\theta=\pi-\arcsin(\frac{-\sqrt{7}/2}{2})\approx 3.86433$
With the proper angles, the Gauss-Green formula can be used to determine the
area of the sector from the point at
$\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$ to the point
$\left(-3,-{\sqrt{7}}/{2}\right)$ in the ellipse of Fig. 1.
$\begin{split}\text{Sector
Area}=&\frac{\left({\theta}_{2}-{\theta}_{1}\right)\cdot A\cdot B}{2}\\\
=&\frac{\left[\left(2\pi-\arccos\left(\frac{-3}{4}\right)\right)-{\arccos\left(\frac{{4}/{\sqrt{5}}}{4}\right)\
}\right]\cdot 4\cdot 2}{2}\\\ \approx&11.0287\end{split}$ (6)
The Gauss-Green formula is sensitive to the direction of integration. For the
larger goal of determining ellipse overlap areas, we define the ellipse sector
area to be calculated from the first point ($x_{1}$, $y_{1}$) to the second
point ($x_{2}$, $y_{2}$) in a counter-clockwise direction along the ellipse.
For example, if the points ($x_{1}$, $x_{1}$) and ($x_{2}$, $y_{2}$) of Fig. 1
were to have their labels switched, then the ellipse sector defined by the new
points will have an area that is complementary to that of the sector in Fig.
1, as shown in Fig. 2.
Figure 2: We define the ellipse sector area to be calculated from the first
point ($x_{1}$, $y_{1}$) to the second point ($x_{2}$, $y_{2}$) in a counter-
clockwise direction along the ellipse.
Switching the point labels, as shown in Fig. 2, also causes the angle labels
to be switched, resulting in the condition that $\theta_{1}>\theta_{2}$. Since
using the definitions in Table 1 will always produce an angle in the range
$0=\theta<2\pi$ for any point on the ellipse, the first angle can be
transformed by subtracting 2$\pi$ to restore the condition that
$\theta_{1}<\theta_{2}$. The sector area formula given above can then be used,
with the integration angle from ($\theta_{1}$ – $2\pi$) through $\theta_{2}$.
With the angle labels shown in Fig. 2, the area of the sector from the point
at $\left(-3,-{\sqrt{7}}/{2}\right)$ to the point at
$\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$ in a counter-clockwise direction
is:
$\begin{split}\text{Sector
Area}=&\frac{\left(\theta_{2}-\left(\theta_{1}-2\pi\right)\right)\cdot A\cdot
B}{2}\\\
=&\frac{\left[\left(2\pi-\arccos\left(\frac{-3}{4}\right)\right)-\left({\arccos\left(\frac{{4}/{\sqrt{5}}}{4}\right)-2\pi\
}\right)\right]\cdot 4\cdot 2}{2}\\\ \approx&14.1040\end{split}$ (7)
The two sector areas shown in Fig. 1 and Fig. 2 are complementary, in that
they add to the total ellipse area. Using the angle labels as shown in Fig. 1
for both sector areas:
$\begin{split}\text{Total Area}=&\frac{\left(\theta_{2}-\theta_{1}\right)\cdot
A\cdot B}{2}+\frac{\left(\theta_{1}-\left(\theta_{2}-2\pi\right)\right)\cdot
A\cdot B}{2}\\\ =&\frac{\left(2\pi\right)\cdot A\cdot B}{2}=\pi\cdot A\cdot
B\\\ =&\pi\cdot 4\cdot 2\\\ \approx&25.1327\end{split}$ (8)
### 2.3 Ellipse Segment Areas
For the overall goal of determining overlap areas between ellipses and other
curves, a useful measure is the area of what we will call an ellipse segment.
A secant line drawn between two points on an ellipse partitions the ellipse
area into two fractions, as shown in Fig. 1 and Fig. 2. We define the ellipse
segment as the area confined by the secant line and the portion of the ellipse
from the first point ($x_{1}$, $y_{1}$) to the second point ($x_{2}$, $y_{2}$)
traversed in a counter-clockwise direction. The segment’s complement is the
second of the two areas that are demarcated by the secant line. For the
ellipse of Fig. 1, the area of the segment defined by the secant line through
the points ($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$) is the area of the sector
minus the area of the triangle defined by the two points and the ellipse
center. To find the area of the triangle, suppose that the coordinates for the
vertices of are known, e.g., as ($x_{1}$, $y_{1}$), ($x_{2}$, $y_{2}$) and
($x_{3}$, $y_{3}$). Then the triangle area can be found by:
$\begin{split}\text{Triangle
Area}=&\frac{1}{2}\cdot\left|det\left(\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\\
y_{1}&y_{2}&y_{3}\\\ 1&1&1\end{array}\right)\right|\\\
=&\frac{1}{2}\cdot\left|x_{1}\cdot\left(y_{2}-y_{3}\right)-x_{2}\cdot\left(y_{1}-y_{3}\right)+x_{3}\cdot\left(y_{1}-y_{2}\right)\right|\end{split}$
(9)
In the case where one vertex, say ($x_{3}$, $y_{3}$), is at the origin, then
the area formula for the triangle can be simplified to:
$\text{Triangle Area}=\frac{1}{2}\cdot\left|x_{1}\cdot y_{2}-x_{2}\cdot
y_{1}\right|$ (10)
For the case depicted in Fig. 1, subtracting the triangle area from the area
of the ellipse sector area gives the area between the secant line and the
ellipse, i.e., the area of the ellipse segment counter-clockwise from
($x_{1}$, $y_{1}$) to ($x_{2}$, $y_{2}$):
$\text{Segment\ Area}=\frac{\left({\theta}_{2}-{\theta}_{1}\right)\cdot A\cdot
B}{2}-\frac{1}{2}\cdot\left|x_{1}\cdot y_{2}-x_{2}\cdot y_{1}\right|$ (11)
For the ellipse of Fig. 1, with the points at
$\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$ and
$\left(-3,-{\sqrt{7}}/{2}\right)$, the area of the segment defined by the
secant line is:
$\begin{split}&\frac{\left[\left(2\pi-\arccos\left(\frac{-3}{4}\right)\right)-{\arccos\left(\frac{{4}/{\sqrt{5}}}{4}\right)\
}\right]\cdot 4\cdot
2}{2}-\frac{1}{2}\cdot\left|\frac{4}{\sqrt{5}}\cdot\frac{-\sqrt{7}}{2}-\frac{4}{\sqrt{5}}\cdot-3\right|\\\
&\approx 9.52865\end{split}$
For the ellipse of Fig. 2, the area of the segment shown is the sector area
plus the area of the triangle.
$\text{Segment
Area}=\frac{\left({\theta}_{2}-\left({\theta}_{1}-2\pi\right)\right)\cdot
A\cdot B}{2}+\frac{1}{2}\cdot\left|x_{1}\cdot y_{2}-x_{2}\cdot y_{1}\right|$
(12)
With the points at $\left(-3,-{\sqrt{7}}/{2}\right)$ and
$\left({4}/{\sqrt{5}},{4}/{\sqrt{5}}\right)$ the area of the segment is:
$\begin{split}&\frac{\left[\left(2\pi-\arccos\left(\frac{-3}{4}\right)\right)-\left({\arccos\left(\frac{{4}/{\sqrt{5}}}{4}\right)-2\pi\
}\right)\right]\cdot 4\cdot
2}{2}+\frac{1}{2}\cdot\left|\frac{4}{\sqrt{5}}\cdot\frac{-\sqrt{7}}{2}-\frac{4}{\sqrt{5}}\cdot-3\right|\\\
&\approx 15.60409411\end{split}$
For the case shown in Fig. 1 and Fig. 2, the sector areas were shown to be
complementary. The segment areas are also complementary, since the triangle
area is added to the sector of Fig. 1, but subtracted from the sector of Fig.
2. Using the angle labels as shown in Fig. 1 for both sector areas:
$\begin{split}\text{Total\
Area}=&\left[\frac{\left({\theta}_{2}-{\theta}_{1}\right)\cdot A\cdot
B}{2}-\frac{1}{2}\cdot\left|x_{1}\cdot y_{2}-x_{2}\cdot y_{1}\right|\right]\\\
+&\left[\frac{\left({\theta}_{1}-\left({\theta}_{2}-2\pi\right)\right)\cdot
A\cdot B}{2}+\frac{1}{2}\cdot\left|x_{1}\cdot y_{2}-x_{2}\cdot
y_{1}\right|\right]\\\ =&\pi\cdot A\cdot B=\pi\cdot 4\cdot 2\approx
25.1327\end{split}$ (13)
The key difference between the cases in Fig. 1 and Fig. 2 that requires the
area of the triangle to be either subtracted from, or added to, the sector
area is the size of the integration angle. If the integration angle is less
than $\pi$, then the triangle area must be subtracted from the sector area to
give the segment area. If the integration angle is greater than $\pi$, the
triangle area must be added to the sector area.
### 2.4 A Core Algorithm for Ellipse Segment Area
A generalization of the cases given in Fig. 1 and Fig. 2 suggests a robust
approach for determining the ellipse segment area defined by a secant line
drawn between two given points on the ellipse. The ellipse is assumed to be
centered at the origin, with its axes parallel to the coordinate axes. We
define the segment area to be demarcated by the secant line and the ellipse
proceeding counter-clockwise from the first given point ($x_{1}$, $y_{1}$) to
the second given point ($x_{2}$, $y_{2}$). The ELLIPSE_SEGMENT algorithm is
outlined in Table 2, with pseudo-code presented in List. 1. The ellipse is
passed to the algorithm by specifying the semi-axes lengths, $A>0$ and $B>0$.
The points are passed to the algorithm as ($x_{1}$, $x_{1}$) and ($x_{2}$,
$y_{2}$), which must be on the ellipse.
ELLIPSE_SEGMENT Area Algorithm:
$1.\ \ \ \ \ \ \ \
\begin{array}[]{c}{\theta}_{1}=\left\\{\begin{array}[]{c}{\arccos\left({x_{1}}/{A}\right)\
}\ \ \ \ \ \ \ \ \ ,\ \ y_{1}\geq 0\\\ 2\pi-{\arccos\left({x_{1}}/{A}\right)\
},\ \ y_{1}<0\end{array}\right.\\\
{\theta}_{2}=\left\\{\begin{array}[]{c}{\arccos\left({x_{2}}/{A}\right)\ }\ \
\ \ \ \ \ \ \ ,\ \ y_{2}\geq 0\\\ 2\pi-{\arccos\left({x_{2}}/{A}\right)\ },\ \
y_{2}<0\end{array}\right.\end{array}$
$2.\ \ \ \ \ \ \ \
{\widehat{\theta}}_{1}=\left\\{\begin{array}[]{c}{\theta}_{1}\ \ \ \ \ \ \ \
,\ \ {{\theta}_{1}<\theta}_{2}\\\ {\theta}_{1}-2\pi,\ \
{{\theta}_{1}>\theta}_{2}\end{array}\right.$
$3.\ \ \ \ \ \ \ \ {\rm
Area}=\frac{\left({\theta}_{2}-{\widehat{\theta}}_{1}\right)\cdot A\cdot
B}{2}+\frac{sign\left({\theta}_{2}-{\widehat{\theta}}_{1}-\pi\right)}{2}\cdot\left|x_{1}\cdot
y_{2}-x_{2}\cdot y_{1}\right|$
---
where:
the ellipse implicit polynomial equation is
$\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1$
A $>$ 0 is the semi-axis length along the x-axis
B $>$ 0 is the semi-axis length along the y-axis
(x1, y1) is the first given point on the ellipse
($x_{2}$, $y_{2}$) is the second given point on the ellipse
$\theta_{1}$ and $\theta_{2}$ are the parametric angles corresponding to the
points
($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$)
Table 2: An outline of the ELLIPSE_SEGMENT area algorithm.
For robustness, the algorithm should avoid divide-by-zero and inverse-
trigonometric errors, so data checks should be included. The ellipse
parameters $A$ and $B$ must be greater than zero. A check is provided to
determine whether the points are on the ellipse, to within some numerical
tolerance, $\varepsilon$. Since the points can only be checked as being on the
ellipse to within some numerical tolerance, it may still be possible for the
$x$-values to be slightly larger than $A$, leading to an error when calling
the inverse trigonometric functions with the argument $x/A$. In this case, the
algorithm checks whether the $x$-value close to $A$ or –$A$, that is within a
distance that is less than the numerical tolerance. If the closeness condition
is met, then the algorithm assumes that the calling function passed a value
that is indeed on the ellipse near the point ($A$, 0) or (–$A$, 0), so the
value of $x$ is nudged back to $A$ or –$A$ to avoid any error when calling the
inverse trigonometric functions. The core algorithm, including all data
checks, is shown in List. 1.
Listing 1: The ELLIPSE_SEGMENT algorithm is shown for calculating the area of
a segment defined by the secant line drawn between two given points ($x_{1}$,
$y_{1}$) and ($x_{2}$, $y_{2}$) on the ellipse $x_{2}/A_{2}+y_{2}/B_{2}=1$. We
define the segment area for this algorithm to be demarcated by the secant line
and the ellipse proceeding counter-clockwise from the first given point
($x_{1}$, $y_{1}$) to the second given point ($x_{2}$, $y_{2}$).
⬇
1 ELLIPSE_SEGMENT (A, B, X1, Y1, X2, Y2)
2 do if (A 0 or B 0)
3 then return (-1, ERROR_ELLIPSE_PARAMETERS) :DATA CHECK
4 2 2 2 2 2 2 2 2
5 do if (|X1 /A + Y1 /B 1| > or |X1 /A + Y1 /B 1| > )
6 then return (-1, ERROR_POINTS_NOT_ON_ELLIPSE) :DATA CHECK
7 do if (|X1|/A > )
8 do if |X1| - A >
9 then return (-1, ERROR_INVERSE_TRIG) :DATA CHECK
10 else do if X1 < 0
11 then X1 -A
12 else X1 A
13 do if (|X2|/A > )
14 do if |X2| - A >
15 then return (-1, ERROR_INVERSE_TRIG) :DATA CHECK
16 else do if X2 < 0
17 then X2 -A
18 else X2 A
19 do if (Y1 < 0) :ANGLE QUADRANT FORMULA (TABLE 1)
20 then 1 2 acos (X1/A)
21 else 1 acos (X1/A)
22 do if (Y2 < 0) :ANGLE QUADRANT FORMULA (TABLE 1)
23 then 2 2 acos (X2/A)
24 else 2 acos (X2/A)
25 do if (1 > 2) :MUST START WITH 1 < 2
26 then 1 1 - 2
27 do if ((2 1) > ) :STORE SIGN OF TRIANGLE AREA
28 then trsgn +1.0
29 else trsgn +1.0
30 area 0.5*(A*B*(2 - 1) trsgn*|X1*Y2 - X2*Y1|)
31 return (area, NORMAL_TERMINATION)
An implementation of the ELLIPSE_SEGMENT algorithm written in c–code is shown
in Appendix 4. The code compiles under Cygwin-1.7.7-1, and returns the
following values for the two test cases presented in Fig. 1 and Fig. 2:
Listing 2: Return values for the test cases in Fig. 1 and Fig. 2
⬇
1 cc call_es.c ellipse_segment.c -o call_es.exe
2 ./call_es
3 Calling ellipse_segment.c
4 Fig. 1: segment area = 9.52864712, return_value = 0
5 Fig. 2: segment area = 15.60409411, return_value = 0
6 sum of ellipse segments = 25.13274123
7 ellipse area by pi*A*B = 25.13274123
## 3 Extending the Core Segment Algorithm to more General Cases
### 3.1 Segment Area for a (Directional) Line through a General Ellipse
The core segment algorithm is based on an ellipse that is centered at the
origin with its axes aligned to the coordinate axes. The algorithm can be
extended to more general ellipses, such as rotated and/or translated ellipse
forms. Start by considering the case for a standard ellipse with semi-major
axis lengths of $A$ and $B$ that is centered at the origin and with its axes
aligned with the coordinate axes. Suppose that the ellipse is rotated through
a counter-clockwise angle $\varphi$, and that the ellipse is then translated
so that its center is at the point ($h$, $k$). The rotated+translated ellipse
could then be defined by the set of parameters ($A$, $B$, $h$, $k$,
$\varphi$), with the understanding that the rotation through $\varphi$ is
performed before the translation through ($h$, $k$). The approach for
extending the core segment area algorithm will be to determine analogs on the
standard ellipse corresponding to any points of intersection between a shape
of interest and the general rotated and translated ellipse. To identify
corresponding points, features of the shape of interest are translated by
(–$h$, –$k$), and then rotated by –$\varphi$. The translated+rotated features
are used to determine any points of intersection with a similar ellipse that
is centered at the origin with its axes aligned to the coordinate axes. Then,
the core segment algorithm can be called with the translated+rotated
intersection points.
Rotation and translation are affine transformations that are also length- and
area-preserving. In particular, the semi-axis lengths in the general rotated
ellipse are preserved by both transformations, and corresponding points on the
two ellipses will demarcate equal partition areas. Fig. 3 illustrates this
idea, showing the ellipse of Fig. 1 which has been rotated counter-clockwise
through an angle $\varphi=3\pi/8$, then translated by $(h,k)=(-6,+3)$.
Figure 3: Translation and rotation are affine transformations that are also
length-and area-preserving. Corresponding points on the two ellipses will
demarcate equal partition areas.
Suppose that we desire to find the area of the rotated+translated ellipse
sector defined by the line $y=-x$, where the line ‘direction’ travels from
lower-right to upper-left, as shown in Fig. 3. We describe an approach for
finding a segment in a rotated+translated ellipse, based on the core ellipse
segment algorithm.
An ellipse that is centered at the origin, with its axes aligned to the
coordinate axes, is defined parametrically by
$\left.\begin{array}[]{c}x=A\cdot\cos(t)\\\
y=B\cdot\sin(t)\end{array}\right\\}\ \ 0\leq t\leq 2\pi$
Suppose the ellipse is rotated through an angle $\varphi$, with counter-
clockwise being positive, and that the ellipse is then to be translated to put
its center is at the point ($h$, $k$). Any point ($x$, $y$) on the standard
ellipse can be rotated and translated to end up in a corresponding location on
the new ellipse by using the transformation:
$\left[\begin{array}[]{c}x_{TR}\\\
y_{TR}\end{array}\right]=\left[\begin{array}[]{cc}{\cos\left(\varphi\right)}&-{\sin\left(\varphi\right)}\\\
{\sin\left(\varphi\right)}&{\cos\left(\varphi\right)}\end{array}\right]\cdot\left[\begin{array}[]{c}x\\\
y\end{array}\right]+\left[\begin{array}[]{c}h\\\ k\end{array}\right]$ (14)
Rotation and translation of the original standard ellipse does not change the
ellipse area, or the semi-axis lengths. One important feature of the
algorithms presented here is that the semi-axis lengths $A$ and $B$ are in the
direction of the $x$\- and y-axes, respectively, in the un-rotated (standard)
ellipse. In its rotated orientation, the semi-axis length $A$ will rarely be
oriented horizontally (in fact, for $\varphi=\pi/4$, the semi-axis length $A$
will be oriented vertically). Regardless of the orientation of the
rotated+translated ellipse, the algorithms presented here assume that the
values of $A$ and $B$ passed into the algorithm represent the semi-axis
lengths along the $x$\- and $y$-axes, respectively, for the corresponding un-
rotated, un-translated ellipse. The angle $\varphi$ is the amount of counter-
clockwise rotation required to put the ellipse into its desired location.
Specifying a negative value for $\varphi$ will rotate the standard ellipse
through a clockwise angle. The angle $\varphi$ can be specified in anywhere in
the range (–8, +8); the working angle in the code will be computed from the
given angle, modulo $2\pi$, to avoid any potential errors (?) when calculating
trigonometric values. The translation ($h$, $k$) is the absolute movement
along the coordinate axes of the ellipse center to move a standard ellipse
into its desired location. Negative values of $h$ move the standard ellipse to
the left; negative values of $k$ move the standard ellipse down.
To find the area between the given line and the rotated+translated ellipse,
the two curve equations can be solved simultaneously to find any points of
intersection. But instead of searching for the points of intersection with the
rotated+translated ellipse, it is more efficient to transform the two given
points that define the line back through the translation (–$h$, –$k$) then
rotation through –$\varphi$. The new line determined by the translated+rotated
points will pass through the standard ellipse at points that are analogous to
where the original line intersects the rotated+translated ellipse.
The transformations required to move the given points ($x_{1}$, $y_{1}$) and
($x_{2}$, $y_{2}$) into an orientation with respect to a standard ellipse that
is analogous to their orientation to the given ellipse are the inverse of what
it took to rotate+translate the ellipse to its desired position. The
translation is performed first, then the rotation:
$\left[\begin{array}[]{c}x_{i_{0}}\\\
y_{i_{0}}\end{array}\right]=\left[\begin{array}[]{cc}{\cos\left(-\varphi\right)}&-{\sin\left(-\varphi\right)}\\\
{\sin\left(-\varphi\right)}&{\cos\left(-\varphi\right)}\end{array}\right]\cdot\left[\begin{array}[]{c}x_{i}-h\\\
y_{i}-k\end{array}\right]$ (15)
Multiplying the vector by the matrix, and simplifying the negative-angle trig
functions gives the following expressions for the translated+rotated points:
$\displaystyle x_{i_{0}}=$ $\displaystyle{\cos\left(\varphi\right)\
}\cdot\left(x_{i}-h\right)+{\sin\left(\varphi\right)\cdot\left(y_{i}-k\right)\
}$ $\displaystyle y_{i_{0}}=$
$\displaystyle{-\sin\left(\varphi\right)\cdot\left(x_{i}-h\right)\
}+{\cos\left(\varphi\right)\ }\cdot\left(y_{i}-k\right)$
The two new points $\left(x_{1_{0}},y_{1_{0}}\right)$ and
$\left(x_{2_{0}},y_{2_{0}}\right)$ can be used to determine a line, e.g., by
the point-slope method:
$y=y_{1_{0}}+\frac{y_{2_{0}}-y_{1_{0}}}{x_{2_{0}}-x_{1_{0}}}\left({x-x}_{1_{0}}\right)$
(16)
The equation can also be formulated in an alternative way to accommodate cases
where the translated+rotated line is vertical, or nearly so:
$x=x_{1_{0}}+\frac{x_{2_{0}}-x_{1_{0}}}{y_{2_{0}}-y_{1_{0}}}\left({y-y}_{1_{0}}\right)$
(17)
Points of intersection are found by substituting the line equations into the
standard ellipse equation, and solving for the remaining variable. For each
case, define the slope as:
$m_{yx}=\frac{y_{2_{0}}-y_{1_{0}}}{x_{2_{0}}-x_{1_{0}}},\;\;m_{xy}=\frac{x_{2_{0}}-x_{1_{0}}}{y_{2_{0}}-y_{1_{0}}}$
(18)
Then the two substitutions proceed as follows:
$\begin{split}y=&y_{1_{0}}+m_{yx}\cdot\left({x-x}_{1_{0}}\right){\rm\ \ into\
\ }\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1\\\
\Longrightarrow&\frac{x^{2}}{A^{2}}+\frac{{\left(y_{1_{0}}+m_{yx}\cdot\left({x-x}_{1_{0}}\right)\right)}^{2}}{B^{2}}=1\\\
\Longrightarrow&\left[\frac{B^{2}+A^{2}\cdot{\left(m_{yx}\right)}^{2}}{A^{2}}\right]\cdot
x^{2}\\\ +&\left[2\cdot\left(y_{1_{0}}\cdot
m_{yx}-{\left(m_{yx}\right)}^{2}\cdot x_{1_{0}}\right)\right]\cdot x\\\
+&\left[{\left(y_{1_{0}}\right)}^{2}-2\cdot m_{yx}\cdot x_{1_{0}}\cdot
y_{1_{0}}+{\left(m_{yx}\cdot x_{1_{0}}\right)}^{2}-B^{2}\right]\\\
=&0\end{split}$ (19)
$\begin{split}x=&x_{1_{0}}+m_{xy}\cdot\left({y-y}_{1_{0}}\right){\rm\ \ into\
\ }\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1\\\
\Longrightarrow&\frac{{\left(x_{1_{0}}+m_{xy}\cdot\left({y-y}_{1_{0}}\right)\right)}^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1\\\
\Longrightarrow&\left[\frac{A^{2}+B^{2}\cdot{\left(m_{xy}\right)}^{2}}{B^{2}}\right]\cdot
y^{2}\\\ +&\left[2\cdot\left(x_{1_{0}}\cdot
m_{xy}-{\left(m_{xy}\right)}^{2}\cdot y_{1_{0}}\right)\right]\cdot y\\\
+&\left[{\left(x_{1_{0}}\right)}^{2}-2\cdot m_{xy}\cdot x_{1_{0}}\cdot
y_{1_{0}}+{\left(m_{xy}\cdot y_{1_{0}}\right)}^{2}-A^{2}\right]\\\
=&0\end{split}$ (20)
If the translated+rotated line is not vertical, then use the first equation to
find the $x$-values for any points of intersection. If the translated+rotated
line is close to vertical, then the second equation can be used to find the
$y$-values for any points of intersection. Since points of intersection
between the line and the ellipse are determined by solving a quadratic
equation $ax^{2}+bx+c$, there are three cases to consider:
1. 1.
$\Delta=b^{2}-4ac<0$: Complex Conjugate Roots (no points of intersection)
2. 2.
$\Delta=b^{2}-4ac=0$: One Double Real Root (1 point of intersection; line
tangent to ellipse)
3. 3.
$\Delta=b^{2}-4ac>0$: Two Real Roots (2 points of intersection; line crosses
ellipse)
For the first two cases, the segment area will be zero. For the third case,
the two points of intersection can be sent to the core segment area algorithm.
However, to enforce a consistency in area measures returned by the core
algorithm, the integration direction is specified to be from the first point
to the second point. As such, the ellipse line overlap algorithm should be
sensitive to the order that the points are passed to the core segment
algorithm. We suggest giving the line a ‘direction’ from the first given point
on the line to the second. The line ‘direction’ can then be used to determine
which is to be the first point of intersection, i.e., the first intersection
point is where the line enters the ellipse based on what ‘direction’ the line
is pointing. The segment area that will be returned from ELLIPSE_SEGMENT by
passing the line’s entry location as the first intersection point is the area
within the ellipse to the right of the line’s path.
The approach outlined above for finding the overlap area between a line and a
general ellipse is implemented in the ELLIPSE_LINE_OVERLAP algorithm, with
pseudo-code shown in List. 3. The ellipse is passed to the algorithm by
specifying the counterclockwise rotation angle $\varphi$ and the translation
($h$, $k$) that takes a standard ellipse and moves it to the desired
orientation, along with the semi-axes lengths, $A>0$ and $>0$. The line is
passed to the algorithm as two points on the line, ($x_{1}$, $y_{1}$) and
($x_{2}$, $y_{2}$). The ‘direction’ of the line is taken to be from ($x_{1}$,
$y_{1}$) toward ($x_{2}$, $y_{2}$). Then, the segment area returned from
ELLIPSE_SEGMENT will be the area within the ellipse to the right of the line’s
path.
Listing 3: The ELLIPSE_LINE_OVERLAP algorithm is shown for calculating the
area of a segment in a general ellipse that is defined by a given line. The
line is considered to have a ‘direction’ that runs from the first given point
($x_{1}$, $y_{1}$) to the second given point ($x_{2}$, $y_{2}$). The line
‘direction’ determines the order in which intersection points are passed to
the ELLIPSE_SEGMENT algorithm, which will return the area of the segment that
runs along the ellipse from the first point to the second in a counter-
clockwise direction. Any routine that calls the algorithm ELLIPSE_LINE_OVERLAP
must be sensitive to the order of points that are passed in.
⬇
1 (Area,Code) $\leftarrow$ ELLIPSE\\_LINE\\_OVERLAP
(A,B,H,K,$\varphi$,X1,Y1,X2,Y2)
2 do if (A $\leq$ 0 or B $\leq$ 0)
3
4 then return (-1, ERROR_ELLIPSE_PARAMETERS) :DATA_CHECK
5
6 do if ( $|\varphi|>2\pi$)
7
8 then $\varphi\leftarrow(\varphi\;\text{modulo}\;2\pi)$ :BRING $\varphi$ INTO
$-2\pi\leq\varphi<2\pi$ (?)
9
10 do if ($|X1|/A>2\pi$)
11
12 then X1 $\leftarrow$ -A
13
14 X10 $\leftarrow\cos(\varphi)*(X1-H)+\sin(\varphi)*(Y1-K)$
15
16 Y10 $\leftarrow-\sin(\varphi)*(X1-H)+\cos(\varphi)*(Y1-K)$
17
18 X20 $\leftarrow\cos(\varphi)*(X2-H)+\sin(\varphi)*(Y2-K)$
19
20 Y20 $\leftarrow-\sin(\varphi)*(X2-H)+\cos(\varphi)*(Y2-K)$
21
22 do if ($|X20-X10|>\varepsilon$ ) :LINE IS NOT VERTICAL
23
24 then m $\leftarrow$ (Y20 - Y10)/(X20 - X10) :STORE QUADRATIC COEFFICIENTS
25
26 a $\leftarrow$ (B2 + (A*m)2)/$A^{2}$
27
28 b $\leftarrow$ (2.0*(Y10*m – $m^{2}$*X10))
29
30 c $\leftarrow$ (Y102 - 2.0*m*Y10*X10 + (m*X10)2 – B2)
31
32 else if (|Y20 – Y10| $>\varepsilon$) :LINE IS NOT HORIZONTAL
33
34 then m $\leftarrow$ (X20 - X10)/(Y20 - Y10) :STORE QUADRATIC COEFFS
35
36 a $\leftarrow$ (A2 + (B*m)2)/B2
37
38 b $\leftarrow$ (2.0*(X10*m – m2*Y10))
39
40 c $\leftarrow$(X102 - 2.0*m*Y10*X10 + (m*Y10)2 – A2)
41
42 else return (-1, ERROR_LINE_POINTS) :LINE POINTS TOO CLOSE
43
44 discrim $\leftarrow$ b2 - 4.0*a*c
45
46 do if (discrim $<$ 0.0) :LINE DOES NOT CROSS ELLIPSE
47
48 then return (0, NO_INTERSECT)
49
50 else if (discrim $>$ 0.0) :TWO INTERSECTION POINTS
51
52 then root1 $\leftarrow$ (-b - sqrt (discrim))/(2.0*a)
53
54 root2 $\leftarrow$ (-b + sqrt (discrim))/(2.0*a)
55
56 else return (0, TANGENT) :LINE TANGENT TO ELLIPSE
57
58 do if ($|X20-X10|>\varepsilon$) :ROOTS ARE X-VALUES
59
60 then do if (X10 $<$ X20) :ORDER PTS SAME AS LINE DIRECTION
61
62 then x1 $\leftarrow$ root1
63
64 x2 $\leftarrow$ root2
65
66 else x1 $\leftarrow$ root2
67
68 x2 $\leftarrow$ root1
69
70 else do if (Y10 $<$ Y20) :ROOTS ARE Y-VALUES
71
72 then y1 $\leftarrow$ root1 :ORDER PTS SAME AS LINE DIRECTION
73
74 y2 $\leftarrow$ root2
75
76 else y1 $\leftarrow$ root2
77
78 y2 $\leftarrow$ root1
79
80 (Area,Code) $\leftarrow$ ELLIPSE_SEGMENT (A,B,x1,y1,x2,y2)
81
82 do if (Code $<$ NORMAL_TERMINATION)
83
84 then return (-1.0, Code)
85
86 else return (Area, TWO_INTERSECTION_POINTS)
An implementation of the ELLIPSE_LINE_OVERLAP algorithm in c-code is shown in
Appendix 5. The code compiles under Cygwin-1.7.7-1, and returns the following
values for the test cases presented above in Fig. 3, with both line
‘directions’:
Listing 4: Return values for the test cases in Fig. 3.
⬇
1 cc call_el.c ellipse_line_overlap.c ellipse_segment.c -o call_el.exe
2
3 ./call_el
4
5Calling ellipse_line_overlap.c
6
7 area = 4.07186819, return_value = 102
8
9 reverse: area = 21.06087304, return_value = 102
10
11 sum of ellipse segments = 25.13274123
12
13 total ellipse area by pi*A*B = 25.13274123
### 3.2 Ellipse-Ellipse Overlap Area
The method described above for determining the area between a line and an
ellipse can be extended to the task of finding the overlap area between two
general ellipses. Suppose the two ellipses are defined by their semi-axis
lengths, center locations and axis rotation angles. Let the two sets of
parameters ($A_{1}$, $B_{1}$, $h_{1}$, $k_{1}$, $\varphi_{1}$) and ($A_{2}$,
$B_{2}$, $h_{2}$, $k_{2}$, $\varphi_{2}$) define the two ellipses for which
overlap area is sought. The approach presented here will be to first translate
both ellipses by an amount (–$h_{1}$, –$k_{1}$) that puts the center of the
first ellipse at the origin. Then, both translated ellipses are rotated about
the origin by an angle –$\varphi_{1}$ that aligns the axes of the first
ellipse with the coordinate axes; see Fig. 4. Intersection points are found
for the two translated+rotated ellipses, using Ferrari’s quartic formula.
Finally, the segment algorithm described above is employed to find all the
pieces of the overlap area.
Figure 4: Intersection points on each curve are used with the ellipse segment
area algorithm to determine overlap area, by calculating the area of
appropriate segments, and polygons in certain cases. For the case of two
intersection points, as shown above, the overlap area can be found by adding
two segments, as shown in Fig. 5.
For example, consider a case of two general ellipses with two (non-tangential)
points of intersection, as shown in Fig. 4. The translation+rotation
transformations that put the first ellipse at the origin and aligned with the
coordinate axes do not alter the overlap area. In the case shown in Fig. 4,
the overlap area consists of one segment from the first ellipse and one
segment from the second ellipse. The segment algorithm presented above can be
used directly for ellipses centered at the origin and aligned with the
coordinate axes. As such, the desired segment from the first ellipse can be
found immediately with the segment algorithm, based on the points of
intersection. To find the desired segment of the second ellipse, the approach
presented here further translates and rotates the second ellipse so that the
segment algorithm can also be used directly. The overlap area for the case
shown in Fig. 4 is equal to the sum of the two segment areas, as shown in Fig.
5. Other cases, e.g. with 3 and 4 points of intersection, can also be handled
using the segment algorithm.
Figure 5: The area of overlap between two intersecting ellipses can be found
by using the ellipse sector algorithm. In the case of two (non-tangential)
intersection points, the overlap area is equal to the sum of two ellipse
sectors. The sector in each ellipse is demarcated by the intersection points.
The overlap area algorithm presented here finds the area of appropriate
sector(s) of each ellipse, which are demarcated by any points of intersection
between the two ellipse curves. To find intersection points, the two ellipse
equations are solved simultaneously. This step can be accomplished by using
the implicit polynomial forms for each ellipse. The first ellipse equation, in
its translated+rotated position is written as an implicit polynomial using the
appropriate semi-axis lengths:
$\frac{x^{2}}{A^{2}_{1}}+\frac{y^{2}}{B^{2}_{1}}=1$ (21)
In a general form of this problem, the translation+rotation that puts the
first ellipse centered at the origin and oriented with the coordinate axes
will typically leave the second ellipse displaced and rotated. The implicit
polynomial form for a more general ellipse that is rotated and/or translated
away from the origin is written in the conventional way as:
$AA\cdot x^{2}+BB\cdot x\cdot y+CC\cdot y^{2}+DD\cdot x+EE\cdot y+FF=0$ (22)
Any points of intersection for the two ellipses will satisfy these two
equations simultaneously. An intermediate goal is to find the implicit
polynomial coefficients in Ellipse Eq. 22 that describe the second ellipse
after the translation+rotation that puts the first ellipse centered at the
origin and oriented with the coordinate axes. The parameters that describe the
second ellipse after the translation+rotation can be determined from the
original parameters for the two ellipses. The first step is to translate the
second ellipse center ($h_{2}$, $k_{2}$) through an amount (–$h_{1}$,
–$k_{1}$), then rotate the center-point through –$\varphi_{1}$ to give a new
center point ($h_{2TR}$, $k_{2TR}$):
$\displaystyle h_{2TR}=$ $\displaystyle{\cos\left({-\varphi}_{1}\right)\
}\cdot\left(h_{2}-h_{1}\right)-{\sin\left({-\varphi}_{1}\right)\cdot\left(k_{2}-k_{1}\right)\
}\ $ $\displaystyle k_{2TR}=$
$\displaystyle{\sin\left(-{\varphi}_{1}\right)\cdot\left(h_{2}-h_{1}\right)\
}+{\cos\left({-\varphi}_{1}\right)\ }\cdot\left(k_{2}-k_{1}\right)\ $
The coordinates for a point ($x_{TR}$, $y_{TR}$) on the second ellipse in its
new translated+rotated position can be found from the following parametric
equations, based on an ellipse with semi-axis lengths $A_{2}$ and $B_{2}$ that
is centered at the origin, then rotated and translated to the desired
position:
$\left.\begin{array}[]{c}x_{TR}=A_{2}\cdot\cos\left(t\right)\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}-B_{2}\cdot\sin\left(t\right)\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}+h_{2_{TR}}\\\
y_{TR}=A_{2}\cdot\cos\left(t\right)\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}+B_{2}\cdot\sin\left(t\right)\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}+k_{2_{TR}}\end{array}\right\\}\ \ 0\leq t\leq 2\pi$
To find the implicit polynomial coefficients from the parametric form, further
transform the locus of points (xTR, yTR) so that they lie on the ellipse
($A_{2}$, $B_{2}$, 0, 0, 0), which is accomplished by first translating
($x_{TR}$, $y_{TR}$) through (–($h_{1}$ – $h_{2}$), –($k_{1}$ – $k_{2}$)) and
then rotating the point through the angle –($\varphi_{1}$ – $\varphi_{2}$):
$\begin{split}x=&{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\cdot\left(x_{TR}-\left(h_{1}-h_{2}\right)\right)-{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left(y_{TR}-\left(k_{1}-k_{2}\right)\right)\
}\ \\\
y=&{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left(x_{TR}-\left(h_{1}-h_{2}\right)\right)\
}+{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\cdot\left(y_{TR}-\left(k_{1}-k_{2}\right)\right)\ \end{split}$
The locus of points ($x$, $y$) should satisfy the standard ellipse equation
with the appropriate semi-axis lengths:
$\frac{x^{2}}{A^{2}_{2}}+\frac{y^{2}}{B^{2}_{2}}=1$ (23)
Finally, the implicit polynomial coefficients for Ellipse Eq. 22 are found by
substituting the expressions for the point ($x$, $y$) into the standard
ellipse equation, yielding the following ellipse equation:
$\begin{split}&\frac{{\left[{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\cdot\left(x_{TR}-\left(h_{1}-h_{2}\right)\right)-{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left(y_{TR}-\left(k_{1}-k_{2}\right)\right)\
}\right]}^{2}}{A^{2}_{2}}\\\
+&\frac{{\left[{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left(x_{TR}-\left(h_{1}-h_{2}\right)\right)\
}+{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\cdot\left(y_{TR}-\left(k_{1}-k_{2}\right)\right)\right]}^{2}}{B^{2}_{2}}\\\
=&1\end{split}$ (24)
where ($xx_{TR}$, $y_{TR}$) are defined as above. Expanding the terms, and
then re-arranging the order to isolate like terms yields the following
expressions for the implicit polynomial coefficients of a general ellipse with
the set of parameters ($A_{2}$, $B_{2}$, $h_{2TR}$, $k_{2TR}$, $\varphi_{2}$ –
$\varphi_{1}$):
$\begin{split}AA=&\frac{{{\cos}^{2}\left({\varphi}_{2}-{\varphi}_{1}\right)\
}}{A^{2}_{2}}+\frac{{{\sin}^{2}\left({\varphi}_{2}-{\varphi}_{1}\right)\
}}{B^{2}_{2}}\\\
BB=&\frac{2\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\
}}{A^{2}_{2}}-\frac{2\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\ }}{B^{2}_{2}}\\\
CC=&\frac{{{\sin}^{2}\left({\varphi}_{2}-{\varphi}_{1}\right)\
}}{A^{2}_{2}}+\frac{{{\cos}^{2}\left({\varphi}_{2}-{\varphi}_{1}\right)\
}}{B^{2}_{2}}\\\
DD=&\frac{-2\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left[h_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)+k_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\ }\right]\ }}{A^{2}_{2}}\\\
+&\frac{2\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left[k_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}-h_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\ }\right]\
}}{B^{2}_{2}}\\\
EE=&\frac{-2\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left[h_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)+k_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\ }\right]\ }}{A^{2}_{2}}\\\
+&\frac{2\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\cdot\left[h_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}-k_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\ }\right]\
}}{B^{2}_{2}}\\\
FF=&\frac{{\left[h_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)+k_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\ }\right]}^{2}}{A^{2}_{2}}\\\
+&\frac{{\left[h_{2_{TR}}\cdot{\sin\left({\varphi}_{2}-{\varphi}_{1}\right)\
}-k_{2_{TR}}\cdot{\cos\left({\varphi}_{2}-{\varphi}_{1}\right)\
}\right]}^{2}}{B^{2}_{2}}-1\end{split}$ (25)
For the area overlap algorithm presented in this paper, the points of
intersection between the two general ellipses are found by solving
simultaneously the two implicit polynomials denoted above as Ellipse Eq. 21
and Ellipse Eq. 22. Solving for $x$ in the first equation:
$\frac{x^{2}}{A^{2}_{1}}+\frac{y^{2}}{B^{2}_{1}}=1\ \ \ \Longrightarrow\ \ \
x=\pm\sqrt{A^{2}_{1}\cdot\left(1-\frac{y^{2}}{B^{2}_{1}}\right)}$ (26)
Substituting these expressions for $x$ into Ellipse Eq. 22 and then collecting
terms yields a quartic polynomial in $y$. It turns out that substituting
either the positive or the negative root gives the same quartic polynomial
coefficients, which are:
$cy\left[4\right]\cdot y^{4}+cy\left[3\right]\cdot y^{3}+cy\left[2\right]\cdot
y^{2}+cy\left[1\right]\cdot y+cy\left[0\right]=0$ (27)
where:
$\begin{split}\frac{cy\left[4\right]}{B_{1}}=&{A^{4}_{1}\cdot
AA}^{2}+B^{2}_{1}\cdot\left[A^{2}_{1}\cdot\left({BB}^{2}-2\cdot AA\cdot
CC\right)+B^{2}_{1}\cdot{CC}^{2}\right]\\\
\frac{cy\left[3\right]}{B_{1}}=&2\cdot B_{1}\cdot\left[B^{2}_{1}\cdot CC\cdot
EE+A^{2}_{1}\cdot\left(BB\cdot DD-AA\cdot EE\right)\right]\\\
\frac{cy\left[2\right]}{B_{1}}=&A^{2}_{1}\cdot\left\\{\left[B^{2}_{1}\cdot\left(2\cdot
AA\cdot CC-{BB}^{2}\right)+{DD}^{2}-2\cdot AA\cdot
FF\right]-2\cdot{A^{2}_{1}\cdot AA}^{2}\right\\}\\\
+&B^{2}_{1}\cdot\left(2\cdot CC\cdot FF+{EE}^{2}\right)\\\
\frac{cy\left[1\right]}{B_{1}}=&2\cdot
B_{1}\cdot\left[A^{2}_{1}\cdot\left(AA\cdot EE-BB\cdot DD\right)+EE\cdot
FF\right]\\\ \frac{cy\left[0\right]}{B_{1}}=&\left[A_{1}\cdot\left(A_{1}\cdot
AA-DD\right)+FF\right]\cdot\left[A_{1}\cdot\left(A_{1}\cdot
AA+DD\right)+FF\right]\end{split}$ (28)
In theory, the quartic polynomial will have real roots if and only if the two
curves intersect. If the ellipses do not intersect, then the quartic will have
only complex roots. Furthermore, any real roots of the quartic polynomial will
represent y-values of intersection points between the two ellipse curves. As
with the quadratic equation that arises in the ellipse-line overlap
calculation, the ellipse-ellipse overlap algorithm should handle all possible
cases for the types of quartic polynomial roots:
1. 1.
Four real roots (distinct or not); the ellipse curves intersect.
2. 2.
Two real roots (distinct or not) and one complex-conjugate pair; the ellipse
curves intersect.
3. 3.
No real roots (two complex-conjugate pairs); the ellipse curves do not
intersect.
For the method we present here, polynomial roots are found using Ferrari’s
quartic formula. A numerical implementation of Ferrari’s formula is given in
[3]. For complex roots are returned, and any roots whose imaginary part is
returned as zero is a real root.
When the polynomial coefficients are constructed as shown above, the general
case of two distinct ellipses typically results in a quartic polynomial, i.e.,
the coefficient $cy[4]$ is non-zero. However, certain cases lead to
polynomials of lesser degree. Fortunately, the solver in [3] is conveniently
modular, providing separate functions BIQUADROOTS, CUBICROOTS and QUADROOTS to
handle all the possible polynomial cases that arise when seeking points of
intersection for two ellipses.
If the polynomial solver returns no real roots to the polynomial, then the
ellipse curves do not intersect. It follows that the two ellipse areas are
either disjoint, or one ellipse area is fully contained inside the other; all
three possibilities are shown in Fig. 6. Each sub-case in Fig. 6 requires a
different overlap-area calculation, i.e. either the overlap area is zero (Case
0-3), or the overlap is the area of the first ellipse (Case 0-2), or the
overlap is the area of the second ellipse (Case 0-1). When the polynomial has
no real roots, geometry can be used to determine which specific sub-case of
Fig. 6 is represented. An efficient logic starts by determining the relative
size of the two ellipses, e.g., by comparing the product of semi-axis lengths
for each ellipse. The area of an ellipse is proportional to the product of its
two semi-axis lengths, so the relative size of two ellipses can be determined
by comparing the product of semi-axis lengths:
$(\pi\cdot A_{1}\cdot B_{1})\;\alpha\;(\pi\cdot A_{2}\cdot B_{2})\ \
\Longrightarrow\ \ \ (A_{1}\cdot B_{1})\;\alpha\;(A_{2}\cdot
B_{2}),\;\;\alpha\in\\{^{\prime}<^{\prime},^{\prime}>^{\prime}\\}$ (29)
Suppose the first ellipse is larger than the second ellipse, then
$A_{1}B_{1}>A_{2}B_{2}$. In this case, if the second ellipse center
($h_{2TR}$, $k_{2TR}$) is inside the first ellipse, then the second ellipse is
wholly contained within the first ellipse (Case 0-1); otherwise, the ellipses
are disjoint (Case 0-3). The logic relies on the fact that there are no
intersection points, which is indicated whenever there are no real solutions
to the quartic polynomial. To test whether the second ellipse center
($h_{2TR}$, $k_{2TR}$) is inside the first ellipse, evaluate the first ellipse
equation at the point $x=h_{2TR}$, and $y=k_{2TR}$; if the result is less than
one, then the point ($h_{2TR}$, $k_{2TR}$) is inside the first ellipse. The
complete logic for determining overlap area when $A_{1}B_{1}>A_{2}B_{2}$ is:
If the polynomial has no real roots, and $A_{1}B_{1}>A_{2}B_{2}$, and
$\frac{h_{2TR}^{2}}{A_{1}^{2}}+\frac{k_{2TR}^{2}}{B_{1}^{2}}<1$, then the
first ellipse wholly contains the second, otherwise the two ellipses are
disjoint.
Figure 6: When the quartic polynomial has no real roots, the ellipse curves do
not intersect. It follows that either one ellipse is fully contained within
the other, or the ellipse areas are completely disjoint, resulting in three
distinct cases for overlap area.
Alternatively, suppose that the second ellipse is larger than the first
ellipse, then $A_{1}B_{1}<A_{2}B_{2}$. If the first ellipse center (0, 0) is
inside the second ellipse, then the first ellipse is wholly contained within
the second ellipse (Case 0-2); otherwise the ellipses are disjoint (Case 0-3).
Again, the logic relies on the fact that there are no intersection points, To
test whether (0, 0) is inside the second ellipse, evaluate the second ellipse
equation at the origin; if the result is less than zero, then the origin is
inside the second ellipse. The complete logic for determining overlap area
when $A_{1}B_{1}<A_{2}B_{2}$ is:
If the polynomial has no real roots, and $A_{1}B_{1}<A_{2}B_{2}$, and $FF<0$,
then the second ellipse wholly contains the first, otherwise the two ellipses
are disjoint.
Suppose that the two ellipses are the same size, i.e.,
$A_{1}B_{1}=A_{2}B_{2}$. In this case, when no intersection points exist, the
ellipses must be disjoint (Case 0-3). It also turns out that the polynomial
solver of [3] will return no real solutions if the ellipses are identical.
This special case is also handled in the overlap area algorithm presented
below. Pseudo-code for a function NOINTPTS that determines overlap area for
the cases depicted in Fig. 6 is shown in Fig. 14.
If the polynomial solver returns either two or four real roots to the quartic
equation, then the ellipse curves intersect. For the algorithm presented here,
all of the various possibilities for the number and type of real roots are
addressed by creating a list of distinct real roots. The first step is to loop
through the entire array of complex roots returned by the polynomial solver,
and retrieve only real roots, i.e., only those roots whose imaginary component
is zero. The algorithm presented here then sorts the real roots, allowing for
an efficient check for multiple roots. As the sorted list of real roots is
traversed, any root that is ‘identical’ to the previous root can be skipped.
Each distinct real root of the polynomial represents a $y$-value where the two
ellipses intersect. Each $y$-value can represent either one or two potential
points of intersection. In the first case, suppose that the polynomial root is
$y=B_{1}$ (or $y=-B_{1}$), then the $y$-value produces a single intersection
point, which is at (0, $B_{1}$) (or (0, -$B_{1}$)). In the second case, if the
$y$-value is in the open interval ($-B_{1}$, $B_{1}$), then there are two
potential intersection points where the $y$-value is on the first ellipse:
$\displaystyle\left(A_{1}\cdot\sqrt{1-\frac{y^{2}}{B^{2}_{1}}},\ y\right){\rm
and}$ $\displaystyle\left({-A}_{1}\cdot\sqrt{1-\frac{y^{2}}{B^{2}_{1}}},\
y\right)$
Each potential intersection point ($x_{i}$, $y_{i}$) is evaluated in the
second ellipse equation:
$AA\cdot x^{2}_{i}+BB\cdot x_{i}\cdot y_{i}+CC\cdot y^{2}_{i}+DD\cdot
x_{i}+EE\cdot y_{i}+FF,\ \ i=1,2$
If the expression evaluates to zero, then the point ($x$, $y$) is on both
ellipses, i.e., it is an intersection point. By checking all points ($x$, $y$)
for each value of $y$ that is a root of the polynomial, a list of distinct
intersection points is generated. The number of distinct intersection points
must be either 0, 1, 2, 3 or 4. The case of zero intersection points is
described above, with all possible sub-cases illustrated in Fig. 6. If there
is only one distinct intersection point, then the two ellipses must be tangent
at that point. The three possibilities for a single tangent point are shown in
Fig. 7.
Figure 7: When only one intersection point exists, the ellipses must be
tangent at the intersection point. As with the case of zero intersection
points, either one ellipse is fully contained within the other, or the ellipse
areas are disjoint. The algorithm for finding overlap area in the case of zero
intersection points can also be used when there is a single intersection
point.
For the purpose of determining overlap area, the cases of 0 or 1 intersection
points can be handled in the same way. When two intersection points exist,
there are three possible sub-cases, shown in Fig. 8. It is possible that both
of the intersection points are tangents (Case 2-1 and Case 2-2). In both of
these sub-cases, one ellipse must be fully contained within the other. The
only other possibility for two intersection points is a partial overlap (Case
2-3).
Figure 8: When two intersection points exist, either both of the points are
tangents, or the ellipse curves cross at both points. For two tangent points,
one ellipse must be fully contained within the other. For two crossing points,
a partial overlap must exist
Each sub-case in Fig. 8 requires a different overlap-area calculation. When
two intersection points exist, either both of the points are tangents, or the
ellipse curves cross at both points. Specifically, when there are two
intersection points, if one point is a tangent, then both points must be
tangents. And, if one point is not a tangent, then neither point is a tangent.
So, it suffices to check one of the intersection points for tangency. Suppose
the ellipses are tangent at an intersection point; then, points that lie along
the first ellipse on either side of the intersection will lie in the same
region of the second ellipse (inside or outside). That is, if two points are
chosen that lie on the first ellipse, one on each side of the intersection,
then both points will either be inside the second ellipse, or they will both
be outside the second ellipse. If the ellipse curves cross at the intersection
point, then the two chosen points will be in different regions of the second
ellipse.
A logic based on testing points that are adjacent to a tangent point can be
implemented numerically to test whether an intersection point is a tangent or
a cross-point. Starting with an intersection point ($x$, $y$), calculate the
parametric angle on the first ellipse, by the rules in Table 1:
$\theta=\begin{cases}\arccos(x/A_{1})&y\geq 0\\\
2\pi-\arccos(x/A_{1})&y<0\end{cases}$ (30)
A small perturbation angle is then calculated. For the method presented here,
we seek to establish an angle that corresponds to a point on the first ellipse
that is a given distance, approximately $2EPS$, away from the intersection
point:
${EPS}_{{\rm Radian}}={\arcsin}\left(\frac{2\cdot{\rm
EPS}}{\sqrt{x^{2}+y^{2}}}\right)$ (31)
The angle ${EPS}_{Radian}$ is then used with the parametric form of the first
ellipse to determine two points adjacent to ($x$, $y$):
$\begin{split}x_{1}=&A_{1}\cdot\cos(\theta+EPS_{\rm Radian})\\\
y_{1}=&B_{1}\cdot\sin(\theta+EPS_{\rm Radian})\\\
x_{2}=&A_{1}\cdot\cos(\theta-EPS_{\rm Radian})\\\
y_{2}=&B_{1}\cdot\sin(\theta-EPS_{\rm Radian})\end{split}$ (32)
Each of the points is then evaluated in the second ellipse equation:
$\text{test}_{i}=AA\cdot x^{2}_{i}+BB\cdot x_{i}\cdot y_{i}+CC\cdot
y^{2}_{i}+DD\cdot x_{i}+EE\cdot y_{i}+FF,\ \ i=1,2$ (33)
If the value of $\text{test}_{i}$ is positive, then the point ($x_{i}$,
$y_{i}$) is outside the second ellipse. It follows that the product of the two
test-point evaluations $\text{test}_{1}\text{test}_{2}$ will be positive if
the intersection point is a tangent, since at a tangent point both test points
will be on the same side of the ellipse. The product of the test-point
evaluations will be negative if the two ellipse curves cross at the
intersection point, since the test points will be on opposite sides of the
ellipse. The function ISTANPT implements this logic to check whether an
intersection point is a tangent or a cross-point; pseudo-code is shown in Fig.
18.
When there are two intersection points, the ISTANPT function can be used to
differentiate the case 2-3 (Fig. 8) from the cases 2-1 and 2-2. Either of the
two known intersection points can be checked with ISTANPT. If the intersection
point is a tangent, then both of the intersection points must be tangents, so
the case is either 2-1 or 2-2, and one ellipse must be fully contained within
the other. For cases 2-1 and 2-2, the geometric logic used for 0 or 1
intersection points can also be used, i.e., the function NOINTPTS can be used
to determine the overlap area for these cases. If the two ellipse curves cross
at the tested intersection point, then the case must be 2-3, representing a
partial overlap between the two ellipse areas.
For case 2-3, with partial overlap between the two ellipses, the approach for
finding overlap area is based on using the two points (${x}_{1}$, $y_{1}$) and
($x_{2}$, $y_{2}$) with segment the algorithm (Table 2; Fig. 2) to determine
the partial overlap area contributed by each ellipse. The total overlap area
is the sum of the two segment areas. The two intersection points divide each
ellipse into two segment areas (see Fig. 5). Only one sector area from each
ellipse contributes to the overlap area. The segment algorithm returns the
area between the secant line and the portion of the ellipse from the first
point to the second point traversed in a counter-clockwise direction. For the
overlap area calculation, the two points must be passed to the segment
algorithm in the order that will return the correct segment area. The default
order is counter-clockwise from the first point ($x_{1}$, $y_{1}$) to the
second point ($x_{2}$, $y_{2}$). A check is made to determine whether this
order will return the desired segment area. First, the parametric angles
corresponding to ($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$) on the first
ellipse are determined, by the rules in Table 1:
$\theta_{1}=\begin{cases}\arccos(x_{1}/A_{1})&y_{1}\geq 0\\\
2\pi-\arccos(x_{1}/A_{1})&y_{1}<0\\\ \end{cases}$ (34)
$\theta_{2}=\begin{cases}\arccos(x_{2}/A_{1})&y_{2}\geq 0\\\
2\pi-\arccos(x_{2}/A_{1})&y_{2}<0\end{cases}$ (35)
Then, a point between ($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$) that is on the
first ellipse is found:
$\begin{split}x_{{\rm
mid}}=&A_{1}\cdot\cos\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\\\ y_{{\rm
mid}}=&B_{1}\cdot\sin\left(\frac{\theta_{1}+\theta_{2}}{2}\right)\end{split}$
(36)
The point ($x_{\rm mid}$, $y_{\rm mid}$) is on the first ellipse between
($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$) when travelling counter- clockwise
from ($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$). If ($x_{\rm mid}$, $y_{\rm
mid}$) is inside the second ellipse, then the desired segment of the first
ellipse contains the point ($x_{\rm mid}$, $y_{\rm mid}$). In this case, the
segment algorithm should integrate in the default order, counterclockwise from
($x_{1}$,$y_{1}$) to ($x_{2}$, $y_{2}$). Otherwise, the order of the points
should be reversed before calling the segment algorithm, causing it to
integrate counterclockwise from ($x_{2}$, $y_{2}$) to ($x_{1}$, $y_{1}$). The
area returned by the segment algorithm is the area contributed by the first
ellipse to the partial overlap.
The desired segment from the second ellipse is found in a manner to the first
ellipse segment. A slight difference in the approach is required because the
segment algorithm is implemented for ellipses that are centered at the origin
and oriented with the coordinate axes; but, in the general case the
intersection points ($x_{1}$, $y_{1}$) and ($x_{2}$, $y_{2}$) lie on the
second ellipse that is in a displaced and rotated location. The approach
presented here translates and rotates the second ellipse to the origin so that
the segment algorithm can be used. It suffices to translate then rotate the
two intersection points by amounts that put the second ellipse centered at the
origin and oriented with the coordinate axes:
$\begin{split}x_{1\rm{TR}}=&(x_{1}-h_{2{\rm
TR}})\cdot\cos(\varphi_{1}-\varphi_{2})+(y_{1}-k_{2{\rm
TR}})\cdot\sin(\varphi_{2}-\varphi_{1})\\\ y_{1{\rm TR}}=&(x_{1}-h_{2{\rm
TR}})\cdot\sin(\varphi_{1}-\varphi_{2})+(y_{1}-k_{2{\rm
TR}})\cdot\cos(\varphi_{1}-\varphi_{2})\\\ x_{2{\rm TR}}=&(x_{2}-h_{2{\rm
TR}})\cdot\cos(\varphi_{1}-\varphi_{2})+(y_{2}-k_{2{\rm
TR}})\cdot\sin(\varphi_{2}-\varphi_{1})\\\ y_{2{\rm TR}}=&(x_{2}-h_{2{\rm
TR}})\cdot\sin(\varphi_{1}-\varphi_{2})+(y_{2}-k_{2{\rm
TR}})\cdot\cos(\varphi_{1}-\varphi_{2})\end{split}$ (37)
The new points ($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) and ($x_{2{\rm TR}}$,
$y_{2{\rm TR}}$) lie on the second ellipse after a translation+rotation that
puts the second ellipse at the origin, oriented with the coordinate axes. The
new points can be used as inputs to the segment algorithm to determine the
overlap area contributed by the second ellipse. As with the first ellipse, the
order of the points must be determined so that the segment algorithm returns
the appropriate area. The default order is counter-clockwise from the first
point ($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) to the second point ($x_{2{\rm TR}}$,
$y_{2{\rm TR}}$). A check is made to determine whether this order will return
the desired segment area. First, the parametric angles corresponding to points
($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) and ($x_{2{\rm TR}}$, $y_{2{\rm TR}}$) on
the second ellipse are determined, by the rules in Table 1:
$\theta_{1}=\begin{cases}\arccos(x_{1{\rm TR}}/A_{2})&y_{1{\rm TR}}\geq 0\\\
2\pi-\arccos(x_{\rm 1{\rm TR}}/A_{2})&y_{1{\rm TR}}<0\end{cases}$ (38)
$\theta_{2}=\begin{cases}\arccos(x_{2{\rm TR}}/A_{2})&y_{2{\rm TR}}\geq 0\\\
2\pi-\arccos(x_{\rm 2{\rm TR}}/A_{2})&y_{2{\rm TR}}<0\end{cases}$ (39)
Then, a point on the second ellipse between ($x_{1{\rm TR}}$, $y_{1{\rm TR}}$)
and ($x_{2{\rm TR}}$, $y_{2{\rm TR}}$) is found:
$\displaystyle x_{{\rm mid}}=$ $\displaystyle
A_{2}\cdot\cos\left(\frac{\theta_{1}+\theta_{2}}{2}\right)$ $\displaystyle
y_{{\rm mid}}=$ $\displaystyle
B_{2}\cdot\sin\left(\frac{\theta_{1}+\theta_{2}}{2}\right)$
The point ($x_{\rm mid}$, $y_{\rm mid}$) is on the second ellipse between
($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) and ($x_{2{\rm TR}}$, $y_{2{\rm TR}}$) when
travelling counter- clockwise from ($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) and
($x_{2{\rm TR}}$, $y_{2{\rm TR}}$). The new point (xmid, ymid) lies on the
centered second ellipse. To determine the desired segment of the second
ellipse, the new point ($x_{\rm mid}$, $y_{\rm mid}$) must be rotated then
translated back to a corresponding position on the once-translated+rotated
second ellipse:
$\displaystyle x_{{\rm mid}{\rm RT}}=$ $\displaystyle x_{{\rm
mid}}\cdot\cos(\varphi_{2}-\varphi_{1})+y_{{\rm
mid}}\cdot\sin(\varphi_{1}-\varphi_{2})+h_{2{\rm TR}}$ $\displaystyle y_{{\rm
mid}{\rm RT}}=$ $\displaystyle x_{{\rm
mid}}\cdot\sin(\varphi_{2}-\varphi_{1})+y_{{\rm
mid}}\cdot\cos(\varphi_{1}-\varphi_{2})+k_{2{\rm TR}}$
If ($x_{{\rm mid}RT}$, $y_{{\rm mid}{\rm RT}}$) is inside the first ellipse,
then the desired segment of the second ellipse contains the point ($x_{\rm
mid}$, $y_{\rm mid}$). In this case, the segment algorithm should integrate in
the default order, counterclockwise from ($x_{1{\rm TR}}$, $y_{1{\rm TR}}$) to
($x_{2{\rm TR}}$, $y_{2{\rm TR}}$). Otherwise, the order of the points should
be reversed before calling the segment algorithm, causing it to integrate
counterclockwise from ($x_{2{\rm TR}}$, $y_{2{\rm TR}}$) to ($x_{1{\rm TR}}$,
$y_{1{\rm TR}}$). The area returned by the segment algorithm is the area
contributed by the second ellipse to the partial overlap. The sum of the
segment areas from the two ellipses is then equal to the ellipse overlap area.
The TWOINTPTS function calculates the overlap area for partial overlap with
two intersection points (Case 2-3); pseudo-code is shown in Fig. 15.
There are two possible sub-cases for three intersection points, shown in Fig.
9. One of the three points must be a tangent point, and the ellipses must
cross at the other two points. The cases are distinct only in the sense that
the tangent point occurs with ellipse 2 on the interior side of ellipse 1
(Case 3-1), or with ellipse 2 on the exterior side of ellipse 1 (Case 3-2).
The overlap area calculation is performed in the same manner for both cases,
by calling the TWOINTPTS function with the two cross-point intersections. The
ISTANPT function can be used to determine which point is a tangent; the
remaining two intersection points are then passed to TWOINTPTS. This logic is
implemented in the THREEINTPTS function, with pseudo-code in Fig. 16.
There is only one possible case for four intersection points, shown in Fig. 9.
The two ellipse curves must cross at all four of the intersection points,
resulting in a partial overlap. The overlap area consists of two segments from
each ellipse, and a central convex quadrilateral. For the approach presented
here, the four intersection points are sorted ascending in a counter-clockwise
order around the first ellipse. The ordered set of intersection points is
($x_{1}$, $y_{1}$), ($x_{2}$, $y_{2}$), ($x_{3}$, $y_{3}$) and ($x_{4}$,
$y_{4}$). The ordering allows a direct calculation of the quadrilateral area.
The standard formula uses the cross-product of the two diagonals:
$\begin{split}{\rm
area}=&\frac{1}{2}\left|\left(x_{3}-x_{1},y_{3}-y_{1}\right)\times\left(x_{4}-x_{2},y_{4}-y_{2}\right)\right|\\\
=&\frac{1}{2}\left|\left(x_{3}-x_{1}\right)\cdot\left(y_{4}-y_{2}\right)-\left(x_{4}-x_{2}\right)\cdot\left(x_{3}-x_{1}\right)\right|\end{split}$
(40)
The point ordering also simplifies the search for the appropriate segments of
each ellipse that contribute to the overlap area.
Suppose that the first two sorted points ($x_{1}$, $y_{1}$) and ($x_{2}$,
$y_{2}$) demarcate a segment of the first ellipse that contributes to the
overlap area, as shown in Fig. 9 and Fig. 10. It follows that the contributing
segments from the first ellipse are between ($x_{1}$, $y_{1}$) and ($x_{2}$,
$y_{2}$), and also between ($x_{3}$, $y_{3}$) and ($x_{4}$, $y_{4}$). In this
case, the contributing segments from the second ellipse are between ($x_{2}$,
$y_{2}$) and ($x_{3}$, $y_{3}$), and between ($x_{4}$, $y_{4}$) and ($x_{1}$,
$y_{1}$). To determine which segments contribute to the overlap area, it
suffices to test whether a point midway between ($x_{1}$, $y_{1}$) and
($x_{2}$, $y_{2}$) is inside or outside the second ellipse. The segment
algorithm is used for each of the four areas, and added to the quadrilateral
to obtain the total overlap area.
Figure 9: When three intersection points exist, one must be a tangent, and the
ellipse curves must cross at the other two points, always resulting in a
partial overlap. When four intersection points exist, the ellipse curves must
cross at all four points, again resulting in a partial overlap Figure 10:
Overlap Area with four intersection points (Case 4-1). The overlap area
consists of two segments from each ellipse, and a central convex
quadrilateral.
An implementation of the ELLIPSE_ELLIPSE_OVERLAP algorithm in c-code is shown
in Appendix6. The code compiles under Cygwin-1.7.7-1, and returns the
following values for the test cases presented above in Fig. 6, Fig. 7 , Fig. 8
and Fig. 9:
Listing 5: Return values for the test cases presented above in Fig. 6, Fig. 7
, Fig. 8 and Fig. 9.
⬇
1 cc call\\_ee.c ellipse\\_ellipse\\_overlap.c -o call\\_ee.exe
2
3 ./call\\_ee
4
5 Calling ellipse\\_ellipse\\_overlap.c
6
7
8
9 Case 0-1: area = 6.28318531, return_value = 111
10
11 ellipse 2 area by pi*a2*b2 = 6.28318531
12
13 Case 0-2: area = 6.28318531, return_value = 110
14
15 ellipse 1 area by pi*a1*b1 = 6.28318531
16
17 Case 0-3: area = 0.00000000, return_value = 103
18
19 Ellipses are disjoint, ovelap area = 0.0
20
21
22
23 Case 1-1: area = 6.28318531, return_value = 111
24
25 ellipse 2 area by pi*a2*b2 = 6.28318531
26
27 Case 1-2: area = 6.28318531, return_value = 110
28
29 ellipse 1 area by pi*a1b1 = 6.28318531
30
31 Case 1-3: area = -0.00000000, return_value = 107
32
33 Ellipses are disjoint, ovelap area = 0.0
34
35
36
37 Case 2-1: area = 10.60055478, return_value = 109
38
39 ellipse 2 area by pi*a2*b2 = 10.60287521
40
41 Case 2-2: area = 6.28318531, return_value = 110
42
43 ellipse 1 area by pi*a1b1 = 6.28318531
44
45 Case 2-3: area = 3.82254574, return_value = 107
46
47
48
49 Case 3-1: area = 7.55370392, return_value = 107
50
51 Case 3-2: area = 5.67996234, return_value = 107
52
53
54
55 Case 4-1: area = 16.93791852, return_value = 109
Listing 6: The ELLIPSE_ELLIPSE_OVERLAP algorithm is shown for calculating the
overlap area between two general ellipses. The algorithm calls several
supporting functions, including the polynomial solvers BIQUADROOTS, CUBICROOTS
and QUADROOTS, from CACM Algorithm 326 [2] . The remaining functions are
outlined in figures below.
⬇
1
2(Area,Code) $\leftarrow$ ELLIPSE_ELLIPSE_OVERLAP
(A1,B1,H1,K1,$\varphi$1,A2,B2,H2,K2,$\varphi$2)
3
4 do if (A1 = 0 or B1 = 0) OR (A2 = 0 or B2 = 0)
5
6 then return (-1, ERROR_ELLIPSE_PARAMETERS) :DATA CHECK
7
8 do if ($|\varphi 1|>2\pi$)
9
10 then $\varphi 1\leftarrow(\varphi 1\text{modulo}\,2\pi$)
11
12 do if ($|\varphi 2|>2\pi$)
13
14 then $\varphi 2\leftarrow(\varphi 2\text{modulo}\,2\pi$)
15
16 H2_TR $\leftarrow$ (H2 - H1)*cos ($\varphi 1$) + (K2 - K1)*sin ($\varphi
1$) :TRANS+ROT ELL2
17
18 K2_TR $\leftarrow$ (H1 - H2)*sin ($\varphi 1$) + (K2 - K1)*cos ($\varphi
1$)
19
20 $\varphi 2R$ $\leftarrow$ $\varphi 2$ – $\varphi 1$
21
22 do if ($|\varphi 2R|>2\pi$)
23
24 then $\varphi 2R\leftarrow(\varphi 2Rmodulo2\pi$)
25
26 AA $\leftarrow$ cos2($\varphi$2R)/A22 + sin2($\varphi$2R)/B22 :BUILD\,
IMPLICIT\, COEFFS ELL2TR
27
28 BB $\leftarrow$ 2*cos ($\varphi$2R)*sin ($\varphi$2R)/A22 – 2*cos
($\varphi$2R)*sin ($\varphi$2R)/B22
29
30 CC $\leftarrow$ sin2($\varphi$2R)/A22 + cos2($\varphi$2R)/B22
31
32 DD $\leftarrow$ -2*cos ($\varphi$2R)*(cos ($\varphi$2R)*H2_TR + sin
($\varphi$2R)*K2_TR)/A22
33
34 - 2*sin ($\varphi$2R)*(sin ($\varphi$2R)*H2_TR - cos
($\varphi$2R)*K2_TR)/B22
35
36EE $\leftarrow$ -2*sin ($\varphi$2R)* (cos ($\varphi$2R)*H2_TR + sin
($\varphi$2R)*K2_TR)/A22
37
38 + 2*cos ($\varphi$2R)* (sin ($\varphi$2R)*H2_TR - cos
($\varphi$2R)*K2_TR)/B22
39
40FF $\leftarrow$ (-cos ($\varphi$2R)*H2_TR - sin ($\varphi$2R)*K2_TR)2/A22
41
42 + (sin ($\varphi$2R)*H2_TR - cos ($\varphi$2R)*K2_TR)2/B22 - 1
43
44 :BUILD QUARTIC POLYNOMIAL COEFFICIENTS FROM THE TWO ELLIPSE EQNS
45
46 cy[4] $\leftarrow$ A14*AA2 + B12*(A12*(BB2 - 2*AA*CC)+ B12*CC2)
47
48 cy[3] $\leftarrow$ 2*B1*(B12*CC*EE + A12*(BB*DD - AA*EE))
49
50 cy[2] $\leftarrow$ A12*((B12*(2*AA*CC – BB2) + DD2 - 2*AA*FF)
51
52 -2*A12*AA2 + B12*(2*CC*FF + EE2)
53
54 cy[1] $\leftarrow$ 2*B1*(A12*(AA*EE – BB*DD) + EE*FF)
55
56 cy[0] $\leftarrow$ (A1*(A1*AA –DD) + FF)*(A1*(A1*AA + DD) + FF)
57
58 py[0] $\leftarrow$ 1
59
60 do if ($|cy[4]|>$ 0) :SOLVE QUARTIC EQ
61
62 then for i $\leftarrow$ 0 to 3 by 1
63
64 py[4-i] $\leftarrow$ cy[i]/cy[4]
65
66 r[][]$\leftarrow$ BIQUADROOTS (py[])
67
68 nroots $\leftarrow$ 4
69
70 else if ($|cy[3]|>0$) :SOLVE CUBIC EQ
71
72 then for i $\leftarrow$ 0 to 2 by 1
73
74 py[3-i] $\leftarrow$ cy[i]/cy[3]
75
76 r[][] $\leftarrow$ CUBICROOTS (py[])
77
78 nroots $\leftarrow$ 3
79
80 else if ($|cy[2]|>0$) :SOLVE QUADRATIC EQ
81
82 then for i $\leftarrow$ 0 to 1 by 1
83
84 py[2-i] $\leftarrow$ cy[i]/cy[2]
85
86 r[][] $\leftarrow$ QUADROOTS (py[])
87
88 nroots $\leftarrow$ 2
89
90 else if ($|cy[1]|>0$) :SOLVE LINEAR EQ
91
92 then r[1][1] $\leftarrow$ (-cy[0]/cy[1])
93
94 r[2][1] $\leftarrow$ 0
95
96 nroots $\leftarrow$ 1
97
98 else :COMPLETELY DEGENERATE EQ
99
100 nroots $\leftarrow$ 0
101
102 nychk $\leftarrow$ 0 :IDENTIFY REAL ROOTS
103
104 for i $\leftarrow$ 1 to nroots by 1
105
106 do if ($|r[2][i]|<\text{EPS}$)
107
108 then nychk $\leftarrow$ nychk + 1
109
110 ychk[nychk] $\leftarrow$ r[1][i]*B1
111
112 for j $\leftarrow$ 2 to nychk by 1 :SORT REAL ROOTS
113
114 tmp0 $\leftarrow$ ychk[j]
115
116 for k $\leftarrow$ (j – 1) to 1 by -1
117
118 do if (ychk[k] = tmp0)
119
120 then break
121
122 else ychk[k+1] $\leftarrow$ ychk[k]
123
124 ychk[k+1] $\leftarrow$ tmp0
125
126 nintpts $\leftarrow$ 0 :FIND INTERSECTION POINTS
127
128 for i $\leftarrow$ 1 to nychk by 1
129
130 do if (($i>1$) and ($|ychk[i]-ychk[i-1]|<$ EPS/2))
131
132 then continue
133
134 do if ($|ychk[i]|>$ -B1)
135
136 then x1 $\leftarrow$ 0
137
138 else x1$\leftarrow$? A1*sqrt (1.0 - ychk[i]2/B12)
139
140 x2 $\leftarrow$ -x1
141
142 do if ($|ellipse2tr(x1,ychk[i],AA,BB,CC,DD,EE,FF)|<$ EPS/2)
143
144 then nintpts $\leftarrow$ nintpts + 1
145
146 do if (nintpts $>$ 4)
147
148 then return (-1, ERROR_INTERSECTION_PTS)
149
150 xint[nintpts] $\leftarrow$ x1
151
152 yint[nintpts] $\leftarrow$ ychk[i]
153
154 do if (($|ellipse2tr(x2,ychk[i],AA,BB,CC,DD,EE,FF)|<$ EPS/2)
155
156 and ($|x2-x1|>$ EPS/2))
157
158 then nintpts $\leftarrow$ nintpts + 1
159
160 do if (nintpts $>$ 4)
161
162 then return (-1, ERROR_INTERSECTION_PTS)
163
164 xint[nintpts] $\leftarrow$ x1
165
166 yint[nintpts] $\leftarrow$ ychk[i]
167
168 switch (nintpts) :HANDLE ALL CASES FOR \\# OF INTERSECTION PTS
169
170 case 0:
171
172 case 1:
173
174 (OverlapArea,Code) $\leftarrow$ NOINTPTS (A1,B1,A2,B2,H1,K1,H2,K2,AA,
175
176 BB,CC,DD,EE,FF)
177
178 return (OverlapArea,Code)
179
180 case 2:
181
182 Code $\leftarrow$ istanpt (xint[1],yint[1],A1,B1,AA,BB,CC,DD,EE,FF)
183
184 do if (Code == TANGENT_POINT)
185
186 then (OverlapArea,Code) $\leftarrow$ NOINTPTS (A1,B1,A2,B2,H1,K1,
187
188 H2,K2,AA,BB,CC,DD,EE,FF)
189
190 else (OverlapArea,Code) $\leftarrow$ TWOINTPTS (xint[],yint[],A1,
191
192 PHI_1,A2,B2,H2_TR,K2_TR,PHI_2,AA,BB,CC,DD,EE,FF)
193
194 return (OverlapArea,Code)
195
196 case 3:
197
198 (OverlapArea,Code) $\leftarrow$ THREEINTPTS (xint,yint,A1,B1,PHI_1,
199
200 A2,B2,H2_TR,K2_TR,PHI_2,AA,BB,CC,DD,EE,FF)
201
202 return (OverlapArea, Code)
203
204 case 4:
205
206 (OverlapArea,Code) $\leftarrow$ FOURINTPTS (xint,yint,A1,B1,PHI_1,
207
208 A2, B2,H2_TR,K2_TR,PHI_2,AA,BB,CC,DD,EE,FF)
209
210 return (OverlapArea,Code)
Listing 7: The NOINTPTS subroutine. If there are either 0 or 1 intersection
points, this function determines whether one ellipse is contained within the
other (Cases 0-1, 0-2, 1-1 and 1-2), or if the ellipses are disjoint (Cases
0-3 and 1-3). The function returns the appropriate overlap area, and a code
describing which case was encountered.
⬇
1
2 (OverlapArea,Code) $\leftarrow$ NOINTPTS (A1,B1,A2,B2,H1,K1,H2_TR,K2_TR,AA,
3
4 BB,CC,DD,EE,FF)
5
6 relsize $\leftarrow$ A1*B1 - A2*B2
7
8 do if (relsize $>$ 0)
9
10 then do if (((H2_TR*H2_TR)/(A1*A1)+(K2_TR*K2_TR)/(B1*B1)) $<$ 1.0)
11
12 then return ($\pi$*A2*B2,ELLIPSE2_INSIDE_ELLIPSE1)
13
14 else return (0, DISJOINT_ELLIPSES)
15
16 else do if (relsize $<$ 0)
17
18 then do if (FF $<$ 0)
19
20 then return ($\pi$*A1*B1,ELLIPSE1_INSIDE_ELLIPSE2)
21
22 else return (0, DISJOINT_ELLIPSES)
23
24 else do if ((H1 = H2_TR) AND (K1 = K2_TR))
25
26 then return ($\pi$*A1*B1, ELLIPSES_ARE_IDENTICAL)
27
28 else return (-1, ERROR_CALCULATIONS
Listing 8: The TWOINTPTS subroutine. If there are 2 intersection points where
the ellipse curves cross (Case 2-3), this function uses the ellipse sector
algorithm to determine the contribution of each ellipse to the total overlap
area. The function returns the appropriate overlap area, and a code indicating
two intersection points.
⬇
1 (OverlapArea,Code) $\leftarrow$ TWOINTPTS
(xint[],yint[],A1,B1,$\varphi$1,A2,B2,H2_TR,
2
3 K2_TR,$\varphi$2,AA,BB,CC,DD,EE,FF)
4
5do if ($|x[1]|>$ A1) :AVOID INVERSE TRIG ERRORS
6
7 then do if (x[1] $<$ 0)
8
9 then x[1] $\leftarrow$ -A1
10
11 else x[1] $\leftarrow$ A1
12
13do if (y[1] $<$ 0) :FIND PARAMETRIC ANGLE FOR (x[1], y[1])
14
15 then $\theta$1 $\leftarrow$ 2$\pi$ – arccos (x[1]/A1)
16
17 else $\theta$1 $\leftarrow$ arccos (x[1]/A1)
18
19do if ($|x[2]|>$ A1) :AVOID INVERSE TRIG ERRORS
20
21 then do if (x[2] $<$ 0)
22
23 then x[2] $\leftarrow$ -A1
24
25 else x[2] $\leftarrow$ A1
26
27do if (y[2] $<$ 0) :FIND PARAMETRIC ANGLE FOR (x[2], y[2])
28
29 then $\theta$2 $\leftarrow$ 2$\pi$ – arccos (x[2]/A1)
30
31 else $\theta$2 $\leftarrow$ arccos (x[2]/A1)
32
33do if ($\theta$1 $>$ $\theta$2) :GO CCW FROM $\theta$1 TO\, $\theta$2
34
35 then tmp $\leftarrow$ $\theta$1, $\theta$1 $\leftarrow$ $\theta$2,
$\theta$2 $\leftarrow$ tmp
36
37xmid $\leftarrow$ A1*cos (($\theta$1 + $\theta$2)/2)
38
39ymid $\leftarrow$ B1*sin (($\theta$1 + $\theta$2)/2)
40
41do if (AA*xmid2+BB*xmid*ymid+CC*ymid2+DD*xmid+EE*ymid+FF $>$ 0)
42
43 then tmp $\leftarrow$ $\theta$1, $\theta$1$\leftarrow$ $\theta$2, $\theta$2
$\leftarrow$ tmp
44
45do if ($\theta$1 $>$ $\theta$2) :SEGMENT ALGORITHM FOR ELLIPSE 1
46
47 then $\theta$1 ? $\theta$1 - 2$\pi$
48
49do if (($\theta$2 - $\theta$1) $>$ $\pi$)
50
51 then trsign $\leftarrow$ 1
52
53 else trsign $\leftarrow$ -1
54
55area1 $\leftarrow$ (A1*B1*($\theta$2 - $\theta$1) + trsign*\textbar
x[1]*y[2] - x[2]*y[1])\textbar /2
56
57x1\\_tr $\leftarrow$ (x[1] - H2\\_TR)*cos($\varphi$1 – $\varphi$2) + (y[1] -
K2\\_TR)*sin($\varphi$2 – $\varphi$1)
58
59y1\\_tr $\leftarrow$ (x[1] - H2\\_TR)*sin($\varphi$1 – $\varphi$2) + (y[1] -
K2\\_TR)*cos($\varphi$1 – $\varphi$2)
60
61x2\\_tr $\leftarrow$ (x[2] - H2\\_TR)*cos($\varphi$1 – $\varphi$2) + (y[2] -
K2\\_TR)*sin($\varphi$2 – $\varphi$1)
62
63y2\\_tr ? (x[2] - H2\\_TR)*sin($\varphi$1 – $\varphi$2) + (y[2] -
K2\\_TR)*cos($\varphi$1 – $\varphi$2)
64
65do if ($|x1\\_tr|>$ A2) :AVOID INVERSE TRIG ERRORS
66
67 then do if (x1_tr $<$ 0)
68
69 then x1_tr $\leftarrow$ -A2
70
71 else x1_tr $\leftarrow$ A2
72
73do if (y1_tr $<$ 0) :FIND PARAMETRIC ANGLE FOR (x1_tr, y1_tr)
74
75 then $\theta$1 $\leftarrow$ 2$\pi$ – arccos (x1_tr/A2)
76
77 else $\theta$1 $\leftarrow$ arccos (x1_tr/A2)
78
79do if ($|x2_{t}r|>$ A2) :AVOID INVERSE TRIG ERRORS
80
81 then do if (x2_tr $<$ 0)
82
83 then x2_tr $\leftarrow$ -A2
84
85 else x2_tr $\leftarrow$ A2
86
87do if (y2_tr $<$ 0) :FIND PARAMETRIC ANGLE FOR (x2_tr, y2_tr)
88
89 then $\theta$2 $\leftarrow$ 2$\pi$ – arccos (x2_tr/A2)
90
91 else $\theta$2 $\leftarrow$ arccos (x2_tr/A2)
92
93do if ($\theta$1 $>$ $\theta$2) :GO CCW FROM $\theta$1 TO\, $\theta$2
94
95 then tmp $\leftarrow$ $\theta$1, $\theta$1 $\leftarrow$ $\theta$2,
$\theta$2 $\leftarrow$ tmp
96
97xmid $\leftarrow$ A2*cos (($\theta$1 + $\theta$2)/2)
98
99ymid $\leftarrow$ B2*sin (($\theta$1 + $\theta$2)/2)
100
101xmid_rt = xmid*cos($\varphi$2 – $\varphi$1) + ymid*sin($\varphi$1 –
$\varphi$2) + H2_TR
102
103ymid_rt = xmid*sin($\varphi$2 – $\varphi$1) + ymid*cos($\varphi$2 –
$\varphi$1) + K2_TR
104
105do if (xmid_rt2/A12 + ymid_rt2/B12 $>$ 1)
106
107 then tmp $\leftarrow$ $\theta$1, $\theta$1 $\leftarrow$ $\theta$2,
$\theta$2 $\leftarrow$ tmp
108
109do if ($\theta$1 $>$ $\theta$2) :SEGMENT ALGORITHM FOR ELLIPSE 2
110
111 then $\theta$1 $\leftarrow$ $\theta$1 - 2$\pi$
112
113do if (($\theta$2 - $\theta$1) $>$ $\pi$)
114
115 then trsign $\leftarrow$ 1
116
117 else trsign $\leftarrow$ -1
118
119area2 $\leftarrow$ (A2*B2*($\theta$2 - $\theta$1)
120
121+ trsign*$|x1_{t}r*y2_{t}r-x2\\_tr*y1_{t}r)|$ /2
122
123 return (area1 + area2, TWO_INTERSECTION_POINTS)
Listing 9: The THREEINTPTS subroutine. When there are three intersection
points, one of the points must be a tangent point, and the ellipses must cross
at the other two points. For the purpose of determining overlap area, the
TWOINTPTS function can be used with the two cross-point intersections. The
ISTANPT function can be used to determine which point is a tangent; the
remaining two intersection points are then passed to TWOINTPTS. The function
returns the appropriate overlap area, and a code indicating three intersection
points.
⬇
1 OverlapArea,Code) $\leftarrow$ THREEINTPTS
(xint[],yint[],A1,B1,$\varphi$1,A2,B2,H2_TR,
2
3 K2_TR,$\varphi$2,AA,BB,CC,DD,EE,FF)
4tanpts $\leftarrow$ 0
5
6for i $\leftarrow$ 1 to nychk by 1
7
8 code $\leftarrow$ ISTANPT ISTANPT (x[i],y[i],A1,B1,AA,BB,CC,DD,EE,FF)
9
10 do if (code = TANGENT_POINT)
11
12 then tanpts $\leftarrow$ tanpts + 1
13
14 tanindex $\leftarrow$ i
15
16do if NOT (tanpts = 1)
17
18 then return (-1, ERROR_INTERSECTION_POINTS)
19
20switch (tanindex) :STORE THE INTERSECTION POINTS
21
22 case 1: :TANGENT POINT IS IN (x[1], y[1])
23
24 xint[1] $\leftarrow$ xint[3]
25
26 yint[1] $\leftarrow$ yint[3]
27
28 case 2: :TANGENT POINT IS IN (x[2], y[2])
29
30 xint[2] $\leftarrow$ xint[3]
31
32 yint[2] $\leftarrow$ yint[3]
33
34(OverlapArea,code) $\leftarrow$ TWOINTPTS
(xint[],yint[],A1,B1,$\varphi$1,A2,B2,H2_TR,
35
36 K2_TR,$\varphi$2,AA,BB,CC,DD,EE,FF)
37
38return (OverlapArea,THREE_INTERSECTION_POINTS)
Listing 10: The FOURINTPTS subroutine. When there are four intersection
points, the ellipse curves must cross at all four points. A partial overlap
area exists, consisting of two segments from each ellipse and a central
quadrilateral. The function returns the appropriate overlap area, and a code
indicating four intersection points.
⬇
1verlapArea,Code) $\leftarrow$ FOURINTPTS
(xint[],yint[],A1,B1,$\varphi$1,A2,B2,H2_TR,
2
3 K2_TR,$\varphi$2,AA,BB,CC,DD,EE,FF)
4
5 for i $\leftarrow$ 1 to 4 by 1 :AVOID INVERSE TRIG ERRORS
6
7 do if ($|xint[i]|>$ A1)
8
9 then do if (xint[i] $<$ 0)
10
11 then xint[i] $\leftarrow$ -A1
12
13 else xint[i] $\leftarrow$ A1
14
15 do if (yint[i] $<$ 0) :FIND PARAMETRIC ANGLES
16
17 then $\theta$[i] $\leftarrow$ 2$\pi$ – arccos (xint[i]/A1)
18
19 else $\theta$[i] $\leftarrow$ arccos (xint[i]/A1)
20
21 for j $\leftarrow$ 2 to 4 by 1 :PUT POINTS IN CCW ORDER
22
23 tmp0 $\leftarrow$ $\theta$[j]
24
25 tmp1 $\leftarrow$ xint[j]
26
27 tmp2 $\leftarrow$ yint[j]
28
29 for k $\leftarrow$ (j-1) to 1 by -1 :INSERTION SORT BY ANGLE
30
31 do if ($\theta$[k] $<$= tmp0)
32
33 then break
34
35 else $\theta$[k+1] $\leftarrow$ $\theta$[k]
36
37 xint[k+1] $\leftarrow$ xint[k]
38
39 yint[k+1] $\leftarrow$ yint[k]
40
41 area1 $\leftarrow$ (|(xint[3] – xint[1])*(yint[4] – yint[2]) –
42
43xint[4] - xint[2])*(yint[3] – yint[1])| /2) :QUAD AREA
44
45 for i $\leftarrow$ 1 to 4 by 1 :TRANSLATE+ROTATE ELLIPSE 2
46
47 xint_tr[i] $\leftarrow$ (xint[i] – H2_TR)*cos ($\varphi$1 – $\varphi$2)
48
49 + (yint[i] – K2_TR)*sin ($\varphi$2 – $\varphi$1)
50
51 yint_tr[i] $\leftarrow$ (xint[i] – H2_TR)*sin ($\varphi$1 – $\varphi$2)
52
53 + (yint[i] – K2_TR)*cos ($\varphi$1 – $\varphi$2)
54
55 do if ($|xint_{t}r[i]|>$ A2) :AVOID INVERSE TRIG ERRORS
56
57 then do if (xint_tr[i] $<$ 0)
58
59 then xint_tr[i] $\leftarrow$ -A2
60
61 else xint_tr[i] $\leftarrow$ A2
62
63 do if (yint_tr[i] $<$ 0) :FIND PARAM ANGLES FOR (xint_tr, yint_tr)
64
65 then $\theta$_tr[i] $\leftarrow$ 2$\pi$ – arccos (xint_tr[i]/A2)
66
67 else $\theta$_tr[i] $\leftarrow$ arccos (xint_tr[i]/A2)
68
69 xmid $\leftarrow$ A1*cos (($\theta$1 + $\theta$2)/2)
70
71 ymid $\leftarrow$ B1*sin (($\theta$1 + $\theta$2)/2)
72
73 do if (AA*xmid2+BB*xmid*ymid+CC*ymid2+DD*xmid+EE*ymid+FF $<$ 0)
74
75 then area2 = (A1*B1*($\theta$[2] - $\theta$[1])
76
77 - |(xint[1]*yint[2] - xint[2]*yint[1])|)/2
78
79 area3 = (A1*B1*($\theta$[4] - $\theta$[3])
80
81 - |(xint[3]*yint[4] - xint[4]*yint[3])|)/2
82
83 area4 = (A2*B2*($\theta$_tr[3] - $\theta$_tr[2])
84
85 - |(xint\\_tr[2]*yint_tr[3] - xint_tr[3]*yint_tr[2])| )/2
86
87 area5 = (A2*B2*($\theta$_tr[1] - $\theta$_tr[4] - twopi))
88
89 - |(xint_tr[4]*yint_tr[1] - xint_tr[1]*yint_tr[4])| /2)
90
91 else area2 = (A1*B1*($\theta$[3] - $\theta$[2])
92
93 - |(xint[2]*yint[3] - xint[3]*yint[2])|)/2
94
95 area3 = (A1*B1*($\theta$[1] - ($\theta$[4] - twopi))
96
97 - |(xint[4]*yint[1] - xint[1]*yint[4])|)/2
98
99 area4 = (A2*B2*($\theta$_tr[2] - $\theta$_tr[1])
100
101 - |(xint_tr[1]*yint_tr[2] - xint_tr[2]*yint_tr[1])|)/2
102
103 area5 = (A2*B2*($\theta$_tr[4] - $\theta$_tr[3])
104
105 - |(xint_tr[3]*yint_tr[4] - xint_tr[4]*yint_tr[3])|)/2
106
107 return (area1+area2+area3+area4+area5, FOUR_INTERSECTION_POINTS)
Listing 11: The ISTANPT subroutine. Given an intersection point (x, y) that
satisfies both Ellipse Eq.21 and Ellipse Eq. 22, the function determines
whether the two ellipse curves are tangent at (x, y), or if the ellipse curves
cross at (x, y).
⬇
1 Code $\leftarrow$ ISTANPT (x,y,A1,B1,AA,BB,CC,DD,EE,FF)
2
3 do if ($|x|>$ A1) :AVOID INVERSE TRIG ERRORS
4
5 then do if x $<$ 0
6
7 then x $\leftarrow$ -A1
8
9 else x $\leftarrow$ A1
10
11 do if (y $<$ 0) :FIND PARAMETRIC ANGLE FOR (x, y)
12
13 then $\theta$ $\leftarrow$ 2$\pi$ – arccos (x/A1)
14
15 else $\theta$ $\leftarrow$ arccos (x/A1)
16
17 branch $\leftarrow$ v(x2 + y2) :DETERMINE PERTURBATION ANGLE
18
19 do if (branch $<$ 100*EPS)
20
21 then eps_radian $\leftarrow$ 2*EPS
22
23 else eps_radian $\leftarrow$ arcsin (2*EPS/branch)
24
25 x1 $\leftarrow$ A1*cos ($\theta$ + eps_radian) :CREATE TEST POINTS ON EACH
SIDE
26
27 y1 $\leftarrow$ B1*cos ($\theta$ + eps_radian) :OF THE INPUT POINT (x, y)
28
29 x2 $\leftarrow$ A1*cos ($\theta$ - eps_radian)
30
31 y2 $\leftarrow$ B1*cos ($\theta$ - eps_radian)
32
33 test1 $\leftarrow$ AA*x12+BB*x1*y1+CC*y12+DD*x1+EE*y1+FF
34
35 test2 $\leftarrow$ AA*x22+BB*x2*y2+CC*y22+DD*x2+EE*y2+FF
36
37 do if (test1*test2 $>$ 0)
38
39 then return TANGENT_POINT
40
41 else return INTERSECTION_POINT
## 4 APPENDIX A
Listing 12: C-SOURCE CODE FOR ELLIPSE_SEGMENT
⬇
1
2
3
/****************************************************************************
4
5 *
6
7 * Function: double ellipse_segment
8
9 *
10
11 * Purpose: Given the parameters of an ellipse and two points that lie on
12
13 * the ellipse, this function calculates the ellipse segment area
14
15 * between the secant line and the ellipse. Points are input as
16
17 * (X1, Y1) and (X2, Y2), and the segment area is defined to be
18
19 * between the secant line and the ellipse from the first point
20
21 * (X1, Y1) to the second point (X2, Y2) in the counter-clockwise
22
23 * direction.
24
25 *
26
27 * Reference: Hughes and Chraibi (2011), Calculating Ellipse Overlap Areas
28
29 *
30
31 * Dependencies: math.h for calls to trig and absolute value functions
32
33 * program_constants.h error message codes and constants
34
35 *
36
37 * Inputs: 1. double A ellipse semi-axis length in x-direction
38
39 * 2. double B ellipse semi-axis length in y-direction
40
41 * 3. double X1 x-value of the first point on the ellipse
42
43 * 4. double Y1 y-value of the first point on the ellipse
44
45 * 5. double X2 x-value of the second point on the ellipse
46
47 * 6. double Y2 y-value of the second point on the ellipse
48
49 *
50
51 * Outputs: 1. int *MessageCode stores diagnostic information
52
53 * integer codes in program_constants.h
54
55 *
56
57 * Return: The value of the ellipse segment area:
58
59 * -1.0 is returned in case of an error with input data
60
61 *
62
63
****************************************************************************/
64
65
66
67
//===========================================================================
68
69 //== INCLUDE ANSI C SYSTEM AND USER-DEFINED HEADER FILES
====================
70
71
//===========================================================================
72
73 #include ”program_constants.h”
74
75
76
77 double ellipse_segment (double A, double B,double X1, double Y1, double X2,
78
79 double Y2, int *MessageCode)
80
81 {
82
83 double theta1; //– parametric angle of the first point
84
85 double theta2; //– parametric angle of the second point
86
87 double trsign; //– sign of the triangle area
88
89 double pi = 2.0 * asin \eqref{GrindEQ__1_0_}; //– a maximum-precision value
of pi
90
91 double twopi = 2.0 * pi; //– a maximum-precision value of 2*pi
92
93
94
95 //– Check the data first
96
97 //– Each of the ellipse axis lengths must be positive
98
99 if (!(A $>$ 0.0) \textbar \textbar !(B $>$ 0.0))
100
101 {
102
103 (*MessageCode) = ERROR_ELLIPSE_PARAMETERS;
104
105 return -1.0;
106
107 }
108
109
110
111 //– Points must be on the ellipse, within EPS, which is defined
112
113 //– in the header file program_constants.h
114
115 if ( (fabs ((X1*X1)/(A*A) + (Y1*Y1)/(B*B) - 1.0) $>$ EPS) textbar textbar
116
117 (fabs ((X2*X2)/(A*A) + (Y2*Y2)/(B*B) - 1.0) $>$ EPS) )
118
119 {
120
121 (*MessageCode) = ERROR_POINTS_NOT_ON_ELLIPSE;
122
123 return -1.0;
124
125 }
126
127
128
129 //– Avoid inverse trig calculation errors: there could be an error
130
131 //– if \textbar X1/A\textbar $>$ 1.0 or \textbar X2/A\textbar $>$ 1.0 when
calling acos()
132
133 //– If execution arrives here, then the point is on the ellipse
134
135 //– within EPS. Try to adjust the value of X1 or X2 before giving
136
137 //– up on the area calculation
138
139 if (fabs (X1)/A $>$ 1.0)
140
141 {
142
143 //– if execution arrives here, already know that \textbar X1\textbar $>$ A
144
145 if ((fabs (X1) - A) $>$ EPS)
146
147 {
148
149 //– if X1 is not close to A or -A, then give up
150
151 (*MessageCode) = ERROR\_INVERSE\_TRIG;
152
153 return -1.0;
154
155 }
156
157 else
158
159 {
160
161 //– nudge X1 back to A or -A, so acos() will work
162
163 X1 = (X1 $<$ 0) ? -A : A;
164
165 }
166
167 }
168
169
170
171 if (fabs (X2)/A $>$ 1.0)
172
173 {
174
175 //– if execution arrives here, already know that \textbar X2\textbar $>$ A
176
177 if ((fabs (X2) - A) $>$ EPS)
178
179 {
180
181 //– if X2 is not close to A or -A, then give up
182
183 (*MessageCode) = ERROR_INVERSE_TRIG;
184
185 return -1.0;
186
187 }
188
189 else
190
191 {
192
193 //– nudge X2 back to A or -A, so acos() will work
194
195 X2 = (X2 $<$ 0) ? -A : A;
196
197 }
198
199 }
200
201
202
203 //– Calculate the parametric angles on the ellipse
204
205 //– The parametric angles depend on the quadrant where each point
206
207 //– is located. See Table 1 in the reference.
208
209 if (Y1 $<$ 0.0) //– Quadrant III or IV
210
211 theta1 = twopi - acos (X1 / A);
212
213 else //– Quadrant I or II
214
215 theta1 = acos (X1 / A);
216
217
218
219 if (Y2 $<$ 0.0) //– Quadrant III or IV
220
221 theta2 = twopi - acos (X2 / A);
222
223 else //– Quadrant I or II
224
225 theta2 = acos (X2 / A);
226
227
228
229 //– need to start the algorithm with theta1 $<$ theta2
230
231 if (theta1 $>$ theta2)
232
233 theta1 -= twopi;
234
235
236
237 //– if the integration angle is less than pi, subtract the triangle
238
239 //– area from the sector, otherwise add the triangle area.
240
241 if ((theta2 - theta1) $>$ pi)
242
243 trsign = 1.0;
244
245 else
246
247 trsign = -1.0;
248
249
250
251 //– The ellipse segment is the area between the line and the ellipse,
252
253 //– calculated by finding the area of the radial sector minus the area
254
255 //– of the triangle created by the center of the ellipse and the two
256
257 //– points. First term is for the ellipse sector; second term is for
258
259 //– the triangle between the points and the origin. Area calculation
260
261 //– is described in the reference.
262
263 (*MessageCode) = NORMAL_TERMINATION;
264
265 return ( 0.5*(A*B*(theta2 - theta1) + trsign*fabs (X1*Y2 - X2*Y1)) );
266
267
268
269 }
## 5 APPENDIX B
Listing 13: C-SOURCE CODE FOR ELLIPSE_LINE_OVERLAP
⬇
1
2
3
/****************************************************************************
4
5 *
6
7 * Function: double ellipse_line_overlap
8
9 *
10
11 * Purpose: Given the parameters of an ellipse and two points on a line,
12
13 * this function calculates the area between the two curves. If
14
15 * the line does not cross the ellipse, or if the line is tangent
16
17 * to the ellipse, then this function returns an area of 0.0
18
19 * If the line intersects the ellipse at two points, then the
20
21 * function returns the area between the secant line and the
22
23 * ellipse. The line is considered to have a direction from
24
25 * the first given point (X1,Y1) to the second given point (X2,Y2)
26
27 * This function determines where the line crosses the ellipse
28
29 * first, and where it crosses second. The area returned is
30
31 * between the secant line and the ellipse traversed counter-
32
33 * clockwise from the first intersection point to the second
34
35 * intersection point.
36
37 *
38
39 * Reference: Hughes and Chraibi (2011), Calculating Ellipse Overlap Areas
40
41 *
42
43 * Dependencies: math.h for calls to trig and absolute value functions
44
45 * program_constants.h error message codes and constants
46
47 * ellipse_segment.c core algorithm for ellipse segment area
48
49 *
50
51 * Inputs: 1. double PHI CCW rotation angle of the ellipse, radians
52
53 * 2. double A ellipse semi-axis length in x-direction
54
55 * 3. double B ellipse semi-axis length in y-direction
56
57 * 4. double H horizontal offset of ellipse center
58
59 * 5. double K vertical offset of ellipse center
60
61 * 6. double X1 x-value of the first point on the line
62
63 * 7. double Y1 y-value of the first point on the line
64
65 * 8. double X2 x-value of the second point on the line
66
67 * 9. double Y2 y-value of the second point on the line
68
69 *
70
71 * Outputs: 1. int *MessageCode returns diagnostic information
72
73 * integer codes in program_constants.h
74
75 *
76
77 * Return: The value of the ellipse segment area:
78
79 * -1.0 is returned in case of an error with the data or
80
81 * calculation
82
83 * 0.0 is returned if the line does not cross the ellipse, or if
84
85 * the line is tangent to the ellipse
86
87 *
88
89
****************************************************************************/
90
91
92
93
//===========================================================================
94
95 //== DEFINE PROGRAM CONSTANTS
===============================================
96
97
//===========================================================================
98
99 #include ”program_constants.h” //– error message codes and constants
100
101
102
103
//===========================================================================
104
105 //== DEPENDENT FUNCTIONS
====================================================
106
107
//===========================================================================
108
109 double textbf{ellipse_segment} (double A, double B,double X1, double Y1,
double X2,
110
111 double Y2, int *MessageCode);
112
113
114
115 double \textbf{ellipse_line_overlap} (double PHI, double A, double B,
double H,
116
117 double K, double X1, double Y1, double X2,
118
119 double Y2, int *MessageCode)
120
121 \\{
122
123 //=======================================================================
124
125 //== DEFINE LOCAL VARIABLES =============================================
126
127 //=======================================================================
128
129 double X10; //– Translated, Rotated x-value of the first point
130
131 double Y10; //– Translated, Rotated y-value of the first point
132
133 double X20; //– Translated, Rotated x-value of the second point
134
135 double Y20; //– Translated, Rotated y-value of the second point
136
137 double cosphi = textbf{cos} (PHI); //– store cos(PHI) to avoid multiple
calcs
138
139 double sinphi = \textbf{sin} (PHI); //– store sin(PHI) to avoid multiple
calcs
140
141 double m; //– line slope, calculated from input line slope
142
143 double a, b, c; //– quadratic equation coefficients a*x\^{}2 + b*x + c
144
145 double discrim; //– quadratic equationdiscriminant b\^{}2 - 4*a*c
146
147 double x1, x2; //– x-values of intersection points
148
149 double y1, y2; //– y-values of intersection points
150
151 double mid_X; //– midpoint of the rotated x-values on the line
152
153 double theta1parm; //– parametric angle of first point
154
155 double theta2parm; //– parametric angle of second point
156
157 double xmidpoint; //– x-value midpoint of secant line
158
159 double ymidpoint; //– y-value midpoint of secant line
160
161 double root1, root2; //– temporary storage variables for roots
162
163 double segment_area; //– stores the ellipse segment area
164
165
166
167 //– Check the data first
168
169 //– Each of the ellipse axis lengths must be positive
170
171 if (!(A $>$ 0.0) \textbar \textbar !(B $>$ 0.0))
172
173 {
174
175 (*MessageCode) = ERROR_ELLIPSE_PARAMETERS;
176
177 return -1.0;
178
179 }
180
181
182
183 //– The rotation angle for the ellipse should be between -2pi and 2pi (?)
184
185 if ( (\textbf{fabs} (PHI) $>$ (2.0*pi)) )
186
187 PHI = \textbf{fmod} (PHI, twopi);
188
189
190
191 //– For this numerical routine, the ellipse will be translated and
192
193 //– rotated so that it is centered at the origin and oriented with
194
195 //– the coordinate axes.
196
197 //– Then, the ellipse will have the implicit (polynomial) form of
198
199 //– x\^{}2/A\^{}2 + y+2/B\^{}2 = 1
200
201
202
203 //– For the line, the given points are first translated by the amount
204
205 //– required to put the ellipse at the origin, e.g., by (-H, -K).
206
207 //– Then, the points are rotated by the amount required to orient
208
209 //– the ellipse with the coordinate axes, e.g., through the angle -PHI.
210
211 X10 = cosphi*(X1 - H) + sinphi*(Y1 - K);
212
213 Y10 = -sinphi*(X1 - H) + cosphi*(Y1 - K);
214
215 X20 = cosphi*(X2 - H) + sinphi*(Y2 - K);
216
217 Y20 = -sinphi*(X2 - H) + cosphi*(Y2 - K);
218
219
220
221 //– To determine if the line and ellipse intersect, solve the two
222
223 //– equations simultaneously, by substituting y = Y10 + m*(x - X10)
224
225 //– and x = X10 + mxy*(y - Y10) into the ellipse equation,
226
227 //– which results in two quadratic equations in x. See the reference
228
229 //– for derivations of the quadratic coefficients.
230
231
232
233 //– If the new line is not close to being vertical, then use the
234
235 //– first derivation
236
237 if (\textbf{fabs} (X20 - X10) $>$ EPS)
238
239 {
240
241 //– ((B\^{}2 + A\^{}2*m\^{}2)/(A\^{}2)) * x\^{}2
242
243 //– 2*(Y10*m - m\^{}2*X10) * x
244
245 //– (Y10\^{}2 - 2*m*Y10*X10 + m\^{}2*X10\^{}2 - B\^{}2)
246
247 m = (Y20 - Y10)/(X20 - X10);
248
249 a = (B*B + A*A*m*m)/(A*A);
250
251 b = 2.0*(Y10*m - m*m*X10);
252
253 c = (Y10*Y10 - 2.0*m*Y10*X10 + m*m*X10*X10 - B*B);
254
255 }
256
257 //– If the new line is close to being vertical, then use the
258
259 //– second derivation
260
261 else if (\textbf{fabs} (Y20 - Y10) $>$ EPS)
262
263 {
264
265 //– ((A\^{}2 + B\^{}2*m\^{}2)/(B\^{}2)) * y\^{}2
266
267 //– 2*(X10*m - m\^{}2*Y10) * y
268
269 //– (X10\^{}2 - 2*m*Y10*X10 + m\^{}2*Y10\^{}2 - A\^{}2)
270
271 m = (X20 - X10)/(Y20 - Y10);
272
273 a = (A*A + B*B*m*m)/(B*B);
274
275 b = 2.0*(X10*m - m*m*Y10);
276
277 c = (X10*X10 - 2.0*m*Y10*X10 + m*m*Y10*Y10 - A*A);
278
279 }
280
281 //– If the two given points on the line are very close together in
282
283 //– both x and y directions, then give up
284
285 else
286
287 {
288
289 (*MessageCode) = ERROR_LINE_POINTS;
290
291 return -1.0;
292
293 }
294
295
296
297 //– Once the coefficients for the Quadratic Equation in x are
298
299 //– known, the roots of the quadratic polynomial will represent
300
301 //– the x- or y-values of the points of intersection of the line
302
303 //– and the ellipse. The discriminant can be used to discern
304
305 //– which case has occurred for the given inputs:
306
307 //– 1. discr $<$ 0
308
309 //– Quadratic has complex conjugate roots.
310
311 //– The line and ellipse do not intersect
312
313 //– 2. discr = 0
314
315 //– Quadratic has one repeated root
316
317 //– The line and ellipse intersect at only one point
318
319 //– i.e., the line is tangent to the ellipse
320
321 //– 3. discr $>$ 0
322
323 //– Quadratic has two distinct real roots
324
325 //– The line crosses the ellipse at two points
326
327 discrim = b*b - 4.0*a*c;
328
329 if (discrim $<$ 0.0)
330
331 {
332
333 //– Line and ellipse do not intersect
334
335 (*MessageCode) = NO_INTERSECTION_POINTS;
336
337 return 0.0;
338
339 }
340
341 else if (discrim $>$ 0.0)
342
343 {
344
345 //– Two real roots exist, so calculate them
346
347 //– The larger root is stored in root2
348
349 root1 = (-b - \textbf{sqrt} (discrim)) / (2.0*a);
350
351 root2 = (-b + \textbf{sqrt} (discrim)) / (2.0*a);
352
353 }
354
355 else
356
357 {
358
359 //– Line is tangent to the ellipse
360
361 (*MessageCode) = LINE_TANGENT_TO_ELLIPSE;
362
363 return 0.0;
364
365 }
366
367
368
369 //– decide which roots go into which x or y values
370
371 if (\textbf{fabs} (X20 - X10) $>$ EPS) //– roots are x-values
372
373 {
374
375 //– order the points in the same direction as X10 -$>$ X20
376
377 if (X10 $<$ X20)
378
379 {
380
381 x1 = root1;
382
383 x2 = root2;
384
385 }
386
387 else
388
389 {
390
391 x1 = root2;
392
393 x2 = root1;
394
395 }
396
397
398
399 //– The y-values can be calculated by substituting the
400
401 //– x-values into the line equation y = Y10 + m*(x - X10)
402
403 y1 = Y10 + m*(x1 - X10);
404
405 y2 = Y10 + m*(x2 - X10);
406
407 }
408
409 else //– roots are y-values
410
411 {
412
413 //– order the points in the same direction as Y10 -$>$ Y20
414
415 if (Y10 $<$ Y20)
416
417 {
418
419 y1 = root1;
420
421 y2 = root2;
422
423 }
424
425 else
426
427 {
428
429 y1 = root2;
430
431 y2 = root1;
432
433 }
434
435
436
437 //– The x-values can be calculated by substituting the
438
439 //– y-values into the line equation x = X10 + m*(y - Y10)
440
441 x1 = X10 + m*(y1 - Y10);
442
443 x2 = X10 + m*(y2 - Y10);
444
445 }
446
447
448
449 //– Arriving here means that two points of intersection have been
450
451 //– found. Pass the ellipse parameters and intersection points to
452
453 //– the ellipse_segment() routine.
454
455 segment_area = \textbf{ellipse_segment} (A, B, x1, y1, x2, y2,
MessageCode);
456
457
458
459 //– The message code will indicate whether the function encountered
460
461 //– any errors
462
463 if ((*MessageCode) $<$ 0)
464
465 {
466
467 return -1;
468
469 }
470
471 else
472
473 {
474
475 (*MessageCode) = TWO_INTERSECTION_POINTS;
476
477 return segment_area;
478
479 }
480
481 }
## 6 APPENDIX C
Listing 14: C-SOURCE CODE FOR ELLIPSE_ELLIPSE_OVERLAP
⬇
1/****************************************************************************
2
3*
4
5* Function: double ellipse_ellipse_overlap
6
7*
8
9* Purpose: Given the parameters of two ellipses, this function calculates
10
11* the area of overlap between the two curves. If the ellipses are
12
13* disjoint, this function returns 0.0; if one ellipse is contained
14
15* within the other, this function returns the area of the enclosed
16
17* ellipse; if the ellipses intersect, this function returns the
18
19* calculated area of overlap.
20
21*
22
23* Reference: Hughes and Chraibi (2011), Calculating Ellipse Overlap Areas
24
25*
26
27* Dependencies: math.h for calls to trig and absolute value functions
28
29* program_constants.h error message codes and constants
30
31*
32
33* Inputs: 1. double PHI_1 CCW rotation angle of first ellipse, radians
34
35* 2. double A1 semi-axis length in x-direction first ellipse
36
37* 3. double B1 semi-axis length in y-direction first ellipse
38
39* 4. double H1 horizontal offset of center first ellipse
40
41* 5. double K1 vertical offset of center first ellipse
42
43* 6. double PHI_2 CCW rotation angle of second ellipse, radians
44
45* 7. double A2 semi-axis length in x-direction second ellipse
46
47* 8. double B2 semi-axis length in y-direction second ellipse
48
49* 9. double H2 horizontal offset of center second ellipse
50
51* 10. double K2 vertical offset of center second ellipse
52
53*
54
55* Outputs: 1. int *rtnCode returns diagnostic information integer code
56
57* integer codes in program_constants.h
58
59*
60
61* Return: The calculated value of the overlap area
62
63* -1 is returned in case of an error with the calculation
64
65* 0 is returned if the ellipses are disjoint
66
67* pi*A*B of smaller ellipse if one ellipse is contained within
68
69* the other ellipse
70
71*
72
73****************************************************************************/
74
75
76
77//===========================================================================
78
79//== DEFINE PROGRAM CONSTANTS
===============================================
80
81//===========================================================================
82
83#include ”program_constants.h” //– error message codes and constants
84
85
86
87//===========================================================================
88
89//== DEPENDENT FUNCTIONS
====================================================
90
91//===========================================================================
92
93double nointpts (double A1, double B1, double A2, double B2, double H1,
94
95 double K1, double H2_TR, double K2_TR, double AA, double BB,
96
97 double CC, double DD, double EE, double FF, int *rtnCode);
98
99
100
101double twointpts (double xint[], double yint[], double A1, double B1,
102
103 double PHI_1, double A2, double B2, double H2_TR,
104
105 double K2_TR, double PHI_2, double AA, double BB,
106
107 double CC, double DD, double EE, double FF, int *rtnCode);
108
109
110
111double threeintpts (double xint[], double yint[], double A1, double B1,
112
113 double PHI_1, double A2, double B2, double H2_TR,
114
115 double K2_TR, double PHI_2, double AA, double BB,
116
117 double CC, double DD, double EE, double FF,
118
119 int *rtnCode);
120
121
122
123double fourintpts (double xint[], double yint[], double A1, double B1,
124
125 double PHI_1, double A2, double B2, double H2_TR,
126
127 double K2_TR, double PHI_2, double AA, double BB,
128
129 double CC, double DD, double EE, double FF, int *rtnCode);
130
131
132
133int istanpt (double x, double y, double A1, double B1, double AA, double
BB,
134
135 double CC, double DD, double EE, double FF);
136
137
138
139double ellipse2tr (double x, double y, double AA, double BB,
140
141 double CC, double DD, double EE, double FF);
142
143
144
145//– functions for solving the quartic equation from Netlib/TOMS
146
147void BIQUADROOTS (double p[], double r[][5]);
148
149void CUBICROOTS (double p[], double r[][5]);
150
151void QUADROOTS (double p[], double r[][5]);
152
153
154
155//===========================================================================
156
157//== ELLIPSE-ELLIPSE OVERLAP
================================================
158
159//===========================================================================
160
161double ellipse_ellipse_overlap (double PHI_1, double A1, double B1,
162
163 double H1, double K1, double PHI_2,
164
165 double A2, double B2, double H2, double K2,
166
167 int *rtnCode)
168
169{
170
171 //=======================================================================
172
173 //== DEFINE LOCAL VARIABLES =============================================
174
175 //=======================================================================
176
177 int i, j, k, nroots, nychk, nintpts, fnRtnCode;
178
179 double AA, BB, CC, DD, EE, FF, H2_TR, K2_TR, A22, B22, PHI_2R;
180
181 double cosphi, cosphi2, sinphi, sinphi2, cosphisinphi;
182
183 double tmp0, tmp1, tmp2, tmp3;
184
185 double cy[5] = {0.0}, py[5] = {0.0}, r[3][5] = {0.0};
186
187 double x1, x2, y12, y22;
188
189 double ychk[5] = {0.0}, xint[5], yint[5];
190
191 double Area1, Area2, OverlapArea;
192
193
194
195 //=======================================================================
196
197 //== DATA CHECK =========================================================
198
199 //=======================================================================
200
201 //– Each of the ellipse axis lengths must be positive
202
203 if ( (!(A1 $>$ 0.0) \textbar \textbar !(B1 $>$ 0.0)) \textbar \textbar
(!(A2 $>$ 0.0) \textbar \textbar !(B2 $>$ 0.0)) )
204
205 {
206
207 (*rtnCode) = ERROR_ELLIPSE_PARAMETERS;
208
209 return -1.0;
210
211 }
212
213
214
215 //– The rotation angles should be between -2pi and 2pi (?)
216
217 if ( (fabs (PHI_1) $>$ (twopi)) )
218
219 PHI_1 = fmod (PHI_1, twopi);
220
221 if ( (fabs (PHI_2) $>$ (twopi)) )
222
223 PHI_2 = fmod (PHI_2, twopi);
224
225
226
227 //=======================================================================
228
229 //== DETERMINE THE TWO ELLIPSE EQUATIONS FROM INPUT PARAMETERS ==========
230
231 //=======================================================================
232
233 //– Finding the points of intersection between two general ellipses
234
235 //– requires solving a quartic equation. Before attempting to solve the
236
237 //– quartic, several quick tests can be used to eliminate some cases
238
239 //– where the ellipses do not intersect. Optionally, can whittle away
240
241 //– at the problem, by addressing the easiest cases first.
242
243
244
245 //– Working with the translated+rotated ellipses simplifies the
246
247 //– calculations. The ellipses are translated then rotated so that the
248
249 //– first ellipse is centered at the origin and oriented with the
250
251 //– coordinate axes. Then, the first ellipse will have the implicit
252
253 //– (polynomial) form of
254
255 //– x\^{}2/A1\^{}2 + y+2/B1\^{}2 = 1
256
257
258
259 //– For the second ellipse, the center is first translated by the amount
260
261 //– required to put the first ellipse at the origin, e.g., by (-H1, -K1)
262
263 //– Then, the center of the second ellipse is rotated by the amount
264
265 //– required to orient the first ellipse with the coordinate axes, e.g.,
266
267 //– through the angle -PHI_1.
268
269 //– The translated and rotated center point coordinates for the second
270
271 //– ellipse are found with the rotation matrix, derivations are
272
273 //– described in the reference.
274
275 cosphi = cos (PHI_1);
276
277 sinphi = sin (PHI_1);
278
279 H2_TR = (H2 - H1)*cosphi + (K2 - K1)*sinphi;
280
281 K2_TR = (H1 - H2)*sinphi + (K2 - K1)*cosphi;
282
283 PHI_2R = PHI_2 - PHI_1;
284
285 if ( (fabs (PHI_2R) $>$ (twopi)) )
286
287 PHI_2R = fmod (PHI_2R, twopi);
288
289
290
291 //– Calculate implicit (Polynomial) coefficients for the second ellipse
292
293 //– in its translated-by (-H1, -H2) and rotated-by -PHI_1 postion
294
295 //– AA*x^{}2 + BB*x*y + CC*y^{}2 + DD*x + EE*y + FF = 0
296
297 //– Formulas derived in the reference
298
299 //– To speed things up, store multiply-used expressions first
300
301 cosphi = cos (PHI_2R);
302
303 cosphi2 = cosphi*cosphi;
304
305 sinphi = sin (PHI_2R);
306
307 sinphi2 = sinphi*sinphi;
308
309 cosphisinphi = 2.0*cosphi*sinphi;
310
311 A22 = A2*A2;
312
313 B22 = B2*B2;
314
315 tmp0 = (cosphi*H2_TR + sinphi*K2_TR)/A22;
316
317 tmp1 = (sinphi*H2_TR - cosphi*K2_TR)/B22;
318
319 tmp2 = cosphi*H2_TR + sinphi*K2_TR;
320
321 tmp3 = sinphi*H2_TR - cosphi*K2_TR;
322
323
324
325 //– implicit polynomial coefficients for the second ellipse
326
327 AA = cosphi2/A22 + sinphi2/B22;
328
329 BB = cosphisinphi/A22 - cosphisinphi/B22;
330
331 CC = sinphi2/A22 + cosphi2/B22;
332
333 DD = -2.0*cosphi*tmp0 - 2.0*sinphi*tmp1;
334
335 EE = -2.0*sinphi*tmp0 + 2.0*cosphi*tmp1;
336
337 FF = tmp2*tmp2/A22 + tmp3*tmp3/B22 - 1.0;
338
339
340
341 //=======================================================================
342
343 //== CREATE AND SOLVE THE QUARTIC EQUATION TO FIND INTERSECTION POINTS ==
344
345 //=======================================================================
346
347 //– If execution arrives here, the ellipses are at least ’close’ to
348
349 //– intersecting.
350
351 //– Coefficients for the Quartic Polynomial in y are calculated from
352
353 //– the two implicit equations.
354
355 //– Formulas for these coefficients are derived in the reference.
356
357 cy[4] = pow (A1, 4.0)*AA*AA + B1*B1*(A1*A1*(BB*BB - 2.0*AA*CC)
358
359 + B1*B1*CC*CC);
360
361 cy[3] = 2.0*B1*(B1*B1*CC*EE + A1*A1*(BB*DD - AA*EE));
362
363 cy[2] = A1*A1*((B1*B1*(2.0*AA*CC - BB*BB) + DD*DD - 2.0*AA*FF)
364
365 - 2.0*A1*A1*AA*AA) + B1*B1*(2.0*CC*FF + EE*EE);
366
367 cy[1] = 2.0*B1*(A1*A1*(AA*EE - BB*DD) + EE*FF);
368
369 cy[0] = (A1*(A1*AA - DD) + FF)*(A1*(A1*AA + DD) + FF);
370
371
372
373 //– Once the coefficients for the Quartic Equation in y are known, the
374
375 //– roots of the quartic polynomial will represent y-values of the
376
377 //– intersection points of the two ellipse curves.
378
379 //– The quartic sometimes degenerates into a polynomial of lesser
380
381 //– degree, so handle all possible cases.
382
383 if (fabs (cy[4]) $>$ 0.0)
384
385 {
386
387 //== QUARTIC COEFFICIENT NONZERO, USE QUARTIC FORMULA ===============
388
389 for (i = 0; i $<$= 3; i++)
390
391 py[4-i] = cy[i]/cy[4];
392
393 py[0] = 1.0;
394
395
396
397 BIQUADROOTS (py, r);
398
399 nroots = 4;
400
401 }
402
403 else if (fabs (cy[3]) $>$ 0.0)
404
405 {
406
407 //== QUARTIC DEGENERATES TO CUBIC, USE CUBIC FORMULA ================
408
409 for (i = 0; i $<$= 2; i++)
410
411 py[3-i] = cy[i]/cy[3];
412
413 py[0] = 1.0;
414
415
416
417 CUBICROOTS (py, r);
418
419 nroots = 3;
420
421 }
422
423 else if (fabs (cy[2]) $>$ 0.0)
424
425 {
426
427 //== QUARTIC DEGENERATES TO QUADRATIC, USE QUADRATIC FORMULA ========
428
429 for (i = 0; i $<$= 1; i++)
430
431 py[2-i] = cy[i]/cy[2];
432
433 py[0] = 1.0;
434
435
436
437 QUADROOTS (py, r);
438
439 nroots = 2;
440
441 }
442
443 else if (fabs (cy[1]) $>$ 0.0)
444
445 {
446
447 //== QUARTIC DEGENERATES TO LINEAR: SOLVE DIRECTLY ==================
448
449 //– cy[1]*Y + cy[0] = 0
450
451 r[1][1] = (-cy[0]/cy[1]);
452
453 r[2][1] = 0.0;
454
455 nroots = 1;
456
457 }
458
459 else
460
461 {
462
463 //== COMPLETELY DEGENERATE QUARTIC: ELLIPSES IDENTICAL??? ===========
464
465 //– a completely degenerate quartic, which would seem to
466
467 //– indicate that the ellipses are identical. However, some
468
469 //– configurations lead to a degenerate quartic with no
470
471 //– points of intersection.
472
473 nroots = 0;
474
475 }
476
477
478
479 //=======================================================================
480
481 //== CHECK ROOTS OF THE QUARTIC: ARE THEY POINTS OF INTERSECTION? =======
482
483 //=======================================================================
484
485 //– determine which roots are real, discard any complex roots
486
487 nychk = 0;
488
489 for (i = 1; i $<$= nroots; i++)
490
491 {
492
493 if (fabs (r[2][i]) $<$ EPS)
494
495 {
496
497 nychk++;
498
499 ychk[nychk] = r[1][i]*B1;
500
501 }
502
503 }
504
505
506
507 //– sort the real roots by straight insertion
508
509 for (j = 2; j $<$= nychk; j++)
510
511 {
512
513 tmp0 = ychk[j];
514
515
516
517 for (k = j - 1; k $>$= 1; k–)
518
519 {
520
521 if (ychk[k] $<$= tmp0)
522
523 break;
524
525
526
527 ychk[k+1] = ychk[k];
528
529 }
530
531
532
533 ychk[k+1] = tmp0;
534
535 }
536
537
538
539 //– determine whether polynomial roots are points of intersection
540
541 //– for the two ellipses
542
543 nintpts = 0;
544
545 for (i = 1; i $<$= nychk; i++)
546
547 {
548
549 //– check for multiple roots
550
551 if ((i $>$ 1) \&\& (fabs (ychk[i] - ychk[i-1]) $<$ (EPS/2.0)))
552
553 continue;
554
555
556
557 //– check intersection points for ychk[i]
558
559 if (fabs (ychk[i]) $>$ B1)
560
561 x1 = 0.0;
562
563 else
564
565 x1 = A1*sqrt (1.0 - (ychk[i]*ychk[i])/(B1*B1));
566
567 x2 = -x1;
568
569
570
571 if (fabs(ellipse2tr(x1, ychk[i], AA, BB, CC, DD, EE, FF)) $<$ EPS/2.0)
572
573 {
574
575 nintpts++;
576
577 if (nintpts $>$ 4)
578
579 {
580
581 (*rtnCode) = ERROR_INTERSECTION_PTS;
582
583 return -1.0;
584
585 }
586
587 xint[nintpts] = x1;
588
589 yint[nintpts] = ychk[i];
590
591 }
592
593
594
595 if ((fabs(ellipse2tr(x2, ychk[i], AA, BB, CC, DD, EE, FF)) $<$ EPS/2.0)
596
597 \&\& (fabs (x2 - x1) $>$ EPS/2.0))
598
599 {
600
601 nintpts++;
602
603 if (nintpts $>$ 4)
604
605 {
606
607 (*rtnCode) = ERROR_INTERSECTION_PTS;
608
609 return -1.0;
610
611 }
612
613 xint[nintpts] = x2;
614
615 yint[nintpts] = ychk[i];
616
617 }
618
619 }
620
621
622
623 //=======================================================================
624
625 //== HANDLE ALL CASES FOR THE NUMBER OF INTERSCTION POINTS ==============
626
627 //=======================================================================
628
629 switch (nintpts)
630
631 {
632
633 case 0:
634
635 case 1:
636
637 OverlapArea = nointpts (A1, B1, A2, B2, H1, K1, H2_TR, K2_TR, AA,
638
639 BB, CC, DD, EE, FF, rtnCode);
640
641 return OverlapArea;
642
643
644
645 case 2:
646
647 //– when there are two intersection points, it is possible for
648
649 //– them to both be tangents, in which case one of the ellipses
650
651 //– is fully contained within the other. Check the points for
652
653 //– tangents; if one of the points is a tangent, then the other
654
655 //– must be as well, otherwise there would be more than 2
656
657 //– intersection points.
658
659 fnRtnCode = istanpt (xint[1], yint[1], A1, B1, AA, BB, CC, DD,
660
661 EE, FF);
662
663
664
665 if (fnRtnCode == TANGENT_POINT)
666
667 OverlapArea = nointpts (A1, B1, A2, B2, H1, K1, H2_TR, K2_TR,
668
669 AA, BB, CC, DD, EE, FF, rtnCode);
670
671 else
672
673 OverlapArea = twointpts (xint, yint, A1, B1, PHI_1, A2, B2,
674
675 H2_TR, K2_TR, PHI_2, AA, BB, CC, DD,
676
677 EE, FF, rtnCode);
678
679 return OverlapArea;
680
681
682
683 case 3:
684
685 //– when there are three intersection points, one and only one
686
687 //– of the points must be a tangent point.
688
689 OverlapArea = threeintpts (xint, yint, A1, B1, PHI_1, A2, B2,
690
691 H2_TR, K2_TR, PHI_2, AA, BB, CC, DD,
692
693 EE, FF, rtnCode);
694
695 return OverlapArea;
696
697
698
699 case 4:
700
701 //– four intersections points has only one case.
702
703 OverlapArea = fourintpts (xint, yint, A1, B1, PHI_1, A2, B2,
704
705 H2_TR, K2_TR, PHI_2, AA, BB, CC, DD,
706
707 EE, FF, rtnCode);
708
709 return OverlapArea;
710
711
712
713 default:
714
715 //– should never get here (but get compiler warning for missing
716
717 //– return value if this line is omitted)
718
719 (*rtnCode) = ERROR_INTERSECTION_PTS;
720
721 return -1.0;
722
723 }
724
725}
726
727
728
729double ellipse2tr (double x, double y, double AA, double BB,
730
731 double CC, double DD, double EE, double FF)
732
733{
734
735 return (AA*x*x + BB*x*y + CC*y*y + DD*x + EE*y + FF);
736
737}
738
739
740
741double nointpts (double A1, double B1, double A2, double B2, double H1,
742
743 double K1, double H2_TR, double K2_TR, double AA, double BB,
744
745 double CC, double DD, double EE, double FF, int *rtnCode)
746
747{
748
749 //– The relative size of the two ellipses can be found from the axis
750
751 //– lengths
752
753 double relsize = (A1*B1) - (A2*B2);
754
755
756
757 if (relsize $>$ 0.0)
758
759 {
760
761 //– First Ellipse is larger than second ellipse.
762
763 //– If second ellipse center (H2_TR, K2_TR) is inside
764
765 //– first ellipse, then ellipse 2 is completely inside
766
767 //– ellipse 1. Otherwise, the ellipses are disjoint.
768
769 if ( ((H2_TR*H2_TR) / (A1*A1)
770
771 + (K2_TR*K2_TR) / (B1*B1)) $<$ 1.0 )
772
773 {
774
775 (*rtnCode) = ELLIPSE2_INSIDE_ELLIPSE1;
776
777 return (pi*A2*B2);
778
779 }
780
781 else
782
783 {
784
785 (*rtnCode) = DISJOINT_ELLIPSES;
786
787 return 0.0;
788
789 }
790
791 }
792
793 else if (relsize $<$ 0.0)
794
795 {
796
797 //– Second Ellipse is larger than first ellipse
798
799 //– If first ellipse center (0, 0) is inside the
800
801 //– second ellipse, then ellipse 1 is completely inside
802
803 //– ellipse 2. Otherwise, the ellipses are disjoint
804
805 //– AA*x^{}2 + BB*x*y + CC*y\^{}2 + DD*x + EE*y + FF = 0
806
807 if (FF $<$ 0.0)
808
809 {
810
811 (*rtnCode) = ELLIPSE1_INSIDE_ELLIPSE2;
812
813 return (pi*A1*B1);
814
815 }
816
817 else
818
819 {
820
821 (*rtnCode) = DISJOINT_ELLIPSES;
822
823 return 0.0;
824
825 }
826
827 }
828
829 else
830
831 {
832
833 //– If execution arrives here, the relative sizes are identical.
834
835 //– Are the ellipses the same? Check the parameters to see.
836
837 if ((H1 == H2_TR) \&\& (K1 == K2_TR))
838
839 {
840
841 (*rtnCode) = ELLIPSES_ARE_IDENTICAL;
842
843 return (pi*A1*B1);
844
845 }
846
847 else
848
849 {
850
851 //– should never get here, so return error
852
853 (*rtnCode) = ERROR_CALCULATIONS;
854
855 return -1.0;
856
857 }
858
859 }//– end if (relsize $>$ 0.0)
860
861}
862
863
864
865//– two distinct intersection points (x1, y1) and (x2, y2) find overlap
area
866
867double twointpts (double x[], double y[], double A1, double B1, double
PHI_1,
868
869 double A2, double B2, double H2_TR, double K2_TR,
870
871 double PHI_2, double AA, double BB, double CC, double DD,
872
873 double EE, double FF, int *rtnCode)
874
875{
876
877 double area1, area2;
878
879 double xmid, ymid, xmid_rt, ymid_rt;
880
881 double theta1, theta2;
882
883 double tmp, trsign;
884
885 double x1_tr, y1_tr, x2_tr, y2_tr;
886
887 double discr;
888
889 double cosphi, sinphi;
890
891
892
893 //– if execution arrives here, the intersection points are not
894
895 //– tangents.
896
897
898
899 //– determine which direction to integrate in the ellipse_segment
900
901 //– routine for each ellipse.
902
903
904
905 //– find the parametric angles for each point on ellipse 1
906
907 if (fabs (x[1]) $>$ A1)
908
909 x[1] = (x[1] $<$ 0) ? -A1 : A1;
910
911 if (y[1] $<$ 0.0) //– Quadrant III or IV
912
913 theta1 = twopi - acos (x[1] / A1);
914
915 else //– Quadrant I or II
916
917 theta1 = acos (x[1] / A1);
918
919
920
921 if (fabs (x[2]) $>$ A1)
922
923 x[2] = (x[2] $<$ 0) ? -A1 : A1;
924
925 if (y[2] $<$ 0.0) //– Quadrant III or IV
926
927 theta2 = twopi - acos (x[2] / A1);
928
929 else //– Quadrant I or II
930
931 theta2 = acos (x[2] / A1);
932
933
934
935 //– logic is for proceeding counterclockwise from theta1 to theta2
936
937 if (theta1 $>$ theta2)
938
939 {
940
941 tmp = theta1;
942
943 theta1 = theta2;
944
945 theta2 = tmp;
946
947 }
948
949
950
951 //– find a point on the first ellipse that is different than the two
952
953 //– intersection points.
954
955 xmid = A1*cos ((theta1 + theta2)/2.0);
956
957 ymid = B1*sin ((theta1 + theta2)/2.0);
958
959
960
961 //– the point (xmid, ymid) is on the first ellipse ’between’ the two
962
963 //– intersection points (x[1], y[1]) and (x[2], y[2]) when travelling
964
965 //– counter- clockwise from (x[1], y[1]) to (x[2], y[2]). If the point
966
967 //– (xmid, ymid) is inside the second ellipse, then the desired segment
968
969 //– of ellipse 1 contains the point (xmid, ymid), so integrate
970
971 //– counterclockwise from (x[1], y[1]) to (x[2], y[2]). Otherwise,
972
973 //– integrate counterclockwise from (x[2], y[2]) to (x[1], y[1])
974
975 if (ellipse2tr (xmid, ymid, AA, BB, CC, DD, EE, FF) $>$ 0.0)
976
977 {
978
979 tmp = theta1;
980
981 theta1 = theta2;
982
983 theta2 = tmp;
984
985 }
986
987
988
989 //– here is the ellipse segment routine for the first ellipse
990
991 if (theta1 $>$ theta2)
992
993 theta1 -= twopi;
994
995 if ((theta2 - theta1) $>$ pi)
996
997 trsign = 1.0;
998
999 else
1000
1001 trsign = -1.0;
1002
1003 area1 = 0.5*(A1*B1*(theta2 - theta1)
1004
1005 + trsign*fabs (x[1]*y[2] - x[2]*y[1]));
1006
1007
1008
1009 //– find ellipse 2 segment area. The ellipse segment routine
1010
1011 //– needs an ellipse that is centered at the origin and oriented
1012
1013 //– with the coordinate axes. The intersection points (x[1], y[1]) and
1014
1015 //– (x[2], y[2]) are found with both ellipses translated and rotated by
1016
1017 //– (-H1, -K1) and -PHI_1. Further translate and rotate the points
1018
1019 //– to put the second ellipse at the origin and oriented with the
1020
1021 //– coordinate axes. The translation is (-H2_TR, -K2_TR), and the
1022
1023 //– rotation is -(PHI_2 - PHI_1) = PHI_1 - PHI_2
1024
1025 cosphi = cos (PHI_1 - PHI_2);
1026
1027 sinphi = sin (PHI_1 - PHI_2);
1028
1029 x1_tr = (x[1] - H2_TR)*cosphi + (y[1] - K2_TR)*-sinphi;
1030
1031 y1_tr = (x[1] - H2_TR)*sinphi + (y[1] - K2_TR)*cosphi;
1032
1033 x2_tr = (x[2] - H2_TR)*cosphi + (y[2] - K2_TR)*-sinphi;
1034
1035 y2_tr = (x[2] - H2_TR)*sinphi + (y[2] - K2_TR)*cosphi;
1036
1037
1038
1039 //– determine which branch of the ellipse to integrate by finding a
1040
1041 //– point on the second ellipse, and asking whether it is inside the
1042
1043 //– first ellipse (in their once-translated+rotated positions)
1044
1045 //– find the parametric angles for each point on ellipse 1
1046
1047 if (fabs (x1_tr) $>$ A2)
1048
1049 x1_tr = (x1_tr $<$ 0) ? -A2 : A2;
1050
1051 if (y1_tr $<$ 0.0) //– Quadrant III or IV
1052
1053 theta1 = twopi - acos (x1_tr/A2);
1054
1055 else //– Quadrant I or II
1056
1057 theta1 = acos (x1_tr/A2);
1058
1059
1060
1061 if (fabs (x2_tr) $>$ A2)
1062
1063 x2_tr = (x2_tr $<$ 0) ? -A2 : A2;
1064
1065 if (y2_tr $<$ 0.0) //– Quadrant III or IV
1066
1067 theta2 = twopi - acos (x2_tr/A2);
1068
1069 else //– Quadrant I or II
1070
1071 theta2 = acos (x2_tr/A2);
1072
1073
1074
1075 //– logic is for proceeding counterclockwise from theta1 to theta2
1076
1077 if (theta1 $>$ theta2)
1078
1079 {
1080
1081 tmp = theta1;
1082
1083 theta1 = theta2;
1084
1085 theta2 = tmp;
1086
1087 }
1088
1089
1090
1091 //– find a point on the second ellipse that is different than the two
1092
1093 //– intersection points.
1094
1095 xmid = A2*cos ((theta1 + theta2)/2.0);
1096
1097 ymid = B2*sin ((theta1 + theta2)/2.0);
1098
1099
1100
1101 //– translate the point back to the second ellipse in its once-
1102
1103 //– translated+rotated position
1104
1105 cosphi = cos (PHI_2 - PHI_1);
1106
1107 sinphi = sin (PHI_2 - PHI_1);
1108
1109 xmid_rt = xmid*cosphi + ymid*-sinphi + H2_TR;
1110
1111 ymid_rt = xmid*sinphi + ymid*cosphi + K2_TR;
1112
1113
1114
1115 //– the point (xmid_rt, ymid_rt) is on the second ellipse ’between’ the
1116
1117 //– intersection points (x[1], y[1]) and (x[2], y[2]) when travelling
1118
1119 //– counterclockwise from (x[1], y[1]) to (x[2], y[2]). If the point
1120
1121 //– (xmid_rt, ymid_rt) is inside the first ellipse, then the desired
1122
1123 //– segment of ellipse 2 contains the point (xmid_rt, ymid_rt), so
1124
1125 //– integrate counterclockwise from (x[1], y[1]) to (x[2], y[2]).
1126
1127 //– Otherwise, integrate counterclockwise from (x[2], y[2]) to
1128
1129 //– (x[1], y[1])
1130
1131 if (((xmid_rt*xmid_rt)/(A1*A1) + (ymid_rt*ymid_rt)/(B1*B1)) $>$ 1.0)
1132
1133 {
1134
1135 tmp = theta1;
1136
1137 theta1 = theta2;
1138
1139 theta2 = tmp;
1140
1141 }
1142
1143
1144
1145 //– here is the ellipse segment routine for the second ellipse
1146
1147 if (theta1 $>$ theta2)
1148
1149 theta1 -= twopi;
1150
1151 if ((theta2 - theta1) $>$ pi)
1152
1153 trsign = 1.0;
1154
1155 else
1156
1157 trsign = -1.0;
1158
1159 area2 = 0.5*(A2*B2*(theta2 - theta1)
1160
1161 + trsign*fabs (x1_tr*y2_tr - x2_tr*y1_tr));
1162
1163
1164
1165 (*rtnCode) = TWO_INTERSECTION_POINTS;
1166
1167 return area1 + area2;
1168
1169}
1170
1171
1172
1173//– three distinct intersection points, must have two intersections
1174
1175//– and one tangent, which is the only possibility
1176
1177double threeintpts (double xint[], double yint[], double A1, double B1,
1178
1179 double PHI_1, double A2, double B2, double H2_TR,
1180
1181 double K2_TR, double PHI_2, double AA, double BB,
1182
1183 double CC, double DD, double EE, double FF,
1184
1185 int *rtnCode)
1186
1187{
1188
1189 int i, tanpts, tanindex, fnRtn;
1190
1191 double OverlapArea;
1192
1193
1194
1195 //– need to determine which point is a tangent, and which two points
1196
1197 //– are intersections
1198
1199 tanpts = 0;
1200
1201 for (i = 1; i $<$= 3; i++)
1202
1203 {
1204
1205 fnRtn = istanpt (xint[i], yint[i], A1, B1, AA, BB, CC, DD, EE, FF);
1206
1207
1208
1209 if (fnRtn == TANGENT_POINT)
1210
1211 {
1212
1213 tanpts++;
1214
1215 tanindex = i;
1216
1217 }
1218
1219 }
1220
1221
1222
1223 //– there MUST be 2 intersection points and only one tangent
1224
1225 if (tanpts != 1)
1226
1227 {
1228
1229 //– should never get here unless there is a problem discerning
1230
1231 //– whether or not a point is a tangent or intersection
1232
1233 (*rtnCode) = ERROR_INTERSECTION_PTS;
1234
1235 return -1.0;
1236
1237 }
1238
1239
1240
1241 //– store the two interesection points into (x[1], y[1]) and
1242
1243 //– (x[2], y[2])
1244
1245 switch (tanindex)
1246
1247 {
1248
1249 case 1:
1250
1251 xint[1] = xint[3];
1252
1253 yint[1] = yint[3];
1254
1255 break;
1256
1257
1258
1259 case 2:
1260
1261 xint[2] = xint[3];
1262
1263 yint[2] = yint[3];
1264
1265 break;
1266
1267
1268
1269 case 3:
1270
1271 //– intersection points are already in the right places
1272
1273 break;
1274
1275 }
1276
1277
1278
1279 OverlapArea = twointpts (xint, yint, A1, B1, PHI_1, A2, B2, H2_TR, K2_TR,
1280
1281 PHI_2, AA, BB, CC, DD, EE, FF, rtnCode);
1282
1283 (*rtnCode) = THREE_INTERSECTION_POINTS;
1284
1285 return OverlapArea;
1286
1287}
1288
1289
1290
1291//– four intersection points
1292
1293double fourintpts (double xint[], double yint[], double A1, double B1,
1294
1295 double PHI_1, double A2, double B2, double H2_TR,
1296
1297 double K2_TR, double PHI_2, double AA, double BB,
1298
1299 double CC, double DD, double EE, double FF, int *rtnCode)
1300
1301{
1302
1303 int i, j, k;
1304
1305 double xmid, ymid, xint_tr[5], yint_tr[5], OverlapArea;
1306
1307 double theta[5], theta_tr[5], cosphi, sinphi, tmp0, tmp1, tmp2;
1308
1309 double area1, area2, area3, area4, area5;
1310
1311
1312
1313 //– only one case, which involves two segments from each ellipse, plus
1314
1315 //– two triangles.
1316
1317 //– get the parametric angles along the first ellipse for each of the
1318
1319 //– intersection points
1320
1321 for (i = 1; i $<$= 4; i++)
1322
1323 {
1324
1325 if (fabs (xint[i]) $>$ A1)
1326
1327 xint[i] = (xint[i] $<$ 0) ? -A1 : A1;
1328
1329 if (yint[i] $<$ 0.0) //– Quadrant III or IV
1330
1331 theta[i] = twopi - acos (xint[i] / A1);
1332
1333 else //– Quadrant I or II
1334
1335 theta[i] = acos (xint[i] / A1);
1336
1337 }
1338
1339
1340
1341 //– sort the angles by straight insertion, and put the points in
1342
1343 //– counter-clockwise order
1344
1345 for (j = 2; j $<$= 4; j++)
1346
1347 {
1348
1349 tmp0 = theta[j];
1350
1351 tmp1 = xint[j];
1352
1353 tmp2 = yint[j];
1354
1355
1356
1357 for (k = j - 1; k $>$= 1; k–)
1358
1359 {
1360
1361 if (theta[k] $<$= tmp0)
1362
1363 break;
1364
1365
1366
1367 theta[k+1] = theta[k];
1368
1369 xint[k+1] = xint[k];
1370
1371 yint[k+1] = yint[k];
1372
1373 }
1374
1375
1376
1377 theta[k+1] = tmp0;
1378
1379 xint[k+1] = tmp1;
1380
1381 yint[k+1] = tmp2;
1382
1383 }
1384
1385
1386
1387 //– find the area of the interior quadrilateral
1388
1389 area1 = 0.5*fabs ((xint[3] - xint[1])*(yint[4] - yint[2])
1390
1391 - (xint[4] - xint[2])*(yint[3] - yint[1]));
1392
1393
1394
1395 //– the intersection points lie on the second ellipse in its once
1396
1397 //– translated+rotated position. The segment algorithm is implemented
1398
1399 //– for an ellipse that is centered at the origin, and oriented with
1400
1401 //– the coordinate axes; so, in order to use the segment algorithm
1402
1403 //– with the second ellipse, the intersection points must be further
1404
1405 //– translated+rotated by amounts that put the second ellipse centered
1406
1407 //– at the origin and oriented with the coordinate axes.
1408
1409 cosphi = cos (PHI_1 - PHI_2);
1410
1411 sinphi = sin (PHI_1 - PHI_2);
1412
1413 for (i = 1; i $<$= 4; i++)
1414
1415 {
1416
1417 xint_tr[i] = (xint[i] - H2_TR)*cosphi + (yint[i] - K2_TR)*-sinphi;
1418
1419 yint_tr[i] = (xint[i] - H2_TR)*sinphi + (yint[i] - K2_TR)*cosphi;
1420
1421
1422
1423 if (fabs (xint_tr[i]) $>$ A2)
1424
1425 xint_tr[i] = (xint_tr[i] $<$ 0) ? -A2 : A2;
1426
1427 if (yint_tr[i] $<$ 0.0) //– Quadrant III or IV
1428
1429 theta_tr[i] = twopi - acos (xint_tr[i]/A2);
1430
1431 else //– Quadrant I or II
1432
1433 theta_tr[i] = acos (xint_tr[i]/A2);
1434
1435 }
1436
1437
1438
1439 //– get the area of the two segments on ellipse 1
1440
1441 xmid = A1*cos ((theta[1] + theta[2])/2.0);
1442
1443 ymid = B1*sin ((theta[1] + theta[2])/2.0);
1444
1445
1446
1447 //– the point (xmid, ymid) is on the first ellipse ’between’ the two
1448
1449 //– sorted intersection points (xint[1], yint[1]) and (xint[2], yint[2])
1450
1451 //– when travelling counter- clockwise from (xint[1], yint[1]) to
1452
1453 //– (xint[2], yint[2]). If the point (xmid, ymid) is inside the second
1454
1455 //– ellipse, then one desired segment of ellipse 1 contains the point
1456
1457 //– (xmid, ymid), so integrate counterclockwise from (xint[1], yint[1])
1458
1459 //– to (xint[2], yint[2]) for the first segment, and from
1460
1461 //– (xint[3], yint[3] to (xint[4], yint[4]) for the second segment.
1462
1463 if (ellipse2tr (xmid, ymid, AA, BB, CC, DD, EE, FF) $<$ 0.0)
1464
1465 {
1466
1467 area2 = 0.5*(A1*B1*(theta[2] - theta[1])
1468
1469 - fabs (xint[1]*yint[2] - xint[2]*yint[1]));
1470
1471 area3 = 0.5*(A1*B1*(theta[4] - theta[3])
1472
1473 - fabs (xint[3]*yint[4] - xint[4]*yint[3]));
1474
1475 area4 = 0.5*(A2*B2*(theta_tr[3] - theta_tr[2])
1476
1477 - fabs (xint_tr[2]*yint_tr[3] - xint_tr[3]*yint_tr[2]));
1478
1479 area5 = 0.5*(A2*B2*(theta_tr[1] - (theta_tr[4] - twopi))
1480
1481 - fabs (xint_tr[4]*yint_tr[1] - xint_tr[1]*yint_tr[4]));
1482
1483 }
1484
1485 else
1486
1487 {
1488
1489 area2 = 0.5*(A1*B1*(theta[3] - theta[2])
1490
1491 - fabs (xint[2]*yint[3] - xint[3]*yint[2]));
1492
1493 area3 = 0.5*(A1*B1*(theta[1] - (theta[4] - twopi))
1494
1495 - fabs (xint[4]*yint[1] - xint[1]*yint[4]));
1496
1497 area4 = 0.5*(A2*B2*(theta[2] - theta[1])
1498
1499 - fabs (xint_tr[1]*yint_tr[2] - xint_tr[2]*yint_tr[1]));
1500
1501 area5 = 0.5*(A2*B2*(theta[4] - theta[3])
1502
1503 - fabs (xint_tr[3]*yint_tr[4] - xint_tr[4]*yint_tr[3]));
1504
1505 }
1506
1507
1508
1509 OverlapArea = area1 + area2 + area3 + area4 + area5;
1510
1511 (*rtnCode) = FOUR_INTERSECTION_POINTS;
1512
1513 return OverlapArea;
1514
1515}
1516
1517
1518
1519//– check whether an intersection point is a tangent or a cross-point
1520
1521int istanpt (double x, double y, double A1, double B1, double AA, double
BB,
1522
1523 double CC, double DD, double EE, double FF)
1524
1525{
1526
1527 double x1, y1, x2, y2, theta, test1, test2, branch, eps_radian;
1528
1529
1530
1531 //– Avoid inverse trig calculation errors: there could be an error
1532
1533 //– if \textbar x1/A\textbar $>$ 1.0 when calling acos(). If execution
arrives here,
1534
1535 //– then the point is on the ellipse within EPS.
1536
1537 if (fabs (x) $>$ A1)
1538
1539 x = (x $<$ 0) ? -A1 : A1;
1540
1541
1542
1543 //– Calculate the parametric angle on the ellipse for (x, y)
1544
1545 //– The parametric angles depend on the quadrant where each point
1546
1547 //– is located. See Table 1 in the reference.
1548
1549 if (y $<$ 0.0) //– Quadrant III or IV
1550
1551 theta = twopi - acos (x / A1);
1552
1553 else //– Quadrant I or II
1554
1555 theta = acos (x / A1);
1556
1557
1558
1559 //– determine the distance from the origin to the point (x, y)
1560
1561 branch = sqrt (x*x + y*y);
1562
1563
1564
1565 //– use the distance to find a small angle, such that the distance
1566
1567 //– along ellipse 1 is approximately 2*EPS
1568
1569 if (branch $<$ 100.0*EPS)
1570
1571 eps_radian = 2.0*EPS;
1572
1573 else
1574
1575 eps_radian = asin (2.0*EPS/branch);
1576
1577
1578
1579 //– determine two points that are on each side of (x, y) and lie on
1580
1581 //– the first ellipse
1582
1583 x1 = A1*cos (theta + eps_radian);
1584
1585 y1 = B1*sin (theta + eps_radian);
1586
1587 x2 = A1*cos (theta - eps_radian);
1588
1589 y2 = B1*sin (theta - eps_radian);
1590
1591
1592
1593 //– evaluate the two adjacent points in the second ellipse equation
1594
1595 test1 = ellipse2tr (x1, y1, AA, BB, CC, DD, EE, FF);
1596
1597 test2 = ellipse2tr (x2, y2, AA, BB, CC, DD, EE, FF);
1598
1599
1600
1601 //– if the ellipses are tangent at the intersection point, then
1602
1603 //– points on both sides will either both be inside ellipse 1, or
1604
1605 //– they will both be outside ellipse 1
1606
1607 if ((test1*test2) $>$ 0.0)
1608
1609 return TANGENT_POINT;
1610
1611 else
1612
1613 return INTERSECTION_POINT;
1614
1615}
1616
1617
1618
1619//===========================================================================
1620
1621//– CACM Algorithm 326: Roots of low order polynomials.
1622
1623//– Nonweiler, Terence R.F., CACM Algorithm 326: Roots of low order
1624
1625//– polynomials, Communications of the ACM, vol. 11 no. 4, pages
1626
1627//– 269-270 (1968). Translated into c and programmed by M. Dow, ANUSF,
1628
1629//– Australian National University, Canberra, Australia.
1630
1631//– Accessed at http://www.netlib.org/toms/326.
1632
1633//– Modified to void functions, integers replaced with floating point
1634
1635//– where appropriate, some other slight modifications for readability
1636
1637//– and debugging ease.
1638
1639//===========================================================================
1640
1641void QUADROOTS (double p[], double r[][5])
1642
1643{
1644
1645 /*
1646
1647 Array r[3][5] p[5]
1648
1649 Roots of poly p[0]*x^{}2 + p[1]*x + p[2]=0
1650
1651 x=r[1][k] + i r[2][k] k=1,2
1652
1653 */
1654
1655 double b,c,d;
1656
1657 b=-p[1]/(2.0*p[0]);
1658
1659 c=p[2]/p[0];
1660
1661 d=b*b-c;
1662
1663 if(d$>$=0.0)
1664
1665 {
1666
1667 if(b$>$0.0)
1668
1669 b=(r[1][2]=(sqrt(d)+b));
1670
1671 else
1672
1673 b=(r[1][2]=(-sqrt(d)+b));
1674
1675 r[1][1]=c/b;
1676
1677 r[2][1]=(r[2][2]=0.0);
1678
1679 }
1680
1681 else
1682
1683 {
1684
1685 d=(r[2][1]=sqrt(-d));
1686
1687 r[2][2]=-d;
1688
1689 r[1][1]=(r[1][2]=b);
1690
1691 }
1692
1693 return;
1694
1695}
1696
1697
1698
1699void CUBICROOTS(double p[], double r[][5])
1700
1701{
1702
1703 /*
1704
1705 Array r[3][5] p[5]
1706
1707 Roots of poly p[0]*x\^{}3 + p[1]*x\^{}2 + p[2]*x + p[3] = 0
1708
1709 x=r[1][k] + i r[2][k] k=1,…,3
1710
1711 Assumes 0$<$arctan(x)$<$pi/2 for x$>$0
1712
1713 */
1714
1715 double s,t,b,c,d;
1716
1717 int k;
1718
1719 if(p[0]!=1.0)
1720
1721 {
1722
1723 for(k=1;k$<$4;k++)
1724
1725 p[k]=p[k]/p[0];
1726
1727 p[0]=1.0;
1728
1729 }
1730
1731 s=p[1]/3.0;
1732
1733 t=s*p[1];
1734
1735 b=0.5*(s*(t/1.5-p[2])+p[3]);
1736
1737 t=(t-p[2])/3.0;
1738
1739 c=t*t*t;
1740
1741 d=b*b-c;
1742
1743 if(d$>$=0.0)
1744
1745 {
1746
1747 d=pow((sqrt(d)+fabs(b)),1.0/3.0);
1748
1749 if(d!=0.0)
1750
1751 {
1752
1753 if(b$>$0.0)
1754
1755 b=-d;
1756
1757 else
1758
1759 b=d;
1760
1761 c=t/b;
1762
1763 }
1764
1765 d=r[2][2]=sqrt\eqref{GrindEQ__0_75_}*(b-c);
1766
1767 b=b+c;
1768
1769 c=r[1][2]=-0.5*b-s;
1770
1771 if((b$>$0.0 \&\& s$<$=0.0) \textbar \textbar (b$<$0.0 \&\& s$>$0.0))
1772
1773 {
1774
1775 r[1][1]=c;
1776
1777 r[2][1]=-d;
1778
1779 r[1][3]=b-s;
1780
1781 r[2][3]=0.0;
1782
1783 }
1784
1785 else
1786
1787 {
1788
1789 r[1][1]=b-s;
1790
1791 r[2][1]=0.0;
1792
1793 r[1][3]=c;
1794
1795 r[2][3]=-d;
1796
1797 }
1798
1799 } /* end 2 equal or complex roots */
1800
1801 else
1802
1803 {
1804
1805 if(b==0.0)
1806
1807 d=atan\eqref{GrindEQ__1_0_}/1.5;
1808
1809 else
1810
1811 d=atan(sqrt(-d)/fabs(b))/3.0;
1812
1813 if(b$<$0.0)
1814
1815 b=2.0*sqrt(t);
1816
1817 else
1818
1819 b=-2.0*sqrt(t);
1820
1821 c=cos(d)*b;
1822
1823 t=-sqrt\eqref{GrindEQ__0_75_}*sin(d)*b-0.5*c;
1824
1825 d=-t-c-s;
1826
1827 c=c-s;
1828
1829 t=t-s;
1830
1831 if(fabs(c)$>$fabs(t))
1832
1833 {
1834
1835 r[1][3]=c;
1836
1837 }
1838
1839 else
1840
1841 {
1842
1843 r[1][3]=t;
1844
1845 t=c;
1846
1847 }
1848
1849 if(fabs(d)$>$fabs(t))
1850
1851 {
1852
1853 r[1][2]=d;
1854
1855 }
1856
1857 else
1858
1859 {
1860
1861 r[1][2]=t;
1862
1863 t=d;
1864
1865 }
1866
1867 r[1][1]=t;
1868
1869 for(k=1;k$<$4;k++)
1870
1871 r[2][k]=0.0;
1872
1873 }
1874
1875 return;
1876
1877}
1878
1879
1880
1881void BIQUADROOTS(double p[],double r[][5])
1882
1883{
1884
1885 /*
1886
1887 Array r[3][5] p[5]
1888
1889 Roots of poly p[0]*x\^{}4 + p[1]*x\^{}3 + p[2]*x\^{}2 + p[3]*x + p[4] = 0
1890
1891 x=r[1][k] + i r[2][k] k=1,…,4
1892
1893 */
1894
1895 double a,b,c,d,e;
1896
1897 int k,j;
1898
1899 if(p[0] != 1.0)
1900
1901 {
1902
1903 for(k=1;k$<$5;k++)
1904
1905 p[k]=p[k]/p[0];
1906
1907 p[0]=1.0;
1908
1909 }
1910
1911 e=0.25*p[1];
1912
1913 b=2.0*e;
1914
1915 c=b*b;
1916
1917 d=0.75*c;
1918
1919 b=p[3]+b*(c-p[2]);
1920
1921 a=p[2]-d;
1922
1923 c=p[4]+e*(e*a-p[3]);
1924
1925 a=a-d;
1926
1927 p[1]=0.5*a;
1928
1929 p[2]=(p[1]*p[1]-c)*0.25;
1930
1931 p[3]=b*b/(-64.0);
1932
1933 if(p[3]$<$0.0)
1934
1935 {
1936
1937 CUBICROOTS(p,r);
1938
1939 for(k=1;k$<$4;k++)
1940
1941 {
1942
1943 if(r[2][k]==0.0 \&\& r[1][k]$>$0.0)
1944
1945 {
1946
1947 d=r[1][k]*4.0;
1948
1949 a=a+d;
1950
1951 if(a$>$=0.0 \&\& b$>$=0.0)
1952
1953 p[1]=sqrt(d);
1954
1955 else if(a$<$=0.0 \&\& b$<$=0.0)
1956
1957 p[1]=sqrt(d);
1958
1959 else
1960
1961 p[1]=-sqrt(d);
1962
1963 b=0.5*(a+b/p[1]);
1964
1965 goto QUAD;
1966
1967 }
1968
1969 }
1970
1971 }
1972
1973 if(p[2]$<$0.0)
1974
1975 {
1976
1977 b=sqrt(c);
1978
1979 d=b+b-a;
1980
1981 p[1]=0.0;
1982
1983 if(d$>$0.0)
1984
1985 p[1]=sqrt(d);
1986
1987 }
1988
1989 else
1990
1991 {
1992
1993 if(p[1]$>$0.0)
1994
1995 b=sqrt(p[2])*2.0+p[1];
1996
1997 else
1998
1999 b=-sqrt(p[2])*2.0+p[1];
2000
2001 if(b!=0.0)
2002
2003 {
2004
2005 p[1]=0.0;
2006
2007 }
2008
2009 else
2010
2011 {
2012
2013 for(k=1;k$<$5;k++)
2014
2015 {
2016
2017 r[1][k]=-e;
2018
2019 r[2][k]=0.0;
2020
2021 }
2022
2023 goto END;
2024
2025 }
2026
2027 }
2028
2029QUAD:
2030
2031 p[2]=c/b;
2032
2033 QUADROOTS(p,r);
2034
2035 for(k=1;k$<$3;k++)
2036
2037 for(j=1;j$<$3;j++)
2038
2039 r[j][k+2]=r[j][k];
2040
2041 p[1]=-p[1];
2042
2043 p[2]=b;
2044
2045 QUADROOTS(p,r);
2046
2047 for(k=1;k$<$5;k++)
2048
2049 r[1][k]=r[1][k]-e;
2050
2051END:
2052
2053 return;
2054
2055}
## 7 APPENDIX D
Listing 15: C-SOURCE CODE FOR UTILITY FUNCTIONS
⬇
1program\_constants.h:
2
3
4
5//===========================================================================
6
7//== INCLUDE ANSI C SYSTEM HEADER FILES =====================================
8
9//===========================================================================
10
11#include $<$math.h$>$ //– for calls to trig, sqrt and power functions
12
13
14
15//==========================================================================
16
17//== DEFINE PROGRAM CONSTANTS ==============================================
18
19//==========================================================================
20
21#define NORMAL_TERMINATION 0
22
23#define NO_INTERSECTION_POINTS 100
24
25#define ONE_INTERSECTION_POINT 101
26
27#define LINE_TANGENT_TO_ELLIPSE 102
28
29#define DISJOINT_ELLIPSES 103
30
31#define ELLIPSE2_OUTSIDETANGENT_ELLIPSE1 104
32
33#define ELLIPSE2_INSIDETANGENT_ELLIPSE1 105
34
35#define ELLIPSES_INTERSECT 106
36
37#define TWO_INTERSECTION_POINTS 107
38
39#define THREE_INTERSECTION_POINTS 108
40
41#define FOUR_INTERSECTION_POINTS 109
42
43#define ELLIPSE1_INSIDE_ELLIPSE2 110
44
45#define ELLIPSE2_INSIDE_ELLIPSE1 111
46
47#define ELLIPSES_ARE_IDENTICAL 112
48
49#define INTERSECTION_POINT 113
50
51#define TANGENT_POINT 114
52
53
54
55#define ERROR_ELLIPSE_PARAMETERS -100
56
57#define ERROR_DEGENERATE_ELLIPSE -101
58
59#define ERROR_POINTS_NOT_ON_ELLIPSE -102
60
61#define ERROR_INVERSE_TRIG -103
62
63#define ERROR_LINE_POINTS -104
64
65#define ERROR_QUARTIC_CASE -105
66
67#define ERROR_POLYNOMIAL_DEGREE -107
68
69#define ERROR_POLYNOMIAL_ROOTS -108
70
71#define ERROR_INTERSECTION_PTS -109
72
73#define ERROR_CALCULATIONS -112
74
75
76
77#define EPS +1.0E-07
78
79#define pi (2.0*asin (1.0)) //– a maximum-precision value of pi
80
81#define twopi (2.0*pi) //– a maximum-precision value of 2*pi
82
83
84
85
86
87
88
89call_es.c:
90
91
92
93#include $<$stdio.h$>$
94
95#include $<$math.h$>$
96
97#include ”program_constants.h”
98
99double ellipse_segment (double A, double B, double X1, double Y1, double X2,
100
101 double Y2, int *MessageCode);
102
103
104
105int main (int argc, char ** argv)
106
107{
108
109 double A, B;
110
111 double X1, Y1;
112
113 double X2, Y2;
114
115 double area1, area2;
116
117 double pi = 2.0 * asin eqref{GrindEQ__1_0_}; //– a maximum-precision value
of pi
118
119 int rtn;
120
121 char msg[1024];
122
123 printf (”Calling ellipse_segment.ctextbackslash n”);
124
125
126
127 //– case shown in Fig. 1
128
129 A = 4.;
130
131 B = 2.;
132
133 X1 = 4./sqrt (5.);
134
135 Y1 = 4./sqrt (5.);
136
137 X2 = -3.;
138
139 Y2 = -sqrt (7.)/2.;
140
141
142
143 area1 = ellipse_segment (A, B, X1, Y1, X2, Y2, &rtn);
144
145 sprintf (msg,”Fig 1: segment area = %15.8f, return_value =
%d\textbackslash n”, area1, rtn);
146
147 printf (msg);
148
149
150
151 //– case shown in Fig. 2
152
153 A = 4.;
154
155 B = 2.;
156
157 X1 = -3.;
158
159 Y1 = -sqrt (7.)/2.;
160
161 X2 = 4./sqrt (5.);
162
163 Y2 = 4./sqrt (5.);
164
165
166
167 area2 = ellipse_segment (A, B, X1, Y1, X2, Y2, &rtn);
168
169 sprintf (msg,”Fig 2: segment area = %15.8f, return_value = %dtextbackslash
n”, area2, rtn);
170
171 printf (msg);
172
173
174
175 sprintf (msg,”sum of ellipse segments = %15.8ftextbackslash n”, area1 +
area2);
176
177 printf (msg);
178
179 sprintf (msg,”total ellipse area by pi*a*b = %15.8ftextbackslash n”,
pi*A*B);
180
181 printf (msg);
182
183
184
185 return rtn;
186
187}
188
189
190
191
192
193call_el.c:
194
195
196
197#include $<$stdio.h$>$
198
199#include $<$math.h$>$
200
201#include ”program_constants.h”
202
203double \textbf{ellipse_segment} (double A, double B, double X1, double Y1,
double X2,
204
205 double Y2, int *MessageCode);
206
207
208
209double \textbf{ellipse_line_overlap} (double PHI, double A, double B,
double H,
210
211 double K, double X1, double Y1, double X2,
212
213 double Y2, int *MessageCode);
214
215
216
217int \textbf{main} (int argc, char ** argv)
218
219{
220
221 double A, B;
222
223 double H, K, PHI;
224
225 double X1, Y1;
226
227 double X2, Y2;
228
229 double area1, area2;
230
231 double pi = 2.0 * \textbf{asin} \eqref{GrindEQ__1_0_}; //– a maximum-
precision value of pi
232
233 int rtn;
234
235 char msg[1024];
236
237 \textbf{printf} (”Calling ellipse_line_overlap.c\textbackslash n”);
238
239
240
241 //– case shown in Fig. 4
242
243 A = 4.;
244
245 B = 2.;
246
247 H = -6;
248
249 K = 3;
250
251 PHI = 3.*pi/8.0;
252
253 X1 = -3.;
254
255 Y1 = 3.;
256
257 X2 = -7.;
258
259 Y2 = 7.;
260
261
262
263 area1 = \textbf{ellipse\_line\_overlap} (PHI, A, B, H, K, X1, Y1, X2, Y2,
\&rtn);
264
265 \textbf{sprintf} (msg,”Fig 4: area = \%15.8f, return_value =
\%d\textbackslash n”, area1, rtn);
266
267 \textbf{printf} (msg);
268
269
270
271 //– case shown in Fig. 4, points reversed
272
273 A = 4.;
274
275 B = 2.;
276
277 H = -6;
278
279 K = 3;
280
281 PHI = 3.*pi/8.0;
282
283 X1 = -7.;
284
285 Y1 = 7.;
286
287 X2 = -3.;
288
289 Y2 = 3.;
290
291
292
293 area2 = \textbf{ellipse\_line\_overlap} (PHI, A, B, H, K, X1, Y1, X2, Y2,
\&rtn);
294
295 \textbf{sprintf} (msg,”Fig 4 reverse: area = %15.8f, return_value =
\%d\textbackslash n”, area2, rtn);
296
297 \textbf{printf} (msg);
298
299
300
301 \textbf{sprintf} (msg,”sum of ellipse segments = %15.8ftextbackslash n”,
area1 + area2);
302
303 \textbf{printf} (msg);
304
305 \textbf{sprintf} (msg,”total ellipse area by pi*a*b = %15.8ftextbackslash
n”, pi*A*B);
306
307 \textbf{printf} (msg);
308
309
310
311 return rtn;
312
313 }
314
315
316
317
318
319 call_ee.c:
320
321
322
323 #include $<$stdio.h$>$
324
325 #include ”program_constants.h”
326
327 double ellipse_ellipse_overlap (double PHI_1, double A1, double B1,
328
329 double H1, double K1, double PHI_2,
330
331 double A2, double B2, double H2, double K2,
332
333 int *rtnCode);
334
335
336
337 int main (int argc, char ** argv)
338
339 {
340
341 double A1, B1, H1, K1, PHI_1;
342
343 double A2, B2, H2, K2, PHI_2;
344
345 double area;
346
347 int rtn;
348
349 char msg[1024];
350
351 printf (”Calling ellipse_ellipse_overlap.c\textbackslash n\textbackslash
n”);
352
353
354
355 //– case 0-1
356
357 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
358
359 A2 = 2.; B2 = 1.; H2 = -.75; K2 = 0.25; PHI_2 = pi/4.;
360
361 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
362
363 PHI_2, A2, B2, H2, K2, \&rtn);
364
365 sprintf (msg,”Case 0-1: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
366
367 printf (msg);
368
369 sprintf (msg,” ellipse 2 area by pi*a2*b2 = \%15.8f\textbackslash n”,
pi*A2*B2);
370
371 printf (msg);
372
373
374
375 //– case 0-2
376
377 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
378
379 A2 = 3.; B2 = 2.; H2 = -.3; K2 = -.25; PHI_2 = pi/4.;
380
381 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
382
383 PHI_2, A2, B2, H2, K2, &rtn);
384
385 sprintf (msg,”Case 0-2: area = %15.8f, return\\_value = \%d\textbackslash
n”, area, rtn);
386
387 printf (msg);
388
389 sprintf (msg,” ellipse 1 area by pi*a1*b1 = \%15.8f\textbackslash n”,
pi*A1*B1);
390
391 printf (msg);
392
393
394
395 //– case 0-3
396
397 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
398
399 A2 = 1.5; B2 = 0.75; H2 = -2.5; K2 = 1.5; PHI_2 = pi/4.;
400
401 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
402
403 PHI_2, A2, B2, H2, K2, &rtn);
404
405 sprintf (msg,”Case 0-3: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
406
407 printf (msg);
408
409 printf (” Ellipses are disjoint, ovelap area = 0.0\textbackslash
n\textbackslash n”);
410
411
412
413 //– case 1-1
414
415 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI\_1 = 0.;
416
417 A2 = 2.; B2 = 1.; H2 = -1.0245209260022; K2 = 0.25; PHI_2 = pi/4.;
418
419 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
420
421 PHI_2, A2, B2, H2, K2, \&rtn);
422
423 sprintf (msg,”Case 1-1: area = \%15.8f, return\\_value = \%d\textbackslash
n”, area, rtn);
424
425 printf (msg);
426
427 sprintf (msg,” ellipse 2 area by pi*a2*b2 = \%15.8f\textbackslash n”,
pi*A2*B2);
428
429 printf (msg);
430
431
432
433 //– case 1-2
434
435 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
436
437 A2 = 3.5; B2 = 1.8; H2 = .22; K2 = .1; PHI_2 = pi/4.;
438
439 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
440
441 PHI_2, A2, B2, H2, K2, \&rtn);
442
443 sprintf (msg,”Case 1-2: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
444
445 printf (msg);
446
447 sprintf (msg,” ellipse 1 area by pi*a1b1 = \%15.8f\textbackslash n”,
pi*A1*B1);
448
449 printf (msg);
450
451
452
453 //– case 1-3
454
455 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
456
457 A2 = 1.5; B2 = 0.75; H2 = -2.01796398085; K2 = 1.25; PHI_2 = pi/4.;
458
459 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
460
461 PHI_2, A2, B2, H2, K2, \&rtn);
462
463 sprintf (msg,”Case 1-3: area = %15.8f, return\\_value = \%d\textbackslash
n”, area, rtn);
464
465 printf (msg);
466
467 printf (” Ellipses are disjoint, ovelap area = 0.0\textbackslash
n\textbackslash n”);
468
469
470
471 //– case 2-1
472
473 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
474
475 A2 = 2.25; B2 = 1.5; H2 = 0.; K2 = 0.; PHI_2 = pi/4.;
476
477 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
478
479 PHI_2, A2, B2, H2, K2, \&rtn);
480
481 sprintf (msg,”Case 2-1: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
482
483 printf (msg);
484
485 sprintf (msg,” ellipse 2 area by pi*a2*b2 = \%15.8f\textbackslash n”,
pi*A2*B2);
486
487 printf (msg);
488
489
490
491 //– case 2-2
492
493 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
494
495 A2 = 3.; B2 = 1.7; H2 = 0.; K2 = 0.; PHI_2 = pi/4.;
496
497 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
498
499 PHI_2, A2, B2, H2, K2, \&rtn);
500
501 sprintf (msg,”Case 2-2: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
502
503 printf (msg);
504
505 sprintf (msg,” ellipse 1 area by pi*a1b1 = \%15.8f\textbackslash n”,
pi*A1*B1);
506
507 printf (msg);
508
509
510
511 //– case 2-3
512
513 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
514
515 A2 = 2.; B2 = 1.; H2 = -2.; K2 = -1.; PHI_2 = pi/4.;
516
517 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
518
519 PHI_2, A2, B2, H2, K2, \&rtn);
520
521 sprintf (msg,”Case 2-3: area = \%15.8f, return\\_value = \%d\textbackslash
n\textbackslash n”, area, rtn);
522
523 printf (msg);
524
525
526
527 //– case 3-1
528
529 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
530
531 A2 = 3.; B2 = 1.; H2 = 1.; K2 = 0.35; PHI_2 = pi/4.;
532
533 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
534
535 PHI_2, A2, B2, H2, K2, \&rtn);
536
537 sprintf (msg,”Case 3-1: area = \%15.8f, return\\_value = \%d\textbackslash
n”, area, rtn);
538
539 printf (msg);
540
541
542
543 //– case 3-2
544
545 A1 = 2.; B1 = 1.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
546
547 A2 = 2.25; B2 = 1.5; H2 = 0.3; K2 = 0.; PHI_2 = pi/4.;
548
549 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
550
551 PHI_2, A2, B2, H2, K2, \&rtn);
552
553 sprintf (msg,”Case 3-2: area = \%15.8f, return\\_value = \%d\textbackslash
n\textbackslash n”, area, rtn);
554
555 printf (msg);
556
557
558
559 //– case 4-1
560
561 A1 = 3.; B1 = 2.; H1 = 0.; K1 = 0.; PHI_1 = 0.;
562
563 A2 = 3.; B2 = 1.; H2 = 1.; K2 = -0.5; PHI_2 = pi/4.;
564
565 area = ellipse_ellipse_overlap (PHI_1, A1, B1, H1, K1,
566
567 PHI_2, A2, B2, H2, K2, \&rtn);
568
569 sprintf (msg,”Case 4-1: area = \%15.8f, return_value = \%d\textbackslash
n”, area, rtn);
570
571 printf (msg);
572
573
574
575 return rtn;
576
577 }
## References
* [1] Kent, S., Kaiser, M. E., Deustua, S. E., Smith, J. A. _Photometric calibrations for 21 st century science_, Astronomy 2010 8 (2009).
* [2] M. Chraibi, A. Seyfried, and A. Schadschneider, _Generalized centrifugal force model for pedestrian dynamics_ , Phys. Rev. E, 82 (2010), 046111.
* [3] Nonweiler, Terence R.F., _CACM Algorithm 326: Roots of low order polynomials_ , Communications of the ACM, vol. 11 no. 4, pages 269-270 (1968). Translated into c and programmed by M. Dow, ANUSF, Australian National University, Canberra, Australia. Accessed at http://www.netlib.org/toms/326.
* [4] Abramowitz, M. and Stegun, I. A. (Eds.). _Solutions of Quartic Equations._
|
arxiv-papers
| 2011-06-19T22:27:32 |
2024-09-04T02:49:19.870402
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gary B. Hughes and Mohcine Chraibi",
"submitter": "Mohcine Chraibi",
"url": "https://arxiv.org/abs/1106.3787"
}
|
1106.3797
|
# Momentum dependence of the symmetry potential and its influence on nuclear
reactions
Zhao-Qing Feng 111Corresponding author. Tel. +86 931 4969215.
_E-mail address:_ fengzhq@impcas.ac.cn (Z.-Q. Feng)
_Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
People’s Republic of China_
Abstract
A Skyrme-type momentum-dependent nucleon-nucleon force distinguishing isospin
effect is parameterized and further implemented in the Lanzhou Quantum
Molecular Dynamics (LQMD) model for the first time, which leads to a splitting
of nucleon effective mass in nuclear matter. Based on the isospin- and
momentum-dependent transport model, we investigate the influence of momentum-
dependent symmetry potential on several isospin-sensitive observables in
heavy-ion collisions. It is found that symmetry potentials with and without
the momentum dependence but corresponding to the same density dependence of
the symmetry energy result in different distributions of the observables. The
mid-rapidity neutron/proton ratios at high transverse momenta and the
excitation functions of the total $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$ yields
are particularly sensitive to the momentum dependence of the symmetry
potential.
_PACS_ : 24.10.Lx, 25.75.-q, 25.70.Pq
_Keywords:_ LQMD model; mass splitting; momentum-dependent potential; isospin-
sensitive observables; symmetry energy
Heavy-ion collisions induced by neutron-rich nuclei at intermediate and
relativistic energies in terrestrial laboratories are a useful tool to extract
the information of nuclear equation of state (EoS) of isospin asymmetric
nuclear matter, which is poorly known for the high-density neutron-rich matter
but has an important application in astrophysics, such as the structure of
neutron star, the cooling of protoneutron stars, the nucleosynthesis during
supernova explosion of massive stars etc [1]. The EoS of nuclear matter is
usually expressed through the energy per nucleon as
$E(\rho,\delta)=E(\rho,\delta=0)+E_{\textrm{sym}}(\rho)\delta^{2}+\textsc{O}(\delta^{2})$
in terms of baryon density $\rho=\rho_{n}+\rho_{p}$, relative neutron excess
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$, energy per nucleon in a
symmetric nuclear matter $E(\rho,\delta=0)$ and bulk nuclear symmetry energy
$E_{\textrm{sym}}=\frac{1}{2}\frac{\partial^{2}E(\rho,\delta)}{\partial\delta^{2}}\mid_{\delta=0}$.
Based on several complementary analysis of available experimental data
associated with transport models, a symmetry energy of
$E_{\textrm{sym}}(\rho)\approx 31.6(\rho/\rho_{0})^{\gamma}$ MeV with
$\gamma=0.69-1.05$ was extracted for densities between 0.1$\rho_{0}$ and
1.2$\rho_{0}$ [2]. However, predictions for high-density symmetry energies
based on various microscopical or phenomenological many-body theories diverge
widely [3, 4, 5]. More realistic approach to extract the information of the
$E_{\textrm{sym}}(\rho)$ is the comparison between transport model
calculations and experimental data. For that, the reliable input potentials
including the isovector (symmetry potential) and isoscalar parts in the
transport models are very necessary for precisely predicting some promising
observables.
The momentum dependence of the isoscalar potential leads to the same nucleon
effective mass (Landau mass) for neutrons and protons in nuclear matter and
has been widely studied in transport models for heavy-ion collisions. The
effective mass splitting of neutrons and protons results from the momentum-
dependent symmetry potential, which has been implemented in the one-body
transport models, such as the isospin Boltzmann-Uehling-Uhlenbeck (IBUU04)
model [6] and the stochastic mean-field (SMF) model [7, 8]. In this work, a
Skyrme-type momentum dependent nucleon-nucleon (NN) force distinguishing
protons and neutrons is parameterized and then included in a N-body approach
(LQMD model) for the first time. Furthermore, effects of the momentum
dependence in heavy-ion collisions are investigated. In particular, its
influence on the isospin sensitive observables to extract the high-density
symmetry energy is discussed.
The LQMD model has been successfully applied to treat the dynamics in heavy-
ion fusion reactions near Coulomb barrier and also to describe the capture of
two heavy colliding nuclides to form a superheavy nucleus [9, 10]. Further
improvements of the LQMD model were performed in order to investigate the
dynamics of pion and strangeness productions in heavy-ion collisions and also
to extract the information of isospin asymmetric EoS at supra-saturation
densities [11, 12, 13, 14]. In the previous versions, we only considered the
scalar part of the momentum-dependent interaction and the density-dependent
symmetry potential. We have included the resonances ($\Delta$(1232), N*(1440),
N*(1535)), hyperons ($\Lambda$, $\Sigma$) and mesons ($\pi$, $K$, $\eta$) in
hadron-hadron collisions and the decays of resonances for treating heavy-ion
collisions in the region of 1A GeV energies. The time evolutions of the
baryons (nucleons and resonances) and mesons in the system under the self-
consistently generated mean-field are governed by Hamilton’s equations of
motion, which read as
$\displaystyle\dot{\mathbf{p}}_{i}=-\frac{\partial
H}{\partial\mathbf{r}_{i}},\quad\dot{\mathbf{r}}_{i}=\frac{\partial
H}{\partial\mathbf{p}_{i}}.$ (1)
We only consider the Coulomb interaction for charged hyperons. The Hamiltonian
of baryons consists of the relativistic energy, the effective interaction
potential and the momentum dependent part as follows:
$H_{B}=\sum_{i}\sqrt{\textbf{p}_{i}^{2}+m_{i}^{2}}+U_{int}+U_{mom}.$ (2)
Here the $\textbf{p}_{i}$ and $m_{i}$ represent the momentum and the mass of
the baryons.
The effective interaction potential is composed of the Coulomb interaction and
the local interaction
$U_{int}=U_{Coul}+U_{loc}.$ (3)
The Coulomb interaction potential is written as
$U_{Coul}=\frac{1}{2}\sum_{i,j,j\neq
i}\frac{e_{i}e_{j}}{r_{ij}}erf(r_{ij}/\sqrt{4L})$ (4)
where the $e_{j}$ is the charged number including protons and charged
resonances. The $r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|$ is the relative
distance of two charged particles.
The local interaction potential is derived directly from the Skyrme energy-
density functional and expressed as
$U_{loc}=\int V_{loc}(\rho(\mathbf{r}))d\mathbf{r}.$ (5)
The local potential energy-density functional reads
$\displaystyle V_{loc}(\rho)=$
$\displaystyle\frac{\alpha}{2}\frac{\rho^{2}}{\rho_{0}}+\frac{\beta}{1+\gamma}\frac{\rho^{1+\gamma}}{\rho_{0}^{\gamma}}+\frac{g_{sur}}{2\rho_{0}}(\nabla\rho)^{2}+\frac{g_{sur}^{iso}}{2\rho_{0}}[\nabla(\rho_{n}-\rho_{p})]^{2}$
(6)
$\displaystyle+E_{sym}^{loc}(\rho)\rho\delta^{2}+g_{\tau}\rho^{8/3}/\rho_{0}^{5/3},$
where the $\rho_{n}$, $\rho_{p}$ and $\rho=\rho_{n}+\rho_{p}$ are the neutron,
proton and total densities, respectively, and the
$\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ is the isospin asymmetry. The
coefficients $\alpha$, $\beta$, $\gamma$, $g_{sur}$, $g_{sur}^{iso}$,
$g_{\tau}$ are related to the Skyrme parameters $t_{0},t_{1},t_{2},t_{3}$ and
$x_{0},x_{1},x_{2},x_{3}$ [10]. The $E_{sym}^{loc}$ is the local part of the
symmetry energy, which can be adjusted to mimic predictions calculated by
microscopical or phenomenological many-body theories and has two-type forms as
follows:
$E_{sym}^{loc}(\rho)=\frac{1}{2}C_{sym}(\rho/\rho_{0})^{\gamma_{s}},$ (7)
and
$E_{sym}^{loc}(\rho)=a_{sym}(\rho/\rho_{0})+b_{sym}(\rho/\rho_{0})^{2}.$ (8)
The parameters $C_{sym}$, $a_{sym}$ and $b_{sym}$ are taken as 52.5 MeV, 43
MeV, -16.75 MeV and 38 MeV, 37.7 MeV, -18.7 MeV for the cases with and without
momentum-dependent interactions, respectively. The values of $\gamma_{s}$=0.5,
1., 2. correspond to the soft, linear and hard symmetry energy, respectively,
and the Eq. (8) gives a supersoft symmetry energy, which cover the largely
uncertain of nuclear symmetry energy, particularly at the supra-saturation
densities.
We have taken the same Skyrme-type form for the momentum-dependent potential
in the Hamiltonian as in Ref. [15] but distinguishing isospin effect, which is
expressed as
$U_{mom}=\frac{1}{2\rho_{0}}\sum_{i,j,j\neq
i}\sum_{\tau,\tau^{\prime}}C_{\tau,\tau^{\prime}}\delta_{\tau,\tau_{i}}\delta_{\tau^{\prime},\tau_{j}}\int\int\int
d\textbf{p}d\textbf{p}^{\prime}d\textbf{r}f_{i}(\textbf{r},\textbf{p},t)[\ln(\epsilon(\textbf{p}-\textbf{p}^{\prime})^{2}+1)]^{2}f_{j}(\textbf{r},\textbf{p}^{\prime},t).$
(9)
The term is also given from the energy-density functional in nuclear matter,
$U_{mom}=\frac{1}{2\rho_{0}}\sum_{\tau,\tau^{\prime}}C_{\tau,\tau^{\prime}}\int\int\int
d\textbf{p}d\textbf{p}^{\prime}d\textbf{r}f_{\tau}(\textbf{r},\textbf{p})[\ln(\epsilon(\textbf{p}-\textbf{p}^{\prime})^{2}+1)]^{2}f_{\tau^{\prime}}(\textbf{r},\textbf{p}^{\prime}).$
(10)
Here $C_{\tau,\tau}=C_{mom}(1+x)$, $C_{\tau,\tau^{\prime}}=C_{mom}(1-x)$
($\tau\neq\tau^{\prime}$) and the isospin symbols $\tau$($\tau^{\prime}$)
represent proton or neutron. The sign of $x$ determines different mass
splitting of proton and neutron in nuclear medium, e.g. positive signs
corresponding to the case of $m^{\ast}_{n}<m^{\ast}_{p}$. The parameters
$C_{mom}$ and $\epsilon$ were determined by fitting the real part of optical
potential as a function of incident energy from the proton-nucleus elastic
scattering data, which determine the nucleon effective mass in isospin
symmetric nuclear matter. In the calculation, we take the values of 1.76 MeV,
500 c2/GeV2 and -0.65 for the $C_{mom}$, $\epsilon$ and $x$, respectively,
which result in the effective mass $m^{\ast}/m$=0.75 in nuclear medium at
saturation density for symmetric nuclear matter. For the cold nuclear matter,
we have the phase-space density
$f_{\tau}(\textbf{r},\textbf{p})=\rho_{\tau}(\textbf{r})\Theta(p_{F}(\tau)-|\textbf{p}|)/(4\pi
p_{F}^{3}(\tau)/3)$ with the Fermi momentum
$p_{F}(\tau)=\hbar(3\pi^{2}\rho_{\tau})^{1/3}$. Implementing the phase-space
distribution into Eq. (10), we get the contribution of the momentum dependence
to the symmetry energy $E_{sym}^{mom}(\rho)$. Therefore, the symmetry energy
per nucleon in the LQMD model is composed of three parts, namely the kinetic
energy, the local part and the momentum dependence of the potential energy as
$E_{sym}(\rho)=\frac{1}{3}\frac{\hbar^{2}}{2m}\left(\frac{3}{2}\pi^{2}\rho\right)^{2/3}+E_{sym}^{loc}(\rho)+E_{sym}^{mom}(\rho).$
(11)
Figure 1 is a comparison of different stiffness of nuclear symmetry energy
after inclusion of the momentum-dependent interactions. One can see that all
cases cross at saturation density with the value of 31.5 MeV. The local part
of the symmetry energy can be adjusted to reflect the uncertain behavior of
the symmetry energy at sub- and supra-normal densities. The contributions of
the local and momentum-dependent interactions of the potential part in the
symmetry energy is shown in Fig. 2. The momentum dependence has a positive
contribution to the symmetry energy in the mass splitting of
$m^{\ast}_{n}<m^{\ast}_{p}$, which is opposite for the case of
$m^{\ast}_{n}>m^{\ast}_{p}$. The local part of the symmetry energy is adjusted
to get the same density dependence of the symmetry energy for a given
stiffness with increasing density.
Combined Eq. (6) and Eq. (10), we get a density, isospin and momentum-
dependent single-particle potential in nuclear matter as follows:
$\displaystyle U_{\tau}(\rho,\delta,\textbf{p})=$
$\displaystyle\alpha\frac{\rho}{\rho_{0}}+\beta\frac{\rho^{\gamma}}{\rho_{0}^{\gamma}}+\frac{8}{3}g_{\tau}\rho^{5/3}/\rho_{0}^{5/3}+E_{sym}^{loc}(\rho)\delta^{2}+\frac{\partial
E_{sym}^{loc}(\rho)}{\partial\rho}\rho\delta^{2}+E_{sym}^{loc}(\rho)\rho\frac{\partial\delta^{2}}{\partial\rho_{\tau}}$
(12) $\displaystyle+\frac{1}{\rho_{0}}C_{\tau,\tau}\int
d\textbf{p}^{\prime}f_{\tau}(\textbf{r},\textbf{p})[\ln(\epsilon(\textbf{p}-\textbf{p}^{\prime})^{2}+1)]^{2}$
$\displaystyle+\frac{1}{\rho_{0}}C_{\tau,\tau^{\prime}}\int
d\textbf{p}^{\prime}f_{\tau^{\prime}}(\textbf{r},\textbf{p})[\ln(\epsilon(\textbf{p}-\textbf{p}^{\prime})^{2}+1)]^{2}.$
Here $\tau\neq\tau^{\prime}$,
$\partial\delta^{2}/\partial\rho_{n}=4\delta\rho_{p}/\rho^{2}$ and
$\partial\delta^{2}/\partial\rho_{p}=-4\delta\rho_{n}/\rho^{2}$. The effective
(Landau) mass in nuclear matter is calculated through the potential as
$m_{\tau}^{\ast}=m_{\tau}/\left(1+\frac{m_{\tau}}{|\textbf{p}|}|\frac{dU_{\tau}}{d\textbf{p}}|\right)$
with the free mass $m_{\tau}$ at Fermi momentum $\textbf{p}=\textbf{p}_{F}$.
Therefore, the nucleon effective mass only depends on the momentum-dependent
interactions. The mass splitting of protons and neutrons in nuclear matter as
functions of density ($\delta$=0.2) and isospin asymmetry
($\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$) at saturation density is
shown in Fig. 3. The left windows are the cases of $m^{\ast}_{n}<m^{\ast}_{p}$
with the parameter $x$=0.65 in the coefficient $C_{\tau,\tau^{\prime}}$ and
the right panels are $m^{\ast}_{n}>m^{\ast}_{p}$. The two sorts of the mass
splitting can be chosen in the LQMD calculations. In accordance with the Lane
potential [16], the symmetry potential can be evaluated from the single-
nucleon potential
$U_{sym}(\rho,\textbf{p})=(U_{n}(\rho,\delta,\textbf{p})-U_{p}(\rho,\delta,\textbf{p}))/2\delta$.
A hard core scattering in two-particle collisions is assumed in the simulation
of the collision processes by Monte Carlo procedures, in which the scattering
of two particles is determined by a geometrical minimum distance criterion
$d\leq\sqrt{0.1\sigma_{tot}/\pi}$ fm weighted by the Pauli blocking of the
final states [17, 18]. Here, the total cross section $\sigma_{tot}$ in mb is
the sum of the elastic and all inelastic cross sections. The probability
reaching a channel in a collision is calculated by its contribution of the
channel cross section to the total cross section as
$P_{ch}=\sigma_{ch}/\sigma_{tot}$. The choice of the channel is done randomly
by the weight of the probability. The primary products in nucleon-nucleon (NN)
collisions in the region of 1A GeV energies are the resonances $\Delta$(1232),
$N^{\ast}$(1440), $N^{\ast}$(1535) and the pions. We have included the
reaction channels as follows:
$\displaystyle NN\leftrightarrow N\triangle,\quad NN\leftrightarrow
NN^{\ast},\quad NN\leftrightarrow\triangle\triangle,$
$\displaystyle\Delta\leftrightarrow N\pi,N^{\ast}\leftrightarrow
N\pi,NN\rightarrow NN\pi(s-state),N^{\ast}(1535)\rightarrow N\eta.$ (13)
At the considered energies, there are mostly $\Delta$ resonances which
disintegrate into a $\pi$ and a nucleon in the evolutions. However, the
$N^{\ast}$ yet gives considerable contribution to the energetic pion yields.
The energy and momentum-dependent decay widths are used in the calculation
[11] for the $\Delta$(1232) and $N^{\ast}$(1440) resonances. We have taken a
constant width $\Gamma$=150 MeV for the $N^{\ast}$(1535) decay. The
strangeness is created by inelastic hadron-hadron collisions [14, 19]. We
included the channels as follows:
$\displaystyle BB\rightarrow BYK,BB\rightarrow BBK\overline{K},B\pi\rightarrow
YK,B\pi\rightarrow NK\overline{K},$ $\displaystyle Y\pi\rightarrow
B\overline{K},\quad B\overline{K}\rightarrow Y\pi,\quad
YN\rightarrow\overline{K}NN.$ (14)
Here the B strands for (N, $\triangle$, N∗) and Y($\Lambda$, $\Sigma$), K(K0,
K+) and $\overline{K}$($\overline{K^{0}}$, K-). The elastic scattering between
strangeness and baryons are considered through the channels $KB\rightarrow
KB$, $YB\rightarrow YB$ and $\overline{K}B\rightarrow\overline{K}B$. The
evolutions of mesons ($\pi$, $K$, $\eta$) are also determined by the
Hamiltonian in which the Coulomb interaction and the in-medium potential were
considered in the model [13, 19].
To check the influence of the momentum dependence of the symmetry potential on
reaction dynamics, we calculated the transverse momentum distributions of the
ratios of neutrons over protons in the mid-rapidity domain and the excitation
functions of charged pion yields as shown in Fig. 4. Inclusion of the
momentum-dependent interaction in the symmetry potential reduces the n/p
yields at high transverse momentum owing to its negative contribution to the
symmetry energy at the case of $m^{\ast}_{n}>m^{\ast}_{p}$ in nuclear medium,
which enforces an attractive force in neutron-neutron collisions and further
increases the collision probabilities. Contrarily, the case without the
isovector of the momentum-dependent interaction quickly squeezes out neutrons,
in particular for the energic neutrons, which nearly appears a flat
distribution at $p_{t}>$0.3 GeV/c. The conclusions are consistent with the
calculations based on the IBUU04 transport model [6]. The charged pion ratios
are slightly changed by the momentum-dependent potential. Therefore, the
neutron and proton transverse emission ratio in the mid-rapidity region is to
be a nice probe of the isovector part of the momentum-dependent interaction.
The pions and strange particles are mainly produced in the dense hadronic
matter formed in heavy-ion collisions. The $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$
ratios can be probes to extract the high-density information of isospin
asymmetric EoS. Shown in Fig. 5 is a comparison of the excitation functions of
the $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$ ratios with and without the momentum
dependence of the symmetry potential in the 197Au+197Au reaction for head-on
collisions. One can see that the momentum-dependent interaction by
distinguishing isospin effect reduces the $\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$
yields, which is more pronounced close to the threshold energies of meson
production. The results are caused from the fact that the interaction enhances
the energic neutron-neutron collisions. Furthermore, the $\pi^{-}$ and $K^{0}$
is produced through the channels $nn\rightarrow p\Delta^{-}$,
$\Delta^{-}\rightarrow n\pi^{-}$ and $nn\rightarrow n\Lambda K^{0}$.
In summary, influence of the momentum dependence of the symmetry potential on
isospin sensitive observables in heavy-ion collisions is investigated by using
an isospin- and momentum-dependent transport model (LQMD). It is found that
the momentum dependence of the symmetry potential plays an important role on
nucleon transverse emissions and the ratios of $\pi^{-}/\pi^{+}$ and
$K^{0}/K^{+}$, which are also as promising probes of high-density symmetry
energy. To precisely constrain the density dependence of the nuclear symmetry
energy, one firstly needs to get the accurate information of the density- and
momentum-dependent symmetry potential. Further experimental data of
determining the momentum-dependent symmetry potential are very necessary to
investigate accurately the dense neutron-rich matter. The updated LQMD model
would to be a useful tool to predict the density dependence of the symmetry
energy from heavy-ion collisions, in particular at supra-normal densities, and
also to analyze experimental data for constraining the symmetry energy.
Acknowledgements
This work was supported by the National Natural Science Foundation of China
under Grant No. 10805061; the Special Foundation of the President Fund and the
West Doctoral Project of Chinese Academy of Sciences.
## References
* [1] A.W. Steiner, M. Prakash, J.M. Lattimer, P. J. Ellis, Phys. Rep. 411 (2005) 325.
* [2] B.A. Li, L.W. Chen, C.M. Ko, Phys. Rep. 464 (2008) 113.
* [3] Dieperink, et al., Phys. Rev. C 68 (2003) 064307.
* [4] L.W. Chen, C.M. Ko, B.A. Li, Phys. Rev. C 72 (2005) 064309; Phys. Rev. C 76 (2007) 054316.
* [5] Z.H. Li, et al., Phys. Rev. C 74 (2006) 047304.
* [6] B.A. Li, C. B. Das, S. Das Gupta, C. Gale, Phys. Rev. C 69 (2004) 011603(R); Nucl. Phys. A 735 (2004) 563.
* [7] M. Di Toro, et al., Prog. Part. Nucl. Phys. 42 (1999) 125.
* [8] V. Baran, M. Colonna, V. Greco, M. Di Toro, Phys. Rep. 410 (2005) 335.
* [9] Z.Q. Feng, F.S. Zhang, G.M. Jin, X. Huang, Nucl. Phys. A 750 (2005) 232; Z.Q. Feng, et al., Chin. Phys. Lett. 22 (2005) 3040.
* [10] Z.Q. Feng, G.M. Jin, F.S. Zhang, Nucl. Phys. A 802 (2008) 91; Z.Q. Feng, G.M. Jin, Phys. Rev. C 80 (2009) 037601.
* [11] Z.Q. Feng, G.M. Jin, Chin. Phys. Lett. 26 (2009) 062501.
* [12] Z.Q. Feng, G.M. Jin, Phys. Lett. B 683 (2010) 140.
* [13] Z.Q. Feng, G.M. Jin, Phys. Rev. C 82 (2010) 044615.
* [14] Z.Q. Feng, G.M. Jin, Phys. Rev. C 82 (2010) 057901.
* [15] J. Aichelin, A. Rosenhauer, G. Peilert, et al., Phys. Rev. Lett. 58 (1987) 1926.
* [16] A.M. Lane, Nucl. Phys. 35 (1962) 676.
* [17] J. Aichelin, Phys. Rep. 202 (1991) 233.
* [18] G.F. Bertsch, S. Das Gupta, Phys. Rep. 160 (1988) 190.
* [19] Z.Q. Feng, arXiv:1102.4696.
Figure 1: Density dependence of the nuclear symmetry energy for the cases of
supersoft, soft, linear and hard trends. Figure 2: Density dependence of the
potential part of nuclear symmetry energy at different mass splitting for hard
(left panel) and supersoft (right panel) symmetry energies. Figure 3: Nucleon
effective mass normalized by vacuum mass as functions of density
($\delta=0.2$) and isospin asymmetry ($\rho=\rho_{0}$). Figure 4: Transverse
momentum distributions of the ratios of neutron/proton within the rapidity bin
$|y/y_{proj}|<$0.3 and $\pi^{-}/\pi^{+}$ in central 124Sn+124Sn collisions at
incident energy 400A MeV. Figure 5: Comparison of excitation functions of the
$\pi^{-}/\pi^{+}$ and $K^{0}/K^{+}$ yields for head-on collisions in the
197Au+197Au reaction with and without the isovector part of momentum
dependence.
|
arxiv-papers
| 2011-06-20T03:27:02 |
2024-09-04T02:49:19.897045
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhao-Qing Feng",
"submitter": "Zhaoqing Feng",
"url": "https://arxiv.org/abs/1106.3797"
}
|
1106.3843
|
# Spin-dependent transport for armchair-edge graphene nanoribbons between
ferromagnetic leads
Benhu Zhou,1 Xiongwen Chen,1 Benliang Zhou,1 Kai-He Ding,2 Guanghui Zhou1,3
ghzhou@hunnu.edu.cn 1Department of Physics and Key Laboratory for Low-
Dimensional Structures and Quantum Manipulation (Ministry of Education), Hunan
Normal University, Changsha 410081, China 2Department of Physics and
Electronic Science, Changsha University of Science and Technology, Changsha
410076, China 3International Center for Materials Physics, Chinese Academy of
Sciences, Shenyang 110015, China
###### Abstract
We theoretically investigate the spin-dependent transport for the system of an
armchair-edge graphene nanoribbon (AGNR) between two ferromagnetic (FM) leads
with arbitrary polarization directions at low temperatures, where a magnetic
insulator is deposited on the AGNR to induce an exchange splitting between
spin-up and -down carriers. By using the standard nonequilibrium Green’s
function (NGF) technique, it is demonstrated that, the spin-resolved transport
property for the system depends sensitively on both the width of AGNR and the
polarization strength of FM leads. The tunneling magnetoresistance (TMR)
around zero bias voltage possesses a pronounced plateau structure for system
with semiconducting 7-AGNR or metallic 8-AGNR in the absence of exchange
splitting, but this plateau structure for 8-AGNR system is remarkably broader
than that for 7-AGNR one. Interestingly, the increase of exchange splitting
$\Delta$ suppresses the amplitude of the structure for 7-AGNR system. However,
the TMR is enhanced much for 8-AGNR system under the bias amplitude comparable
to splitting strength. Further, the current-induced spin transfer torque (STT)
for 7-AGNR system is systematically larger than that for 8-AGNR one. The
findings here suggest the design of GNR-based spintronic devices by using a
metallic AGNR, but it is more favorable to fabricate a current-controlled
magnetic memory element by using a semiconducting AGNR.
###### pacs:
73.63.-b, 85.35.-p, 72.25.Hg
## I Introduction
Spintronics utilizes the electron spin degree of freedom to carry information
in electronic devices and functionalities. Its central task is to control
electron spin for information storage and precessing.1 Graphene, a single
layer of carbon, has low intrinsic spin-orbit and hyperfine couplings,2 long
spin diffusion lengths ($\sim$2$\mu$ m).3 These features suggest that graphene
is a promising candidate for the potential applications in spintronics.
Triggered by these findings, a large number of works have been devoted to the
spin-dependent transport in graphene and graphene-FM heterostructures, as
prototypical spintronic devices. For example, a graphene spin-valve device has
been successfully fabricated showing that spin-polarized currents can be
injected by means of magnetic (cobalt, permalloy, etc.) electrodes,4-9 which
shows high spin-polarized injection efficiency. Meanwhile, a 10%
magnetoresistance (MR) is observed 10 in a graphene contacted by two soft
magnetic electrodes. The further experiments have found that a inserted tunnel
film at the graphene/electrode interface favors MR which can reach up to
$\sim$12%.11 Additionally, it is expected12 that the spin dependence of the
electron energy can be also produced by a FM insulator substrate which acts as
an effective magnetic field parallel to the graphene layer.13,14.
Theoretically, using the tight-binding model, Brey and Fertig15 found that the
MR is rather small due to the weak dependence of the graphene conductivity on
the electronic parameters of FM leads. Using a continuous model by NGF, Ding
$et$ $al$.16 have demonstrated that the TMR exhibits a cusp around zero bias
in the absence of external magnetic field for a similar device. Recently,
special attentions have been paid to zigzag-edge graphene nanoribbons (ZGNRs)
spin-valve devices since ZGNRs exhibit fascinating phenomenon due to the size
confinement and the edge state.17-19 For example, some theoretical studies
have predicated that ZGNR-based spin-valve devices have a very large TMR.20-25
On the other hand, the leads (electrodes) for spin-valve devices in the
previous works have been supposed to be two-dimensional (of honeycomb- or
square-lattice) type, but more realistically they should be modeled as three-
dimensional semi-infinite slabs. Further, the investigation on the TMR for
GNR-magnetic junctions is still lacking so far, especially, the influence of
an exchange field on the spin-dependent transport. Another important
phenomenon in spintronics is the reverse effect to TMR, i.e., spin-polarized
electrons passing from the left FM layer into the right layer, where the
magnetization deviates the left by an angle, may exert a torque to the right
FM lead. This effect is the so-called26,27 STT which has been extensively
studied both theoretically28-32 and experimentally33,34 for conventional
magnetic junction systems. A demonstration of the spin-transfer phenomenon is
the current-induced magnetic switching, which has been confirmed
experimentally in spin-valves35,36 and magnetic tunnel junctions (MTJs).37
Thus, current-induced magnetic switching provides a powerful new tool for the
study of spin transport in magnetic nanostructures. In addition, it offers the
intriguing possibility of manipulating high-density nonvolatile magnetic-
device elements, such as magnetoresistive random access memory (MRAM), without
applying cumbersome magnetic fields.38 However, the previous works on spin-
dependent transport through a graphene-based system are mainly focused on the
tunnel current and the TMR effect.10-16,20-25 The investigation on the
current-induced STT in such system is sparsely reported. Although we have
addressed this issue for a FM/graphene/FM device,39 in this paper we extend it
to a FM/AGNR/FM system.
FIG. 1: Schematic illustration of the system consisting of an AGNR between two
FM leads, where a FM insulator layer with a control gate is deposited on AGNR.
A unit cell of AGNR represented by two dashed lines contains $n$ numbers of
$A$ and $B$ sublattice sites labeled as 1$A$, 1$B$, $\cdot\cdot\cdot$, $nA$,
$nB$. Two additional hard walls are imposed on both edges at chains $j$=0 and
$n$+1. The magnetic moment of the two leads is aligned at a relative angle
$\theta$.
In this paper, we present a theoretical investigation on the fully spin-
dependent transport through an AGNR between two FM leads with arbitrary
polarization direction at low temperatures, where a magnetic insulator is
deposited on AGNRs to induce an exchange splitting. By using the Keldysh NGF
method,40,41 the density of state (DOS), the linear conductance, the
differential conductance, and the current for the system have been calculated
separately for spin-up and -down channels, and consequently the TMR dependence
on applied bias has been calculated. Further, the dependence of STT on the
bias and the polarization angle of the two leads have also been examined. It
is demonstrated that, the spin-resolved transport property of the system
depends sensitively on both the widths of AGNRs and the polarization strength
of FM leads. Around zero bias the TMR versus bias voltage possesses a
pronounced plateau structure for both semiconducting 7-AGNR and metallic
8-AGNR systems without exchange splitting. The plateau structure for 8-AGNR
system is remarkably broader than that for 7-AGNR one. Interestingly, the
increase of the exchange splitting strength suppresses the amplitude of this
structure for 7-AGNR system. However, the TMR is enhanced at bias that ranged
from $-\Delta$ to $\Delta$ for 8-AGNR system. Further, the current-induced STT
for 7-AGNR system is systematically larger than that for 8-AGNR system.
The rest of the paper is organized as follows. In Sec. II, we derive the
analytical expressions for the spin-dependent DOS, the conductance, the
current, and the STT, starting from the system Hamiltonian by NGF approach.
Some numerical examples and the discussions for the results are demonstrated
in Sec. III. Finally, Sec. IV concludes the paper.
## II Model and Method
The geometry of the system considered in this paper is shown in Fig. 1, where
an AGNR contains two sublattices denoted by $A$ and $B$. We use $n$, the
number of $A(B)$-site atoms in a unit cell, to denote AGNR with different
width. Ideal (perfect) AGNRs with $n$=3$j$-1 with positive integer $j$ are
metallic, otherwise are semiconducting.18,19 From the top down, atoms in a
unit cell are labeled as 1$A$, 1$B$,$\cdots$, $n$$A$, $n$$B$. As shown in Fig.
1, the hard-wall condition is imposed on both edges at chains $j$=0 and $n+1$.
The definition of GNRs in this work is in accord with the previous
convention,19 i.e., an AGNR is identified by the number of carbon zigzag
chains forming the width of ribbon, and an AGNR with $n$ carbon chains is
named as $n$-AGNR, therefore its width is $W$=$(n-1)\sqrt{3}a/2$, where
$a$=1.42 ${\AA}$ is the C$-$C bond length.
The system under consideration here is composed of a semiconducting 7-AGNR or
a metallic 8-AGNR connected to two FM leads, which can be considered as a MTJ
(or a spin-valve) device. A layer of magnetic insulator is deposited on the
top of AGNR sample to induce an exchange splitting between spin-up and -down
carriers.12 The gate allows us to control the Fermi level locally, i.e., to
create a tunable barrier in AGNR. In this way, controlling on both charge and
spin carrier concentrations can be achieved. The magnetic moment $M_{L}$ of
the left FM lead is assumed to be parallel to the $y$-axis, while the moment
$M_{R}$ of the right FM deviates from the $y$-direction by a relative angle
$\theta$. The tunneling current flows along the $x$-axis [see Fig. 1]. We
assume that the magnitude of exchange splitting can be modulated by the
magnetic insulator. It seems a reasonable assumption that the magnetic
insulator is made of different materials from FM leads.
The total Hamiltonian for the system considered reads
$\displaystyle H=H_{G}+H_{L}+H_{R}+H_{T},$ (1)
where $H_{G}$ describes the central AGNR region, $H_{L(R)}$ is the Hamiltonian
for the left (right) FM lead, and $H_{T}$ for the coupling between AGNR and
leads. Here, spin-orbit interaction in graphene or GNRs is neglected because
it is too weak and is of the order of 3$-$4 meV,42,43 it opens up a gap of the
order of 10-3 meV at the Dirac point. In the tight-binding approximation,
these partial Hamiltonians can be respectively written as following:
$\displaystyle H_{G}=\sum_{\beta
i,\sigma}(\epsilon_{0}+s\Delta)a^{{\dagger}}_{\beta i,\sigma}a_{\beta
i,\sigma}+t\sum_{<ij>,\sigma}(a^{{\dagger}}_{Ai,\sigma}a_{Bj,\sigma}+\text{H.c}),$
(2)
where $a^{{\dagger}}_{\beta i,\sigma}$ ($a_{\beta i,\sigma}$) creates
(annihilates) an electron on site $i$ with spin $\sigma$ and sublattice index
$\beta$=$A(B)$, $\sigma$=$\uparrow$ ($\downarrow$) represents the spin-up
(-down) state of electrons, $\epsilon_{0}$ is the on-site energy, $t$
($\approx$2.75 eV) the nearest-neighbor hopping energy, and $\Delta$ the
exchange splitting energy induced by the FM insulator on the top of AGNR,
s=$\pm 1$ stands for the electron spin parallel ($+$) or antiparallel ($-$) to
the exchange field. The sum over $\langle i,j\rangle$ is restricted to the
nearest-neighbor atoms. Hamiltonian (2) can also be rewritten in momentum
space
$\displaystyle
H_{G}=\sum_{\beta,\textbf{q}\sigma}(\epsilon_{0}+s\Delta)a^{\dagger}_{\beta,\textbf{q}\sigma}a_{\beta,\textbf{q}\sigma}+\sum_{\textbf{q}\sigma}[\phi(\textbf{q})a^{\dagger}_{A,\textbf{q}\sigma}a_{B,\textbf{q}\sigma}+\text{H.c.}],$
(3)
where
$\phi(\textbf{q})=-t[2e^{iq_{x}a/2}\text{cos}(\frac{\sqrt{3}a}{2}q_{y})+e^{-iq_{x}a}]$
is the structural factor with wave-vector in the $x$-direction within the
first Brillouin zone (0$\leq$$|q_{x}|$$\leq$$\pi/(3a)$) and the discretized
wave-vector in the $y$-direction $q_{y}$=$\frac{2}{\sqrt{3}a}\frac{m\pi}{n+1}$
($m$=1,2, $\cdots$, $n$).
The two Hamiltonians
$\displaystyle
H_{L}=\sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}L\sigma}c_{\textbf{k}L\sigma}^{{\dagger}}c_{\textbf{k}L\sigma}$
(4)
and
$\displaystyle H_{R}=\sum_{\textbf{k},\sigma}[\epsilon_{R}(\textbf{k})-\sigma
M_{R}\cos\theta]c_{\textbf{k}R\sigma}^{{\dagger}}c_{\textbf{k}R\sigma}-M_{R}\sin\theta
c_{\textbf{k}R\sigma}^{{\dagger}}c_{\textbf{k}R\bar{\sigma}}$ (5)
are respectively for the left and right lead, where
$\epsilon_{\textbf{k}L\sigma}$=$\epsilon_{L}(\textbf{k})-eV-\sigma M_{L}$ with
applied bias voltage $V$ and magnetic momentum $M_{L}$,
$\epsilon_{L,(R)}(\textbf{k})$ is the single-particle dispersion for the left
(right) lead, $c^{{\dagger}}_{\textbf{k}L(R)\sigma}(c_{\textbf{k}L(R)\sigma})$
is the creation (annihilation) operator of an electron with wavevector k in
the left (right) lead, $\bar{\sigma}$ denotes the opposite spin-polarization
with respect to $\sigma$. Note that here the coupling between the left lead
and AGNR involves the $B$ sublattice, while it between the right lead and AGNR
involves the $A$ sublattice (see Fig. 1). However, Hamiltonian (5) will be
slightly different if the coupling at two interfaces involves the same
sublattice, but the result is unchanged.
The coupling Hamiltonian
$\displaystyle H_{T}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\textbf{k}L\textbf{q}\sigma}(T_{\textbf{k}L\textbf{q}}c_{\textbf{k}L\sigma}^{{\dagger}}a_{A,\textbf{q}\sigma}+\text{H.c.})$
(6)
$\displaystyle+\frac{1}{\sqrt{N}}\sum_{\textbf{k}R\textbf{q}\sigma}(T_{\textbf{k}R\textbf{q}}c_{\textbf{k}R\sigma}^{{\dagger}}a_{B,\textbf{q}\sigma}+\text{H.c.}),$
where $T_{\textbf{k}L\textbf{q}}$ ($T_{\textbf{k}R\textbf{q}}$) is the
coupling matrix between left (right) lead and AGNR, $N$ is the number of sites
in sublattice $A$ or $B$. It should be pointed out that the annihilation
operator $a_{B,\textbf{q}\sigma}$ in Eq. (6) should be replaced by
$a_{A,\textbf{q}\sigma}$ if the coupling between right lead and AGNR involves
the $A$ sublattice.
Now we employ the usually defined retarded, advanced and correlated Green’s
function (GF)40,41
$G_{\textbf{q}a,\textbf{q}^{\prime}a}^{\sigma\sigma^{\prime},r}(t_{2},t_{1})=-i\theta(t_{2}-t_{1})\langle{\\{a_{\textbf{q}\sigma}(t_{2}),a_{\textbf{q}^{\prime}\sigma^{\prime}}^{\dagger}{(t_{1})}}\\}\rangle,$
$G_{\textbf{q}a,\textbf{q}^{\prime}a}^{\sigma\sigma^{\prime},a}(t_{2},t_{1})=i\theta(-t_{2}$+$t_{1})\langle{\\{a_{\textbf{q}\sigma}(t_{2}),a_{\textbf{q}^{\prime}\sigma^{\prime}}^{\dagger}{(t_{1})}}\\}\rangle$,
and
$G_{\textbf{q}a,\textbf{q}^{\prime}a}^{\sigma\sigma^{\prime},<}(t_{2},t_{1})=i\langle{a_{\textbf{q}^{\prime}\sigma^{\prime}}^{\dagger}(t_{1}),a_{\textbf{q}\sigma}{(t_{2})}}\rangle$
to the total Hamiltonian in spin space. When the unperturbed GF
$g^{r,a}_{\textbf{q}a,\textbf{q}a}(\omega)$ of the AGNR is obtained, one can
obtain $G^{r,a}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega)$ and
$G^{<}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega)$ from the standard GF
technique and the Keldysh equation40
$\displaystyle
G^{r,a}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega)=\delta_{\textbf{q}\textbf{q}^{\prime}}g^{r,a}_{\textbf{q}a,\textbf{q}a}(\omega)+g^{r,a}_{\textbf{q}a,\textbf{q}a}(\omega)T^{r,a}(\omega)g^{r,a}_{\textbf{q}^{\prime}a,\textbf{q}^{\prime}a}(\omega),$
(7) $\displaystyle
G^{<}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega)=G^{r}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega)\Sigma^{<}(\omega)G^{a}_{\textbf{q}a,\textbf{q}^{\prime}a}(\omega),$
(8)
where
$T^{r,a}(\omega)=\Sigma^{r,a}(\omega)/[1$$-$$g^{r,a}_{aa}(\omega)\Sigma^{r,a}(\omega)]$
with
$g^{r,a}_{aa}(\omega)=1/N{\sum_{\textbf{q}}}g^{r,a}_{\textbf{q}a,\textbf{q}a}(\omega)$
and $\Sigma^{r,a}(\omega)=\mp
i/2[\bm{\Gamma}_{L\textbf{q}\textbf{q}^{\prime}}(\omega)+R\bm{\Gamma}_{R\textbf{q}\textbf{q}^{\prime}}R^{\dagger}]$
is the retarded/advanced self-energy, and
$\Sigma^{<}(\omega)=i[f_{L}(\omega)\bm{\Gamma}_{L\textbf{q}\textbf{q}^{\prime}}(\omega)+f_{R}(\omega)R\bm{\Gamma}_{R\textbf{q}\textbf{q}^{\prime}}R^{\dagger}]$
is the correlated self-energy. In the above equations, the matrix
$\displaystyle
R=\left(\begin{array}[]{cc}\text{cos}(\theta/2)&-\text{sin}(\theta/2)\\\
\text{sin}(\theta/2)&\text{cos}(\theta/2)\\\ \end{array}\right),$
$f_{L(R)}(\omega)$=$1/[{e^{(\omega-\mu_{L(R)})/k_{B}T}+1}]$ is the Fermi-Dirac
distribution function of the left (right) FM lead, the linewidth function
$\displaystyle\bm{\Gamma}_{\alpha\textbf{q}\textbf{q}^{\prime}}(\omega)=\left(\begin{array}[]{cc}\bm{\Gamma}_{\alpha\textbf{q}\textbf{q}^{\prime}\uparrow}(\omega)&0\\\
0&\bm{\Gamma}_{\alpha\textbf{q}\textbf{q}^{\prime}\downarrow}(\omega)\\\
\end{array}\right)$
with
$\bm{\Gamma}^{\sigma}_{\alpha\textbf{q}\textbf{q}^{\prime}}(\omega)$=$2\pi\sum_{\textbf{k}}T^{*}_{\textbf{k}\alpha\textbf{q}}T_{\textbf{k}\alpha\textbf{q}^{\prime}}\delta(\omega-\varepsilon_{\textbf{k}\alpha\sigma})$
describes the influence of the leads. Here, we consider electrons near the
Fermi level $E_{F}$ which contribute predominantly to the tunneling. In this
case one may assume that the coupling matrix is independent of energy and set
$\bm{\Gamma}^{\sigma}_{\alpha\textbf{q}\textbf{q}^{\prime}}$=$\bm{\Gamma}^{\sigma}_{\alpha}$.
Furthermore, the isolated retarded GF for an AGNR is defined as
$\displaystyle
g^{r}_{aa}(\omega)=\sum_{q_{x},q_{y},\pm}\frac{|\psi\rangle_{\pm\pm}\langle\psi|}{[\omega-(\epsilon_{0}+s\Delta)]-E_{\pm}+i\eta^{+}}$
(11)
with a positive infinitesimal $\eta^{+}$, the dispersion relation
$E_{\pm}$=$\pm|\phi(\textbf{q})|$ and the wave function
$|\psi\rangle_{\pm}$=$\frac{\sqrt{2}}{2}(|\psi\rangle_{A}\pm\sqrt{\frac{\phi^{*}(\textbf{q})}{\phi(\textbf{q})}}|\psi\rangle_{B})$,
where the plus (minus) sign applies to the upper (lower) $\pi^{*}$ ($\pi$)
band,
$|\psi\rangle_{A}$=$\sqrt{\frac{2}{N_{x}(n+1)}}\sum^{n}_{m^{\prime}=1}\sum_{x_{A_{m^{\prime}}}}e^{iq_{x}x_{A_{m^{\prime}}}}\text{sin}(\frac{\sqrt{3}q_{y}a}{2}m^{\prime})|A_{m^{\prime}}\rangle$
and
$|\psi\rangle_{B}$=$\sqrt{\frac{2}{N_{x}(n+1)}}\sum^{n}_{m^{\prime}=1}\sum_{x_{B_{m^{\prime}}}}e^{iq_{x}x_{B_{m^{\prime}}}}\text{sin}(\frac{\sqrt{3}q_{y}a}{2}m^{\prime})|B_{m^{\prime}}\rangle$
with the number of unit cells $N_{x}$ along the $x$-direction.19 A direct
calculation yields expressions for the GF of an AGNR
$\displaystyle g^{r}_{aa}(\omega)$ $\displaystyle=$
$\displaystyle\frac{12a}{N_{x}(n+1)}\frac{\omega-(\epsilon_{0}+s\Delta)+i\eta^{+}}{2\pi}\int^{\frac{\pi}{3a}}_{\frac{-\pi}{3a}}dq_{x}\sum^{n}_{m,m^{\prime}=1}\frac{\text{sin}^{2}(\frac{m\pi}{n+1})m^{\prime}}{[\omega-(\epsilon_{0}+s\Delta)+i\eta^{+}]^{2}-t^{2}[1+4\text{cos}^{2}\frac{m\pi}{n+1}+4\text{cos}\frac{m\pi}{n+1}\text{cos}(3q_{x}/2)]}.$
(12)
The exchange term splits the system into two separate spin subsystems.
Therefore, the DOS, the linear conductance, and the current for spin-up and
-down channels can be respectively obtained as
$\displaystyle\rho_{\uparrow(\downarrow)}(\omega)=-\frac{1}{\pi}\text{Im}G_{a\uparrow\uparrow(\downarrow\downarrow)}^{r}(\omega),$
(13) $\displaystyle
G_{\uparrow(\downarrow)}=\frac{e^{2}}{h}X_{\uparrow\uparrow(\downarrow\downarrow)}(\omega)|_{\omega=E_{F}},$
(14) $\displaystyle
I_{\uparrow(\downarrow)}(V)=\frac{e}{\hbar}\int_{-eV/2}^{eV/2}\frac{d\omega}{2\pi}(f_{R}-f_{L})X_{\uparrow\uparrow(\downarrow\downarrow)}(\omega),$
(15)
where $V$ is the applied bias voltage, and the matrix elements in Eqs.
(11)-(13) are defined as
$\displaystyle
G_{a}^{r}(\omega)=\sum_{\textbf{q}\textbf{q}^{\prime}}G_{\textbf{q}a,\textbf{q}^{\prime}a}^{r(a)}(\omega)=\left(\begin{array}[]{cc}G_{a\uparrow\uparrow}^{r}(\omega)&G_{a\uparrow\downarrow}^{r}(\omega)\\\
G_{a\downarrow\uparrow}^{r}(\omega)&G_{a\downarrow\downarrow}^{r}(\omega)\\\
\end{array}\right)$ (18)
and
$\displaystyle
X(\omega)=[G_{a}^{r}(\omega)(R\bm{\Gamma}_{R}R^{{\dagger}})]G_{a}^{a}(\omega)\bm{\Gamma}_{L}=\left(\begin{array}[]{cc}X_{\uparrow\uparrow}(\omega)&X_{\uparrow\downarrow}(\omega)\\\
X_{\uparrow\downarrow}(\omega)&X_{\downarrow\downarrow}(\omega)\\\
\end{array}\right).$ (21)
On the other hand, the STT is the time evolution rate of the total spin of
left or right FM lead.26,27 Here, we ignored the interlayer exchange coupling,
so the out-of-plane torque is zero.44 By means of the NGF method, the current-
induced in-plane STT along $x^{\prime}$-direction in the
$(x^{\prime},y^{\prime})$ coordinate frame [see Fig. 1] exerting on the right
FM lead can be obtained32
$\displaystyle\tau_{R}^{x^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int
d\omega(f_{R}-f_{L})\text{Tr}[\text{G}_{a}^{r}(\omega)\Gamma_{L}\text{G}_{a}^{a}(\omega)R\Gamma_{R}R^{{\dagger}}$
(22) $\displaystyle\cdot(-\cos\theta\sigma_{x}+\sin\theta\sigma_{z})],$
where $\sigma_{x}$ and $\sigma_{z}$ are the Pauli matrices.
## III Results and Discussions
In what follows, we present some numerical examples of
$\rho_{\uparrow(\downarrow)}$, $G_{\uparrow(\downarrow)}$,
$G_{d,\uparrow(\downarrow)}$, TMR, and $\tau_{R}^{x^{\prime}}$ for the system
with 7- or 8-AGNR at low temperatures according to Eqs. (11)-(16). In the
calculation, the two FM leads are assumed to be made of the same materials,
i.e., $p_{L}$=$p_{R}$=$p$, where
$p_{L(R)}$=$(\Gamma_{L(R)\uparrow}-\Gamma_{L(R)\downarrow})/(\Gamma_{L(R)\uparrow}+\Gamma_{L(R)\downarrow})$
is the polarization strength of the left (right) FM. Under the wide bandwidth
approximation,
$\Gamma_{L\uparrow(\downarrow)}$=$\Gamma_{R\uparrow(\downarrow)}$=$\Gamma_{0}(1\pm
p)$. The TMR is conventionally defined as TMR=$(I_{P}-I_{AP})/I_{P}$, where
$I_{P}$ and $I_{AP}$ are the total current for the parallel and antiparallel
configuration, respectively. The coupling $\Gamma_{0}$ between the AGNR and
leads without internal magnetization is taken as the energy unit. The chemical
potentials for the two FM leads are set as $\mu_{L,R}$=$E_{F}\pm$0.5 eV with
the Fermi energy $E_{F}$=0. The on-site energy $\epsilon_{0}$=0, and
$\Gamma_{0}$=$t$ because that they are should be in the same order.45
### III.1 Spin-dependent transport
FIG. 2: (Color online) DOS for spin-up [(a) and (c)], and -down [(b) and (d)]
channels as a function of energy $\omega$ (in units of $\Gamma_{0}$) with
different polarization $p$, where the parameters are taken as
$\epsilon_{0}$=0, $\theta$=0 and $\Delta$=$0.05$. For comparison, the DOS for
bare ($\Delta$=$\Gamma_{0}$=$0$) 7-AGNR (a) and 8-AGNR (c) is shown in (dark)
solid lines.
In Fig. 2, we present the spin-dependent DOS versus energy $\omega$ (in units
of $\Gamma_{0}$) in parallel configuration ($\theta$=0) with exchange
splitting $\Delta$=0.05 and different polarization strength $p$ for the system
with 7- and 8-AGNR, respectively. In the case of bare AGNR
($\Delta$=$\Gamma$=$0$), a zero value plateau in DOS for 7-AGNR appears
symmetrically with respect to the Fermi level [see (dark) solid line in Fig.
2(a)] due to the presence of the energy gap for semiconducting 7-AGNR, while
for 8-AGNR a nonzero symmetrical plateau with respect to the Fermi level
appears [see (dark) solid line in Fig. 2(c)] because of the gapless energy
band in metallic 8-AGNR. However, when a magnetic deposition is applied, the
positions of plateaus for both 7-AGNR and 8-AGNR systems shift with the
exchange field strength $\Delta$: the spin-up (-down) DOS deviates the origin
point by the magnitude of $\Delta$ towards the positive (negative) direction
of the energy. The reason for this phenomenon is that the exchange interaction
in the magnetic deposition behaves as an effective in-plane magnetic field,
and acts on the electrons in AGNRs, thus leading to the shift of the energy
level related to the electron spin. For larger $\omega$, $\rho_{\sigma}$ shows
an oscillation behavior with sharp peaks, which indicates the buildup of Van
Hove singularities (VHSs) at subband edge. Additionally, we notice that as $p$
increases from zero [(red) dashed lines for normal leads] to 0.6 [(green)
dash-dotted lines] in Fig. 2, both 7- and 8-AGNR systems are almost immune
from $p$, which is different from the bulk graphene tunneling junction.39 This
stems from the fact the VHSs induced by the edge effect of AGNRs are
independent on $p$.
FIG. 3: (Color online) The linear conductance as a function of exchange
splitting $\Delta$ with different polarization $p$ for spin-up [(a) and (c)],
and spin-down [(b) and (d)] channels. The other parameters are the same as
those of Fig. 2.
Figure 3 shows the spin-dependent linear conductance (in units of $e^{2}/h$)
versus exchange splitting $\Delta$ (in units of $\Gamma_{0}$) with different
polarization $p$ in parallel alignment at the Fermi energy for the system with
7- and 8-AGNR, respectively. It is seen that for both 7-AGNR and 8-AGNR
systems, $G_{\uparrow}$ increases [see Figs. 3(a) and 3(c)] while
$G_{\downarrow}$ decreases [see Figs. 3(b) and 3(d)] with the increase of
polarization $p$, which is due to the conventional spin-valve effect in the
magnetic tunneling junction. Namely, with increasing $p$, the proportion of
spin-up electrons increases and that of spin-down electrons decreases in
AGNRs. Consequently, it becomes easier for spin-up electrons to tunnel through
the barrier, but harder for spin-down ones due to the presence of the inverse
spin-direction for spin-down electrons in the tunneling process, thus we
conclude that $G_{\uparrow}$ increases while $G_{\downarrow}$ decreases with
the increase of $p$. In addition, it is found that for the small $\Delta$
(below the full spin polarization46), the conductance value is zero for 7-AGNR
system, while nonzero constant for 8-AGNR system. This is related to the
presence (absence) of energy gap in semiconducting 7-AGNR (metallic 8-AGNR).
For a larger $\Delta$ (above the full spin polarization), the conductance for
either 7- or 8-AGNR system displays an oscillation enhancement behavior with
shape peaks, which stems from the resonant tunneling through different
subbands due to the edge effect.47,48
FIG. 4: (Color online) The I-V curve with different polarization $p$ for spin-
up [(a) and (c)], and spin-down [(b) and (d)] channels. The parameters are the
same as those of Fig. 2.
The bias voltage dependence of current (in units of $e/h\Gamma_{0}$) for the
7-AGNR and 8-AGNR systems with different polarization $p$ under the parallel
configuration is demonstrated in Fig. 4. It is clearly seen that the current
has a step-like structure as a function of bias voltage, which results from
the constant DOS at the Fermi energy and resonance due to VHSs. In particular,
the dependence of the current on the polarization $p$ exhibits different
behaviors for the 7-AGNR and 8-AGNR systems. With increasing $p$, the current
increases in the entire bias voltage for the 8-AGNR system, while for the
7-AGNR system, the current rises only in the large bias, and almost remains
zero at small bias voltage. The explanation for this phenomenon is as follows:
the DOS for the 7-AGNR system vanishes near the Fermi level, thus the 7-AGNR
system resembles a insulator-like barrier at the low energy. In this case, the
electrons will difficultly tunnel through the 7-AGNR system at low bias.
FIG. 5: (Color online) The differential conductance as a function of bias $V$
with different polarization $p$ for spin-up [(a) and (c)], and spin-down [(b)
and (d)] channels. The other parameters are the same as those of Fig. 2. For
comparison, the differential conductance for 7- and 8-AGNR with $\Delta$=$p$=0
is also shown in (dark) solid lines.
In Fig. 5, we demonstrate the dependence of differential conductance
$G_{d}$=$dI/dV$ (in units of $e^{2}/h$) on the bias voltage for the system
with different polarization $p$ under the parallel configuration. For
comparison, we plot the differential conductance of the 7- and 8-AGNR systems
for $\Delta$=$p$=$0$ in (dark) solid lines. The differential conductance
exhibits the successive oscillation peaks corresponding to the resonant
tunneling through the edge-induced subbands. In addition, we find that there
exists a respective plateau structure at small bias for the 7- and 8-AGNR
systems. In the range of the plateau, the conductance approaches to zero for
the 7-AGNR system, while a finite constant for 8-AGNR system. An interesting
characteristic is that the plateau structure for the 8-AGNR system is broader
than that for the 7-AGNR system since 8-AGNR has a larger interval between the
Fermi energy and the lowest subband [see Fig. 2]. When $\Delta$ is applied,
the peak splits into two peaks located at the two sides of the original peak
for both 7-AGNR and 8-AGNR systems, and the position of the peaks depends on
the magnitude of the exchange splitting. This behavior is due to the exchange
splitting of the edge-induced subbands. From Fig. 5, $G_{d,\uparrow}$
increases while $G_{d,\downarrow}$ decreases for both 7- and 8-AGNR systems
with increasing $p$, which is a typical spin-valve effect.
FIG. 6: (Color online) The dependence of TMR on bias $V$ for (a) 7-AGNR, (b)
8-AGNR system with different polarization $p$ and splitting $\Delta$.
Figure 6 shows the TMR versus the bias voltage $V$ for 7- and 8-AGNR systems
with different polarization $p$ and exchange splitting $\Delta$. Firstly, it
is noted that the TMR for the 7- and 8-AGNR systems has a high value relative
to the conventional magnetic junction; a pronounced plateau for 7- and 8-AGNR
systems appears at lower bias due to the constant DOS near the Fermi energy
[see Fig. 2]. These results are quite consistent with the previous
studies.39,44 For high bias, the TMR exhibits the successive oscillation peaks
resided near the edge-induced subbands. The amplitude of these peaks can even
approach to 1 indicating an perfect spin-valve effect. A high TMR for the
system with an AGNR has also been obtained in Refs. [49,50]. It should be
pointed out that the result here is different from that of Ref. [23] where the
TMR for the AGNRs systems is very small in the vicinity of the Fermi energy.
The plateau structure for 8-AGNR system is systematically broader than that
corresponding to 7-AGNR system, which may be very useful to overcome a well-
known shortcoming of MRAM, which is the large decrease in TMR with applied
bias voltage. Interestingly, the increase of $\Delta$ suppresses the amplitude
of this structure for 7-AGNR system. However, the TMR is enhanced within bias
range from -$\Delta$ to $\Delta$ for 8-AGNR system. This is caused by the
exchange splitting influence on the DOS as mentioned in Fig. 2. The results
are quantitatively different from our previous work for FM/graphene/FM system
due to the finite size for AGNRs.16,39 When the polarization changes to
$p$=0.3, as shown by the (blue) dotted lines and (green) dash-dotted lines in
Fig. 6, the TMR for both 7-AGNR and 8-AGNR systems becomes obviously smaller
because the leads become less spin-polarized with decreasing $p$.51 This
result is different from Ref. [15], where the TMR is nearly independent of the
electronic details of the leads. We speculate that the structure in the Ref.
[15] is very different from the structure considered here. Our results suggest
that it is more favorable to fabricate a graphene-based spin-valve device by
using a metallic AGNR.
### III.2 Spin transfer torque
FIG. 7: (Color online) The dependence of current-induced STT for 7-AGNR on (a)
relative orientation angle $\theta$ at the bias voltage $V$=$0.5\Gamma_{0}/e$,
(b) bias voltage $V$ at $\theta$=$\pi/3$, and (c) exchange splitting $\Delta$
at $\theta$=$\pi/3$ and $V$=$0.5\Gamma_{0}/e$ with different polarization $p$.
The other parameters are the same as in Fig. 2. FIG. 8: (Color online) The
dependence of current-induced STT for 8-AGNR on (a) relative orientation angle
$\theta$ at the bias voltage $V$=$0.5\Gamma_{0}/e$, (b) bias voltage $V$ at
$\theta$=$\pi/3$, and (c) exchange splitting $\Delta$ at $\theta$=$\pi/3$ and
$V$=$0.5\Gamma_{0}/e$ with different polarization $p$. The other parameters
are the same as in Fig. 2.
Figures 7 and 8 show the current-induced STT ($\tau_{R}^{x^{\prime}}$) as
functions of angular $\theta$, bias voltage and exchange splitting strength
for 7-AGNR and 8-AGNR systems with different polarization $p$. It is seen that
$\tau_{R}^{x^{\prime}}$ versus the angle $\theta$ shows a sine-like behavior
for 7-AGNR and 8-AGNR systems, which is in line with the previous
findings.28,39 A similar result has also been obtained for a FM/NM/FM trilayer
system discussed in Ref. [52], although the transport mechanisms are
different. The present result can be easily understood because the spin torque
is proportional to $\mathbf{S_{R}\times(S_{L}\times S_{R})}$, where
$\mathbf{S_{L}}$ and $\mathbf{S_{R}}$ are the spin moments of the left and
right FMs, respectively. Thus we can conclude that $\tau_{R}^{x^{\prime}}$
vanishes when the relative alignment of magnetization of the two FM leads is
parallel ($\theta$=0) or antiparallel ($\theta$=$\pi$). Furthermore, the STT
for 7-AGNR system is quantitatively larger than that for 8-AGNR system,
suggesting that a semiconducting AGNR is a better choice for a current-
controlled magnetic memory element.
In addition, it is evident that with increasing polarization $p$, the STT is
enhanced, as displayed in Fig. 7(a) and Fig. 8(a). This is in line with the
statement that $\tau_{R}^{x^{\prime}}$ is proportional to the polarization
strength of FM.28 Similar result is obtained in the conventional spin-valve
device.26 From Figs. 7(b) and 8(b), one can notice the step-like feature in
the STT as a function of bias voltage for both 7- and 8-AGNR systems, which is
reminiscent of the current versus the bias. The reason is that the in-plane
$\tau_{R}^{x^{\prime}}$ is proportional to the difference of spin-polarized
current ($I^{s}(0)-I^{s}(\pi))$,28,31 where
$I^{s}(\theta)=I_{\uparrow}(\theta)-I_{\downarrow}(\theta)$ is the spin-
current densities for the angle $\theta$. In practice, $I^{s}(\pi)$ (not shown
here) for both 7- and 8-AGNR systems vanishes identically due to the absence
of minority states available for tunneling. Thereby, the bias dependence of
$\tau_{R}^{x^{\prime}}$ remains the similar feature to that of $I^{s}(0)$.
This is also the reason why the anomalous bias dependence of the STT has not
been predicted in this paper according to Ref. [31], where anomalous bias
dependence of the STT only appears in the case of
$I^{s}(0)$$\leq$$I^{s}(\pi)$. Another reason for this is that the result is
obtained here under the wide bandwidth approximation. From Figs. 7(c) and
8(c), it can be found that $\tau_{R}^{x^{\prime}}$ with the exchange splitting
$\Delta$ displays a oscillation behavior for the 7-AGNR and 8-AGNR systems.
This result stems from the fact that the exchange splitting shifts the VHSs
[cf. Figs. 2(a) and (c)], thus leads to the edge-induced subband crossing of
the bias windows causing the tunneling resonance.
## IV Summary and Conclusion
In conclusion, we have demonstrated the effects of the AGNR width, the
polarization strength of FM leads with arbitrary polarization directions and
the exchange splitting on the spin-resolved transport properties for both
semiconducting and metallic AGNRs. In contrast to other related theoretical
works, the leads in our system are not supposed to be two-dimensional of
honeycomb- or square-lattice type, but is more realistical modeled as three-
dimensional semi-infinite FM slabs. By means of Keldysh NGF method, it is
found that the spin-resolved transport property for the system depends
sensitively on both the AGNR type and the polarization strength of FM leads.
Around zero bias the TMR versus bias voltage possesses a pronounced plateau
structure for both semiconducting 7-AGNR and metallic 8-AGNR systems without
exchange splitting. This phenomenon may be very useful to overcome a well-
known shortcoming of MRAM, which is the large decrease in TMR with applied
bias voltage. Remarkably, the plateau structure for 8-AGNR system is much
broader than that for 7-AGNR system. Interestingly, the increase of the
exchange splitting $\Delta$ suppresses the amplitude of this structure for
7-AGNR system. However, the TMR is enhanced at bias that ranged from -$\Delta$
to $\Delta$ for 8-AGNR system. So it may be useful in the design of spin-valve
device and graphene nanoribbons-based spintronic devices by using a metallic
AGNR. In addition, the current-induced STT for 7-AGNR system is systematically
larger that that for 8-AGNR system, which may be more favorable to fabricate a
current-controlled magnetic memory element by using a semiconducting AGNR. As
a development of graphene fabrication technology, several tens of nanometer
wide GNRs can be obtained by patterning graphene into a narrow ribbon, and the
large exchange splitting can also be realizable in GNRs, thus the model
considered in this paper may have potential application in spintronic devices.
###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China
(Grant Nos. 10974052 and 10574042), the Program for Changjiang Scholars and
Innovative Research Team in University (PCSIRT, No. IRT0964), and the
Scientific Research Fund of Hunan Provincial Education Department (Grant
No.09B079).
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|
arxiv-papers
| 2011-06-20T08:59:29 |
2024-09-04T02:49:19.905117
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Benhu Zhou, Xiongwen Chen, Benliang Zhou, Kai-He Ding, and Guanghui\n Zhou",
"submitter": "Guanghui Zhou",
"url": "https://arxiv.org/abs/1106.3843"
}
|
1106.3871
|
Dipartimento di Fisica dell Universit‘a di Padova, I-35131 Padova, Italy
INFN Sezione di Padova, I-35131 Padova, Italy 13.15.+g, 14.60.Pq, 29.40Gx,
29.40.Rg…
# Search for $\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations in appearance mode
in the OPERA experiment
U. Koseins:pd On behalf of OPERA Collaboration, umut.kose@cern.chins:pdins:pd
###### Abstract
The OPERA experiment in the underground Gran Sasso Laboratory (LNGS) has been
designed to perform the first detection of neutrino oscillations in direct
appearance mode in the muon neutrino to tau neutrino channel. The detector is
hybrid, being made of an emulsion/lead target and of electronic detectors. It
is placed in the CNGS neutrino beam 730 $km$ away from the neutrino source.
Runs with CNGS neutrinos were successfully carried out in 2008, 2009, and
2010. After a brief description of the beam and the experimental setup, we
report on event analysis of a sample of events corresponding to $1.89\times
10^{19}$ $p.o.t.$ in the CERN CNGS $\nu_{\mu}$ beam that yielded the
observation of a first candidate $\nu_{\tau}$ CC interaction. The topology and
kinematics of this candidate event are described in detail. The background
sources are explained and the significance of the candidate is assessed.
## 1 Introduction
Two types of experimental methods can be used to detect neutrino oscillations:
observing the appearance of a neutrino flavour initially absent in the beam or
measuring the disappearance rate of the initial flavour. In the latter case,
one must know the flux of the beam precisely. In this type of experiment one
explores whether less than the expected number of neutrinos of a produced
flavour arrives at a detector or whether the spectral shape changes if
observed at various distances from a source. Since the final state is not
observed, disappearance experiments cannot tell into which flavor a neutrino
has oscillated. An appearance experiment searches for possible new flavours of
neutrino, which does not exist in the original beam, or for an enhancement of
an existing neutrino flavour. The identification of the flavour relies on the
detection of the corresponding lepton produced in its charged current (CC)
interactions: $\nu_{l}N\rightarrow{l}^{-}X$ with $l=e,\mu,\tau$ and where X
denotes the hadronic final state.
In the past two decades, several experiments carried out with atmospheric and
accelerator neutrinos, as well as with solar and reactor neutrinos, have
established the picture of a three-neutrino oscillation scenario with two
large mixing angles. Atmospheric sector flavor conversion was first
established by the Super-Kamiokande [1] and MACRO [2] experiments and then
confirmed by the K2K [3] and MINOS [4] longbaseline experiments. The CHOOZ [5]
and Palo Verde [6] reactor experiments excluded indirectly the
$\nu_{\mu}\rightarrow\nu_{e}$ channel as the dominant process in the
atmospheric sector. However, the direct observation of flavour transition
through the detection of the corresponding lepton has never been observed.
Appearance of $\nu_{\tau}$ will prove unambiguously that
$\nu_{\mu}\rightarrow\nu_{\tau}$ oscillation is the dominant transition
channel at the atmospheric scale.
The OPERA experiment [7] has been designed to directly observe the appearance
of $\nu_{\tau}$ in a pure $\nu_{\mu}$ beam on an event by event basis. The
$\nu_{\tau}$ signature is given by the decay topology and kinematics of the
short lived $\tau^{-}$ leptons produced in the interaction of
$\nu_{\tau}N\rightarrow\tau^{-}X$ and decaying to one prong ($\mu,e$ or
$hadron$) or three prongs, which are [8]:
$\displaystyle\tau^{-}\rightarrow\mu^{-}\nu_{\mu}\bar{\nu}_{\tau}\hskip
28.45274ptwith\hskip 14.22636ptBR=17.36\pm{0.05}\%\hskip 122.34692pt$
$\displaystyle\tau^{-}\rightarrow e^{-}\nu_{e}\bar{\nu}_{\tau}\hskip
28.45274ptwith\hskip 14.22636ptBR=17.85\pm{0.05}\%\hskip 125.19194pt$
$\displaystyle\tau^{-}\rightarrow h^{-}(n\pi^{0})\bar{\nu}_{\tau}\hskip
14.22636ptwith\hskip 14.22636ptBR=49.52\pm{0.07}\%\hskip 122.34692pt$
$\displaystyle\tau^{-}\rightarrow 2h^{-}h^{+}(n\pi^{0})\bar{\nu}_{\tau}\hskip
14.22636ptwith\hskip 14.22636ptBR=15.19\pm{0.08}\%.\hskip 102.43008pt$
## 2 The Neutrino Beam
The CNGS $\nu_{\mu}$ beam produced by the CERN-SPS is directed towards the
OPERA detector, located in the Gran Sasso underground laboratory (LNGS) [9] in
Italy, 730 km away from the neutrino source at CERN. In order to study
$\nu_{\mu}\rightarrow{\nu}_{\tau}$ oscillations in appearance mode as
indicated in the atmospheric neutrino sector, the CERN Neutrinos to GranSasso
(CNGS) neutrino beam [10] was designed and optimized by maximizing the number
of $\nu_{\tau}$CC interactions at the LNGS.
The average $\nu_{\mu}$ beam energy is 17 $GeV$, well above tau production
energy treshold. The $\bar{\nu}_{\mu}$ contamination is $\sim 4\%$ in flux,
$2.1\%$ in terms of interactions. The $\nu_{e}$ and $\bar{\nu}_{e}$
contaminations are lower than $1\%$, while the number of prompt $\nu_{\tau}$
from $D_{s}$ decay is negligible. The average $L/E_{\nu}$ ratio is 43
$km/GeV$, suitable for oscillation studies at atmospheric $\Delta{m^{2}}$. Due
to the Earth’s curvature neutrinos from CERN enter the LNGS halls with an
angle of about $3^{\circ}$ with respect to the horizontal plane.
With a nominal CNGS beam intensity of $4.5\times 10^{19}$ protons on target
($p.o.t.$) per year, and assuming $\Delta{m}^{2}_{23}=2.5\times 10^{-3}eV^{2}$
and full mixing, about 10 $\nu_{\tau}$ events are expected to be observed in
OPERA in 5 years of data taking, with selection criteria reducing the
background to 0.75 events.
The goal is to accumulate a statistics of neutrino interactions correspomding
to $22.5\times 10^{19}$ $p.o.t.$ in 5 years. The 2008, 2009 and 2010 runs
achieved a total intensity of $1.78\times 10^{19}$, $3.52\times 10^{19}$ and
$4.04\times 10^{19}$ $p.o.t.$ respectively. Within these three years,
neutrinos produced 9637 beam events. The processing of these events,
particularly the scanning of emulsion films, is continuously going on. The
2011 run started on May 2011 and is still in progress.
At the CNGS energies the average $\tau^{-}$ decay length is submillimetric, so
OPERA uses nuclear emulsion films as high precision tracking device in order
to be able to detect such short decays. Emulsion films are interspaced with
$1mm$ thick lead plates, which act as neutrino target and form the largest
part of the detector mass. This technique is called Emulsion Cloud Chamber
(ECC). It was successfully used to establish the first evidence for charm in
cosmic rays interactions [11] and in the DONUT experiment [12] for the first
direct observation of the $\nu_{\tau}$. To date, nine $\nu_{\tau}$ CC
interactions have been observed by DONUT produced by a fixed target 800 $GeV$
proton beam configuration.
## 3 The OPERA Detector
OPERA is a hybrid detector made of two identical Super Modules (SM1 and SM2),
each one formed by a target section and a muon spectrometer as shown in Figure
1. Each target section is organized in 31 vertical ”$walls$”, transverse to
the beam direction. Walls are filled with ”$ECC$ $bricks$” with an overall
mass of 1.25 $kton$. They are followed by double layers of scintillator planes
acting as Target Trackers (TT) that are used to locate neutrino interactions
occurred within the target. A target brick consists of 56 lead plates of 1 mm
thickness interleaved with 57 emulsion films. The lead plates serve as
neutrino interaction target and the emulsion films as 3-dimensional tracking
detectors providing track coordinates with a sub-micron accuracy and track
angles with a few mrad accuracy. The material of a brick along the beam
direction corresponds to about 10 radiation length and 0.33 interaction
length. The brick size is $10cm\times 12.5cm\times 8cm$ and its weight is
about $8.3kg$.
Figure 1: View of the OPERA detector; the neutrino beam enters from the left.
Arrows show the position of detector components, the VETO planes, the target
and TT, the drift tubes (PT) laid out along the XPC, the magnets and the RPC
installed between the magnet iron slabs. The Brick Manipulator System (BMS) is
partly shown.
In order to reduce the emulsion scanning load, Changeable Sheets (CS) [13]
film interfaces have been used. They consist in tightly packed doublets of
emulsion films glued to the downstream face of each brick. Charged particles
from a neutrino interaction in a brick cross the CS and produce signals in the
TT that allow the corresponding brick to be identified and extracted by an
automated Brick Manipulator System (BMS).
The spectrometers consist of a dipolar magnet instrumented with active
detectors, planes of RPCs (Internal Tracker, IT) and drift tubes (Precision
Tracker, PT). Tasks of the spectrometers are muon identification and charge
measurement in order to minimize the background. For muon momenta between 2.5
$GeV/c$ and 45 $GeV/c$, the fraction of events with wrong charge determination
is $1.2\%$. The $\mu^{+}$ to $\mu^{-}$ events ratio, within the selected
momentum range, obtained from data can be directly compared with predictions
based on Monte Carlo simulations: $3.92\pm 0.37(stat.)\%$ for data, $3.63\pm
0.13(stat.)\%$ for MC. Figure 2 left-side shows the momentum and momentum
times charge distribution for data and MC.
Figure 2: Right: Muon charge comparison (momentum$\times$charge): data (black
dots with error bars) and MC (solid line) are normalised to one. Left:
Bjorken-y variable reconstructed in data (dots with error bars) and MC (shaded
areas). The MC distributions are normalised to data. The different
contributions of the MC are shown in different colours: QE + RES contribution
in light grey, DIS contribution in grey and the NC contamination in dark grey.
In Figure 2 right-side, Bjorken-y distribution is shown for the events with at
least a muon track. The agreement between data and MC simulation is
reasonable. The sum of the QE and RES processes can be clearly seen as a peak
at low y values. The NC contribution shows up at values of Bjorken-y close to
one. The NC contribution becomes negligible when a track with its momentum
measured by the spectrometer is required.
A detailed description of the complete detector can be found in [7]. Event
reconstruction procedures and a performances of the OPERA electronic detectors
can be found in more detail in [14].
## 4 Neutrino interaction location
Neutrino event analysis starts with the pattern recognition in the electronic
detectors. Charged particle tracks produced in a neutrino interaction generate
signals in the TT and in the muon spectrometer. A brick finding algorithm is
applied in order to select the brick which has the maximum probability to
contain the neutrino interaction. The brick with the highest probability is
extracted from the detector for analysis. The efficiency of this procedure
reaches $83\%$ in a subsample where up to 4 bricks per event were processed.
After extraction of the brick predicted by the electronic detectors, its
validation comes from the analysis of the CS films. The measurement of
emulsion films is performed through high-speed automated microscopes [15, 16]
with a sub-micrometric position resolution and angular resolution of the order
of one milliradian. If no expected charged track related to the event is found
in the CS, the brick is returned back to the detector with another CS doublet
attached. If any track originating from the interaction is detected in the CS,
the brick is exposed to cosmic rays (for alignment purposes) and then
depacked. The emulsion films are developed and sent to the scanning
laboratories of the Collaboration for event location studies and decay search
analysis.
All the track information of the CS is then used for a precise prediction of
the tracks in the most downstream films of the brick (with an accuracy of
about $100\mu{m}$). When found in this films, tracks are followed upstream
from film to film. The scan-back procedure is stopped when no track candidate
is found in three consecutive films and the lead plate just upstream the last
detected track segment is defined as the vertex plate. In order to study the
located vertices and reconstruct the events, a general scanning volume is
defined with a transverse area of $1\times 1cm^{2}$ for 5 films upstream and
10 films downstream of the stopping point. All track segments in this volume
are collected and analysed. After rejection of the passing through tracks
related to cosmic rays and of the tracks due to low energy particles, the
tracks produced by the neutrino interaction can be selected and reconstructed.
The present overall location efficiency averaged over NC and CC events, from
the electronic detector predictions down to the vertex confirmation, is about
$60\%$.
## 5 Decay Search
Once the neutrino interaction is located, a decay search procedure is applied
to detect possible decay or interaction topologies on tracks attached to the
primary vertex. The main signature of a secondary vertex (decay or nuclear
inetaraction) is the observation of a track with a significant impact
parameter (IP) relative to the neutrino interaction vertex. The IP of primary
tracks is smaller than 10$\mu{m}$ after excluding tracks produced by low
momentum particles. When secondary vertices are found in the event, a
kinematical analysis is performed, using particle angles and momenta measured
in the emulsion films. For charged particles up to about 6 $GeV/c$, momenta
can be determined using the angular deviations produced by Multiple Coulomb
Scattering (MCS) of tracks in the lead plates [17] with a resolution better
than $22\%$. For higher momentum particles, the measurement is based on the
position deviations. The resolution is better than $33\%$ on 1/p up to 12
$GeV/c$ for particles passing through an entire brick.
A $\gamma$-ray search is performed in the whole scanned volume by checking all
tracks having an IP with respect to the primary or secondary vertices lower
than 800$\mu{m}$. The angular acceptance is $\pm 500$ $mrad$. The $\gamma$-ray
energy is estimated by a Neural Network algorithm that uses the number of
segments, the shape of the electromagnetic shower and also the MCS of the
leading tracks.
## 6 Data analysis
In the following, the analysis results [18] of about $35\%$ of the 2008 and
2009 data sample, corresponding to the $1.89\times 10^{19}$ $p.o.t$ are
presented. The decay search procedure was applied to a sample of 1088 events
of which 901 were classified as CC interactions. In the sample of CC
interactions, 20 charm decay candidates were observed, in good agreement with
the expectations from the Monte Carlo simulation, $16\pm 2.9$. Out of them 3
have a 1-prong topology where $0.8\pm 0.2$ was expected. The background for
the total charm sample is about 2 events. Several $\nu_{e}$-induced events
have also been observed.
Moreover, a first CC $\nu_{\tau}$ candidate has been detected. The expected
number of $\nu_{\tau}$ events detected in the analysed sample is about
$0.54\pm 0.13(syst.)$ at $\Delta{m^{2}}_{23}=2.5\times 10^{-3}$ $eV^{2}$ and
full mixing.
## 7 The first tau neutrino candidate
In this section, the first tau neutrino candidate [18] will be described. The
location and decay search procedure yielded a neutrino interaction vertex with
7 tracks. One track exhibits a visible kink with an angular change of $41\pm
2mrad$ after a path length of $1335\pm 35\mu{m}$. The kink daughter momentum
is estimated to be $12^{+6}_{-3}$ $GeV/c$ by MCS measurement and its
transverse momentum to the parent direction is $470^{+230}_{-120}$ $MeV/c$.
The event is displayed in Figures 4 and 4.
Figure 3: Display of the $\nu_{\tau}$ candidate event. Left: view transverse
to the neutrino direction. Right: same view zoomed on the vertices.The short
track named ”4 parent” is the $\tau^{-}$ candidate.
Figure 4: Longitudinal view of the $\nu_{\tau}$ candidate event.
All the tracks from the neutrino interaction vertex were followed until they
stop or interact. The probability that one of them is left by a muon is
estimated to be less than $10^{-3}$. The residual probability for being a
$\nu_{\mu}$CC event, with a possibly undetected large angle $\mu$ track, is
about $1\%$; a nominal value of $5\%$ is assumed. None of the tracks is
compatible with being an electron.
Two electromagnetic showers caused by $\gamma$-rays, associated with the
event, have been located and studied. The energy of $\gamma 1$ is $(5.6\pm
1.0(stat.)\pm 1.7(syst.))$ $GeV$ and it is clearly pointing to the decay
vertex. The $\gamma 2$ has an energy of $1.2\pm 0.4(stat.)\pm 0.4(syst.)$
$GeV$ and it is compatible with pointing to either vertex, with a
significantly larger probability to the decay vertex.
All the selection cuts used in the analysis were those described in detail in
the experiment proposal [19] and its addendum [20]. All the kinematical
variables of the event and the cut applied are given in Table 1.
Table 1: Kinematical variables of $\nu_{\tau}$ candidate event. Variable | Measured | Selection criteria
---|---|---
Kink angle ($mrad$) | $42\pm 2$ | >20
Decay length ($\mu{m}$) | $1335\pm 35$ | Within 2 plates
P daughter ($GeV/c$) | $12^{+6}_{-3}$ | >2
PT daughter ($MeV/c$) | $470^{+230}_{-120}$ | >300 ($\gamma$ attached)
Missing PT ($MeV/c$) | $570^{+320}_{-170}$ | <1000
Angle $\phi$ ($deg$) | $173\pm 2$ | >90
The invariant mass of the two observed $\gamma$-rays is $120\pm 20(stat)\pm
35(syst)$ supporting the hypothesis that they are emitted in a $\pi^{0}$
decay. The invariant mass of the charged decay daughter assumed to be a
$\pi^{-}$ and of the two $\gamma$-rays amount to
$640^{+125}_{-80}(stat)^{+100}_{-90}(syst)$ $MeV/c$, which is compatible with
the $\rho(770)$ mass. So the decay mode of the candidate is consistent with
the hypothesis $\tau^{-}\rightarrow\rho^{-}\nu_{\tau}$ (where the branching
ratio is about $25\%$).
## 8 Background Estimation
The two main sources of background to the
$\tau^{-}\rightarrow{h}(n\pi^{0})\nu_{\tau}$ channel where a similar final
state may be produced are:
* •
the decays of charmed particles produced in $\nu_{\mu}$ CC interactions where
the primary muon is not identified as well as the $c\bar{c}$ pair production
in $\nu_{\mu}$ NC interactions where one charm particle is not identified and
the other decays to a 1-prong hadron channel;
* •
the 1-prong inelastic interactions of primary hadrons produced in
$\nu_{\mu}$CC interactions where the primary muon is not identified or in
$\nu_{\mu}$ NC interactions and in which no nuclear fragment can be associated
with the secondary interaction.
The Monte Carlo expectation of the first background source is $0.007\pm
0.004(syst.)$ event, the fraction produced in $\nu_{e}$ CC interactions is
less than $10^{-3}$ events, The second type of background amounts to $0.011\pm
0.006(syst.)$ event. The total background in the decay channel to a single
charged hadron is $0.018\pm 0.007(syst)$ events. The probability that this
background events fluctuate to one event is $1.8\%$ $(2.36\sigma)$. As the
search for $\tau^{-}$ decays is extended to all four channels, the total
background then becomes $0.045\pm 0.023(syst)$. The probability that this
expected background to all searched decay channels of the $\tau^{-}$
fluctuates to one event is $4.5\%$ $(2.01\sigma)$. At $\Delta{m}^{2}=2.5\times
10^{-3}$ $eV^{2}$ and full mixing, the expected number of observed $\tau^{-}$
events with the present analyzed statistics is $0.54\pm 0.13(syst)$ of which
$0.16\pm 0.04(syst)$ in the one-prong hadron topology, compatible with the
observation of one event.
## 9 Conclusions
During 2008, 2009 and 2010 runs, a total intensity of $1.78\times 10^{19}$,
$3.52\times 10^{19}$ and $4.04\times 10^{19}$ $p.o.t.$ respectively, was
achieved. Within these three years, 9637 beam events have been collected
within the OPERA target. The neutrino interaction location and decay search
are going on.
A first candidate $\nu_{\tau}$ CC interaction in the OPERA detector at LNGS
was detected after analysis of a sample of events corresponding to $1.89\times
10^{19}$ $p.o.t.$ in the CERN CNGS $\nu_{\mu}$ beam. The expected number of
$\nu_{\tau}$ events in the analysed sample is $0.54\pm 0.13(syst.)$. The
candidate event passes all selection criteria, it is assumed to be a
$\tau^{-}$ lepton decaying into $h^{-}(n\pi^{0})\nu_{\tau}$. The observation
of one possible tau candidate in the decay channel $h^{-}(\pi^{0})\nu_{\tau}$
has a significance of $2.36\sigma$ of not being a background fluctuation.
## References
* [1] Y. Fukuda et al. Phys. Rev. Lett.8119981562.
* [2] M. Ambrosio et al. Eur. Phys. J. C362004323.
* [3] M.H. Ahn et al. Phys. Rev. D742006072003.
* [4] D.G. Michael et al. Phys. Rev. Lett.1012008131802.
* [5] M. Apollonio et al. Eur. Phys. J. C272003331.
* [6] A. Piepke Prog. Part. Nucl. Phys.482002113.
* [7] R. Acquafredda et al. JINST42009P04018.
* [8] K. Nakamura et al. J. Phys. G372010075021.
* [9] LNGS web site http://www.lngs.infn.it/.
* [10] CNGS project http://proj-cngs.web.cern.ch/prj-cngs/.
* [11] K. Niu, E. Mikumo, Y. Maeda Prog. Theor. Phys.4619711644.
* [12] K. Kodama et al. Phys. Lett. B5042001218; Phys. Rev. D782008052002.
* [13] T. Anokhina et al. JINST32008P07005.
* [14] N. Agafonova et al. New J. Phys.132011053051
* [15] N. Armenise et al. Nucl. Instrum. Methods A5512005261;
M. De Serio et al. Nucl. Instrum. Methods A5542005247;
L. Arrabito et al. Nucl. Instrum. Methods A5682006578
* [16] K. Morishima, T. Nakano JINST52010P04011.
* [17] M. De Serio et al. Nucl. Instrum. Methods A5122003539;
M. Besnier PhD. Thesis, Universite de Savoie, 2008, LAPP-T-2008-02.
* [18] N. Agafonova et al. Phys. Lett. B6912010138.
* [19] M. Guler et al. CERN-SPSC-2000-028; LNGS P25/2000.
* [20] M. Guler et al. CERN-SPSC-2001-025; LNGS-EXP 30/2001 add. 1/01.
|
arxiv-papers
| 2011-06-20T11:47:39 |
2024-09-04T02:49:19.913982
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Umut Kose",
"submitter": "Umut Kose",
"url": "https://arxiv.org/abs/1106.3871"
}
|
1106.3931
|
# Stabilization of Navier–Stokes equations
by oblique boundary feedback controllers
Viorel Barbu
Octav Mayer Institute of Mathematics of Romanian Academy, Iaşi, Romania
Supported by CNCSIS project PNID-/2011.
###### Abstract
One designs a linear stabilizable boundary feedback controller for the
Navier–Stokes system on a bounded and open domain
$\mathcal{O}\subset{\mathbb{R}}^{d}$, $d=2,3$, of the form
$u=\eta\displaystyle\sum^{N}_{j=1}\mu_{j}\left<y,\psi_{j}\right>\left(\displaystyle\frac{\partial\phi_{j}}{\partial
n}\,(x)+{\alpha}(x)\vec{n}(x)\right),$ where $\psi_{j},\phi_{j}$ are related
to the eigenfunction system for the adjoint Stokes–Oseen system, $\vec{n}$ is
the normal to ${\partial}\mathcal{O}$, $\left<\cdot,\cdot\right>$ is the
scalar product in$(L^{2}(\mathcal{O}))^{d}$ and ${\alpha}$ is any continuous
function with circulation zero on ${\partial}\mathcal{O}$.
2000 Mathematics Subject Classification. Primary: 93B52, 93C20, 93D15;
Secondary: 35Q30, 76D55, 76D15.
Keywords and phrases. Navier–Stokes system, Stokes–Oseen operator, feedback
controller, eigenvalue.
## 1 Introduction
Consider the Navier–Stokes system
(1.1) $\begin{array}[]{ll}\displaystyle\frac{\partial Y}{\partial t}-\nu\Delta
Y+(Y\cdot{\nabla})Y={\nabla}p+f_{e}&\mbox{ in }\
(0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot\ Y=0&\mbox{ in
}\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ Y=v&\mbox{ on }\
(0,{\infty})\times{\partial}\mathcal{O},\vspace*{1,5mm}\\\ Y(0)=y_{0}&\mbox{
in }\ \mathcal{O},\end{array}$
in a bounded open domain $\mathcal{O}\subset{\mathbb{R}}^{d}$, $d=2,3$, with a
smooth boundary ${\partial}\mathcal{O}$ which, for simplicity, we assume
simple connected. Here $\nu>0$, $f_{e}$ is a given smooth function and $v$ is
a boundary input. If $y_{e}$ is an equilibrium solution to (1.1), then (1.1)
can be, equivalently, written as
(1.2) $\begin{array}[]{ll}\lx@intercol\displaystyle\frac{\partial Y}{\partial
t}-\nu\Delta
Y+(y_{e}\cdot{\nabla})Y+(Y\cdot{\nabla})y_{e}+(Y\cdot{\nabla})Y={\nabla}p\hfil\lx@intercol\vspace*{1,5mm}\\\
&\mbox{ in }\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot\
Y=0&\mbox{ in }\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ Y=u&\mbox{
on }\ (0,{\infty})\times{\partial}\mathcal{O},\vspace*{1,5mm}\\\
Y(0)=y_{0}-y_{e}&\mbox{ in }\ \mathcal{O}.\end{array}$
Our main concern here is the design of an oblique boundary feedback controller
which stabilizes exponentially the equilibrium state $y_{e}$, or,
equivalently, the zero solution to (1.2). The main step toward this end is the
stabilization of the linear system corresponding to (1.2) or, more generally,
of the Oseen–Stokes system
(1.3) $\begin{array}[]{ll}\displaystyle\frac{\partial Y}{\partial t}-\nu\Delta
Y+(Y\cdot{\nabla})a+(b\cdot{\nabla})Y={\nabla}p&\mbox{ in }\
(0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot\ Y=0&\mbox{ in
}\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ Y=u&\mbox{ on }\
(0,{\infty})\times{\partial}\mathcal{O},\end{array}$
where $a,b\in(C^{2}(\overline{\mathcal{O}}))^{d},\ {\nabla}\cdot
a={\nabla}\cdot b=0$ in $\mathcal{O}$. Besides its significance as first order
linear approximation of (1.2), this system models the dynamics of a Stokes
flow with inclusion of a convection acceleration $(b\cdot{\nabla})Y$ and also
the disturbance flow induced by a moving body in a Stokes fluid flow.
In its complex form, the main result of this work, Theorem 2.1, amounts to
saying that there is a boundary feedback controller of the form
(1.4)
$\begin{array}[]{ll}u(t,x)=\eta\displaystyle\sum^{N}_{j=1}\mu_{j}\left(\int_{\mathcal{O}}Y(t,x)\overline{\varphi}^{*}_{j}(x)dx\right)(\phi_{j}(x)+{\alpha}(x)\vec{n}(x)),\\\
\hfill t\geq 0,\ x\in{\partial}\mathcal{O},\end{array}$
which stabilizes exponentially system (1.1). Here $\phi_{j}={\rm
lin}\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{i}}{\partial
n}\right\\}^{N}_{i=1}$ and $\\{{\varphi}^{*}_{j}\\}^{N}_{j=1}$ is an
eigenfunction system for the adjoint of the Stokes–Oseen operator
(1.5)
$\begin{array}[]{rcll}\mathcal{L}{\varphi}&=&-\nu\Delta{\varphi}+(a\cdot{\nabla}){\varphi}+({\varphi}\cdot{\nabla})b-{\nabla}p,\
{\varphi}\in D(\mathcal{L}),\vspace*{1,5mm}\\\
D(\mathcal{L})&=&\\{{\varphi}\in(H^{2}(\mathcal{O}))^{d}\cap(H^{1}_{0}(\mathcal{O}))^{d};\
{\nabla}\cdot{\varphi}=0\ \mbox{ in }\ \mathcal{O}\\}.\end{array}$
It turns out (see Theorem 2.3) that this feedback controller also stabilizes
the Navier–Stokes system (1.2) in a neighborhood of the origin.
In (1.4), $N$ is the number of the eigenvalue ${\lambda}_{j}$ of $\mathcal{L}$
with ${\rm Re}\,{\lambda}_{j}<0$, ${\alpha}\in C(\overline{\mathcal{O}})$ is
an arbitrary function with zero circulation on ${\partial}\mathcal{O}$, that
is,
(1.6)
$\begin{array}[]{ll}\displaystyle\int_{\mathcal{O}}{\alpha}(x)dx=0\end{array}$
$\mbox{and }\phi_{j}={\rm lin\
span}\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial
n}\right\\}^{N}_{i=1}.$ (In its real version given in Theorem 3.1, the
stability controller $u$ is of the form (1.4) but with
$\\{{\varphi}^{*}_{j}\\}$ replaced by $\\{{\rm Re}\,{\varphi}_{j},{\rm
Im}\,{\varphi}_{j}\\}$.)
Taking into account that the vectors
$\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\ (x)$ are
tangential at each $x\in{\partial}\mathcal{O}$, that is,
$\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial\nu}=\nu\vec{\tau}$,
where $\vec{\tau}$ is the tangent vector (see [11], p. 35), we see that $u$ is
an oblique vector field on ${\partial}\mathcal{O}$.
More precisely, we have
(1.7) $\displaystyle u(t,x)\cdot\vec{n}(x)={\alpha}(x),\ \ \forall
x\in{\partial}\mathcal{O},$ (1.8)
$\displaystyle|\cos\left<u(t,x),\vec{n}(x)\right>|\geq
1-\displaystyle\frac{C}{C+|{\alpha}(x)|}\,,\ \ \forall
x\in{\partial}\mathcal{O},$
where $C>0$ is independent of ${\alpha}$. This means that the ”stabilizable
boundary controller $u$ can be chosen ”almost” normal to
${\partial}\mathcal{O}$. However, for technical reasons the limit case
$|{\alpha}|\equiv+{\infty}$, that is, $u$ normal is excluded from our
discussion.
It should be said that in the stabilization literature, only in a few
situations was designed a normal stabilizable controller for equation (1.1)
and this for periodic flows in $2{-}D$ channels (see, e.g., [1], [2], [3],
[24], [25], [26]). However, even in this case, the feedback controller is not
given in explicit form and sometimes one assumes restrictive conditions on
$\nu$ or on the spectrum of the operator $\mathcal{L}$.
It should be said that there is a large body of results obtained in recent
years on boundary stabilization of system (1.1) and here the works [10], [11],
[13], [14], [16], [17], [19], [20] should be primarily cited. (See, also, [7],
[14], [15], [21].) The approach used in these works can be described in a few
words as follows; one decomposes system (1.1) in a finite-dimensional unstable
part which is exactly controllable and an infinite-dimensional part which is
exponentially stable and proves so its stabilization by open loop boundary
controller with finite-dimensional structure. Then one designs in a standard
way a stabilizable feedback controller via the infinite-dimensional algebraic
Riccati equation associated with an infinite horizon quadratic optimal control
problem. Our construction of boundary stabilizable controller for (1.1) avoids
the Riccati equation based approach which though provides a robust controller
it is, however, untreatable from computational point of view. Instead, we
propose an explicit feedback controller of the form (1.4) easy to implement
into system. It should be said that this construction resembles the form of
stabilizable noise controllers recently designed in the author’s works [4],
[5], [6], [8], [9], which seem to be, however, more robust to stochastic
perturbations.
The plan of the paper is the following. In Section 2, we present the main
stabilization result which will be proved in Section 3. In Section 4, we shall
give an application to stabilization of Stokes–Oseen periodic flows in a $2-D$
channel.
Everywhere in the following, we shall use the standard notation for spaces of
functions on $\mathcal{O}\subset{\mathbb{R}}^{d}$. In particular,
$C^{k}(\overline{\mathcal{O}})$, $k=0,1,...$, is the space of
$k$-differentiable functions on $\overline{\mathcal{O}}$ and
$H^{k}(\mathcal{O})$, $k=1,2,$ $H^{1}_{0}(\mathcal{O})$ are Sobolev spaces on
$\mathcal{O}$.
## 2 The main result
### 2.1 Notation
Everywhere in the following, $\mathcal{O}$ is a bounded and open domain of
${\mathbb{R}}^{d}$, $d=2,3$, with smooth and simply connected boundary
${\partial}\mathcal{O}$.
We set
$H=\\{y\in(L^{2}(\mathcal{O}))^{d};\ {\nabla}\cdot y=0\ \mbox{ in }\
\mathcal{O},\ \ y\cdot\vec{n}=0\ \mbox{ on }\ {\partial}\mathcal{O}\\}$
and denote by $\Pi:(L^{2}(\mathcal{O}))^{d}\to H$ the Leray projector on $H$.
We consider the operator $A:D(A)\subset H\to H$,
${\mathcal{A}}:D({\mathcal{A}})\subset H\to H$,
(2.1) $\displaystyle\quad Ay$ $\displaystyle=$ $\displaystyle-\nu\Pi(\Delta
y),\ \forall y\in
D(A)=(H^{1}_{0}(\mathcal{O}))^{d}\cap(H^{2}(\mathcal{O}))^{d}\cap H,$
$\displaystyle\quad Ay$ $\displaystyle=$ $\displaystyle\Pi(-\nu\Delta
y+(y\cdot{\nabla})a+(b\cdot{\nabla})y)$ $\displaystyle=$ $\displaystyle
Ay+\Pi((y\cdot{\nabla})a+(b\cdot{\nabla})y),$
$\displaystyle\qquad\qquad\forall y\in D({\mathcal{A}})=D(A).$
We denote by $\widetilde{H}$ the complexified space $\widetilde{H}=H+iH$ and
consider the extension $\widetilde{\mathcal{A}}$ of ${\mathcal{A}}$ to
$\widetilde{H}$, that is,
$\widetilde{\mathcal{A}}(y+iz)={\mathcal{A}}y+i{\mathcal{A}}z$ for all $y,z\in
D({\mathcal{A}})$.
The scalar product of $H$ and of $\widetilde{H}$ are denoted by
$\left<\cdot,\cdot\right>$ and $\left<\cdot,\cdot\right>_{\widetilde{H}}$,
respectively. The corresponding norms are denoted by $|\cdot|_{H}$ and
$|\cdot|_{\widetilde{H}}$, respectively.
For simplicity, we denote in the following again by ${\mathcal{A}}$ the
operator $\widetilde{\mathcal{A}}$ and the difference will be clear from the
content. The operator ${\mathcal{A}}$ has a compact resolvent
$({\lambda}I-{\mathcal{A}})^{-1}$ (see, e.g., [7], p 92). Consequently,
${\mathcal{A}}$ has a countable number of eigenvalues
$\\{{\lambda}_{j}\\}^{\infty}_{j=1}$ with corresponding eigenfunctions
${\varphi}_{j}$ each with finite algebraic multiplicity $m_{j}$. In the
following, each eigenvalue ${\lambda}_{j}$ is repeated according to its
algebraic multiplicity $m_{j}$.
Note also that there is a finite number of eigenvalues
$\\{{\lambda}_{j}\\}^{N}_{j=1}$ with ${\rm Re}\,{\lambda}_{j}{\leq}0$ and that
the spaces $X_{u}={\rm lin\
span}\\{{\varphi}_{j}\\}^{N}_{j=1}=P_{N}\widetilde{H},$
$X_{s}=(I-P_{N})\widetilde{H}$ are invariant with respect to ${\mathcal{A}}$.
Here, $P_{N}$ is the algebraic projection of $\widetilde{H}$ on $X_{u}$ and is
defined by
(2.3) $P_{N}=\frac{1}{2\pi
i}\int_{\Gamma}({\lambda}I-{\mathcal{A}})^{-1}d{\lambda},$
where ${\Gamma}$ is a closed curve which contains in interior the eigenvalues
$\\{{\lambda}_{j}\\}^{N}_{j=1}$.
If we set ${\mathcal{A}}_{u}={\mathcal{A}}|_{X_{u}},\
{\mathcal{A}}_{s}={\mathcal{A}}|_{X_{s}}$, then we have
$\sigma({\mathcal{A}}_{u})=\\{{\lambda}_{j}:{\rm Re}\,{\lambda}_{j}\leq 0\\},\
\ \sigma({\mathcal{A}}_{s})=\\{{\lambda}_{j}:{\rm Re}\,{\lambda}_{j}>0\\}.$
We recall that the eigenvalue ${\lambda}_{j}$ is called semisimple if its
algebraic multiplicity $m_{j}$ coincides with its geometric multiplicity
$m^{g}_{j}$. In particular, this happens if ${\lambda}_{j}$ is simple and it
turns out that the property of the eigenvalues ${\lambda}_{j}$ to be all
simple is generic (see [7], p. 164). The dual operator ${\mathcal{A}}^{*}$ has
the eigenvalues $\overline{\lambda}_{j}$ with the eigenfunctions
${\varphi}^{*}_{j}$, $j=1,...\,.$
For the time being, the following hypotheses will be assumed.
* (H1)
The eigenvalues ${\lambda}_{j},\ j=1,...,N,$ are semisimple.
This implies that
(2.4) ${\mathcal{A}}{\varphi}_{j}={\lambda}_{j}{\varphi}_{j},\ \
{\mathcal{A}}^{*}{\varphi}^{*}_{j}=\overline{\lambda}_{j}{\varphi}^{*}_{j},\ \
\ j=1,...,N,$
and so we can choose systems $\\{{\varphi}_{j}\\},\\{{\varphi}^{*}_{j}\\}$ in
such a way that
(2.5)
$\left<{\varphi}_{j},{\varphi}^{*}_{k}\right>_{\widetilde{H}}={\delta}_{jk},\
\ j,k=1,...,N.$
Next hypothesis is a unique continuation assumption on normal derivatives
$\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\,,$ $j=1,...,N.$
* (H2)
The system $\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ is linearly independent on ${\partial}\mathcal{O}$.
We note that, in the special case $N=1$, hypothesis (H2) reduces to:
$\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\not\equiv 0$ for
all $j=1,...,N.$ It is not known if this unique continuation property is
always satisfied, but it holds, however, for ”almost all $a,b$” in the generic
sense (see [12]). In specific examples, however, this assumption might be
easily checked and we shall see later on in Section 4 that it holds for
systems in a $2-D$ channel $\mathcal{O}=\\{(x,y)\in{\mathbb{R}}\times(0,1)\\}$
with periodic conditions in $x$.
### 2.2 The main stabilization result
Consider the feedback boundary controller
(2.6)
$u=\eta\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n}),$
where
(2.7) $\displaystyle\mu_{j}$ $\displaystyle=$
$\displaystyle\frac{k+{\lambda}_{j}}{k+{\lambda}_{j}-\nu\eta}\,\ \ j=1,...,N,$
(2.8) $\displaystyle\phi_{j}$ $\displaystyle=$
$\displaystyle\sum^{N}_{i=1}{\alpha}_{ij}\
\displaystyle\frac{\partial{\varphi}^{*}_{i}}{\partial n}\,,\ \ j=1,...,N,$
and the matrix $\mathcal{X}=\|{\alpha}_{ij}\|^{N}_{i,j=1}$ is given by
(2.9) $\mathcal{X}=\mathcal{F}^{-1},\
\mathcal{F}=\left\|\int_{{\partial}\mathcal{O}}\displaystyle\frac{\partial{\varphi}^{*}_{i}}{\partial
n}\cdot\displaystyle\frac{\partial\overline{\varphi}^{*}_{j}}{\partial n}\
dx\right\|^{N}_{i,j=1}.$
In virtue of hypothesis (H2), $\mathcal{F}$ is invertible and so $\mathcal{X}$
is well defined.
###### Theorem 2.1
Assume that (H1), (H2), (1.6) hold and that ${\rm Re}\,{\lambda}_{j}<0$ for
$j=1,...,N$, ${\rm Re}\ {\lambda}_{j}>0$ for $j>N.$ Let $k>0$ sufficiently
large and $\eta>0$ be such that
(2.10) $(|k+{\lambda}_{j}|^{2}-\eta k\nu){\rm Re}\,{\lambda}_{j}-\eta\nu\ {\rm
Re}\,{\lambda}_{j}^{2}>0.$
Then the feedback controller (2.6) stabilizes exponentially system (1.3), that
is, the solution $Y$ to the closed loop system
(2.11) $\begin{array}[]{ll}\displaystyle\frac{\partial Y}{\partial
t}-\nu\Delta Y+(Y\cdot{\nabla})a+(b\cdot{\nabla})Y={\nabla}p&\mbox{ in }\
(0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot Y=0&\mbox{ in
}\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\
Y=\eta\displaystyle\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n})&\mbox{
on }\ (0,{\infty})\times{\partial}\mathcal{O},\end{array}$
satisfies
(2.12) $|Y(t)|_{\widetilde{H}}\leq Ce^{-{\gamma}t}|Y(0)|_{\widetilde{H}},\ \
\forall t\geq 0,$
for some ${\gamma}>0.$
As noticed earlier, by (2.8) it follows that, in each
$x\in{\partial}\mathcal{O}$, $\phi_{j}(x)$ are tangent to
${\partial}\mathcal{O}$ and so, for $|{\alpha}|$ large enough, the controller
$u$ is ”almost” normal. Moreover, since ${\rm Re}\,{\lambda}_{j}<0$ for
$j=1,...,N$, by (2.7) it is easily seen that (2.10) holds for $\eta>0$ and
$k>0$ sufficiently large and suitable chosen.
It should be observed that, if assumption (H2) is strengthen to all $j=1,...,$
and so (2.5) holds for all $i,j=1,...,$ then
$\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}=\left<\psi,{\varphi}^{*}_{j}\right>_{\widetilde{H}}$
for all $j$ and so the controller (2.6) reduces to
$u=\eta\sum^{N}_{j=1}\mu_{j}\left<Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n}).$
If ${\lambda}_{j}$ are complex valued, then the controller (2.6) is complex
valued too and plugged into system (1.3) leads to a real closed loop system in
$({\rm Re}\,Y,{\rm Im}\,Y)$. In order to avoid this situation, we shall
construct in Section 3.3 a real stabilizable feedback controller of the form
(2.6) which has a similar stabilization effect. (See Theorem 3.1.)
###### Remark 2.1
With choice of $\phi_{j}$ we have
$\displaystyle\int_{{\partial}\mathcal{O}}\overline{\phi}_{j}\
\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\ dx={\delta}_{ij},\
i,j=1,...,N,$ and, as seen later on, this is essential in the proof of Theorem
2.1. However, this can be also achieved for $\phi_{j}$ of the form
$\phi_{j}=\displaystyle\sum^{N}_{i=1}\alpha_{ij}\chi_{i}$
where $\\{\chi_{i}\\}$ are suitably chosen.
To find such $\chi_{i}$ and $\alpha_{ij}$, it sufficed to assume instead (H2)
that all $\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\not\equiv
0$ on ${\partial}\mathcal{O}$.
### 2.3 Stabilizable controllers with support in
${\Gamma}_{0}\subset{\partial}\mathcal{O}$
Consider system (1.1) with a boundary controller $u$ with support in an open
and smooth subset ${\Gamma}_{0}\subset{\partial}\mathcal{O}$, that is,
(2.13) $\begin{array}[]{ll}\displaystyle\frac{\partial Y}{\partial
t}-\nu\Delta Y+(Y\cdot{\nabla})a+(a\cdot{\nabla})Y={\nabla}p&\mbox{ in }\
(0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot Y=0\ \ \mbox{
in }\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\
Y=1\\!\\!\\!\;\mathrm{l}_{{\Gamma}_{0}}u&\mbox{ on }\
(0,{\infty})\times{\partial}\mathcal{O},\end{array}$
where $1\\!\\!\\!\;\mathrm{l}_{{\Gamma}_{0}}$ is the characteristic function
of ${\Gamma}_{0}$.
In this case, instead of (H2) we assume that
* (H2)′
The system $\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ is linearly independent on ${\Gamma}_{0}$.
We assume also that
(2.14) $\int_{{\Gamma}_{0}}{\alpha}(x)dx=0.$
We choose $\widetilde{\phi}_{j},\ j=1,...,N$, of the form
(2.15) $\widetilde{\phi}_{j}=\sum^{N}_{k=1}\widetilde{\alpha}_{jk}\
\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\,,$
where the matrix $\|\widetilde{\alpha}_{jk}\|^{N}_{j,k=1}$ is given by
$\left(\left\|\int_{{\partial}\mathcal{O}}\displaystyle\frac{\partial}{\partial
n}\ {\varphi}^{*}_{j}\cdot\displaystyle\frac{\partial}{\partial n}\
{\varphi}^{*}_{k}\,dx\right\|^{N}_{i,j=1}\right)^{-1}.$
Consider the feedback controller
(2.16)
$u_{{\Gamma}_{0}}=\eta\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\widetilde{\phi}_{j}+{\alpha}\vec{n}),$
where $\eta,\mu_{j}$ are chosen as in Theorem 2.1.
We have
###### Theorem 2.2
The controller $u_{{\Gamma}_{0}}$ stabilizes exponentially system (1.3).
### 2.4 Stabilization of system (1.1) ((1.2))
We set $W=(H^{\frac{1}{2}-{\varepsilon}}(\mathcal{O}))^{d}\cap H$ for
$Z=(H^{\frac{3}{2}+{\varepsilon}}(\mathcal{O}))^{d}\cap H)$, $d=2.$
###### Theorem 2.3
Let $d=2$. Then, under the assumptions of Theorem 2.1, the feedback boundary
controller (2.6) stabilizes exponentially system (1.2) in a neighborhood
$\mathcal{W}=\\{y_{0}\in W;\ \|y_{0}\|_{W}<\rho\\}$. More precisely, the
solution $y\in C([0,{\infty});W)\cap L^{2}(0,{\infty};Z)$ to the closed loop
system
(2.17) $\begin{array}[]{ll}\displaystyle\frac{\partial Y}{\partial
t}-\nu\Delta
Y+(Y\cdot{\nabla})a+(b\cdot{\nabla})Y+(Y\cdot{\nabla})Y={\nabla}p\\\
\hfill\mbox{ in }\ (0,{\infty})\times\mathcal{O},\vspace*{1,5mm}\\\
Y=\eta\displaystyle\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n})\
\mbox{ on }\ (0,{\infty})\times{\partial}\mathcal{O},\end{array}$
satisfies for $Y(0)\in\mathcal{W}$ and $\rho$ sufficiently small
(2.18) $\|Y(t)\|_{W}\leq Ce^{-{\gamma}t}\|Y(0)\|_{W},\ \ \forall t\geq 0,$
for some ${\gamma}>0$.
In particular, it follows that the boundary feedback controller
(2.19)
$u=\eta\sum^{N}_{j=1}\mu_{j}\left<P_{N}(Y-y_{e}),{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n})$
stabilizes exponentially the equilibrium solution $y_{e}$ to (1.1) in a
neighborhood $\\{y_{0}\in W;\ \|y_{0}-y_{e}\|_{W}<\rho\\}$.
## 3 Proofs
### 3.1 Proof of Theorem 2.1
We set
$U^{0}=\left\\{u\in(L^{2}({\partial}\mathcal{O}))^{d};\int_{{\partial}\mathcal{O}}u(x)\cdot\vec{n}(x)dx=0\right\\}.$
Then, for $k>0$ sufficiently large, there is a unique solution
$y\in(H^{\frac{1}{2}}(\mathcal{O}))^{d}$ to the equation
$\begin{array}[]{l}-\nu\Delta
y+(y\cdot{\nabla})a+(b\cdot{\nabla})y+ky={\nabla}p\ \mbox{ in }\
\mathcal{O},\vspace*{1,5mm}\\\ {\nabla}\cdot y=0\ \mbox{ in }\ \mathcal{O},\ \
y=u\ \mbox{ on }\ {\partial}\mathcal{O}.\end{array}$
(See, e.g., [23], p. 365.) We set $y=Du$ and note that (see, e.g., [11], p.
102),
$D\in L((H^{s}({\partial}\mathcal{O}))^{d}\cap
U^{0};(H^{s+\frac{1}{2}})\mathcal{O}))^{d}),\mbox{ for
}s\geq-\frac{1}{2}\,\cdot$
In terms of the Dirichlet map $D$, system(1.3) can be written as
(3.1) $\begin{array}[]{l}\displaystyle\frac{d}{dt}\
Y(t)+{\mathcal{A}}(Y(t)-Du(t))=0,\ \ t\geq 0,\vspace*{1,5mm}\\\
Y(0)=y_{0}.\end{array}$
Equivalently,
(3.2) $\begin{array}[]{l}\displaystyle\frac{d}{dt}\
z(t)+{\mathcal{A}}z(t)=-\Pi\left(D\ \displaystyle\frac{du}{dt}\ (t)\right),\ \
t\geq 0,\vspace*{1,5mm}\\\ z(0)=y_{0}-Du(0),\end{array}$ (3.3)
$z(t)=Y(t)-Du(t),\ \ t\geq 0.$
In the following, we fix $k>0$ sufficiently large and $\eta>0$ such that
(2.10) holds. In particular, we also have
(3.4) ${\lambda}_{i}+k-\nu\eta\neq 0\ \mbox{ for }\ i=1,2,...,N.$
We note fist that in terms of $z$ the controller (2.6) can be, equivalently,
expressed as
(3.5)
$u(t)=\eta\sum^{N}_{j=1}\left<P_{N}z(t),{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n}).$
Indeed, by (3.3) and (3.5), we have
(3.6)
$\begin{array}[]{lcl}u(t)&=&\displaystyle\eta\sum^{N}_{j=1}\left<P_{N}Y(t),{\varphi}^{*}_{j}\right>)_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n})\vspace*{1,5mm}\\\
&&-\eta\displaystyle\sum^{N}_{j=1}\left<u(t),D^{*}{\varphi}^{*}_{j}\right>_{(L^{2}({\partial}\mathcal{O}))^{d}}(\phi_{j}+{\alpha}\vec{n}),\end{array}$
where $D^{*}$ is the adjoint of $D$.
On the other hand, if we set $\psi=D(\phi_{j}+{\alpha}\vec{n})$ and recall
that
$\begin{array}[]{rclcrcl}\mathcal{L}^{*}{\varphi}^{*}_{i}-\overline{\lambda}_{i}{\varphi}^{*}_{i}&=&{\nabla}p_{i}\
\mbox{ in }\ \mathcal{O},&&{\varphi}^{*}_{i}&=&0\ \mbox{ on }\
{\partial}\mathcal{O},\vspace*{1,5mm}\\\
\mathcal{L}\psi+k\psi&=&{\nabla}\widetilde{p}\ \mbox{ in }\
\mathcal{O},&&\psi&=&\phi_{j}+{\alpha}\vec{n}\ \mbox{ on }\
{\partial}\mathcal{O},\end{array}$
where $\mathcal{L}$ is the Stokes–Oseen operator (1.5) and $\mathcal{L}^{*}$
is its formal adjoint, we get via Green’s formula
(3.7)
$\begin{array}[]{r}\left<\phi_{j}{+}{\alpha}\vec{n},D^{*}{\varphi}^{*}_{i}\right>_{(L^{2}({\partial}\mathcal{O}))^{d}}{=}\\!\displaystyle\int_{\mathcal{O}}\\!\\!\psi\cdot\overline{\varphi}^{*}_{i}dx=-\displaystyle\frac{\nu}{{\lambda}_{i}{+}k}\\!\int_{{\partial}\mathcal{O}}\\!\\!(\phi_{j}{+}{\alpha}\vec{n}){\cdot}\displaystyle\frac{\partial\overline{\varphi}^{*}_{i}}{\partial
n}\ dx\vspace*{1,5mm}\\\
\qquad\displaystyle=-\displaystyle\frac{\nu}{{\lambda}_{i}{+}k}\
{\delta}_{ij},\ \forall i,j=1,...,N,\end{array}$
because $\vec{n}\cdot\displaystyle\frac{\partial{\varphi}^{*}_{i}}{\partial
n}=0,$ a.e. on ${\partial}\mathcal{O}$ (see [11], Lemma 3.3) and, by (2.5),
(2.8), (2.9), we have
(3.8)
$\int_{{\partial}\mathcal{O}}\phi_{j}\cdot\displaystyle\frac{\partial\overline{\varphi}^{*}_{i}}{\partial
n}\ dx={\delta}_{ij},\ \ i,j=1,...,N.$
Then, by (3.6), (3.7), we see that
$\left<u(t),D^{*}{\varphi}^{*}_{i}\right>_{(L^{2}({\partial}\mathcal{O}))^{d}}=\displaystyle\frac{-\eta\nu}{k+{\lambda}_{i}-\nu\eta}\
\left<P_{N}Y,{\varphi}^{*}_{i}\right>_{\widetilde{H}}$
and, substituting into (3.6), we get (2.6) as claimed.
Now, substituting (3.5) into (3.2), we obtain that
(3.9)
$\begin{array}[]{l}\displaystyle\frac{dz}{dt}+{\mathcal{A}}z=-\eta\displaystyle\sum^{N}_{j=1}\left<P_{N}\
\displaystyle\frac{d}{dt}z(t),{\varphi}^{*}_{j}\right>\Pi
D(\phi_{j}+{\alpha}\vec{n}),\vspace*{1,5mm}\\\
z(0)=z_{0}=y_{0}-Du(0).\end{array}$
We write (3.9) as
(3.10) $\displaystyle\quad\frac{dz_{u}}{dt}+{\mathcal{A}}_{u}z_{u}$
$\displaystyle=$ $\displaystyle-\eta
P_{N}\displaystyle\sum^{N}_{j=1}\left<P_{N}\displaystyle\frac{dz}{dt}\,,{\varphi}^{*}_{j}\right>_{\widetilde{H}}\Pi
D(\phi_{j}+{\alpha}\vec{n}),$ (3.11)
$\displaystyle\quad\frac{dz_{s}}{dt}+{\mathcal{A}}_{s}z_{s}$ $\displaystyle=$
$\displaystyle-\eta(I-P_{N})\displaystyle\sum^{N}_{j=1}\left<P_{N}\displaystyle\frac{dz}{dt}\,,{\varphi}^{*}_{j}\right>_{\widetilde{H}}\Pi
D(\phi_{j}+{\alpha}\vec{n}),$
$z=z_{u}+z_{s},\ z_{u}\in X_{u},$ $z_{s}\in X_{s}$ and $P_{N}$ is given by
(2.3). If we represent $z_{u}$ as
$z_{u}=\displaystyle\sum^{N}_{j=1}z_{j}{\varphi}_{j},$
and recall (3.7), we rewrite (3.10) as
(3.12) $z^{\prime}_{j}+{\lambda}_{j}z_{j}=\frac{\eta\nu}{k+{\lambda}_{j}}\
z^{\prime}_{j},\ \ t\geq 0.$
By (2.10) we have
${\rm
Re}\left[{\lambda}_{j}\left(1-\frac{\eta\nu}{k+{\lambda}_{j}}\right)^{-1}\right]>0.$
Then, by (3.12) we see that we have for some ${\gamma}_{0}>0$
(3.13) $|z_{j}(t)|\leq e^{-{\gamma}_{0}t}|z_{j}(0)|,\ \ j=1,...,N.$
On the other hand, by (3.11) we have
(3.14)
$\frac{dz_{s}}{dt}+{\mathcal{A}}_{s}z_{s}=\eta(I-P_{N})\sum^{N}_{j=1}z_{j}\Pi
D(\phi_{j}+{\alpha}\vec{n}),$
and since
$\|e^{-{\mathcal{A}}_{s}t}\|_{L(\widetilde{H},\widetilde{H})}\leq
Ce^{-{\gamma}_{1}t},\ \ \forall t\geq 0,$
for some ${\gamma}_{1}>0$, we see that
$|z_{s}(t)|_{\widetilde{H}}\leq
C\exp(-{\gamma}_{0}t)|z_{s}(0)|_{\widetilde{H}},\ \ \forall t\geq 0,$
which together with (3.13) yields
(3.15) $|z(t)|_{\widetilde{H}}\leq
C\exp(-{\gamma}_{0}t)|z(0)|_{\widetilde{H}},\ \ \forall t\geq 0.$
Now, recalling (3.3) and (3.5), we obtain (2.12), thereby completing the
proof.
### 3.2 Proof of Theorem 2.2
The proof is exactly the same as that of Theorem 2.1 except that the Dirichlet
map $D$ is taken for the boundary condition
$y=1\\!\\!\\!\;\mathrm{l}_{{\Gamma}_{0}}.$ The details are omitted.
### 3.3 Proof of Theorem 2.3
We shall apply Theorem 1.2.1 from [10] (see, also, Theorem 5.1 in [11]).
In fact, system (1.2) with the feedback controller
$u=FY=\eta\displaystyle\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}(\phi_{j}+{\alpha}\vec{n})$
can be written as
$\begin{array}[]{l}\displaystyle\frac{dY}{dt}+{\mathcal{A}}(Y-DFY)+BY=0,\
t>0,\vspace*{1,5mm}\\\ Y(0)=y_{0},\end{array}$
where $BY=\Pi(Y\cdot{\nabla})Y).$
By Theorem 2.1, it is easily seen that the operator
$A_{F}={\mathcal{A}}(I-DF):W\to W$ with $D(A_{F})=\\{y\in W;\
{\mathcal{A}}(y-DFy)\in W\\}$ generates an analytic $C_{0}$-semigroup on $W$
which is exponentially stable on $W$.
Moreover, coming back to system (3.9)-(3.11), we see that besides (3.15) we
have also
$\int^{\infty}_{0}|A^{\frac{3}{4}}z(t|^{2}dt\leq C\|z(0)\|^{2}_{W}$
and recalling that $Y=e^{-A_{F}t}y_{0}$ is given by
$Y=z+DFY=z+\eta\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,{\varphi}^{*}_{j}\right>_{\widetilde{H}}D(\phi_{j}+{\alpha}\vec{n})$
we infer that
$\int^{\infty}_{0}\|e^{-A_{F}t}y_{0}\|^{2}_{Z}dt\leq C\|y_{0}\|^{2}_{W},\ \
\forall y_{0}\in W.$
Then, by Theorem 1.2.1 from [10], we infer that the conclusion of Theorem 2.3
holds.
### 3.4 Real stabilizable feedback controllers
We shall construct here a real stabilizable feedback controller of the form
(3.5). To this purpose, we consider in the space $H$ the system
$\\{{\rm Re}\,{\varphi}_{j},{\rm
Im}\,{\varphi}_{j}\\}^{N}_{j=1}=\\{\psi_{j}\\}^{N}_{j=1}.$
We set $X^{*}_{u}={\rm lin\ span}\\{{\rm Re}\,{\varphi}_{j},{\rm Im}\
{\varphi}_{j}\\}^{N}_{j=1}$, $j=1,...,N.$ We decompose the space
$H=X^{*}_{u}\oplus X^{*}_{s}$ and note that the real operator ${\mathcal{A}}$
leaves invariant both spaces $X^{*}_{s}$ and $X^{*}_{u}$ and
${\mathcal{A}}^{*}_{s}={\mathcal{A}}|_{X^{*}_{s}}$ generates an exponential
stable semigroup on $X^{*}_{s}\subset H$.
We have
(3.16) ${\mathcal{A}}\psi_{j}=({\rm Re}\,{\lambda}_{j})\psi_{j}-({\rm
Im}\,{\lambda}_{j})\psi_{j+1},\ {\mathcal{A}}\psi_{j+1}=({\rm
Im}\,{\lambda}_{j})\psi_{j}+({\rm Re}\,{\lambda}_{j})\psi_{j+1}.$
We may assume via Schmidt’s ortogonalization algorithm that the system
$\\{\psi_{j}\\}^{N}_{j=1}$ is orthonormal.
Then, we construct the feedback controller
(3.17)
$u^{*}=\eta\sum^{N}_{j=1}\mu_{j}\left<P_{N}Y,\psi_{j}\right>(\phi^{*}_{j}+{\alpha}\vec{n}),$
where $\phi^{*}_{j}$ is of the form
(3.18) $\phi^{*}_{j}=\sum^{N}_{i=1}{\alpha}^{*}_{ij}\
\displaystyle\frac{\partial\psi_{i}}{\partial n}\,,\ \ j=1,...,N,$
and ${\alpha}^{*}_{ij}$ are chosen in a such a way that
$\sum^{N}_{i=1}{\alpha}^{*}_{ij}\left<\widetilde{D}\
\displaystyle\frac{\partial\psi_{i}}{\partial
n}\,,\psi_{\ell}\right>={\delta}_{j\ell},\ j,\ell=1,...,N,$
where $\widetilde{D}$ is the Dirichlet map corresponding to the operator
${\mathcal{A}}^{*}_{k}$. Keeping in mind that
$\left<\widetilde{D}\chi_{i},{\mathcal{A}}_{k}\psi_{j}\right>=-\nu\int_{{\partial}\mathcal{O}}\chi_{i}\
\displaystyle\frac{\partial}{\partial n}\ \psi_{j}\,dx,\ i,j=1,...,N,$
we see by (3.16) that, for $k$ large enough, we have for
$\chi_{i}=\displaystyle\frac{\partial\psi_{i}}{\partial n}\,,$
$\begin{array}[]{rcl}\left<\widetilde{D}\chi_{i},\psi_{j}\right>&=&-\displaystyle\frac{\nu(b^{i}_{j}\
{\rm Re}\,{\lambda}_{j}+b^{i}_{j+1}\ {\rm Im}\,{\lambda}_{j})}{|z_{j}|^{2}+k\
{\rm Re}\,{\lambda}_{j}},\vspace*{1,5mm}\\\
\left<\widetilde{D}\chi_{i},\psi_{j+1}\right>&=&-\displaystyle\frac{\nu(({\rm
Re}\,{\lambda}_{j}+k)b^{i}_{j+1}-{\rm
Im}\,{\lambda}_{j}b^{i}_{j})}{|z_{j}|^{2}+k\ {\rm
Re}\,{\lambda}_{j}},\end{array}$
where $b^{i}_{j}=\displaystyle\int_{{\partial}\mathcal{O}}\chi_{i}\
\displaystyle\frac{\partial\psi_{j}}{\partial n}\ dx$. Then, assuming that
* (H2)∗
The system $\left\\{\displaystyle\frac{\partial{\varphi}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ is linearly independent on ${\partial}{\Omega}$.
it follows that so is $\left\\{\displaystyle\frac{\partial\psi_{j}}{\partial
n}\right\\}^{N}_{j=1}$ and this implies that such a choice of
${\alpha}^{*}_{ij}$ is possible. Then, arguing exactly as in the proof of
Theorem 2.1, we see that, for $\eta$ and $\mu_{j}$ suitable chosen, the real
controller (3.17) stabilizes exponentially system 1.3. We have, therefore,
###### Theorem 3.1
Under assumptions (H1), (H2)∗ and (1.6), there is a boundary feedback
controller $u^{*}$ of the form (3.17) which stabilizes exponentially system
(1.3).
The proof is exactly the same as that of Theorem 2.1 and so it is omitted.
We note, however, that if instead of (H2)∗ we assume only that
$\displaystyle\frac{\partial{\varphi}_{j}}{\partial n}\not\equiv 0$ on
${\partial}\mathcal{O}$ for $j=1,...,N,$ then Theorem 3.1 still remains valid
with $\phi^{*}=\displaystyle\sum^{N}_{i=1}\alpha_{ij}\chi_{i}$, where
$\chi_{i}$ are chosen in such a way that
$\displaystyle\sum^{N}_{i=1}\alpha^{*}_{ij}\left<\widetilde{D}\chi_{i},\psi_{\ell}\right>=\delta_{j\ell}$,
$j,\ell=1,...,N.$ Note also that Theorem 2.3 remains true in the present
situation.
## 4 Boundary stabilization of a periodic flow
in a $2{-}D$ channel
Consider a laminar flow in a two-dimensional channel with the walls located at
$y=0,1.$ We shall assume that the velocity field $(u(t,x,y),v(t,x,y))$ and the
pressure $p(t,x,y)$ are $2\pi$ periodic in $x\in(-{\infty},+{\infty})$.
The dynamic of flow is governed by the incompressible $2-D$ Navier–Stokes
equations
(4.1) $\begin{array}[]{ll}u_{t}-\nu\Delta u+uu_{x}+vu_{y}=p_{x},\quad
x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\ v_{t}-\nu\Delta
v+uv_{x}+vv_{y}=p_{y},\quad x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\
u_{x}+v_{y}=0,\vspace*{1,5mm}\\\ u(t,x+2\pi,y)\equiv u(t,x,y),\ \
v(t,x+2\pi,y)\equiv v(t,x,y),\ \ y\in(0,1).\end{array}$
Consider a steady-state flow governed by (4.1) with zero vertical velocity
component, i.e., $(U(x,y),0)$. Since the flow is freely divergent, we have
$U_{x}\equiv 0$ and so $U(x,y)\equiv U(y)$. This yields
(4.2) $U(y)=C(y^{2}-y),\ \ \forall y\in(0,1),$
where $C\in{\mathbb{R}}^{-}$. In the following, we take
$C=-\displaystyle\frac{a}{2\nu}$ where $a\in{\mathbb{R}}^{+}$.
The linearization of (4.1) around the steady-state flow $(U(y),0)$ leads to
the following system
(4.3) $\begin{array}[]{ll}u_{t}-\nu\Delta u+u_{x}U+vU^{\prime}=p_{x},\ \
y\in(0,1),\ x,t\in{\mathbb{R}},\vspace*{1,5mm}\\\ v_{t}-\nu\Delta
v+v_{x}U=p_{y},\vspace*{1,5mm}\\\ u_{x}+v_{y}=0,\vspace*{1,5mm}\\\
u(t,x+2\pi,y)\equiv u(t,x,y),\ \ v(t,x+2\pi,y)\equiv v(t,x,y).\end{array}$
Here we apply Theorem 2.1 to construct an oblique boundary feedback controller
for system(4.3). To this aim, we recall first the Fourier functional setting
for description of periodic fluid flows in the channel
$(-{\infty},+{\infty})\times(0,1).$
Let $L^{2}_{\pi}(Q)$, $Q=(0,2\pi)\times(0,1)$ be the space of all the
functions $u\in L^{2}_{\rm loc}(R\times(0,1))$ which are $2\pi$-periodic in
$x$. These functions are characterized by their Fourier series
$u(x,y)=\displaystyle\sum_{k}a_{k}(y)e^{ikx},\ a_{k}=\bar{a}_{-k},$ $a_{0}=0$,
$\displaystyle\sum_{k}\int^{1}_{0}|a_{k}|^{2}dy<{\infty}.$
Similarly, $H^{1}_{\pi}(Q),\ H^{2}_{\pi}(Q)$ are defined. For instance,
$\begin{array}[]{l}H^{1}_{\pi}(Q)=\left\\{u\in L^{2}_{\pi}(Q);\
u\in\displaystyle\sum_{k}a_{k}e^{ikx},\ a_{k}=\bar{a}_{-k},\
a_{0}=0,\right.\vspace*{1,5mm}\\\
\qquad\qquad\qquad\left.\displaystyle\sum_{k}\int^{1}_{0}(k^{2}|a_{k}|^{2}+|a^{\prime}_{k}|^{2})dy<{\infty}\right\\},\
k\to j.\end{array}$
We set
$H=\\{(u,v)\in(L^{2}_{\pi}(Q))^{2};\ u_{x}+v_{y}=0,\ v(x,0)=v(x,1)=0\\}.$
If $u_{x}+v_{y}=0$, then the trace of $(u,v)$ at $y=0,1$ is well defined as an
element of $H^{-1}(0,2\pi)\times H^{-1}(0,2\pi)$ (see, e.g., [22]).
We also set
$V=\\{(u,v)\in H\cap H^{1}_{\pi}(Q);\ u(x,0)=u(x,1)=v(x,0)=v(x,1)=0\\}.$
As defined above, the space $L^{2}_{\pi}(Q)$ is, in fact, the factor space
$L^{2}_{\pi}(Q)/Z.$ The space $H$ can be defined equally as
$\begin{array}[]{lcl}H&=&\Big{\\{}u=\displaystyle\sum_{k\neq
0}u_{k}(y)e^{ikx},\ v=\displaystyle\sum_{k\neq 0}v_{k}(y)e^{ikx},\
v_{k}j(0)=v_{k}(1)=0,\vspace*{1,5mm}\\\ &&\ \ \ \ \ \displaystyle\sum_{k\neq
0}\int^{1}_{0}(|u_{k}|^{2}+|v_{k}|^{2})dy<{\infty},\
iku_{k}(y)+v^{\prime}_{k}(y)=0,\vspace*{1,5mm}\\\ &&\hfill\ \ \ \ \mbox{ a.e.
}y\in(0,1),\ k\in{\mathbb{R}}\Big{\\}},\ k\to j.\end{array}$
Let $\Pi:L^{2}_{\pi}(Q)\to H$ be the Leray projector and
${\mathcal{A}}:D({\mathcal{A}})\subset H\to H^{\prime}$ the operator
(4.4) $\begin{array}[]{r}{\mathcal{A}}(u,v)=\Pi\\{-\nu\Delta
u+u_{x}U+vU^{\prime},\ -\nu\Delta v+v_{x}U\\},\vspace*{1,5mm}\\\
\forall(u,v)\in D({\mathcal{A}})=(H^{2}((0,2\pi)\times(0,1)).\end{array}$
We associate with (4.3) the boundary value conditions
(4.5) $\begin{array}[]{lll}u(t,x,0)=u^{0}(t,x),&u(t,x,1)=u^{1}(t,x),&t\geq 0,\
x\in{\mathbb{R}},\vspace*{1,5mm}\\\
v(t,x,0)=v^{0}(t,x),&v(t,x,1)=v^{1}(t,x),&t\geq 0,\
x\in{\mathbb{R}},\end{array}$
and, for $k>0$ sufficiently large, we consider the Dirichlet map $D:X\to
L^{2}_{\pi}(Q)$ defined by $D(u^{*},v^{*})=(\widetilde{u},\widetilde{v})$,
(4.6)
$\begin{array}[]{ll}-\nu\Delta\widetilde{u}+\widetilde{u}_{\lambda}U+\widetilde{v}U^{\prime}+k\widetilde{u}=p_{x},\
x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\
-\nu\Delta\widetilde{v}+\widetilde{v}_{x}U+k\widetilde{v}=p_{y},\
x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\
\widetilde{u}_{x}+\widetilde{v}_{y}=0,\
\widetilde{u}(x+2\pi,y)=\widetilde{u}(x,y),\
\widetilde{v}(x+2\pi,y)=\widetilde{v}(x,y),\vspace*{1,5mm}\\\
\widetilde{u}(x,y)=u^{*}(x,y),\ \widetilde{v}(x,y)=v^{*}(x,y),\
y=0,1.\end{array}$
Here
$\begin{array}[]{lcl}X&=&\Big{\\{}(u^{*},v^{*})\in
L^{2}((0,2\pi)\times{\partial}(0,1));\
u^{*}(x+2\pi,y)=u^{*}(x,y),\vspace*{1,5mm}\\\ &&\ \
v^{*}(x+2\pi,y)=v^{*}(x,y),\
\displaystyle\int^{2\pi}_{0}v^{*}(x,0)dx=\displaystyle\int^{2\pi}_{0}v^{*}(x,1)dx\Big{\\}}.\end{array}$
Then system (4.3) with boundary conditions (4.4) can be written as
(4.7) $\begin{array}[]{l}\displaystyle\frac{d}{dt}\
Y(t)+{\mathcal{A}}(Y(t)-DU^{*}(t))=0,\ \ t\geq 0,\vspace*{1,5mm}\\\
Y(0)=(u_{0},v_{0}),\end{array}$
where $Y=(u,v),\ U^{*}=(u^{*},v^{*}).$
In order to apply Theorem 2.1, we shall check hypothesis (H2) in this case.
To this end, we denote again by ${\mathcal{A}}$ the extension of
${\mathcal{A}}$ on the complexified space $\widetilde{H}$ and by
${\lambda}_{j},{\varphi}_{j}$ the eigenvalues and corresponding eigenvectors
of the operator ${\mathcal{A}}$. By ${\varphi}^{*}_{j}$, we denote the
eigenvector to the dual operator ${\mathcal{A}}^{*}$.
###### Lemma 4.1
For all $j=1,2,...,N,$ we have
(4.8) $\displaystyle\frac{\partial{\varphi}_{j}}{\partial n}\ (x,y)\not\equiv
0,\ \ x\in(0,2\pi),\ y=0,1,$
and
(4.9) $\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial n}\
(x,y)\not\equiv 0,\ \ x\in(0,2\pi),\ y=0,1.$
Proof. If we represent ${\varphi}_{j}=(u^{j},v^{j}),$ then (4.8) reduces to
(4.10) $\left|\displaystyle\frac{\partial}{\partial y}\
v^{j}(x,y)\right|+\left|\displaystyle\frac{\partial}{\partial y}\
u^{j}(x,y)\right|>0,\ x\in(0,2\pi),\ y=0,1.$
We set ${\lambda}={\lambda}_{j}$ and ${\varphi}_{j}=(u,v)$. This means that,
if ${\lambda}$ is semisimple, then
(4.11) $\begin{array}[]{ll}-\nu\Delta u+u_{x}U+vU^{\prime}={\lambda}u+p_{x},\
x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\ -\nu\Delta
v+v_{x}U={\lambda}v+p_{y},\ x\in{\mathbb{R}},\ y\in(0,1),\vspace*{1,5mm}\\\
u_{x}+v_{y}=0,\vspace*{1,5mm}\\\ u(x+2\pi,y)=u(x,y),\
v(x+2\pi,y)=v(x,y).\end{array}$
If we represent $u,v,p$ as Fourier series,
(4.12)
$u(x,y){=}\sum_{k}e^{ikx}u_{k}(y),\,v(x,y){=}\displaystyle\sum_{k}e^{ikx}v_{k}(y),\,p(x,y){=}\displaystyle\sum_{k}e^{ikx}p_{k}(y),$
we reduce (4.11) to (see, e.g., [7], p. 144)
$\begin{array}[]{l}-\nu u^{\prime\prime}_{k}+(\nu
k^{2}+ikU)u_{k}+U^{\prime}v_{k}=ikp_{k}+{\lambda}u_{k},\
y\in(0,1),\vspace*{1,5mm}\\\ -\nu v^{\prime\prime}_{k}+(\nu
k^{2}+ikU)v_{k}=p^{\prime}_{k}+{\lambda}v_{k},\ iku_{k}+v^{\prime}_{k}=0\
\mbox{ in }\ (0,1),\vspace*{1,5mm}\\\ u_{k}(0)=u_{k}(1)=0,\ \
v_{k}(0)=v_{k}(1)=0.\end{array}$
Equivalently,
(4.13) $\begin{array}[]{l}-\nu v^{\rm iv}_{k}{+}(2\nu
k^{2}{+}ikU)v^{\prime\prime}_{k}{-}k(\nu
k^{3}{+}ik^{2}U{+}iU^{\prime\prime})v_{k}{-}{\lambda}(v^{\prime\prime}_{k}{-}k^{2}v_{k})=0,\\\
\hfill y\in(0,1),\vspace*{1,5mm}\\\ v_{k}(0)=v_{k}(1)=0,\
v^{\prime}_{k}(0)=v^{\prime}_{k}(1)=0,\ \ \forall k\neq 0.\end{array}$
Now, let us check (4.9) or, equivalently, (4.10). We have for $u=u^{j}$
$\displaystyle\frac{\partial}{\partial n}\
u(x,y)=-i\displaystyle\sum_{k}\frac{e^{ikx}}{k}v^{\prime\prime}_{k}(y),\ \
\forall x,y\in 0,1,$
and so (4.10) reduces to
(4.14) $|v^{\prime\prime}_{k}(0)|+|v^{\prime\prime}_{k}(1)|>0\mbox{\ \ for all
$k$.}$
Assume that $v^{\prime\prime}_{k}(0)=v^{\prime\prime}_{k}(1)=0$ for all $k$
and lead from this to a contradiction. To this end we set
$W_{k}=v^{\prime\prime}_{k}-k^{2}v_{k}$ and rewrite (4.13) as
(4.15) $\begin{array}[]{c}-\nu W^{\prime\prime}_{k}+(\nu
k^{2}+ikU-{\lambda})W_{k}=ikU^{\prime\prime}v_{k}\ \mbox{ in }\
(0,1),\vspace*{1,5mm}\\\ W_{k}(0)=W_{k}(1)=0.\end{array}$
If we multiply (4.15) by $\overline{W}_{k}$, integrate on $(0,1)$ and take the
real part, we obtain that
$\int^{1}_{0}(\nu|W^{\prime}_{k}|^{2}+(\nu k^{2}-{\rm
Re}\,{\lambda})|W_{k}|^{2})dy=0,\ \forall k$
and since ${\rm Re}\,{\lambda}={\rm Re}\,{\lambda}_{j}\leq 0$ for all
$j=1,...,N$, we get $W_{k}\equiv 0$, and so $v_{k}\equiv 0.$ The contradiction
we arrived at proves (4.14) and (4.8). Arguing similarly for the dual system
with eigenfunctions $\\{u^{*}_{k},v^{*}_{k}\\}$, we get for
$W^{*}_{k}=(v^{*}_{k})^{\prime\prime}-k^{2}v^{*}_{k}$
$\begin{array}[]{c}L^{*}_{k}v^{*}_{k}\equiv-\nu(W^{*}_{k})^{\prime\prime}+(\nu
k^{2}-ikU-\overline{\lambda})W^{*}_{k}-2ik(v^{*}_{k})^{\prime}U^{\prime}=0\
\mbox{ in }(0,1),\vspace*{1,5mm}\\\ v^{*}_{k}(0)=v^{*}_{k}(1)=0,\ \
(v^{*}_{k})^{\prime}(0)=(v^{*}_{k})^{\prime}(1)=0,\ \
(v^{*}_{k})^{\prime\prime}(0)=(v^{*}_{k})^{\prime\prime}(1)=0,\end{array}$
and argue from this to a contradiction. We set $\mathcal{X}^{*}=\\{{\varphi};\
L^{*}_{k}{\varphi}=0,\ {\varphi}(0)={\varphi}(1)=0,\
{\varphi}^{\prime}(0)={\varphi}^{\prime}(1)=0\\}$ and let
$\check{\varphi}(y)={\varphi}(1-y),$ $y\in[0,1]$. Since
$\dim\mathcal{X}^{*}\leq 2$ and $\check{U}\equiv U$, we infer that each
${\varphi}\in\mathcal{X}^{*}$ is either symmetric (that is,
${\varphi}\equiv\check{\varphi})$ or antisymmetric (that is,
${\varphi}\equiv-\check{\varphi})$. Assume that $v^{*}_{k}$ is symmetric. By
(4.15) we see via integration by parts that $\int^{1}_{0}|v^{*}_{k}|^{2}dy=0$
if there is ${\varphi}$ such that
(4.16) $L_{k}{\varphi}=v^{*}_{k}\ \mbox{ in }(0,1);\ \ \
{\varphi}(0)={\varphi}(1)=0.$
In order to show that there is such a function ${\varphi}$, we shall prove
that there is ${\varphi}_{1}$ such that
(4.17) $L_{k}{\varphi}_{1}=0\mbox{ in }(0,1),\ \ \ \
{\varphi}_{1}(0)+{\varphi}_{1}(1)\neq 0.$
Indeed, if such a ${\varphi}_{1}$ exists, by replacing ${\varphi}_{1}$ by
${\varphi}+\check{\varphi}_{1}$, we may assume that ${\varphi}_{1}$ is
symmetric. If ${\varphi}_{2}$ is a symmetric solution to
$L_{k}{\varphi}_{2}=v^{*}_{k}$, then clearly
${\varphi}={\varphi}_{2}-{\varphi}_{2}(0)({\varphi}_{1}(0))^{-1}{\varphi}_{1}$
satisfies (4.16) because, by (4.17), ${\varphi}_{1}(0)\neq 0$.
Now, to prove the existence in (4.17), we shall argue as in [20] and assume
that $\mathcal{X}=\\{\psi;\ L_{k}\psi=0\\}\equiv\\{\psi;\ L_{k}\psi=0,\
\psi(0)+\psi(1)=0\\}$ and argue from this to a contradiction. We set
$\mathcal{X}_{1}=\\{\psi\in\mathcal{X};\
\psi^{\prime\prime}(0)+\psi^{\prime\prime}(1)=0\\}$ and prove that
$\psi=-\check{\psi}$ for each $\psi\in\mathcal{X}_{1}$. Indeed,
$\theta=\psi+\check{\psi}$ satisfies $\theta(0)=\theta(1)=0$,
$\theta^{\prime\prime}(0)=\theta^{\prime\prime}(1)=0$ and
$W=\theta^{\prime\prime}-k^{2}\theta$ satisfies (4.15) with $v_{k}=\theta$.
Then, we obtain as above that $W\equiv 0,$ $\theta\equiv 0.$ The spaces
$\mathcal{X}_{1}=\\{\psi\in\mathcal{X};\ \psi\equiv-\check{\psi}\\}$ and
$\mathcal{X}_{2}=\\{\psi\in\mathcal{X};\ \psi\equiv\check{\psi}\\}$ are
orthogonal and both have dimension 2 because
$\begin{array}[]{lcl}\mathcal{X}_{1}&=&\\{\psi\in\mathcal{X};\
\psi^{\prime}(\frac{1}{2})=\psi^{\prime\prime\prime}(\frac{1}{2})=0\\},\vspace*{1,5mm}\\\
\mathcal{X}_{2}&=&\\{\psi\in\mathcal{X};\
\psi(\frac{1}{2})=\psi^{\prime\prime}(\frac{1}{2})=0\\}.\end{array}$
Hence, $\mathcal{X}=\mathcal{X}_{1}\oplus\mathcal{X}_{2}$. On the other hand,
the space $\\{\psi\in\mathcal{X};\ \psi^{\prime\prime}(0)=0\\}$ which has
dimension 3, has nonempty intersection with $\mathcal{X}_{2}$. Hence, there is
$\psi\in\mathcal{X}$ symmetric such that
$\psi^{\prime\prime}(0)=\psi^{\prime\prime}(1)=0$. Clearly,
$\psi\in\mathcal{X}_{1}$, which is absurd. This completes the proof.
As regards (H1), it is not clear if it is always satisfied in the present
situation and so we keep it.
###### Lemma 4.2
The systems $\left\\{\displaystyle\frac{\partial{\varphi}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ and
$\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ are linearly independent on ${\partial}\mathcal{O}$.
Proof. It suffices to prove the independence of
$\left\\{\displaystyle\frac{\partial{\varphi}_{j}}{\partial
n}\right\\}^{N}_{j=1}$, ${\varphi}_{j}=(u^{j},v^{j})$. We have as above
$u^{j}=\\{u^{j}_{k}\\}_{k},$ $v^{j}=\\{v^{j}_{k}\\}$, $j=1,...,N.$ If
$\\{{\varphi}_{j}\\}$ are eigenvectors corresponding to the same eigenvalue,
the independence follows by Lemma 4.1. Assume that
${\mathcal{A}}{\varphi}_{1}={\lambda}_{1}{\varphi}_{1}$,
${\mathcal{A}}{\varphi}_{2}={\lambda}_{2}{\varphi}_{2}$, where
${\lambda}_{1}\neq{\lambda}_{2}$, and that
$\displaystyle\frac{\partial{\varphi}_{1}}{\partial
n}+\displaystyle\frac{\partial{\varphi}_{2}}{\partial n}=0\ \mbox{ in }\
y=0,1.$
We set $\widetilde{v}_{k}=v^{1}_{k}+v^{2}_{k}$ and
$\widetilde{W}_{k}=\widetilde{v}^{\prime\prime}_{k}-k^{2}\widetilde{v}_{k}.$
Then, we have as above (see (4.15))
$\begin{array}[]{l}-\nu\widetilde{W}^{\prime\prime}_{k}+(\nu
k^{2}+ikU-{\lambda}_{1})\widetilde{W}_{k}-({\lambda}_{2}-{\lambda}_{1})((v^{2}_{k})^{\prime\prime}-k^{2}v^{2}_{k})=ikU^{\prime\prime}\widetilde{v}_{k},\vspace*{1,5mm}\\\
-\nu\widetilde{W}^{\prime\prime}_{k}+(\nu
k^{2}+ikU-{\lambda}_{2})\widetilde{W}_{k}-({\lambda}_{1}-{\lambda}_{2})((v^{1}_{k})^{\prime\prime}-k^{2}v^{1}_{k})=ikU^{\prime\prime}\widetilde{v}_{k}.\end{array}$
This yields
$\begin{array}[]{l}\displaystyle\int^{1}_{0}(\nu|\widetilde{W}^{\prime}_{k}|^{2}+(\nu
k^{2}-{\rm Re}\,{\lambda}_{1})|\widetilde{W}_{k}|^{2})dy-{\rm
Re}\left[({\lambda}_{2}-{\lambda}_{1})\displaystyle\int^{1}_{0}((v^{2}_{k})^{\prime\prime}-k^{2}v^{2}_{k})\overline{\widetilde{W}}_{k})dy\right]=0,\vspace*{1,5mm}\\\
\displaystyle\int^{1}_{0}(\nu|\widetilde{W}^{\prime}_{k}|^{2}+(\nu k^{2}-{\rm
Re}\,{\lambda}_{2})|\widetilde{W}_{k}|^{2})dy-{\rm
Re}\left[({\lambda}_{1}-{\lambda}_{2})\displaystyle\int^{1}_{0}(v^{1}_{k})^{\prime\prime}-k^{2}v^{2}_{k})\widetilde{W}_{k}dy\right]=0,\end{array}$
and, therefore,
$\int^{1}_{0}(\nu|\widetilde{W}^{\prime}_{k}|^{2}+(\nu k^{2}-{\rm
Re}\,{\lambda}_{1}-{\rm Re}\,{\lambda}_{2})|\widetilde{W}_{k}|^{2})dy=0.$
Hence, $\widetilde{W}_{k}\equiv 0$, $\widetilde{v}_{k}=0$, which is absurd
because $v^{1}_{k},v^{2}_{k}$ are independent.
By induction with respect to $j$, one proves the independence of
$\left\\{\displaystyle\frac{\partial{\varphi}_{j}}{\partial
n}\right\\}^{N}_{j=1}$. The case
$\left\\{\displaystyle\frac{\partial{\varphi}^{*}_{j}}{\partial
n}\right\\}^{N}_{j=1}$ is completely similar.
Now, following the general case (3.5), we can design a feedback controller
$(u^{0},v^{0})$ for system (4.3), (4.5). We set
${\varphi}^{*}_{j}=(u^{*}_{j},v^{*}_{j}),\ \ j=1,...,N,$
where ${\varphi}^{*}_{j}$ are eigenvectors of the dual operator
${\mathcal{A}}^{*}$ with corresponding eigenvalues $\overline{\lambda}_{j}$
and ${\rm Re}\,{\lambda}_{j}<0$ for $j=1,...,N.$
We consider the feedback controller
(4.18)
$\begin{array}[]{rcll}u^{0}(t,x,y)&{=}&\eta\displaystyle\sum^{N}_{j=1}\mu_{j}v_{j}(t)\phi^{1}_{j}(x,y),\
x\in{\mathbb{R}},\ y=0,1,\vspace*{1,5mm}\\\
v^{0}(t,x,y)&{=}&\eta\displaystyle\sum^{N}_{j=1}\mu_{j}v_{j}(t)(\phi^{2}_{j}(x,y)+{\alpha}H(y)),\
x\in{\mathbb{R}},\ y=0,1,\vspace*{1,5mm}\\\
v_{j}(t)&{=}&\displaystyle\int^{2\pi}_{0}(u(t,x,y)u^{*}_{j}(x,y)+v(t,x,y)v^{*}_{j}(x,y))dx\,dy\vspace*{1,5mm}\\\
&{=}&\displaystyle\sum_{k}(u_{k}(t,y)(\bar{u}^{*}_{j})_{k}(y)+v_{k}(t,y)(\bar{v}^{*}_{k})_{k}(y)).\end{array}$
Here ${\alpha}$ is an arbitrary constant, $H(0)=-1$, $H(1)=1$, $\mu_{j}$ are
defined as (2.7) and, according to (2.8), $\phi^{i}_{j}$, $i=1,2,$ are of the
form
$\phi^{1}_{j}=\sum^{N}_{j=1}{\alpha}_{ij}(u^{*}_{i})^{\prime}(y),\ \
\phi^{2}_{j}=\displaystyle\sum^{N}_{i=1}{\alpha}_{ij}(v^{*}_{j})^{\prime}(y),$
where ${\alpha}_{ij}$ are chosen as in Section 2.2. Then, by Theorem 2.1, we
have
###### Theorem 4.3
For each ${\alpha}\in{\mathbb{R}}$ and $\eta$ suitable chosen, the feedback
boundary controller (4.18) stabilizes exponentially system (4.3).
We note that condition (1.6) automatically holds in this case for any constant
${\alpha}$. However, by Theorem 2.2, it follows also the stabilization with a
controller $(u^{0},v^{0})$ with support in $\\{y=0\\}$ or $\\{y=1\\}$ if
${\alpha}={\alpha}(x,y)$ is taken in such a way that
$\displaystyle\int^{2\pi}_{0}{\alpha}(x)dx=0$.
We note also that, by Theorem 4.3, we infer that the feedback controller
(4.18) is exponentially stabilizable in the Navier–Stokes equation (4.1).
###### Remark 4.1
The boundary stabilization of (4.1) was studied in [1], [2], [3], [5], [24],
[25]. In [3] and [18] it is proved the existence of a normal stabilizing
controller $\\{u_{k},v_{k}\\}$ such that $u_{k}\equiv v_{k}\equiv 0$ for
$|k|\geq M,$ which is, apparently, a stronger result than Theorem 4.3.
However, the advantage of the present result is the explicit design of the
feedback controller.
Note also that, by Theorem 2.3, the feedback controller is stabilizable in
Navier–Stokes equation (4.1). Also, as in Theorem 3.1, it can be replaced by a
real feedback controller.
## References
* [1] Aamo O.M., Krstic M., Bewley T.R., Control of mixing by boundary feedb ack in $2D$-channel, Automatica, 39 (2003), 1597-1606.
* [2] Balogh A., Liu W.-L., Krsstic M., Stability enhancement by b oundary control in $2D$ channel flow, IEEE Trans. Autom. Control, 11 (2001), 1696-1711.
* [3] Barbu V., Stabilization of a plane channel flow by wall normal controllers, Nonlinear Anal. Theory – Methods Appl., 56 (2007), 145-168.
* [4] Barbu V., The internal stabilization by noise of the linearlized Navier–Stokes equation, ESAIM COCV, 2 (2009).
* [5] Barbu V., Stabilization of a plane periodic channel flow by noise wall normal controllers, Syst. Control Lett., 50 (10) (2010), 608-618.
* [6] Barbu V., Exponential stabilization of the linearized Navier–Stokes equation by pointwise feedback controllers, Automatica, 46 (12) (2010), 1890-1895.
* [7] Barbu V., Stabilization of Navier–Stokes Flows, Communications and Control Engineering, Springer, London, 2010.
* [8] Barbu V., Internal stabilization of the Oseen–Stokes equations by Stratonovitch noise, Systems & Control Letters, doi:10.1016/j.sycont.2011.04.019
* [9] Barbu V., Da Prato G., Internal stabilization by noise of the Navier–Stokes equation, SIAM J. Control Opitm., 49 (1) (2010), 1-21.
* [10] Barbu V., Lasiecka I., Triggiani R., Abstract setting for tangential boundary stabilization of Navier–Stokes equations by high and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746.
* [11] Barbu V., Lasiecka I., Triggiani R., Tangential boundary stabilization of Navier–Stokes equations, Mem. Am. Math. Soc., 852 (2006), 1-145.
* [12] Barbu V., Lasiecka I., Triggiani R., The unique continuations property of eigenfunctions to Stokes–Oseen operator (submitted).
* [13] Barbu V., Triggiani R., Internal stabilitation of Navier–Stokes equations with finite dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.
* [14] Badra M., Feedback stabilization of the $2{-}D$ and $3{-}D$ Navier–Stokesequations based on an exten ded system, ESAIM COCV, 15 (2009), 934-968.
* [15] Badra M., Lyapunov functions and local feedback boundary stabilization of the Navier–Stokesequations, SIAM J. Control Optim., 48 (2009), 1797-1830.
* [16] Fursikov A.V., Real processes of the $3{-}D$ Navier–Stokes systems and its feedback stabilization from the boundary. In: Agranovic M.S., Shubin M.A. (eds.), AMS Translations. Partial Differential Equations, M. Vishnik Seminar, 2002, 95-123.
* [17] Fursikov A.V., Stabilization for the $3{-}D$ Navier–Stokes systems by feedback boundary control, Discrete Contin. Dyn. Syst., 10 (2004), 289-314.
* [18] Munteanu I., Normal feedback stabilization of periodic flows in a $2{-}D$ channel, JOTA (to appear).
* [19] Raymond J.P., Feedback boundary stabilization of the two dimensional Navier–Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828.
* [20] Raymond J.P., Feedback boundary stabilization of the three dimensional inequations, J. Math. Pures Appl., 87 (2007), 627-669.
* [21] Raymond J.P., Thevenet L., Boundary feedback stabilization of the two dimensional Naier–Stokes equations with finite dimensional controllers (to appear).
* [22] Temam R., Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, 1985\.
* [23] Temam R., Navier–Stokes Equations, North–Holland, Amsterdam, 1979.
* [24] Triggiani R., Stability enhancement of a $2{-}D$ linear Navier Stokes channel flow by a $2{-}D$, wall-normal boundary controller, Discrete Contin. Dyn. Syst., Series B (DCDS-B) 8 (2) (2007), 279-314.
* [25] Vazquez R., Krstic M., A closed-form feedback controller for stabilization of linearized Navier–Stokes equations: The $2D$ Poisseuille flow, IEEE Trans. Automatica, 2005, 10.1109/CDC.2005.1583349
* [26] Vazquez R., Tvelat E., Coron J.M., Control for fast and stable Laminar-to-High-Reynolds-Number transfer in a $2D$ channel flow, Discrete Contin. Dyn. Syst., SB 10 (2008).
|
arxiv-papers
| 2011-06-20T15:04:11 |
2024-09-04T02:49:19.921899
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Viorel Barbu",
"submitter": "Viorel Barbu",
"url": "https://arxiv.org/abs/1106.3931"
}
|
1106.3966
|
# Magnetic order of the hexagonal rare-earth manganite Dy0.5Y0.5MnO3
Joel S. Helton1,∗ Deepak K. Singh1,2 Harikrishnan S. Nair3 Suja Elizabeth4
1NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, MD 20899, USA 2Department of Materials Science and
Engineering, University of Maryland, College Park, MD 20742, USA 3Jülich
Centre for Neutron Science, Forschungszentrum Jülich, Outstation at FRM-II,
D-85747 Garching, Germany 4Department of Physics, Indian Institute of
Science, Bangalore 560012, India
###### Abstract
Hexagonal Dy0.5Y0.5MnO3, a multiferroic rare-earth manganite with
geometrically frustrated antiferromagnetism, has been investigated with
single-crystal neutron diffraction measurements. Below 3.4 K magnetic order is
observed on both the Mn (antiferromagnetic) and Dy (ferrimagnetic) sublattices
that is identical to that of undiluted hexagonal DyMnO3 at low temperature.
The Mn moments undergo a spin reorientation transition between 3.4 K and 10 K,
with antiferromagnetic order of the Mn sublattice persisting up to 70 K; the
antiferromagnetic order in this phase is distinct from that observed in
undiluted (h)DyMnO3, yielding a qualitatively new phase diagram not seen in
other hexagonal rare-earth manganites. A magnetic field applied parallel to
the crystallographic $c$ axis will drive a transition from the
antiferromagnetic phase into the low-temperature ferrimagnetic phase with
little hysteresis.
###### pacs:
75.25.-j, 75.85.+t, 75.50.Ee, 75.47.Lx
## I INTRODUCTION
The crystalline structure of rare-earth manganites ($R$MnO3 with $R$ = Y, Sc,
or a lanthanide) is determined by the ionic radius of the $R^{3+}$ cation.
Materials with a large $R^{3+}$ ionic radius ($R$ = La through Tb) crystallize
in an orthorhombic perovskite structure, while materials with a smaller
$R^{3+}$ ionic radius ($R$ = Y, Sc, or Ho through Lu)Fröhlich et al. (1999);
A. Muñoz et al. (2000); Tomuta et al. (2001); Sugie et al. (2002); A. Muñoz et
al. (2001); Yen et al. (2005) crystallize in a hexagonal structure. DyMnO3
typically crystallizes in the orthorhombic structure,Prokhnenko et al. (2007)
but with proper growth conditions hexagonal (h)DyMnO3 can be stabilized.V. Yu.
Ivanov et al. (2006); Harikrishnan et al. (2009) The hexagonal rare-earth
manganites, (h)$R$MnO3, are paraelectric at very high temperatures but display
a structural transition ($T_{C}$ $\approx$ 1000 K) to a ferroelectric phase
with the noncentrosymmetric $P6_{3}cm$ space group.Van Aken et al. (2004) The
(h)$R$MnO3 materials feature slightly distorted triangular lattice planes of
Mn ions; the antiferromagnetic nearest-neighbor exchange interaction leads to
geometrically frustrated magnetism. Below a Néel temperature of $\approx$100 K
these materials are magnetically ordered, with easy-plane antiferromagnetic
order of the Mn sublattice coexisting with the ferroelectric order. Many
hexagonal rare-earth manganites display one or more spin reorientation
transitions of the Mn moments at lower temperaturesVajk et al. (2005); Th.
Lonkai et al. (2002) or under an applied field;Lorenz et al. (2005); Vajk et
al. (2006) the $R$ ions also form distorted triangular lattice planes and, for
magnetic ions, will order along the $c$ axis. Several of these materials have
attracted interest because of significant magnetoelectricLottermoser et al.
(2004); Ueland et al. (2010) or magnetoelasticLee et al. (2008) effects.
Despite the structural similarities between the various members of the
(h)$R$MnO3 family, the magnetically ordered structures often differ, with
structures that transform according to each of the four one-dimensional
irreducible representations of the point group observed in at least one
material. The dependence of the magnetic ordering of the Mn sublattice on the
rare-earth element has been attributed to the ionic radius of the $R^{3+}$
cationKozlenko et al. (2007) or a biquadratic 3$d$-4$f$ magnetic
coupling.Wehrenfennig et al. (2010)
Hexagonal (h)DyMnO3, with the largest rare-earth ionic radius of this series,
features an interesting magnetic phase diagram.Harikrishnan et al. (2009);
Nandi et al. (2008a); Wehrenfennig et al. (2010) In the low-temperature phase
(below $\approx$8 K) both the Mn and Dy sublattices are magnetically ordered
according to the $\Gamma_{2}$ irreducible representation. At a temperature of
8 K the Mn moments undergo a spin reorientation transition and are ordered in
the $\Gamma_{4}$ representation up to 68 K.Wehrenfennig et al. (2010) X-ray
resonant magnetic scattering measurements also find a weak Dy moment in this
temperature range, ordered according to the $\Gamma_{3}$ representation.Nandi
et al. (2008a) This incompatible order, with different irreducible
representations present on the two sublattices, calls into question long
standing assumptions about the rigidity of the 3$d$-4$f$ interaction in
(h)$R$MnO3 materials.Wehrenfennig et al. (2010) YMnO3 features magnetism only
on the Mn sublattice and is known to order in the $\Gamma_{3}$ irreducible
representation below $T_{N}$ $\approx$ 75 K.Fiebig et al. (2000) YMnO3 has
also been reported to feature large magnetoelastic effectsLee et al. (2008)
and coupling between electric and magnetic domains.Fiebig et al. (2002)
Compounds where magnetic rare-earth ions are partially replaced with
nonmagnetic Y3+ ions, such as Ho1-xYxMnO3Zhou et al. (2007); Vasic et al.
(2007) and Er1-xYxMnO3,Sekhar et al. (2005); Vajk et al. (2011) allow for
further examination of the role that the rare-earth ions play in determining
the magnetic structure and open up the possibility of novel phase diagrams not
observed in undoped compounds. We report single-crystal neutron diffraction
studies of hexagonal Dy0.5Y0.5MnO3 (DYMO) and find evidence for a spin
reorientation transition not seen in other hexagonal rare-earth manganites at
zero field.
## II EXPERIMENT
Large, high-quality single-crystal samples of Dy0.5Y0.5MnO3 were prepared as
previously reported.Harikrishnan S. Nair et al. (2011) DYMO crystallizes in
the hexagonal $P6_{3}cm$ space group (#185) with lattice parameters (at 300 K)
of $a$ = $b$ = 6.161(1) Å and $c$ = 11.446(2) Å. As in other hexagonal rare-
earth manganites, Mn3+ ions ($S$ = 2) occupy the $6c$ positions at ($x$, 0, 0)
with $x$ $\approx$ $\frac{1}{3}$. A material with $x$ = $\frac{1}{3}$ would
feature perfect triangular lattice planes; in DYMO $x$ = 0.3379(4) yielding a
slightly distorted triangular lattice. The rare-earth $R^{3+}$ ions occupy two
crystallographically distinct sites, at the $2a$ and 4$b$ Wyckoff positions,
and also form distorted triangular lattice planes. In DYMO the rare-earth
sites are occupied with equal probability by nonmagnetic Y3+ ions and
${}^{6}H_{15/2}$ Dy3+ ions ($gJ$ = 10 $\mu_{B}$), as determined by x-ray
powder diffraction on crushed single crystals.Harikrishnan S. Nair et al.
(2011) Previously reported specific heat measurements found peaks at 3 K and
68 K; analogously with other hexagonal rare-earth manganites such as DyMnO3 it
was suggested that these peaks correspond to the onset of antiferromagnetic
order of the Mn lattice, $T_{N}^{\textrm{Mn}}$ $\approx$ 68 K, and
ferrimagnetic order of Dy moments on the rare-earth lattice,
$T_{N}^{\textrm{Dy}}$ $\approx$ 3 K.
Neutron diffraction experiments were carried out using the BT9 thermal triple-
axis spectrometer at the NIST Center for Neutron Research. The neutron initial
and final energies were selected using the (0 0 2) reflection of the pyrolytic
graphite (PG) monochromator and analyzer. We used 40′-47′-40′-open collimation
as well as PG filters to reduce contamination of the beam with higher-order
neutron wavelengths. The detailed temperature dependence of four magnetic
reflections (shown in Fig. 1) was measured using a large ($\approx$1 g) single
crystal mounted in the ($H$ 0 $L$) scattering plane at a fixed neutron energy
of 30.5 meV ($\lambda$ = 1.64 Å). Refinement of the ordered moments utilized
diffraction measurements taken at temperatures of 1.6 K, 25 K, and 120 K at a
fixed neutron energy of 30.5 meV. In order to minimize absorption of neutrons
by the sample, these measurements were taken with small single crystals: an
8-mg sample mounted in the ($H$ 0 $L$) scattering plane and a 6-mg sample
mounted in the ($H$ $K$ 0) scattering plane. Some magnetic reflections also
featured weak nuclear contributions which were removed by subtracting the 120
K intensity; the Debye-Waller factor has been ignored, which is justifiable,
given that the intensities of nuclear Bragg peaks with no magnetic intensity
[such as (0 0 $L$) where $L$ is even] remained constant within the statistical
uncertainties between 1.6 K and 120 K. The scattering intensity in absolute
units, and from this the value of the ordered moments, was determined by
normalizing the intensities of nuclear Bragg peaks measured at 120 K to the
calculated nuclear intensities. The intensities of the (1 0 0) and (3
$\bar{1}$ 0) reflections in a magnetic field were measured on the 6-mg sample
mounted in the ($H$ $K$ 0) scattering plane; the sample was placed inside a
helium flow dewar with a minimum temperature of 4 K inserted into a 7-T
vertical field superconducting magnet. These data were taken at a fixed
neutron energy of 14.7 meV ($\lambda$ = 2.36 Å). All neutron diffraction data
are reported in terms of the integrated intensity, integrated over a rocking
curve ($\theta$ scan) through the peak position.
The allowed magnetic structures in DYMO correspond to the six irreducible
representations of the $P6_{3}cm$ space group with propagation vector
$\vec{k}$ = 0; as in other hexagonal rare-earth manganites, only the four one-
dimensional irreducible representations (designated as $\Gamma_{1}$ through
$\Gamma_{4}$) are required to describe the observed structures.Sikora and
Syromyatnikov (1986); Munawar and Curnoe (2006) For each of these
representations the Mn sublattice displays antiferromagnetic order within the
$ab$ plane, with the three spins around any triangle oriented 120∘ apart. The
$R^{3+}$ sublattice is ordered along the crystallographic $c$ axis. The rare-
earth 4$b$ sites are antiferromagnetically ordered for the $\Gamma_{1}$,
$\Gamma_{3}$, and $\Gamma_{4}$ representations; the 2$a$ rare-earth sites are
antiferromagnetically ordered in the $\Gamma_{3}$ representation and
paramagnetic for the $\Gamma_{1}$ and $\Gamma_{4}$ representations. The 2$a$
and 4$b$ sites are each ferromagnetically ordered in the $\Gamma_{2}$
representation, with the coupling between the two sites yielding either
ferromagnetism or ferrimagnetism. In phases where only the Mn sublattice is
magnetically ordered a hexagonal rare-earth manganite with $x$ = $\frac{1}{3}$
(featuring undistorted triangular lattice planes) would have perfectly
identical neutron scattering structure factors in either the $\Gamma_{1}$ or
$\Gamma_{3}$ representations, as well as in the $\Gamma_{2}$ or $\Gamma_{4}$
representations. Unambiguous differentiation between these homometric magnetic
structures requires complementary measurement techniques such as optical
second harmonic spectroscopy.Fröhlich et al. (1999); Fiebig et al. (2003,
2000) The distortion of the triangular planes leads to some variation in the
neutron scattering structure factors; however, these results, from single-
crystal diffraction measurements on a strongly absorbing sample, will not be
able to distinguish between the $\Gamma_{1}$ and $\Gamma_{3}$ structures.
## III RESULTS
### III.1 Zero-field magnetic order
Figure 1 displays the temperature dependence of the integrated intensities of
the (1 0 0), (1 0 1), (2 0 0), and (2 0 1) Bragg reflections measured while
warming from 2.1 K to 130 K. The background and any high temperature nuclear
contributions to the intensities have been subtracted. At 2.1 K all four
reflections display magnetic intensity, with the (1 0 1) reflection the
strongest. Between 2.1 K and 3.4 K the measured intensities of the ($H$ 0 1)
reflections remain relatively constant while those of the ($H$ 0 0)
reflections steadily decrease with increasing temperature, with an intensity
at 3.4 K that is about 68% of the base temperature intensity. On increasing
the temperature beyond 3.4 K, the intensities of the ($H$ 0 0) peaks steadily
increase while the intensities of the ($H$ 0 1) peaks decrease so that the (1
0 0) reflection becomes the strongest measured; this rapid change in intensity
persists only to around 10 K. This will be shown to correspond to a spin
reorientation transition of the Mn moments and perhaps explains the 10 K
anomaly reportedHarikrishnan S. Nair et al. (2011) in the derivative of
1/$\chi$. Above 10 K the intensities of all peaks slowly decrease until
reaching zero at 70 K. The intensity can be fit to an order parameter form:
$I(T)\,\propto\,(|T-T_{N}|/T_{N})^{2\beta}$. The dotted red line in Fig. 1 is
a fit of the (1 0 0) intensity to this form with $\beta$ = 0.25 $\pm$ 0.05.
Figure 1: (Color online) Integrated magnetic intensities of the (1 0 0), (1 0 1), (2 0 0), and (2 0 1) Bragg reflections as a function of temperature. All data were measured while warming. The inset shows a closer view of the low temperature portion of the graph. Transitions are observed at $T_{N}^{\textrm{Mn}}$ = 70 K and $T_{N}^{\textrm{Dy}}$ = 3.4 K; these temperatures are designated with dashed vertical lines. The red dotted line reflects a fit of the (1 0 0) intensity to an order parameter form with $\beta$ = 0.25 $\pm$ 0.05. Error bars throughout this article are statistical in nature and represent one standard deviation. Table 1: Calculated magnetic scattering intensities for the reflections displayed in Fig. 1. The most intense reflection in each column has been normalized to 100. In the low-temperature phase below 3.4 K the data are consistent with $\Gamma_{2}$ order for both the Mn and Dy sublattice. In the antiferromagnetic phase between 10 K and 70 K the data are consistent with Mn sublattice order in either the $\Gamma_{1}$ or $\Gamma_{3}$ irreducible representation. | Mn order | | Dy order
---|---|---|---
| | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $\Gamma_{4}$ | | $\Gamma_{2}$
(1 0 0) | | 100 | 0 | 100 | 0 | | 100
(1 0 1) | | 17 | 100 | 15 | 100 | | 0
(2 0 0) | | 32 | 0 | 27 | 0 | | 38
(2 0 1) | | 1 | 26 | 1 | 33 | | 0
Table 1 displays the calculated magnetic intensities for the reflections shown
in Fig. 1 with the largest reflection in each column normalized to 100. The
reported intensities reflect the calculated magnetic cross section corrected
for the resolution function of a triple-axis neutron spectrometer. Calculated
intensities are shown for Mn moments ordered in each of the four one-
dimensional irreducible representations, as well as for Dy moments ordered in
the $\Gamma_{2}$ representation. Previous magnetization measurements on
DYMOHarikrishnan S. Nair et al. (2011) with $\vec{H}$ $||$ $\vec{c}$ revealed
a low-temperature state with a spontaneous magnetization of $\approx$0.5
$\mu_{B}$ per formula unit (measured at 2 K). Of the four irreducible
representations present in hexagonal rare-earth manganites only the
$\Gamma_{2}$ structure is consistent with either ferrimagnetism or
ferromagnetism; undoped hexagonal DyMnO3 is ferrimagnetic at low temperatures
and is known to order in the $\Gamma_{2}$ representation.Nandi et al. (2008a)
When ordering in the $\Gamma_{2}$ structure the magnetic cross sections of the
($H$ 0 1) reflections depend only on the ordered moment of the Mn ions while
those of the ($H$ 0 0) reflections depend only on the ordered moment of the Dy
ions. The Mn ordered moment is therefore found to be almost temperature
independent below 3.4 K, while the Dy ordered moment decreases continuously
with increasing temperature between 2.1 K and 3.4 K.
Assuming a ferrimagnetic structure with ordered moments on the 2$a$ and 4$b$
sites that are antiparallel but equal in magnitude, a refinement of magnetic
reflections measured at 1.6 K reveals ordered moments of 3.7 $\pm$ 0.4
$\mu_{B}$ for the Mn ions and 3.1 $\pm$ 0.3 $\mu_{B}$ for the Dy ions (1.6
$\pm$ 0.2 $\mu_{B}$ for each rare-earth site). This structure is displayed in
Fig. 2(a). (The crystal structures shown in Figure 2 were produced using
_VESTA_.Momma and Izumi (2008)) When magnetic domains are fully aligned this
ordered moment will lead to a net magnetization of 3.1 $\pm$ 0.3 $\mu_{B}$ per
unit cell (or 0.52 $\pm$ 0.05 $\mu_{B}$ per formula unit) along the
crystallographic $c$ axis, which is consistent with the reported bulk
spontaneous magnetization.Harikrishnan S. Nair et al. (2011) Allowing for
different ordered moments on the Dy $4b$ and $2a$ sites did not appreciably
improve the fit of the 1.6 K data, nor could allowing for different moments on
these sites produce a comparably good fit while yielding a net moment
consistent with magnetization measurements. A ferromagnetic ground state can
likewise be excluded. A ferromagnetic state with equal ordered moments on the
2$a$ and 4$b$ sites would give no intensity for the ($H$ 0 0) reflections;
when ordering in the $\Gamma_{2}$ structure the intensities of these
reflections depend on the difference in moment at the rare-earth sites:
$I(\vec{Q}_{H00})~{}\propto~{}|\vec{M}_{4b}-\vec{M}_{2a}|^{2}$, where the
ordered moments on the 2$a$ and 4$b$ sites are $\vec{M}_{2a}$ and
$\vec{M}_{4b}$. A ferromagnetic state with different ordered moments on the
2$a$ and 4$b$ sites is consistent with the neutron diffraction data only while
yielding a net moment inconsistent with magnetization measurements. The Mn
ordered moment at 1.6 K is close to the full moment value; however, the Dy
ordered moment is considerably reduced from the full 10 $\mu_{B}$ value of the
${}^{6}H_{15/2}$ Dy3+ ions. The low temperature magnetization in undoped
DyMnO3 would suggest a similarly reduced ordered moment for the Dy ions in the
ferrimagnetic phase.V. Yu. Ivanov et al. (2006)
Figure 2: (Color online) (a) Magnetic structure in the low-temperature
ferrimagnetic phase, present below $T_{N}^{\textrm{Dy}}$ = 3.4 K. The
structure belongs to the $P6_{3}c^{\prime}m^{\prime}$ magnetic space group
($\Gamma_{2}$ irreducible representation). Mn ions at $z$ = 0 are shown in
purple, Mn ions at $z$ = 1/2 are shown in orange, and the rare-earth ions are
shown in blue. The refined ordered moments are 3.7 $\pm$ 0.4 $\mu_{B}$ on the
Mn ions and 3.1 $\pm$ 0.3 $\mu_{B}$ on the Dy ions. (b) The two possible
magnetic structures for the antiferromagnetic phase: $P6_{3}cm$ magnetic space
group ($\Gamma_{1}$ irreducible representation) on the left and
$P6^{\prime}_{3}cm^{\prime}$ magnetic space group ($\Gamma_{3}$) on the right.
The Mn ordered moment is 3.5 $\pm$ 0.4 $\mu_{B}$. Any Dy ordered moment in
this temperature range is too small to be measured by neutron diffraction. The
structural unit cell is outlined in gray
The magnetic structure in the antiferromagnetic phase (10 K $<$ $T$ $<$ 70 K)
is consistent with an ordered moment on only the Mn sublattice, with order
according to either the $\Gamma_{1}$ or $\Gamma_{3}$ irreducible
representation. These magnetic structures are shown in Fig. 2(b). It should be
noted that neither of these possibilities are consistent with the structure of
the Mn moments in (h)DyMnO3, which order in the $\Gamma_{4}$ representation
above the low-temperature ferrimagnetic phase.Wehrenfennig et al. (2010) A
refinement of magnetic reflections measured at 25 K gives an ordered moment of
3.5 $\pm$ 0.4 $\mu_{B}$ for the Mn sublattice. The ordered moment at 25 K is
comparable in size to that determined at 1.6 K, suggesting a spin
reorientation transition where Mn spins rotate between 3.4 K and 10 K with
little change in magnitude. Element specific x-ray resonant magnetic
scattering measurements on hexagonal DyMnO3Nandi et al. (2008a) and
HoMnO3Nandi et al. (2008b) have reported a weak rare-earth moment at
comparable temperatures, induced by a splitting of the ground-state crystal
field doublet. While DYMO might similarly feature a weak Dy ordered moment in
this temperature range, neutron scattering measurements will be sensitive to
only the much larger ordered moment of the Mn sublattice and the refinement is
not improved by allowing for an ordered moment on the Dy sublattice at 25 K.
The magnetic structure of YMnO3 is known to be the $\Gamma_{3}$
representation.Fiebig et al. (2000) If the antiferromagnetic phase of DYMO
were likewise $\Gamma_{3}$, the spin reorientation transition into the low-
temperature $\Gamma_{2}$ phase would differ from the spin reorientation
transitions previously observed in hexagonal rare-earth manganites. A magnetic
structure in the $\Gamma_{1}$ irreducible representation can be transformed
into the $\Gamma_{2}$ representation through a 90∘ counterclockwise rotation
of all Mn moments. However, a transformation of a structure in the
$\Gamma_{3}$ representation as shown in Fig. 2(b) into the $\Gamma_{2}$
representation would require a 180∘ rotation of only the Mn moments in the $z$
= $\frac{1}{2}$ plane. In other hexagonal rare-earth manganites, it has been
suggestedFiebig et al. (2003) that spin reorientation transitions between the
four irreducible representations occur via in-phase or antiphase rotations
where the moments in adjacent Mn layers rotate with an equal (in-phase) or
opposite (antiphase) direction. For example, the Mn moments in HoMnO3 display
an in-phase reorientation transitionVajk et al. (2005) ($\Gamma_{4}$ to
$\Gamma_{3}$) at $T$ $\approx$ 40 K and an antiphase transition ($\Gamma_{3}$
to $\Gamma_{1}$) at $T$ $\approx$ 8 K, while the Mn moments in (h)DyMnO3
display an antiphase rotationWehrenfennig et al. (2010) ($\Gamma_{4}$ to
$\Gamma_{2}$) at $T$ $\approx$ 8 K. Further, while materials such as ScMnO3,A.
Muñoz et al. (2000) ErxY1-xMnO3,Sekhar et al. (2005) and YMnO3 under
pressureKozlenko et al. (2007) have been reported to feature broad temperature
ranges where the magnetic structure is not one of the four irreducible
representations, the observed structures can still be described as an in-phase
or antiphase rotation of all Mn moments away from one of the principal
structures.
### III.2 Field dependence of the magnetic order
Below $T_{N}^{\textrm{Dy}}$ = 3.4 K DYMO is ferrimagnetic with a spontaneous
net magnetization. Above $T_{N}^{\textrm{Dy}}$, $M(H)$ curves display
symmetric magnetization steps at a critical field value that increases with
temperature; the size of these steps decreases with increasing temperature,
becoming immeasurably small around 40 K.Harikrishnan S. Nair et al. (2011)
This behavior is remarkably similar to that previously reported in
(h)DyMnO3,V. Yu. Ivanov et al. (2006) but the moment increase associated with
the magnetization steps in DYMO is about one half of the increase in
(h)DyMnO3. Figure 3 displays the intensity of the (1 0 0) magnetic Bragg
reflection as a function of field with $\vec{H}$ $||$ $\vec{c}$ at
temperatures between 4 K and 50 K, measured while increasing the field after
zero field cooling. For each of these temperatures, we find a sudden drop in
the (1 0 0) intensity at a critical field value that increases with
temperature. As was shown in Fig. 1, the intensity of the (1 0 0) reflection
will be much higher in the antiferromagnetic phase than in the low-temperature
phase, such that this behavior is easily associated with a field-induced
transition into the ferrimagnetic phase. The data definitively show a spin
reorientation of the Mn moments along with the field-induced ferrimagnetic
ordering of the Dy moments, as the intensity of this reflection would not drop
if the Mn moments remained ordered in the original structure. The intensity of
this reflection in the field-induced phase is weaker than in zero field at low
temperature [the dotted horizontal line represents the expected intensity of
the (1 0 0) reflection at 2 K and zero field]. This likely arises from a
considerably smaller Dy ordered moment, consistent with the smaller magnitude
of the magnetization steps at higher temperatures in the $M(H)$ curves.
Interestingly, neutron diffraction is sensitive enough that this transition is
still clearly measurable in the 50 K data, while it could not be measured
above 40 K in the magnetization data.Harikrishnan S. Nair et al. (2011)
Figure 3: (Color online) Integrated intensity of the (1 0 0) peak as a
function of field. The data were taken while increasing field. Lines are a
guide to the eye. The dotted horizonal line represents the expected intensity
at 2 K in zero field.
The field dependence of the intensities at the (1 0 0) and (3 $\bar{1}$ 0)
reflections at $T$ = 15 K is displayed in Fig. 4. In Fig. 4(a), both
reflections show the expected transition at $\mu_{0}H_{c}$ $\approx$ 1 T; data
taken in rising and falling field show only a slight hysteresis, consistent
with the small hysteresis of the magnetization steps in the bulk magnetization
data. In addition to DYMO, undoped (h)DyMnO3 displays little hysteresis while
magnetization steps in ErMnO3Fiebig et al. (2001) display considerable
hysteresis. The (3 $\bar{1}$ 0) reflection displays scattering in the low-
temperature $\Gamma_{2}$ phase and very little scattering in the
antiferromagnetic phase, such that this change in intensity is likewise
consistent with a field-induced transition between the two. A more detailed
view of the (1 0 0) intensity in the high-field region is shown in Fig. 4(b).
Interestingly, the intensity of this peak rises slowly with increasing field
up to about 4 T but then decreases with increasing field beyond 5 T. This
likely reflects the evolution of the system from ferrimagnetic to
ferromagnetic as the field is increased, as the fully polarized state
(ferromagnetic with an equal moment for the 4$a$ and 2$b$ sites) would display
no scattering at (1 0 0).
Figure 4: (Color online) (a) Integrated intensities of the (1 0 0) (left
scale) and (3 $\bar{1}$ 0) (right scale) peaks measured at $T$ = 15 K. Solid
symbols represent measurements taken with increasing field, and open symbols
represent those taken with a decreasing field. (b) Detail of the high field
region for the (1 0 0) intensity.
## IV SUMMARY
Below $T_{N}^{\textrm{Dy}}$ = 3.4 K Dy0.5Y0.5MnO3 is magnetically ordered in
the $\Gamma_{2}$ irreducible representation with antiferromagnetic Mn order
and ferrimagnetic Dy order. Between 3.4 K and 10 K we observe a spin
reorientation transition of the Mn moments. Antiferromagnetic order of the Mn
sites persists up to $T_{N}^{\textrm{Mn}}$ = 70 K. The magnetic structure of
this phase is either $\Gamma_{1}$ or $\Gamma_{3}$, inconsistent with the
$\Gamma_{4}$ order observed in undiluted (h)DyMnO3.Wehrenfennig et al. (2010)
In the antiferromagnetic phase, a magnetic field applied parallel to the $c$
axis will drive a field-induced transition into the low-temperature
ferrimagnetic phase with little hysteresis. Further knowledge of the different
magnetic structures and reorientation transitions displayed by the Mn moments
in various (h)$R$MnO3 materials should lead to improved understanding of the
unusual 3$d$-4$f$ magnetic coupling present in these materials.Wehrenfennig et
al. (2010) While the magnetic structure of the antiferromagnetic phase cannot
be definitively determined from this experiment, it is clear that
Dy0.5Y0.5MnO3 displays a zero-field spin reorientation transition not
previously observed in any other hexagonal rare-earth manganite. If the
magnetic structure is $\Gamma_{3}$, then the $\Gamma_{3}$ to $\Gamma_{2}$ spin
reorientation transition would involve only half of the Mn ions in a manner
not observed in other (h)$R$MnO3 materials. The $H$-$T$ phase diagram of
HoMnO3 features a boundary between $\Gamma_{1}$ and $\Gamma_{2}$ phasesLorenz
et al. (2005); Munawar and Curnoe (2006) at $\mu_{0}H$ $\approx$ 2 T, yet a
zero field $\Gamma_{1}$ to $\Gamma_{2}$ spin reorientation transition has not
been reported. Further work will be necessary to distinguish between the
$\Gamma_{1}$ and $\Gamma_{3}$ structures in the antiferromagnetic phase and to
determine if the domain walls arising from this unique spin reorientation will
display magnetoelectric coupling as is observed in compounds such as
HoMnO3Ueland et al. (2010); Lottermoser and Fiebig (2004) or YMnO3.Fiebig et
al. (2002)
## ACKNOWLEDGEMENTS
We thank J.W. Lynn for guidance and helpful discussions. J.S.H. acknowledges
support from the NRC/NIST Postdoctoral Associateship Program. This work was
supported in part by the National Science Foundation under Agreement No.
DMR-0944772.
* email: joel.helton@nist.gov
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|
arxiv-papers
| 2011-06-20T16:55:47 |
2024-09-04T02:49:19.931202
|
{
"license": "Public Domain",
"authors": "Joel S. Helton, Deepak K. Singh, Harikrishnan S. Nair, and Suja\n Elizabeth",
"submitter": "Joel Helton",
"url": "https://arxiv.org/abs/1106.3966"
}
|
1106.4082
|
RANK-BASED SLOCC CLASSIFICATION FOR ODD $N$ QUBITS
Xiangrong Lia, Dafa Lib
a Department of Mathematics, University of California, Irvine, CA 92697-3875,
USA
b Dept of mathematical sciences, Tsinghua University, Beijing 100084 CHINA
###### Abstract
We study the entanglement classification under stochastic local operations and
classical communication (SLOCC) for odd $n$-qubit pure states. For this
purpose, we introduce the rank with respect to qubit $i$ for an odd $n$-qubit
state. The ranks with respect to qubits $1,2,\cdots,n$ give rise to the
classification of the space of odd $n$ qubits into $3^{n}$ families.
keywords: Entanglement classification, SLOCC equations, odd $n$ qubits
## 1 Introduction
Quantum entanglement plays a crucial role in quantum computation and quantum
information processing. If two states can be obtained from each other by means
of local operations and classical communication with nonzero probability
(SLOCC), then the two states are said to have the same kind of entanglement
[1] and suited to do the same tasks of quantum information theory [2].
The complete classification for three qubit pure states has been achieved [2].
While there are six SLOCC equivalence classes for pure states of three qubits,
two of which are genuine entanglement classes: the $|GHZ\rangle$ class and the
$|W\rangle$ class, the number of SLOCC equivalent classes for four or more
qubits is infinite. An important first step in tackling the classification
problem for four or more qubits is to divide the infinite SLOCC classes into a
finite number of families, using some type of criteria to determine which
family an arbitrary state belongs to. Many efforts have been devoted to the
SLOCC entanglement classifications for pure states of four qubits which result
in different finite number of families or classes, including those based on
Lie group theory [3], on hyperdeterminant [4], on inductive approach [5], and
on string theory [6]. Polynomial invariants for four and five qubits [7, 8, 9]
as well as for $n$ qubits [10, 11] have been discussed, and several attempts
have been made for SLOCC classification via the vanishing or not of the
polynomial invariants [11, 12, 13, 14, 15, 16], Recently, entanglement
classification for the symmetric $n$-qubit states has been achieved by
introducing two parameters called the diversity degree and the degeneracy
configuration [17].
In this paper, we investigate SLOCC classification of odd $n$-qubit pure
states. To this end, we introduce the rank with respect to qubit $i$ for an
odd $n$-qubit state and establish its invariance under SLOCC. The rank with
respect to qubit $i$ ranges over the values 0, 1, 2, and therefore gives rise
to the classification of the space of odd $n$ qubits into 3 families, as
exemplified here. Furthermore, the ranks with respect to qubits
$1,2,\cdots,n$, permit the partitioning of the space of the pure states of odd
$n\geq 5$ qubits into $3^{n}$ inequivalent families under SLOCC. We also
characterize pure biseparable states and genuinely entangled states in terms
of the ranks.
The paper is organized as follows. In section 2, we introduce the rank with
respect to qubit $i$ for any state of odd $n\geq 3$ qubits. In section 3, we
investigate SLOCC classification of odd $n$ qubits. We give the brief
discussion in section 4 and the conclusion in section 5.
## 2 Rank of a state with respect to qubit $i$
For odd $n$ qubits, let the state
$|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$, where $|i\rangle$ are basis
states and $a_{i}$ are coefficients. Let the $2\times 2$ matrix
$M(|\psi\rangle)=\left(\begin{tabular}[]{cc}$P(|\psi\rangle)$&$T(|\psi\rangle)$\\\
$T(|\psi\rangle)$&$Q(|\psi\rangle)$\end{tabular}\right),$ (2.1)
where $T(|\psi\rangle)$, $P(|\psi\rangle)$ and $Q(|\psi\rangle)$ are three
quantities defined on the space of pure states of odd $n$ qubits:
$\displaystyle T(|\psi\rangle)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}a_{i}a_{2^{n}-i-1},$ (2.2)
$\displaystyle P(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2i}a_{2^{n-1}-2i-1},$ (2.3)
$\displaystyle Q(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
2\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+2i}a_{2^{n}-2i-1}.$ (2.4)
Here $N(i)$ is the parity of $i$ (i.e. the number of 1’s in the binary
representation of $i$). Clearly $M(|\psi\rangle)$ is symmetric and the rank of
$M(|\psi\rangle)$ ranges over the values 0, 1, 2. We refer to the rank of
$M(|\psi\rangle)$ as the rank of the state $|\psi\rangle$ with respect to
qubit 1.
As the quantities $T(|\psi\rangle)$, $P(|\psi\rangle)$ and $Q(|\psi\rangle)$
vary under transpositions $(1,i)$ on qubits 1 and $i$ ($2\leq i\leq n$), so in
general does the rank of the state $|\psi\rangle$ with respect to qubit 1. The
variance allows one to define the rank of a state with respect to qubit $i$
($2\leq i\leq n$). For this purpose, we first let $T^{(i)}(|\psi\rangle)$,
$P^{(i)}(|\psi\rangle)$ and $Q^{(i)}(|\psi\rangle)$ be obtained from
$T(|\psi\rangle)$, $P(|\psi\rangle)$ and $Q(|\psi\rangle)$, respectively,
under transposition $(1,i)$ on qubits 1 and $i$, namely
$\displaystyle T^{(i)}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle(1,i)T(|\psi\rangle),$ (2.5) $\displaystyle
P^{(i)}(|\psi\rangle)$ $\displaystyle=$ $\displaystyle(1,i)P(|\psi\rangle),$
(2.6) $\displaystyle Q^{(i)}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle(1,i)Q(|\psi\rangle),$ (2.7)
for $i=1,2,\cdots,n$. It is trivial to see that
$T^{(1)}(|\psi\rangle)=T(|\psi\rangle)$,
$P^{(1)}(|\psi\rangle)=P(|\psi\rangle)$, and
$Q^{(1)}(|\psi\rangle)=Q(|\psi\rangle)$.
Analogously, we can construct $M^{(i)}(|\psi\rangle)$ as
$M^{(i)}(|\psi\rangle)=\left(\begin{tabular}[]{cc}$P^{(i)}(|\psi\rangle)$&$T^{(i)}(|\psi\rangle)$\\\
$T^{(i)}(|\psi\rangle)$&$Q^{(i)}(|\psi\rangle)$\end{tabular}\right).$ (2.8)
Note that $M^{(i)}(|\psi\rangle)$ can also be obtained from $M(|\psi\rangle)$
by taking transpositions $(1,i)$ on qubits 1 and $i$. Clearly,
$M^{(i)}(|\psi\rangle)$ is a symmetric matrix and
$M^{(1)}(|\psi\rangle)=M(|\psi\rangle)$. The rank of the matrix
$M^{(i)}(|\psi\rangle)$ in Eq. (2.8) is referred to as the rank of the state
$|\psi\rangle$ with respect to qubit $i$ and denoted as
$rank^{(i)}(|\psi\rangle)$.
For example, for three qubits, we obtain $rank^{(i)}(|W\rangle)=1$ for $i=1$,
$2$, $3$, whereas for any odd $n$ qubits, we find that
$rank^{(i)}(|GHZ\rangle)=2$ and $rank^{(i)}(|0\cdots 0\rangle)=0$ for
$i=1,\cdots,n$.
Next, we establish the invariance of the rank for any state of odd $n$ qubits
under SLOCC. Let $|\psi^{\prime}\rangle$ be another odd $n$-qubit state with
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle$. Recall that if two
states $|\psi\rangle$ and $|\psi^{\prime}$ are SLOCC equivalent, then there
exist invertible local operators ${\mathcal{A}}_{1}$,
${\mathcal{A}}_{2},\cdots$ and ${\mathcal{A}}_{n}$ ($\det(A_{i})\not=0$) such
that [2]
$|\psi\rangle=\underbrace{{\mathcal{A}}_{1}\otimes{\mathcal{A}}_{2}\otimes\cdots\otimes{\mathcal{A}}_{n}}_{n}|\psi^{\prime}\rangle.$
(2.9)
Then, we assert that if $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are SLOCC
equivalent then the following SLOCC matrix equation holds (see Appendix A for
the proof):
$M^{(i)}(|\psi\rangle)={\mathcal{A}}_{i}M^{(i)}(|\psi^{\prime}\rangle){\mathcal{A}}_{i}^{T}\det({\mathcal{A}}_{1})\cdots\det({\mathcal{A}}_{i-1})\det({\mathcal{A}}_{i+1})\cdots\det({\mathcal{A}}_{n}),$
(2.10)
where $M^{(i)}(|\psi^{\prime}\rangle)$ is obtained from
$M^{(i)}(|\psi\rangle)$ by replacing $|\psi\rangle$ by
$|\psi^{\prime}\rangle$.
It follows from Eq. (2.10) that the rank of the matrix $M^{(i)}(|\psi\rangle)$
in Eq. (2.8) is invariant under SLOCC, thereby revealing that the rank of the
state $|\psi\rangle$ with respect to qubit $i$ is an inherent property. Then
the following result holds: if two states are SLOCC equivalent, then they have
the same rank with respect to the same qubit $i$. It should be noted that the
converse does not hold, i.e., two states with the same rank with respect to
the same qubit are not necessarily equivalent.
To exemplify, we consider the $n$-qubit symmetric Dicke states
$|\ell,n\rangle$ with $\ell$ excitations, $1\leq\ell\leq(n-1)$ [18]:
$|\ell,n\rangle=\left({}_{\ell}^{n}\right)^{-1/2}\sum\limits_{k}P_{k}|1_{1},1_{2},\cdots,1_{\ell},0_{\ell+1},\cdots,0_{n}\rangle,$
(2.11)
where $\\{P_{k}\\}$ is the set of all distinct permutations of the spins. For
any odd $n\geq 3$ qubits, a straightforward calculation yields
$rank^{(i)}(|(n-1)/2,n\rangle)=1$, $i=1,\cdots,n$. For any odd $n\geq 5$
qubits, $rank^{(i)}(|\ell,n\rangle)=0$ (note that $rank^{(i)}(|W\rangle)=0$ as
well, since $n$-qubit $|W\rangle$ state is identical with $|1,n\rangle$) for
$1\leq\ell<(n-1)/2$ and $i=1,\cdots,n$. Since the Dicke states
$|\ell,n\rangle$ and $|(n-\ell),n\rangle$ are SLOCC equivalent, the rank for
any Dicke state can be determined.
Now consider pure biseparable states, i.e., those that are separable under
some bipartition. By virtue of Theorem 3.4 of [19], we arrive at a necessary
condition for a pure state to be biseparable: if $|\psi\rangle$ is a pure
biseparable state of odd $n$ qubits, then $rank^{(i)}(|\psi\rangle)=0$ or $1$
for some $i$ with $1\leq i\leq n$.
In view of the above condition and the fact that a pure state of $n$ qubits is
genuinely entangled if it is not biseparable, we obtain the following
sufficient condition for a pure state to be genuinely entangled: for any pure
state $|\psi\rangle$ of odd $n$ qubits, if $rank^{(i)}(|\psi\rangle)=2$ for
any $1\leq i\leq n$, then $|\psi\rangle$ is genuinely entangled.
Remark. If we take the absolute value of the determinant of
$M^{(i)}(|\psi\rangle)$ given in Eq. (2.8), then we obtain the $n$-tangle with
respect to qubit $i$ of odd $n$ qubits $\tau_{12\cdots n}^{(i)}$ given in [20]
(up to a constant factor). In particular, when $n=3$, $|\det M(|\psi\rangle)|$
is, up to a constant factor, equal to the 3-tangle [21] (we refer the reader
to [10] for more details). Further, taking the determinants of both sides of
Eq. (2.10) yields
$\det M^{(i)}(|\psi\rangle)=\det
M^{(i)}(|\psi^{\prime}\rangle)[\det({\mathcal{A}}_{1})\cdots\det({\mathcal{A}}_{n})]^{2}.$
(2.12)
Note that for $i=1$, we recover Eq. (2.16) of [10]. It follows from Eq. (2.12)
that if one of $\det M^{(i)}(|\psi\rangle)$ and $\det
M^{(i)}(|\psi^{\prime}\rangle)$ vanishes while the other does not, then the
state $|\psi\rangle$ is not equivalent to $|\psi^{\prime}\rangle$ under SLOCC.
Clearly, the SLOCC invariance of the rank of $M^{(i)}(|\psi\rangle)$ is
stronger than the invariance of the determinant.
## 3 SLOCC classification of odd $n$ qubits
### 3.1 Three families based on the rank with respect to qubit $i$
The rank with respect to qubit $i$ permits the partitioning of the space of
the pure states of odd $n$ qubits into the following three families:
$F_{r_{i}}^{(i)}=\\{|\psi\rangle:rank^{(i)}(|\psi\rangle)=r_{i}\\}$,
$r_{i}\in\\{0,1,2\\}$. For example, the rank with respect to qubit $1$ divides
the space of the pure states of odd $n$ qubits into three families:
$F_{0}^{(1)}=\\{|\psi\rangle:rank^{(1)}(|\psi\rangle)=0\\}$,
$F_{1}^{(1)}=\\{|\psi\rangle:rank^{(1)}(|\psi\rangle)=1\\}$, and
$F_{2}^{(1)}=\\{|\psi\rangle:rank^{(1)}(|\psi\rangle)=2\\}$.
It is not hard to see that two states belong to the same family if and only if
they have the same rank with respect to the same qubit. Accordingly, if two
states are SLOCC equivalent then they belong to the same family
$F_{r_{i}}^{(i)}$. However, the converse does not hold, i.e., the states in
the same family may be inequivalent under SLOCC. It is further noted that the
aforementioned three SLOCC families $F_{0}^{(i)}$, $F_{1}^{(i)}$ and
$F_{2}^{(i)}$ form a complete partition of the space of odd $n$ qubits. That
is, any state of odd $n$ qubits belongs to one and only one of the above three
families.
We exemplify the result for the six SLOCC equivalent classes for three qubits:
$|GHZ\rangle$, $|W\rangle$, $A$-$BC$, $B$-$AC$, $C$-$AB$ and $A$-$B$-$C$ [2].
The rank with respect to qubit $i$ permits the partitioning of the space of
three qubits into three families $F_{0}^{(i)}$, $F_{1}^{(i)}$ and
$F_{2}^{(i)}$, as illustrated in Table 1.
Table 1: The three partitions for three qubits
qubit $i$ family SLOCC classes $F_{2}^{(1)}$ $|GHZ\rangle$ $i=1$ $F_{1}^{(1)}$
$|W\rangle$, $A-BC$ $F_{0}^{(1)}$ $A-B-C$, $B-AC$, $C-AB$ $F_{2}^{(2)}$
$|GHZ\rangle$ $i=2$ $F_{1}^{(2)}$ $|W\rangle$, $B-AC$ $F_{0}^{(2)}$ $A-B-C$,
$A-BC$, $C-AB$ $F_{2}^{(3)}$ $|GHZ\rangle$ $i=3$ $F_{1}^{(3)}$ $|W\rangle$,
$C-AB$ $F_{0}^{(3)}$ $A-B-C$, $A-BC$, $B-AC$
We also revisit the examples in the last section. Clearly, for any odd $n\geq
5$ qubits, $|GHZ\rangle$ belongs to family $F_{2}^{(i)}$, the Dicke state
$|(n-1)/2,n\rangle$ belongs to family $F_{1}^{(i)}$, whereas all the full
separable states and all the Dicke states $|\ell,n\rangle$ (including
$n$-qubit $|W\rangle$ state) for $1\leq\ell<(n-1)/2$, belong to family
$F_{0}^{(i)}$, $i=1,\cdots,n$.
### 3.2 Nine families based on the ranks with respect to qubits $1$ and $2$
As discussed in the previous section, the rank with respect to qubit $1$
divides the space of odd $n$ qubits into three families $F_{0}^{(1)}$,
$F_{1}^{(1)}$ and $F_{2}^{(1)}$. For odd $n\geq 5$ qubits, based on the rank
with respect to qubit 2 each family $F_{r_{1}}^{(1)}$, $r_{1}\in\\{0$, $1$,
$2\\}$, can be further divided into three different families:
$F_{r_{1},r_{2}}^{(1,2)}=F_{r_{1}}^{(1)}\cap F_{r_{2}}^{(2)}$, $r_{2}\in\\{0$,
$1$, $2\\}$. Here, each family $F_{r_{1},r_{2}}^{(1,2)}$ is the intersection
of the families $F_{r_{1}}^{(1)}$ and $F_{r_{2}}^{(2)}$. More specifically,
the family $F_{2}^{(1)}$ is divided into three families $F_{2,0}^{(1,2)}$,
$F_{2,1}^{(1,2)}$ and $F_{2,2}^{(1,2)}$, the family $F_{1}^{(1)}$ into three
families $F_{1,0}^{(1,2)}$, $F_{1,1}^{(1,2)}$ and $F_{1,2}^{(1,2)}$, and the
family $F_{0}^{(1)}$ into three families $F_{0,0}^{(1,2)}$, $F_{0,1}^{(1,2)}$
and $F_{0,2}^{(1,2)}$. For odd $n\geq 5$ qubits, we list the representative
states of the families $F_{r_{1},r_{2}}^{(1,2)}$ in Table 2.
Table 2: The nine families for odd $n\geq 5$ qubits based on the ranks with
respect to qubits 1, 2
family representative state $F_{2,2}^{(1,2)}$ $|GHZ\rangle$ $F_{2,1}^{(1,2)}$
$\frac{1}{\sqrt{6}}[(|0\cdots 0\rangle+|1\cdots 1\rangle)+(|010\cdots
010\rangle+|101\cdots 101\rangle)+(|0\cdots 0110\rangle-|101\cdots
10001\rangle)]$ $F_{2,0}^{(1,2)}$ $\frac{1}{2}[(|0\cdots 0\rangle+|1\cdots
1\rangle)+(|010\cdots 010\rangle+|101\cdots 101\rangle)]$ $F_{1,2}^{(1,2)}$
$\frac{1}{\sqrt{5}}[(|001\cdots 1\rangle-|010\cdots 0\rangle)+(|0110\cdots
0\rangle+|10\cdots 0\rangle)+|1101\cdots 1\rangle]$ $F_{1,1}^{(1,2)}$
$|(n-1)/2,n\rangle$ $F_{1,0}^{(1,2)}$ $\frac{1}{\sqrt{2}}(|0\cdots
0\rangle+|01\cdots 1\rangle)$ $F_{0,2}^{(1,2)}$ $\frac{1}{2}(|0\cdots
0\rangle+|1\cdots 1\rangle+|010\cdots 01\rangle-|101\cdots 10\rangle)$
$F_{0,1}^{(1,2)}$ $\frac{1}{\sqrt{2}}(|0\cdots 0\rangle+|101\cdots 1\rangle)$
$F_{0,0}^{(1,2)}$ $|0\cdots 0\rangle$
Consequently, the ranks with respect to qubits 1 and 2 divide the space of odd
$n\geq 5$ qubits into nine different families. Note furthermore that the nine
SLOCC families form a complete partition of the space of odd $n\geq 5$ qubits.
That is, any state of odd $n$ qubits belongs to one and only one of the nine
families.
Continuing with the example for three qubits, we see that the six SLOCC
equivalence classes are divided into five families based on the ranks with
respect to qubits 1 and 2, see Table 3.
Table 3: Partition for three qubits based on the ranks with respect to qubits
1, 2
family SLOCC equivalent class $F_{2,2}^{(1,2)}$ $|GHZ\rangle$
$F_{1,1}^{(1,2)}$ $|W\rangle$ $F_{1,0}^{(1,2)}$ $A-BC$ $F_{0,1}^{(1,2)}$
$B-AC$ $F_{0,0}^{(1,2)}$ $A-B-C$, $C-AB$
### 3.3 $3^{n}$ families based on the ranks with respect to qubits
$1,\cdots,n$
Now, assume that the ranks with respect to qubits $1,\cdots,(\ell-1)$ permit
the partitioning of the space of odd $n\geq 5$ qubits into $3^{(\ell-1)}$
families: $F_{r_{1},r_{2},\cdots,r_{(\ell-1)}}^{(1,2,\cdots,(\ell-1))}$,
$r_{1},r_{2},\cdots,r_{(\ell-1)}\in\\{0,1,2\\}$. Then, each family
$F_{r_{1},r_{2},\cdots,r_{(\ell-1)}}^{(1,2,\cdots,(\ell-1))}$ can be further
divided into three families:
$F_{r_{1},r_{2},\cdots,r_{\ell}}^{(1,2,\cdots,\ell)}=F_{r_{1},r_{2},\cdots,r_{(\ell-1)}}^{(1,2,\cdots,(\ell-1))}\cap
F_{r_{\ell}}^{(\ell)}$, $r_{\ell}\in\\{0,1,2\\}$ based on the rank with
respect to qubit $\ell$. Clearly, each family
$F_{r_{1},r_{2},\cdots,r_{\ell}}^{(1,2,\cdots,\ell)}$ is associated with the
sequence $\\{r_{1},\cdots,r_{\ell}\\}$, $r_{1},\cdots,r_{\ell}\in\\{0,1,2\\}$,
and different sequences correspond to different families. Consequently, in
total there are $3^{\ell}$ SLOCC families based on the ranks with respect to
qubits $1,\cdots,\ell$. In particular, there are $3^{n}$ SLOCC families based
on the ranks with respect to qubits $1,\cdots,n$. It should be noted that at
least one family contains an infinite number of SLOCC classes.
It is readily seen that $n$-qubit $|GHZ\rangle$ state belongs to family
$F_{2,\cdots,2}^{(1,\cdots,n)}$, the Dicke states $|(n-1)/2,n\rangle$ and
$|(n+1)/2,n\rangle$ belong to family $F_{1,\cdots,1}^{(1,\cdots,n)}$, whereas
all the full separable states and all the Dicke states $|\ell,n\rangle$
(including $n$-qubit $|W\rangle$ state), with $1\leq\ell<(n-1)/2$ and $n\geq
5$, belong to family $F_{0,\cdots,0}^{(1,\cdots,n)}$. It is worth pointing out
that all the states in the family $F_{2,\cdots,2}^{(1,\cdots,n)}$ are
genuinely entangled as discussed in section 2.
For any state $|\psi\rangle$ of odd $n$ qubits, by computing
$rank^{(i)}(|\psi\rangle)$, $i=1,\cdots,\ell$ $(\leq n)$, we can determine
which family the state $|\psi\rangle$ belongs to. It is plain to see that two
states belong to the same family if and only if they have the same ranks with
respect to qubits $1,\cdots,\ell$ $(\leq n)$. Thus, if two states are SLOCC
equivalent then they belong to the same family.
Consider once again the example for three qubits. A straightforward
calculation demonstrates that the six SLOCC equivalence classes of three
qubits are divided into six families based on the ranks with respect to qubits
1, 2 and 3, i.e., each family is just a single SLOCC class, see Table 4.
Table 4: Partition for three qubits based on the ranks with respect to qubits
1, 2, 3
family SLOCC equivalent class $F_{2,2,2}^{(1,2,3)}$ $|GHZ\rangle$
$F_{1,1,1}^{(1,2,3)}$ $|W\rangle$ $F_{1,0,0}^{(1,2,3)}$ $A-BC$
$F_{0,1,0}^{(1,2,3)}$ $B-AC$ $F_{0,0,1}^{(1,2,3)}$ $C-AB$
$F_{0,0,0}^{(1,2,3)}$ $A-B-C$
## 4 Discussion
In [22], the “filter” approach was used to separate SLOCC orbits and it was
shown that the following four five-qubit states
$\displaystyle|\Phi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|11111\rangle+|00000\rangle),$ (4.1)
$\displaystyle|\Phi_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}(|11111\rangle+|11100\rangle+|00010\rangle+|00001\rangle),$
(4.2) $\displaystyle|\Phi_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(\sqrt{2}|11111\rangle+|11000\rangle+|00100\rangle+|00010\rangle+|00001\rangle),$
(4.3) $\displaystyle|\Phi_{4}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(\sqrt{3}|11111\rangle+|10000\rangle+|01000\rangle+|00100\rangle+|00010\rangle+|00001\rangle)$
(4.4)
are in different orbits.
We now classify the above four states using our framework. Based on the ranks
with respect to qubits 1, 2 and 3, $|\Phi_{1}\rangle$ belongs to
$F_{2,2,2}^{(1,2,3)}$, $|\Phi_{2}\rangle$ belongs to $F_{0,0,0}^{(1,2,3)}$,
$|\Phi_{3}\rangle$ belongs to $F_{0,0,1}^{(1,2,3)}$, and $|\Phi_{4}\rangle$
belongs to $F_{1,1,1}^{(1,2,3)}$, in agreement with [22] that the four states
are in different orbits.
Also note that the space of five qubits is divided into nine different
families based on the ranks with respect to qubits 1 and 2\. We list the
representatives of the nine families in Table 5.
Table 5: The nine families for five qubits based on the ranks with respect to
qubits 1, 2
family representative state $F_{2,2}^{(1,2)}$ $|GHZ\rangle$ $F_{2,1}^{(1,2)}$
$\frac{1}{\sqrt{6}}(|00000\rangle+|11111\rangle+|01010\rangle+|10101\rangle+|00110\rangle-|10001\rangle)$
$F_{2,0}^{(1,2)}$
$\frac{1}{2}(|00000\rangle+|11111\rangle+|01010\rangle+|10101\rangle)$
$F_{1,2}^{(1,2)}$
$\frac{1}{\sqrt{5}}(|00111\rangle-|01000\rangle+|01100\rangle+|10000\rangle+|11011\rangle)$
$F_{1,1}^{(1,2)}$ $|2,5\rangle$ $F_{1,0}^{(1,2)}$
$\frac{1}{\sqrt{2}}(|00000\rangle+|01111\rangle)$ $F_{0,2}^{(1,2)}$
$\frac{1}{2}(|00000\rangle+|11111\rangle+|01001\rangle-|10110\rangle)$
$F_{0,1}^{(1,2)}$ $\frac{1}{\sqrt{2}}(|00000\rangle+|10111\rangle)$
$F_{0,0}^{(1,2)}$ $|00000\rangle$
## 5 Conclusion
In this paper, we have introduced the rank with respect to qubit $i$ for any
state of odd $n$ qubits and established its invariance under SLOCC. That is,
if two states are SLOCC equivalent then they have the same ranks with respect
to qubits $1,\cdots,n$. The ranks with respect to qubits $1,\cdots,n$ permit
the partitioning of the space of odd $n\geq 5$ qubits into $3^{n}$
inequivalent families. It is straightforward to know that two states belong to
the same family if and only if they have the same ranks with respect to qubits
$1,\cdots,n$. In other words, all the states of a family have the same ranks
with respect to qubits $1,\cdots,n$. As a consequence, if two states are SLOCC
equivalent then they belong to the same family. Furthermore, each family
corresponds to the sequence $\\{r_{1},\cdots,r_{n}\\}$, $r_{i}\in\\{0,1,2\\}$,
and different families correspond to different sequences. In terms of the
ranks, we have given a necessary condition for a pure state to be biseparable
and a sufficient condition for a pure state to be genuinely entangled. The
classification based on the ranks of states may possess more physical meaning.
As a final note, we would like to mention that the SLOCC invariance of the
rank for odd $n$ qubits does not hold for even $n$ qubits.
Acknowledgement. We would like to thank Thierry Bastin for helpful discussions
and the reviewers for the useful comments. This work was supported by NSFC
(Grant No. 10875061) and Tsinghua National Laboratory for Information Science
and Technology.
## Appendix
We here give the proof of Eq. (2.10). We distinguish two cases: $i=1$ and
$2\leq i\leq n$.
Case 1. $i=1$.
In this case, Eq. (2.10) becomes
$M(|\psi\rangle)={\mathcal{A}}_{1}M(|\psi^{\prime}\rangle){\mathcal{A}}_{1}^{T}\det({\mathcal{A}}_{2})\cdots\det({\mathcal{A}}_{n}).$
(5.1)
Let $|\psi\rangle$ and $|\psi^{\prime}\rangle$ be related by Eq. (2.11), and
${\mathcal{A}}_{1}=\left(\begin{tabular}[]{ll}$\alpha_{1}$&$\alpha_{2}$\\\
$\alpha_{3}$&$\alpha_{4}$\end{tabular}\right)$. It is easy to verify that Eq.
(5.1) holds if and only if the following three SLOCC equations hold together:
$\displaystyle T(|\psi\rangle)$ $\displaystyle=$
$\displaystyle[P(|\psi^{\prime}\rangle)\alpha_{1}\alpha_{3}+T(|\psi^{\prime}\rangle)(\alpha_{2}\alpha_{3}+\alpha_{1}\alpha_{4})+Q(|\psi^{\prime}\rangle)\alpha_{2}\alpha_{4}]$
(5.2)
$\displaystyle\times\det({\mathcal{A}}_{2})\cdots\det({\mathcal{A}}_{n}),$
$\displaystyle P(|\psi\rangle)$ $\displaystyle=$
$\displaystyle[P(|\psi^{\prime}\rangle)\alpha_{1}^{2}+2T(|\psi^{\prime}\rangle)\alpha_{1}\alpha_{2}+Q(|\psi^{\prime}\rangle)\alpha_{2}^{2}]$
(5.3)
$\displaystyle\times\det({\mathcal{A}}_{2})\cdots\det({\mathcal{A}}_{n}),$
$\displaystyle Q(|\psi\rangle)$ $\displaystyle=$
$\displaystyle[P(|\psi^{\prime}\rangle)\alpha_{3}^{2}+2T(|\psi^{\prime}\rangle)\alpha_{3}\alpha_{4}+Q(|\psi^{\prime}\rangle)\alpha_{4}^{2}]$
(5.4)
$\displaystyle\times\det({\mathcal{A}}_{2})\cdots\det({\mathcal{A}}_{n}).$
Notice that
${\mathcal{A}}_{1}\otimes{\mathcal{A}}_{2}\otimes\cdots\otimes{\mathcal{A}}_{n}$
can be written as
$({\mathcal{A}}_{1}\otimes{\mathcal{I}}_{2}\otimes\cdots\otimes{\mathcal{I}}_{n})\circ({\mathcal{I}}_{1}\otimes{\mathcal{A}}_{2}\otimes{\mathcal{I}}_{3}\otimes\cdots\otimes{\mathcal{I}}_{n})\circ\cdots\circ({\mathcal{I}}_{1}\otimes\cdots\otimes{\mathcal{I}}_{n-1}\otimes{\mathcal{A}}_{n})$,
then Eqs. (5.2), (5.3) and (5.4) follow immediately from the two lemmas below.
Lemma 1. For odd $n$ qubits, if $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are
related by
$|\psi\rangle=\underbrace{{\mathcal{A}}_{1}\otimes{\mathcal{I}}_{2}\otimes\cdots\otimes{\mathcal{I}}_{n}}_{n}|\psi^{\prime}\rangle,$
(5.5)
then
$\displaystyle T(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
P(|\psi^{\prime}\rangle)\alpha_{1}\alpha_{3}+T(|\psi^{\prime}\rangle)(\alpha_{2}\alpha_{3}+\alpha_{1}\alpha_{4})+Q(|\psi^{\prime}\rangle)\alpha_{2}\alpha_{4},$
(5.6) $\displaystyle P(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
P(|\psi^{\prime}\rangle)\alpha_{1}^{2}+2T(|\psi^{\prime}\rangle)\alpha_{1}\alpha_{2}+Q(|\psi^{\prime}\rangle)\alpha_{2}^{2},$
(5.7) $\displaystyle Q(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
P(|\psi^{\prime}\rangle)\alpha_{3}^{2}+2T(|\psi^{\prime}\rangle)\alpha_{3}\alpha_{4}+Q(|\psi^{\prime}\rangle)\alpha_{4}^{2}.$
(5.8)
Proof. We only prove Eq. (5.6). The proofs for Eqs. (5.7) and (5.8) are
analogous. By Eq. (5.5), we obtain
$a_{i}=\alpha_{1}b_{i}+\alpha_{2}b_{2^{n-1}+i},\quad
a_{2^{n-1}+i}=\alpha_{3}b_{i}+\alpha_{4}b_{2^{n-1}+i},$ (5.9)
for $0\leq i\leq 2^{n-1}-1$. By substituting Eq. (5.9) into Eq. (2.2), we
obtain
$T(|\psi\rangle)=\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}(\alpha_{1}b_{i}+\alpha_{2}b_{2^{n-1}+i})(\alpha_{3}b_{2^{n-1}-i-1}+\alpha_{4}b_{2^{n}-i-1}).$
(5.10)
Note that $T(|\psi^{\prime}\rangle)$, $P(|\psi^{\prime}\rangle)$, and
$Q(|\psi^{\prime}\rangle)$ can be rewritten as:
$\displaystyle T(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}b_{2^{n-1}+i}b_{2^{n-1}-i-1},$
(5.11) $\displaystyle P(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}b_{i}b_{2^{n-1}-i-1},$ (5.12)
$\displaystyle Q(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2^{n-1}-1}(-1)^{N(i)}b_{2^{n-1}+i}b_{2^{n}-i-1}.$
(5.13)
Expanding Eq. (5.10) and using Eqs. (5.11), (5.12) and (5.13) yield Eq. (5.6).
Lemma 2. For odd $n$ qubits, if $|\psi\rangle$ and $|\psi^{\prime}\rangle$ are
related by
$|\psi\rangle=\underbrace{{\mathcal{I}}_{1}\otimes\cdots\otimes{\mathcal{I}}_{k-1}\otimes{\mathcal{A}}_{k}\otimes{\mathcal{I}}_{k+1}\otimes\cdots\otimes{\mathcal{I}}_{n}}_{n}|\psi^{\prime}\rangle,$
(5.14)
then
$\displaystyle T(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
T(|\psi^{\prime}\rangle)\det({\mathcal{A}}_{k}),$ (5.15) $\displaystyle
P(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
P(|\psi^{\prime}\rangle)\det({\mathcal{A}}_{k}),$ (5.16) $\displaystyle
Q(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
Q(|\psi^{\prime}\rangle)\det({\mathcal{A}}_{k}),$ (5.17)
for $2\leq k\leq n$.
Proof. We only prove Eq. (5.15). The proofs for Eqs. (5.16) and (5.17) can be
given analogously. It is sufficient to consider $k=2$. Let
${\mathcal{A}}_{2}=\left(\begin{tabular}[]{cc}$\beta_{1}$&$\beta_{2}$\\\
$\beta_{3}$&$\beta_{4}$\end{tabular}\right)$. Then, by Eq. (5.14), we obtain
$\displaystyle a_{i}$ $\displaystyle=$
$\displaystyle\beta_{1}b_{i}+\beta_{2}b_{2^{n-2}+i},$ (5.18) $\displaystyle
a_{2^{n-2}+i}$ $\displaystyle=$
$\displaystyle\beta_{3}b_{i}+\beta_{4}b_{2^{n-2}+i},$ (5.19) $\displaystyle
a_{2^{n-1}+i}$ $\displaystyle=$
$\displaystyle\beta_{1}b_{2^{n-1}+i}+\beta_{2}b_{2^{n-1}+2^{n-2}+i},$ (5.20)
$\displaystyle a_{2^{n-1}+2^{n-2}+i}$ $\displaystyle=$
$\displaystyle\beta_{3}b_{2^{n-1}+i}+\beta_{4}b_{2^{n-1}+2^{n-2}+i},$ (5.21)
for $0\leq i\leq 2^{n-2}-1$. We may rewrite $T(|\psi\rangle)$ in Eq. (2.2) as
$\displaystyle T(|\psi\rangle)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{i}a_{2^{n}-i-1}$ (5.22)
$\displaystyle-\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-2}+i}a_{2^{n-1}+2^{n-2}-i-1}$
$\displaystyle-\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+i}a_{2^{n-1}-i-1}$
$\displaystyle+\sum_{i=0}^{2^{n-2}-1}(-1)^{N(i)}a_{2^{n-1}+2^{n-2}+i}a_{2^{n-2}-i-1}.$
Substituting Eqs. (5.18), (5.19), (5.20) and (5.21) into Eq. (5.22) yields the
desired result Eq. (5.15).
Case 2. $2\leq i\leq n$.
We give a brief proof here. After a tedious but straightforward calculation,
the following identity holds:
$(1,i)\circ({\mathcal{A}}_{1}\otimes\cdots\otimes{\mathcal{A}}_{n})\circ(1,i)={\mathcal{A}}_{i}\otimes{\mathcal{A}}_{2}\otimes\cdots{\mathcal{A}}_{i-1}\otimes{\mathcal{A}}_{1}\otimes{\mathcal{A}}_{i+1}\otimes\cdots\otimes{\mathcal{A}}_{n}.$
(5.23)
Letting $M^{(i)}=M\circ(1,i)$ and using Eq. (2.11), we have
$\displaystyle M^{(i)}(|\psi\rangle)$ $\displaystyle=$ $\displaystyle
M^{(i)}({\mathcal{A}}_{1}\otimes\cdots\otimes{\mathcal{A}}_{n}|\psi^{\prime}\rangle)$
(5.24) $\displaystyle=$ $\displaystyle
M\circ(1,i)\circ({\mathcal{A}}_{1}\otimes\cdots\otimes{\mathcal{A}}_{n})\circ(1,i)((1,i)|\psi^{\prime}\rangle).$
By substituting Eq. (5.23) into Eq. (5.24), then using Eq. (5.1), we obtain
that
$M^{(i)}(|\psi\rangle)={\mathcal{A}}_{i}M((1,i)|\psi^{\prime}\rangle){\mathcal{A}}_{i}^{T}\det({\mathcal{A}}_{1})\cdots\det({\mathcal{A}}_{i-1})\det({\mathcal{A}}_{i+1})\cdots\det({\mathcal{A}}_{n}),$
(5.25)
and then Eq. (2.10) follows immediately.
## References
* [1] C.H. Bennett, S. Popescu, D. Rohrlich, J.A. Smolin, and A.V. Thapliyal (2000), Exact and asymptotic measures of multipartite pure-state entanglement, Phys. Rev. A, 63, pp. 012307.
* [2] W. Dür, G. Vidal and J.I. Cirac (2000), Three qubits can be entangled in two inequivalent ways, Phys. Rev. A, 62, pp. 062314.
* [3] F. Verstraete, J. Dehaene, B. De Moor and H. Verschelde (2002), Four qubits can be entangled in nine different ways, Phys. Rev. A, 65, pp. 052112.
* [4] A. Miyake (2003), Classification of multipartite entangled states by multidimensional determinants, Phys. Rev. A, 67, pp. 012108.
* [5] L. Lamata, J. León, D. Salgado, and E. Solano (2007), Inductive entanglement classification of four qubits under stochastic local operations and classical communication, Phys. Rev. A, 75, pp. 022318.
* [6] L. Borsten, D. Dahanayake, M.J. Duff, A. Marrani, and W. Rubens (2010), Four-qubit entanglement classification from string theory, Phys. Rev. Lett., 105, pp. 100507.
* [7] J.-G. Luque and J.-Y. Thibon (2003), Polynomial invariants of four qubits, Phys. Rev. A, 67, pp. 042303\.
* [8] J.-G. Luque and J.-Y. Thibon (2006), Algebraic invariants of five qubits, J. Phys. A: Math. Gen., 39, pp. 371-377.
* [9] D.Ž. Doković and A. Osterloh (2009), On polynomial invariants of several qubits, J. Math. Phys., 50, pp. 033509.
* [10] D. Li, X. Li, H. Huang, and X. Li (2007), Stochastic local operations and classical communication invariant and the residual entanglement for n qubits, Phys. Rev. A, 76, pp. 032304.
* [11] X. Li and D. Li (2011), Stochastic local operations and classical communication equations and classification of even n qubits, J. Phys. A: Math. Gen., 44, pp. 155304.
* [12] D. Li, X. Li, H. Huang, and X. Li (2007), Classification of four-qubit states by means of a stochastic local operation and the classical communication invariant and semi-invariants, Phys. Rev. A, 76, pp. 052311.
* [13] D. Li, X. Li, H. Huang, and X. Li (2009), SLOCC classification for nine families of four-qubits, Quantum Inf. Comput., 9, pp. 0778-0800.
* [14] D. Li, X. Li, H. Huang, and D. Li (2009), Stochastic local operations and classical communication properties of the n-qubit symmetric Dicke states, Europhys. Lett., 87, pp. 20006.
* [15] X. Zha and G. Ma (2011), Classification of four-qubit states by means of a stochastic local operation and the classical communication invariant, Chin. Phys. Lett., 28, pp. 020301.
* [16] O. Viehmann, C. Eltschka, and J. Siewert (2011), Polynomial invariants for discrimination and classification of four-qubit entanglement, quant-ph/1101.5558.
* [17] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano (2009), Operational families of entanglement classes for symmetric n-qubit states, Phys. Rev. Lett., 103, pp. 070503.
* [18] J.K. Stockton, J.M. Geremia, A.C. Doherty, and H. Mabuchi (2003), Characterizing the entanglement of symmetric many-particle spin-1/2 systems, Phys. Rev. A, 67, pp. 022112\.
* [19] D. Li, X. Li, H. Huang, and X. Li (2009), An entanglement measure for n qubits, J. Math. Phys., 50, pp. 012104\.
* [20] D. Li (2009), The n-tangle of odd n qubits, quant-ph/0912.0812.
* [21] V. Coffman, J. Kundu, and W.K. Wootters (2000), Distributed entanglement, Phys. Rev. A, 61, pp. 052306.
* [22] A. Osterloh and J. Siewert (2006), Entanglement monotones and maximally entangled states for multipartite qubit systems, Int. J. Quant. Inf., 4, pp. 531-540.
|
arxiv-papers
| 2011-06-21T03:45:35 |
2024-09-04T02:49:19.942452
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangrong Li, Dafa Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/1106.4082"
}
|
1106.4221
|
myctr
# Understanding opinions. A cognitive and formal account
Francesca Giardini Department of Cognitive Science, Central European
University,
Budapest, Hungary, email: GiardiniF@ceu.hu 111Central European University,
Hungary email: GiardiniF@ceu.hu
GiardiniF@ceu.hu Walter Quattrociocchi Department of Mathematics and
Computer Sciences,
University Of Siena, Italy 222University of Siena, Italy email:
walter.quattriocchi@unisi.it
walter.quattriocchi@unisi.it Rosaria Conte LABSS,
CNR - Institute of Cognitive Sciences and Technologies, Rome, Italy 333Labss-
ISTC-CNR, Italy email: rosaria.conte@istc.cnr.it
rosaria.conte@istc.cnr.it
((received date); (revised date))
###### Abstract
The study of opinions, their formation and change, is one of the defining
topics addressed by social psychology, but in recent years other disciplines,
as computer science and complexity, have addressed this challenge. Despite the
flourishing of different models and theories in both fields, several key
questions still remain unanswered. The aim of this paper is to challenge the
current theories on opinion by putting forward a cognitively grounded model
where opinions are described as specific mental representations whose main
properties are put forward. A comparison with reputation will be also
presented.
###### keywords:
opinion dynamics; social influence; gossip; media; agenda-setting
## 1 Introduction
Opinions represent a conspicuous part of our mental representations. A large
part of our social time is spent in exchanging, evaluating, revising and
comparing our opinions. We also say, about many different issues, that we have
opinions and we try to convince others about the groundedness of our own
opinions. Since the beginning of the last century, social psychologists have
been interested in understanding the specificity of opinions, as compared to
other kinds of mental representations, by focusing their attention on the
multiplicity of dimensions, including attitudes, beliefs and evaluations, that
take part within this phenomenon. Also political science has always been very
attentive to what is considered as a way to measure people’s preferences and
beliefs about publicly relevant issues. Many of these contributions have been
directed towards understanding the so-called _public opinion_ and the
processes through which it is possible to influence it, manipulating people’s
awareness and tendencies ([18]). More recently, other disciplines have shown a
great interest regarding such an issue, ranging from computer science passing
through socio-physics ([7, 13]) up to complexity science ([19]).
Despite the large amount of studies on opinions, the term itself and the
underlying concept are poorly specified and too general, since there are at
least two classes of mental representations that can be termed “opinions” but
they differ with regard to important aspects. Moreover, relevant contributions
coming from social psychology and computer science try to model distinct
issues, thus making the analysis of opinions quite difficult. This lack of
sound theoretical contributions is often compensated by giving more
preeminence to transmission and communication processes, thus partially
putting aside the “ontological” issue. In this work we propose a theoretical
account in which, starting from a critical review of approaches coming from
social psychology and computer science, the necessity of a cognitive approach
is claimed. Defining the specific cognitive features that characterize an
opinion, thus distinguishing it from other mental representations, and
introducing also two different kinds of opinions, evaluative and factual, we
will claim for the necessity of investigating the mental roots of opinions, in
order to understand how they are transformed and manipulated within and
between minds. This means that an opinion is specific with regard to other
mental representations, that has special features and is transformed through
specific mental processes. Defining an opinion in terms of its mental
ingredients permits to predict opinion change, its persistence, the effects of
contrasting forces and alternative paths of diffusion, because the different
forces are endogenously determined by specific rules. Understanding opinions,
describing how they are generated and revised, and how fare opinions travel
over the social space both as a consequence of social influence and as one of
the main means through which social influence unfolds, is crucial for grasping
a deeper understanding of human social cognition and behaviors. Moreover, our
cognitive analysis is supported by a preliminary formal description, in which
a new tool called Time Varying Graphs [8] is presented. This formalism has
been developed to deal with dynamically evolving systems[24, 23], and it
allows to overcome some of the limitations imposed by other instruments -e.g.
metrics, formalisms – that are not suited to account for a) the relationships
between opinions and other epistemic representations and b) their dynamics
both at social and individual level. In section 2, a critical introduction to
some of the main contributions about opinions is provided. Section 3 is
devoted to the description of our model, in which a definition of opinions as
specific mental representations and cognitively founded hypotheses about their
diffusion and change will be put forward. In section 4 a preliminary formal
account of how opinions are generated and how they can change is provided. In
section 5 some conclusions are drawn and future directions are suggested.
## 2 A critique to existing approaches
The understanding of opinions requires to take into account two levels of
explanation: the individual and the social level. As mental representations,
opinions are created within agents’ minds and they need to be integrated with
the existing network of beliefs, data, information, memories and evaluations.
However, in opinion change the social influence plays a major role and the
sharedness of an opinion can heavily affect its persistence and resistance to
change. These two dimensions are tightly linked and their interplay is one of
the defining features of opinions, but social psychology and computer science
are usually interested in tackling only one of these two aspects, without
studying them in combination. We claim that developing a cognitive theory of
opinions allows us to combine the micro- and the macro-level, understanding
how macro-social phenomena emerge, unintentionally, from micro-elements and
their interactions. In this way we can see that opinions derive from agents’
cognitive representations and states but they also exist in the social space,
in which they are transmitted and shared, and this social process affects, in
turn, individuals’ opinions. This complex loop requires a non-reductionist
approach in order to deal with both levels, without giving preeminence to one
or the other.
Social psychology mainly focuses on the individual side, trying to describe
how opinions are generated within the mind, devoting much attention to define
attitudes and evaluations, but paying little attention to the socially
interactive dimension of opinions. On the other hand, scholars from computer
science and physics have tried to explain how different opinions can coexist
or how they are modified through communication, treating opinions as mere
objects that are exchanged and revised according to certain mechanisms that
are quite far from the reality of cognitive and social processes. In both
cases there is a reductionist fallacy that works in different ways but in both
cases results in a downgrading of a complex issue into either a set of
unrelated specific elements or a unidimensional object that is far from the
complexity of a cognitive representation.
### 2.1 Social psychology: individualistic fallacy
Social psychologists have devoted much attention to the study of opinions’
formation and spreading, but a comprehensive and definite model allowing for
an operational and generative account is still missing. Providing a
comprehensive review of social psychology literature is beyond the scope of
this work, but in this section we will discuss some of the main theories in
order to underline how partial is the picture of opinions emerging from these
studies.
In general, opinions are treated as synonyms for different mental objects, as
beliefs [21], or more frequently, attitudes. Opinions are often conceptualized
as attitudes [20], [16], [22] or they are used as interchangeable terms that
have in common the fact of being affected by social influence and persuasion
[26]. It is worth noticing that many contributions are specifically oriented
to understand ”public opinion” [14], as the integration of opinions and
attitudes coming from different sources and exposed to different kinds of
influencing. Another general feature of the social psychology approach to
opinions is the preeminence given to measuring opinions, rather than on
conceptualizing them. As a result, many studies (for a review, see Schwarz N,
Sudman S, eds. 1996. Answering Questions: Methodology for Determining
Cognitive and Communicative Procesess in Survey Research. San Francisco:
Jossey- Bass) tried to develop reliable and fine-tuned ways to measure
people’s approaches to general questions, partially abandoning the issue of
defining what an opinion is and focusing on how it should be measured.
Allport [3] recognizes the difference between attitudes and opinions but he
nonetheless considers the measurement of opinions as one way of identifying
the strength and value of personal attitudes. An alternative view contrasts
the affective content of attitudes with the more cognitive quality of opinions
that involve some kind of conscious judgments [12]. In general, it is possible
to identify two main trends in the relevant literature: one more focused on
attitudes and the other more centered on conscious reasoning and judgment.
Crespi [9] considers individual opinions as ” judgmental outcomes of an
individual’s transactions with the surrounding world” (p.19), emphasizing the
interplay between what he calls an attitudinal system and the external world
characterized by the presence of other agents and different subjective
perceptions. Opinions are the outcomes of a judging process but this does not
mean that they are necessarily rational or reasoned, although Crespi
recognizes that they need to be consistent with the individual’s beliefs,
values and affective states.
As other authors already pointed out [1], many models of opinion and social
influence do not provide careful definitions of what an opinion is and how it
is affected by social influence. This happens to be true also for theories of
persuasion, like the social impact theory [17], a static theory of how social
processes operate at the level of the individual at a given point in time.
Part of this theory has been developed using computational modeling by Nowak,
Szamrej and Latané [2]. In their model, individuals change their attitudes as
a consequence of other individuals’ influence. In parallel with the idea that
social influence is proportional to a multiplicative function of the strength,
immediacy, and number of sources in a social force field [17], [14] suggest
that each attitude within a cognitive structure is jointly determined by the
strength, immediacy, and number of linked attitudes as individuals seek
harmony, balance, or consistency among them. Although very interesting, this
account fails to distinguish between attitudes and beliefs and does not
explain how inconsistencies can be resolved. The effect of communication on
opinion formation has been addressed by different disciplines from within the
social and the computational sciences, as well as complex systems science (for
a review on attitude change models, see [1]). One of the first works on this
topic has focused on polarization, i.e. the concentration of opinions by means
of interaction, as one main effect of the ”social influence” [11], whereas the
Social Impact Theory’ [2] proposes a more dynamic account, in which the amount
of influence depends on the distance, number, and strength (i.e.,
persuasiveness) of influence sources. As stated in ([7]), an important
variable, poorly controlled in current studies, is structure topology.
Interactions are invariably assumed as either all-to-all or based on a spatial
regular location (lattice), while more realistic scenarios are ignored.
Although very interesting, these studies fail to address the specificity of
opinions, treating them as generic mental objects that change as a consequence
of social influence, as it happens also to beliefs, or even goals. The
question about what an opinion is and what its main features are remains
unanswered, as well as their relationships with attitudes and their resistance
to influencing.
### 2.2 Computer science and complex systems: hyper-simplification fallacy
Turning our attention to complex systems science, one of the most popular
model applied to the aggregation of opinions is the bounded confidence model,
presented in [10]. Much like previous studies, in this work agents exchanging
information are modeled as likely to adjust their opinions only if the
preceding and the received information are close enough to each other. Such an
aspect is modeled by introducing a real number $\epsilon$, which stands for
tolerance or uncertainty ([7]) such that an agent with opinion $x$ interacts
only with agents whose opinions is in the interval $]x-\epsilon,x+\epsilon[$.
This hyper-simplification helps in making this complex phenomenon more
tractable using computational tools but, at the same time, reduces it to a
simple exchange of values that stand for mental objects, without any kind of
relationship with mental representations. An analogous attempt to model social
influence has been done by Axelrod (1997), who focused on the spreading of
given cultural features through communication. Again, agents do not have
internal representations of what they transmit, and final results are mainly
due to initial topology and to the distribution of traits, without a real
exchange among agents.
The model we present in this paper extends the bounded confidence model by
providing a cognitively plausible definition of opinion as mental
representations and identifying their constitutive elements and their
relationships.
We claim that opinions are highly dynamical representations resulting from the
interplay of different mental objects and affected by the mental states of
other individuals in the same network. Aim of this work is to provide an
interdisciplinary account to describe how social influence leads to opinion
formation, evolution and change. Moving from a characterization of opinions as
mental representations with specific features, we will try to model how
opinions are generated within the agents’ minds (micro-level) and how they
spread within a network of agents (macro-level). When explaining the emergence
of macro-social phenomena we need to know what happens at the micro-level,
i.e. what drives human actions and decisions in order to understand how
individuals’ representations and behaviors can give rise to socially complex
phenomena and how those affect agents’ actions. Without explaining how
opinions are formed and manipulated within the individuals’ minds, it is very
difficult to account for the way in which they change as an effect of social
influence. Our aim is to understand whether and how heterogeneous agents,
endowed with different beliefs and goals, may come to share a given viewpoint
and what consequences this sharing has on agents’ behaviors. We are interested
in providing answers, at least partially, to the following questions: What is
an opinion? What mechanisms lead people to change their opinions? How can
individuals resist to changes? What are the mechanisms of influence acting
within and between individual minds? How does social impact affect agents’
elaboration of new or contrasting information?
## 3 A Cognitive Theory of Opinions
This work aims at outlining a non-reductionist cognitive model of opinions and
their dynamics. Differently from the models reviewed above, we first provide a
definition of opinions as mental representations presenting specific features
that make their revision and updating more or less easy and enduring.
Moreover, grounding opinions in the minds allow us to take into account not
only direct processes of revision triggered by the comparison with others’
different opinions, i.e. social influence, but also revisions based upon
changing in other mental representations supporting that opinion.
The computational model introduced in this paper is intended to provide a
preliminary unifying framework to define opinions and to characterize their
dynamics in an easy but non-reductionist approach. Opinions in several models
of opinion dynamics are considered to change according to social influence, we
try to outline what is social influence and the way the social network
structure affects the agents’ opinions.
### 3.1 Facts and evaluations: two kinds of opinions
In everyday language the word “opinion” is often confronted with “fact”,
stressing the difference between something objective because it happened and
there are proofs of it, like in the latter case, and something that does not
have any reference in the external reality. This distinction is important,
because it points to a prominent feature of opinions, i.e. their being
regarded as uncertain and not grounded in any external proof. Opinions can be
debated, compared, discussed, argumented, but they can not be proven to be
true, contrary to what happens with facts. However, individuals continuously
resort to their opinions as less stable but more versatile mental objects
whose relevance is not reduced because of their being uncertain. This feature
is specific of opinions and it also explains why opinions are more prone to
change and revision, especially when confronted with others’ opinions.
Moreover, identifying this and other traits as specific, allows us to place
opinions among other kinds of mental representations, describing the kinds of
relationships opinions have with epistemic and motivational mental objects.
Opinions can be described as configurations of an individual’s beliefs, values
and feelings that can be conditionally activated. Conditional activation
points to the flexible and dynamic nature of these representations that are
not grounded in certainty and that usually come out from the merging and
elaboration of other representations and attitudes. Opinions are not only
conditional, but also compositional. This means that, for instance, starting
from my feeling of aversion toward mathematics and as a consequence of having
met a rude friend of friends who happened to teach math at school, when asked
about my opinion on the time kids should spend in studying mathematics, I can
form or, better, activate an opinion according to which the less time they
spend the better it is.
Opinions stem from the conditional activation of different kinds of mental
representations, that can have a propositional content or, as in the case of
attitudes and feelings, they can be more evaluative. However, there is a
specific feature that distinguishes an opinion from other kinds of mental
objects. An opinion is an epistemic representation in which the truth-value is
deemed to be uncertain. Opinions refer to objects of the external world that
can not be told to be either true or false. This impossibility (or
irrelevance) to say whether the content of a representation is true or false,
but only if it makes sense according to what someone believes and knows is
what makes a mental representation an opinion. This essential feature accounts
for the fact that opinions can be easily influenced not only by social
influence, i.e. an external force, but that they can also be easily revised
according to the change in one’s own mental representations.
This basic feature can be paired with the presence of an attitude, i.e. an
evaluative component that specifies whether the individual likes or dislikes
the topic. In general, attitudes are present when the topic is somehow
involving for the subject, so he is positively or negatively inclined toward
it.
When this is not the case, we have ”factual opinions”, like in the following
example. If someone is required to say when Mozart died, he can know the
correct answer or not, but this is not a moot point. On the contrary, the
causes of Mozart’s death are debatable because without knowing where he was
buried it is impossible to analyze the bones and to ascertain what killed him.
This means that we know that Mozart died in 1791 but there are contrasting
opinions about the causes of his death, and, even if there exist one true
opinion, none can tell which is the truth. On the other hand, when opinions
involve also evaluative components or facts, the opinions result from the
activation of a pattern of related representations like beliefs, knowledge,
other opinions, but also goals. This view allows us to describe opinions as
non-static patterns of relationships in which different representations are
linked through a variety of different linkages. This work is meant to address
the origin and changing of opinions thanks to these inter-relationships.
### 3.2 A tripartite model of opinion: truth-value, confidence and sharedness
An opinion is characterized by the three following features. First, the truth
value can not be verified (or it is not relevant). In general, opinions are
representations whose truth value can not be assessed through direct
experience. The topic of the opinion can not be experienced and then it is
impossible to say whether a given object is true or false. If I ask someone
about his opinion on the military intervention in Afghanistan, he can not tell
me that his opinion, whether positive or negative, is true, because it is not
possible to test an alternative state of the world in which the intervention
has not taken place and then asses which state was the best. Nonetheless, he
can tell me that he has a strong opinion or that he is very confident in it
because he has many supporting beliefs (e.g. Talibans’ regime had to be
fighted, civilians needed the intervention, the world is a safer place after
the intervention, etc) and even some goals (for instance, feeling safer)
related with that opinion. We can have strong or weak opinions, but our
confidence does not depend on the fact that something is known to be true,
given the impossibility to assess its truth-value. In other cases, assessing
the truth-value is not relevant, because the attitude and the supporting
mental representations are stronger enough to support the opinion, without
caring for its being true or false. Going back to the example about the time
spent in studying math, I can build upon my negative experience at school,
supporting it with my negative attitude and recalling my experience with the
unfriendly friend of my friends who happens to be a math teacher, to build up
my negative opinion. Furthermore, notwithstanding the existence of statistics
or experts that can support or confute my opinion, I do not care about them,
because they are not relevant to me. A creationist’s opinion about Darwin and
the theory of natural selection is not affected by the proofs of its validity,
because he does not care for those proofs and focus his attention on other
kinds of knowledge (like that coming from the Bible, for instance).
The second feature is the degree of confidence which is a subjective measure
of the strength of belief and it expresses the exent to which one’s opinion is
resistant to change. This is to say that the lack of an assessable truth value
is totally independent from the confidence one has in his opinions.The degree
of confidence depends on the number of supporting representations, and the
higher this number the stronger an opinion will be. Castelfranchi, Poggi [6]
made a distinction between confidence coming from the source and confidence
coming from the degree of compatibility that a given belief has with pre-
existing beliefs. It is interesting to notice that representations do not need
to be about the same topic or to belong to the same set to form a coherent
network. If we take the Afghanistan example, we can easily imagine that a
negative opinion about the military intervention could be supported by a
general belief about the right of other countries to intervene in internal
disputes or by negative evaluations about the US foreign policy, or even by
knowledge about the roles played by URSS and US in Afghanistan during the Cold
War. These beliefs are not exclusively related to the target opinion and they
can have stronger or weaker connections with other opinions. The stronger the
confidence in these beliefs and the higher their number, the stronger will be
the confidence in that opinion.
The degree of confidence can also vary in accordance with the configuration
activated by a certain opinion. Since opinions are dynamic configurations
emerging from the conditional activation of other representations, the path
followed to link different beliefs, goals, data and memories can result in
opinions that have the same content but different degrees of confidence. I can
be against the military intervention in Afghanistan because I feel empathic
with the civilians, thus focusing on the attitudinal and evaluative aspects,
or because I have strong beliefs about the US foreign policy. In this latter
case, my opinion is supported by facts and follows a specific argumentative
line, and it could lead me to be more confident.
Finally, the sharing of an opinion, i.e. the extent to which a given opinion
is considered shared, is another crucial feature. The sharing may heavily
affect the degree of confidence, making people feel more confident because
many other individuals have the same opinion. The sharing is the outcome of a
process of social influence, through which agents’ opinion are circulated
within the social space and they can become more or less shared. This
dimension is crucial, but it is also true that it carachterizes other social
beliefs, like reputation.
It is worth noticing that there are other kinds of beliefs that are really
close to opinions but, at a closer investigation, there are some important
differences. Reputation can be one of these, because it is shared and it is
also carachterized by a varying degree of confidence. But, unlikely opinions,
reputation has a truth value because it refers to someone’s behaviors or
actions that were actually exhibited (or that were reported as such, but we do
not want to address here the issue of lying) and reported to other people.
Reality matters in reputation, whereas it is much less relevant in opinions,
as witnessed also by the fact that reputation does not have to be convincing
(i.e. supported by some reasoning or arguments), whereas opinions need.
## 4 Toward a Formal Definition
### 4.1 Preliminaries
#### 4.1.1 Time Varying Graphs
As mentioned in previous section the temporal aspects of our opinion model is
based upon Time-Varying Graphs (TVG) formalism, an algorithmic framework [8]
designed to deal with the temporal dimension of networked data and to express
their dynamics from an interaction-centric point of view [27].
Consider a set of entities $V$ (or nodes), a set of relations $E$ between
these entities (edges), and an alphabet $L$ accounting for any property such
that a relation could have (label); that is, $E\subseteq V\times V\times L$.
$L$ can contain multi-valued elements.
The relations (interactions) among entities are assumed to take place over a
time dimension (continuos or discrete) $\mathcal{T}$ the lifetime of the
system which is generally a subset of $\mathbb{N}$ (discrete-time systems) or
$\mathbb{R}$ (continuous-time systems). The dynamics of the system can
subsequently be described by a time-varying graph, or TVG,
$\mathcal{G}=(V,E,\mathcal{T},\rho,\zeta)$, where
* •
$\rho:E\times\mathcal{T}\rightarrow\\{0,1\\}$, called presence function,
indicates whether a given edge or node is available at a given time.
* •
$\zeta:E\times\mathcal{T}\rightarrow\mathbb{T}$, called latency function,
indicates the time it takes to cross a given edge if starting at a given date
(the latency of an edge could vary in time).
#### 4.1.2 The underlying graph
Given a TVG $\mathcal{G}=(V,E,\mathcal{T},\rho,\zeta)$, the graph $G=(V,E)$ is
called underlying graph of $\mathcal{G}$. This static graph should be seen as
a sort of footprint of $\mathcal{G}$, which flattens the time dimension and
indicates only the pairs of nodes that have relations at some time in a given
time interval $\mathcal{T}$. In most studies and applications, $G$ is assumed
to be connected; in general, this is not necessarily the case. Note that the
connectivity of $G=(V,E)$ does not imply that $\mathcal{G}$ is connected at a
given time instant; in fact, $\mathcal{G}$ could be disconnected at all times.
The lack of relationship, with regards to connectivity, between $\mathcal{G}$
and its footprint $G$ is even stronger: the fact that $G=(V,E)$ is connected
does not even imply that $\mathcal{G}$ is “connected over time”.
#### 4.1.3 Edge-centric evolution
From an edge point of view (relationships within epistemic representations),
the evolution derives from variations of the availability. TVG defines the
available dates of an edge $e$, noted $\mathcal{I}(e)$, as the union of all
dates at which the edge is available, that is,
$\mathcal{I}(e)=\\{t\in\mathcal{T}:\rho(e,t)=1\\}$. Given a multi-interval of
availability $\mathcal{I}(e)=\\{[t_{1},t_{2})\cup[t_{3},t_{4})...\\}$, the
sequence of dates $t_{1},t_{3},...$ is called appearance dates of $e$, noted
$App(e)$, and the sequence of dates $t_{2},t_{4},...$ is called disappearance
dates of $e$, noted $Dis(e)$. Finally, the sequence $t_{1},t_{2},t_{3},...$ is
called characteristic dates of $e$, noted $\mathcal{S}_{\mathcal{T}}(e)$.
#### 4.1.4 Graph-centric evolution
From a global standpoint, the evolution of the system can be derived by a
sequence of (static) graphs $\mathcal{S}_{\mathcal{G}}=G_{1},G_{2}..$ where
every $G_{i}$ corresponds to a static snapshot of $\mathcal{G}$ such that
$e\in E_{G_{i}}\iff\rho_{[t_{i},t_{i}+1)}(e)=1$, with two possible meanings
for the $t_{i}$s: either the sequence of $t_{i}$s is a discretization of time
(for example $t_{i}=i$); or it corresponds to the set of particular dates when
topological events occur in the graph, in which case this sequence is equal to
$sort(\cup\\{\mathcal{S}_{\mathcal{T}}(e):e\in E\\})$. In the latter case, the
sequence is called characteristic dates of $\mathcal{G}$, and noted
$\mathcal{S}_{\mathcal{T}}(\mathcal{G})$.
### 4.2 Modeling Epistemic Representations
An opinion is an epistemic representation of a state of the world with respect
to a given object $p$. It is defined on a three dimensional space defined by:
a) the objective truth value $T_{o}$, a subjective truth value, namely $T_{s}$
and a degree of confidence $d_{c}$ with respect to the object $p$.
More formally we can state that:
###### Definition 4.1.
an epistemic representation of a state of the world $m\in M$ is a quadruplet
${p,T_{o},T_{s},d_{c}}$ defined by a preposition $p$ related to a given object
$O$, and two variable $T_{o}$ and $T_{s}$ defined on $\mathbb{R}$. The
$d_{c}\in\mathbb{R}$ respectively quantifying the “real“ truth value of an
information, namely the objective truth value, the perceived truth values, and
the degree of confidence, with respect to the preposition $p$.
By varying the dimensions of the domain of $T_{o}$ and $T_{s}$, we can define
a taxonomy of the epistemic representation of the world that can be summarised
as follows:
###### Definition 4.2.
An epistemic representation $m_{k}=\\{p,T_{o},T_{s},d_{c}\\}$ is knowledge
when $T_{o}=T_{s}$.
###### Definition 4.3.
An epistemic representation $m_{b}=\\{p,T_{o},T_{s},d_{c}\\}$ is a belief when
$0<T_{o}<1\wedge 0\leq T_{s}\leq 1$ .
###### Definition 4.4.
An epistemic representation $m_{o}=\\{p,T_{o},T_{s},d_{c}\\}$ is an opinion
when $0\leq T_{o}<1\wedge 0\leq T_{s}\leq 1$.
### 4.3 Opinions and Individuals
We can define an epistemic representation graph as a network of epistemic
representation immerged in a dynamic network in a given time interval and the
links state the correlation among them. Let us consider a set $V$ of mental
representation (or nodes), interacting with one another over time. Each
relation among the mental representation can be formalized by a quadruplet
$c=\\{u,v,t_{1},t_{2}\\}$, where $u$ and $v$ are the involved mental
representations (either beliefs, or knowledge or an opinion), $t_{1}$ is the
time at which the correlation occurs, and $t_{2}$ the time at which the
relation terminates. A given pair of nodes can naturally be subject to several
such interactions over time (and for generality, we allow these interactions
to overlap). Given a time interval $\mathcal{T}=[t_{a},t_{b})\subseteq{\cal
T}$ (where $t_{a}$ and $t_{b}$ may be either two dates, or one date and one
infinity, or both infinities), the set $C(\mathcal{T})$ (or simply $C$) of all
interactions occurring during that time interval defines a set of
intermittently-available edges $E(\mathcal{T})\subseteq V\times V$, such that:
$\begin{split}\forall u,v\in V,(u,v)\in E(\mathcal{T})\\\ \iff\exists
t^{\prime}\in[t_{a},t_{b}),(u,v,t_{1},t_{2})\in C(\mathcal{T})\ :\ t_{1}\leq
t^{\prime}<t_{2}\end{split}$ (1)
that is, an edge $(u,v)$ exists iff at least one interaction between $u$ and
$v$ occurs, or terminates, between $t_{a}$ and $t_{b}$. The intermittent
availability of an edge $e=(u,v)\in E(\mathcal{T})$ is described by the
presence function $\rho:E(\mathcal{T})\times\mathcal{T}\rightarrow\\{0,1\\}$
such that $\forall t\in\mathcal{T},e\in E(\mathcal{T})$:
$\rho(e,t)=1\iff\exists(u,v,t_{1},t_{2})\in C:t_{1}\leq t<t_{2}$ (2)
The triplet $\mathcal{G}=(V,E,\rho)$ is called an epistemic representation
graph, and the temporal domain $\mathcal{T}=[t_{a},t_{b})$ of the function
$\rho$, is the lifetime of $\mathcal{G}$. We denote by
$\mathcal{G}_{[t,t^{\prime})}$ the mental representation subgraph of
$\mathcal{G}$ covering the period $[t_{a},t_{b})\cap[t,t^{\prime})$
Hence, a sequence of couples
$\mathcal{J}=\\{(e_{1},t_{1}),(e_{2},t_{2}),...\\}$, with $e_{i}\in E$ and
$t_{i}\in\mathcal{T}$ for all $i$, is called a journey in $\mathcal{G}$ iff
$\\{e_{1},e_{2},...\\}$ is a walk in $G$ and for all $i$,
$\rho(e_{i},t_{i})=1$ and $t_{i+1}\geq t_{i}$. Journeys can be thought of as
paths over time from a source node to a destination node (if the journey is
finite).
Let us denote by $\mathcal{J}^{*}_{\mathcal{G}}$ the set of all possible
journeys in an epistemic representation system $\mathcal{G}$. We will say that
$\mathcal{G}$ admits a journey from a node $u$ to a node $v$, and note
$\exists\mathcal{J}_{(u,v)}\in\mathcal{J}^{*}_{\mathcal{G}}$, if there exists
at least one possible journey from $u$ to $v$ in $\mathcal{G}$.
### 4.4 Opinion Dynamics and Society
One of the most famous formalisms aimed at describing the process of
persuasion is the “Bounded Confidence Model” (BCM) where agents exchanging
information are modeled as likely to adjust their opinions only if the
preceding and the received information are close enough to each other. Such an
aspect is modeled by introducing a real number $\epsilon$ , which stands for
tolerance or uncertainty such that an agent with opinion $x$ interacts only
with agents whose opinions is in the interval ]x − $\epsilon$ , x + $\epsilon$
[. Neverthless the wide, massive and cross-disciplinary use of the BCM ([19,
15]) ranging from “viral marketing” to to the Italians’ opinions distortion
played by controlled mass media ([25, 4, 5, 15]). Such a model does not
provide an explanation of the phenomena yielding to the tolerance value, it is
just assumed as a static value.
In this work we will outline which are the factors affecting the acceptance or
the refuse of one another opinion. In particular, how can we formalize
comparison of two or more opinions? Recalling that a mental representation is
a preposition with the truth value defined by two variable
$T_{o},T_{s}\in\mathbb{R}$ and $d_{c}\in\mathbb{R}$ respectively quantifying
the “real” and the perceived truth value and the degree of confidence with
respect to a given object or proposition. And considering that such mental
representations are modeled as set of time connected entities of the form
$\mathcal{G}=(V,E,\rho)$ we can now provide some definitions aimed at
describing the process of persuasion.
Assuming that an epistemic representation system, which is by nature adaptive,
when facing with external events, reacts to the stimulus by activating only a
subset of its components. For instance, consider the example where an agent
$x$ is questioned by an agent $y$ about his opinion on a given target.
What does happen in the $x$’s mental representation system? How can we
quantify $x$’s attitudes to change or not is opinions regarding a given matter
of fact?
According to our model the epistemic representation system of $x$, as reaction
to the external stimulus posed by the $y$’s question, will perform $journey$
within the elements that in its mind are related with the target of the
question and on this base will be able to compare its opinion with the one
owned by $y$.
###### Definition 4.5.
(relational-)connected component induced by an external event in
$\mathcal{G}_{x}$ is defined as a set of nodes $V^{\prime}\subseteq V$ such
that $\forall u,v\in
V^{\prime},\exists\mathcal{J}_{(u,v)}\in\mathcal{J}^{*}_{\mathcal{G}}$. Then
$\mathcal{G}$ is said connected if it is itself a connected component
($V^{\prime}=V$).
Since all nodes in $V^{\prime}$ are defined by an objective truth value $T$
and a degree of confidence (perceived truth value) $d_{g}$ it is obvious that
the resistence to an opinion to change is denoted by these values in all the
nodes in $V^{\prime}$.
## 5 Conclusions
In this preliminary work we tried to sketch a cognitively grounded dynamic
model of opinions, in which we defined these mental representations as
carachterized by the presence of three specific features. Differently than
psychological theories of opinions that usually provide rich definitions that
are too complex to be reduced to measurable variables, we isolated three main
constitutive elements that characterize this kind of mental representations.
On the other hand, we tried to overcome the reductionist approach of opinion
dynamic models, in which the richness of human cognitive processes is
substituted by easy-to-compute factors poorly related to actual human
behaviors. For this reason, we proposed to apply time-varying-graph to develop
a formal model able to account for the way in which opinions are generated and
change as a function of the presence and opinions of other agents in the
network.
We are perfectly aware of the complexity of this issue and this work
represents a preliminary attempt to merge the cognitive complexity of opinions
with a rigorous formal approach, but there are many problems that we need to
address. First, the cognitive model should be refined and specific hypotheses
about opinion revision and diffusion should be put forward. Moreover, the
robustness of the formal model will be tested and such a model will be
implemented in cognitive multi-agent system in order to explore the parameter
space upon which our model has been defined. Our ultimate aim is to build up a
simulation environment in which agents endowed with heterogeneous
representations of the external world interact and this leads to the creation
of new opinions, the disappearing of some of the previous ones and, in
general, to different distributions of representations in the population.
## 6 Acknowledgements
This work was supported by the European Community under the FP6 programme
(eRep project CIT5-028575). A particular thanks to Ilvo Diamanti, Federica
Mattei, Mario Paolucci, Federico Cecconi, Stefano Picascia, Geronimo Stilton
and the Hypnotoad. In addition we are grateful to the biggest Italian anomaly
and the Italian media for the inspirations and insights.
## References
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|
arxiv-papers
| 2011-06-21T15:00:33 |
2024-09-04T02:49:19.951817
|
{
"license": "Public Domain",
"authors": "Francesca Giardini, Walter Quattrociocchi, Rosaria Conte",
"submitter": "Walter Quattrociocchi",
"url": "https://arxiv.org/abs/1106.4221"
}
|
1106.4317
|
A Self-Organizing State-Space-Model Approach for Parameter Estimation in
Hodgkin-Huxley-Type Models of Single Neurons
Dimitrios V. Vavoulis1,∗, Volko A. Straub2, John A. D. Aston3, Jianfeng
Feng1,∗
1 Dept. of Computer Science, University of Warwick, Coventry, UK
2 Dept. of Cell Physiology and Pharmacology, University of Leicester,
Leicester, UK
3 Dept. of Statistics, University of Warwick, Coventry, UK
$\ast$ E-mail: Dimitris.Vavoulis@dcs.warwick.ac.uk,
Jianfeng.Feng@warwick.ac.uk
## Abstract
Traditional approaches to the problem of parameter estimation in biophysical
models of neurons and neural networks usually adopt a global search algorithm
(for example, an evolutionary algorithm), often in combination with a local
search method (such as gradient descent) in order to minimize the value of a
cost function, which measures the discrepancy between various features of the
available experimental data and model output. In this study, we approach the
problem of parameter estimation in conductance-based models of single neurons
from a different perspective. By adopting a hidden-dynamical-systems
formalism, we expressed parameter estimation as an inference problem in these
systems, which can then be tackled using a range of well-established
statistical inference methods. The particular method we used was Kitagawa’s
self-organizing state-space model, which was applied on a number of Hodgkin-
Huxley-type models using simulated or actual electrophysiological data. We
showed that the algorithm can be used to estimate a large number of
parameters, including maximal conductances, reversal potentials, kinetics of
ionic currents, measurement and intrinsic noise, based on low-dimensional
experimental data and sufficiently informative priors in the form of pre-
defined constraints imposed on model parameters. The algorithm remained
operational even when very noisy experimental data were used. Importantly, by
combining the self-organizing state-space model with an adaptive sampling
algorithm akin to the Covariance Matrix Adaptation Evolution Strategy, we
achieved a significant reduction in the variance of parameter estimates. The
algorithm did not require the explicit formulation of a cost function and it
was straightforward to apply on compartmental models and multiple data sets.
Overall, the proposed methodology is particularly suitable for resolving high-
dimensional inference problems based on noisy electrophysiological data and,
therefore, a potentially useful tool in the construction of biophysical neuron
models.
## Author Summary
Parameter estimation is a problem of central importance and, perhaps, the most
laborious task in biophysical modeling of neurons and neural networks. An
emerging trend is to treat parameter estimation in this context as yet another
statistical inference problem, which can be tackled using well-established
methods from Computational Statistics. Inspired by these recent advances, we
adopted a self-organizing state-space-model approach augmented with an
adaptive sampling algorithm akin to the Covariance Matrix Adaptation Evolution
Strategy in order to estimate a large number of parameters in a number of
Hodgkin-Huxley-type models of single neurons. Parameter estimation was based
on noisy electrophysiological data and involved the maximal conductances,
reversal potentials, levels of noise and, unlike most mainstream work, the
kinetics of ionic currents in the examined models. Our main conclusion was
that parameters in complex, conductance-based neuron models can be inferred
using the aforementioned methodology, if sufficiently informative priors
regarding the unknown model parameters are available. Importantly, the use of
an adaptive algorithm for sampling new parameter vectors significantly reduced
the variance of parameter estimates. Flexibility and scalability are
additional advantages of the proposed method, which is particularly suited to
resolve high-dimensional inference problems.
## Introduction
Among several tools at the disposal of neuroscientists today, data-driven
computational models have come to hold an eminent position for studying the
electrical activity of single neurons and the significance of this activity
for the operation of neural circuits[1, 2, 3, 4]. Typically, these models
depend on a large number of parameters, such as the maximal conductances and
kinetics of gated ion channels. Estimating appropriate values for these
parameters based on the available experimental data is an issue of central
importance and, at the same time, the most laborious task in single-neuron and
circuit modeling.
Ideally, all unknown parameters in a model should be determined directly from
experimental data analysis. For example, based on a set of voltage-clamp
recordings, the type, kinetics and maximal conductances of the voltage-gated
ionic currents flowing through the cell membrane could be determined[5] and,
then, combined in a conductance-based model, which replicates the activity of
the biological neuron of interest under current-clamp conditions with
sufficient accuracy. Unfortunately, this is not always possible, especially
for complex compartmental models, which contain a large number of ionic
currents.
A first problem arises from the fact that not all parameters can be estimated
within an acceptable error margin, especially for small currents and large
levels of noise. A second problem arises from the practice of estimating
different sets of parameters based on data collected from different neurons of
a particular type, instead of estimating all unknown parameters using data
collected from a single neuron. Different neurons of the same type may have
quite different compositions of ionic currents[6, 7, 8, 9] (but, see also
[10]). This implies that combining ionic currents measured from different
neurons in the same model or even using the average of several parameters
calculated over a population of neurons of the same type will not necessarily
result in a model that expresses the experimentally recorded patterns of
electrical activity under current-clamp conditions. Usually, only some
parameters are well characterized, while others are difficult or impossible to
measure directly. Thus, most modeling studies rely on a mixture of
experimentally determined parameters and estimates of the remaining unknown
ones using automated optimization methodology (see, for example, [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22]). Typically, these methods require the
construction of a cost function (for measuring the discrepancy between various
features of the experimental data and the output of the model) and an
automated parameter selection method, which iteratively generates new sets of
parameters, such that the value of the cost function progressively decreases
during the course of the simulation (see [23] for a review). Popular choices
of such methods are evolutionary algorithms, simulated annealing and gradient
descent methods. Often, a global search method (i.e. an evolutionary
algorithm) is combined with local search (gradient descent) for locating
multiple minima of the cost function with high precision. Since a poorly
designed cost function (for example, one that merely matches model and
experimental membrane potential trajectories) can seriously impede
optimization, the construction of this function often requires particular
attention (see, for example, [24]). Nevertheless, these computationally
intensive methodologies have gained much popularity, particularly due to the
availability of powerful personal computers at consumer-level prices and the
development of specialized optimization software (e.g. [25]).
Alternative approaches also exist as, for example, methods based on the
concept of synchronization between model dynamics and experimental data[26].
An emerging trend in parameter estimation methodologies for models in
Computational Biology is to recast parameter estimation as an inference
problem in hidden dynamical systems and then adopt standard Computational
Statistics techniques to resolve it[27, 28]. For example, a particular study
following this approach makes use of Sequential Monte Carlo methods (particle
filters) embedded in an Expectation Maximization (EM) framework[28]. Given a
set of electrophysiological recordings and a set of dynamic equations that
govern the evolution of the hidden states, at each iteration of the algorithm
the expected joint log-likelihood of the hidden states and the data is
approximated using particle filters (Expectation Step). At a second stage
during each iteration (Maximization Step), the log-likelihood is locally
maximized with respect to the unknown parameters. The advantage of these
methods, beyond the fact that they recast the estimation problem in a well-
established statistical framework, is that they can handle various types of
noisy biophysical data made available by recent advances in voltage and
calcium imaging techniques.
Inspired by this emerging approach, we present a method for estimating a large
number of parameters in Hodgkin-Huxley-type models of single neurons. The
method is a version of Kitagawa’s self-organizing state-space model[29]
combined with an adaptive algorithm for selecting new sets of model
parameters. The adaptive algorithm we have used is akin to the Covariance
Matrix Adaption (CMA) Evolution Strategy[30], but other methods (e.g.
Differential Evolution as described in [31]) may be used instead. We
demonstrate the applicability of the algorithm on a range of models using
simulated or actual electrophysiological data. We show that the algorithm can
be used successfully with very noisy data and it is straightforward to apply
on compartmental models and multiple datasets. An interesting result from this
study is that by using the self-organizing state-space model in combination
with a CMA-like algorithm, we managed to achieve a dramatic reduction in the
variance of the inferred parameter values. Our main conclusion is that a large
number of parameters in a conductance-based model of a neuron (including
maximal conductances, reversal potentials and kinetics of gated ionic
currents) can be inferred from low-dimensional experimental data (typically, a
single or a few recordings of membrane potential activity) using the
algorithm, if sufficiently informative priors are available, for example in
the form of well-defined ranges of valid parameter values.
## Methods
### Modeling Framework
We begin by presenting the current conservation equation that describes the
time evolution of the membrane potential for a single-compartment model
neuron:
$\frac{dV}{dt}=\frac{I_{ext}-G_{L}(V-E_{L})-\sum_{i}I_{i}}{C_{m}}$ (1)
where $V$, $I_{ext}$ and $I_{i}$ are all functions of time. In the above
equation, $C_{m}$ is the membrane capacitance, $V$ is the membrane potential,
$I_{ext}$ is the externally applied (injected) current, $G_{L}$ and $E_{L}$
are the maximal conductance and reversal potential of the leakage current,
respectively, and $I_{i}$ is the $i^{th}$ transmembrane ionic current. A
voltage-gated current $I_{i}$ can be modeled according to the Hodgkin-Huxley
formalism, as follows:
$I_{i}=G_{i}m_{i}^{p_{i}}h_{i}(V-E_{i})$ (2)
where $m_{i}$ and $h_{i}$ are both functions of time. In the above expression,
$G_{i}$ and $E_{i}$ are the maximal conductance and reversal potential of the
$i^{th}$ ionic current, $m_{i}$ and $h_{i}$ are dynamic gating variables,
which model the voltage-dependent activation and inactivation of the current,
and $p_{i}$ is a small positive integer power (usually, not taking values
larger than 4). The product $m_{i}^{p_{i}}h_{i}$ is the proportion of open
channels in the membrane that carry the $i^{th}$ current. The gating variables
$m_{i}$ and $h_{i}$ obey first-order relaxation kinetics, as shown below:
$\frac{dm_{i}}{dt}=\frac{m_{\infty,i}-m_{i}}{\tau_{m_{i}}}\qquad,\qquad\frac{dh_{i}}{dt}=\frac{h_{\infty,i}-h_{i}}{\tau_{h_{i}}}$
(3)
where the steady states ($m_{\infty,i}$ , $h_{\infty,i}$) and relaxation times
($\tau_{m_{i}}$ , $\tau_{h_{i}}$) are all functions of voltage.
Using vector notation, we can write the above system of Ordinary Differential
Equations (ODEs) in more concise form:
$\frac{d\mathbf{x}(t)}{dt}=\mathbf{f}(\mathbf{x}(t),t)$ (4)
where the state vector $\mathbf{x}(t)$ is composed of the time-evolving state
variables $V$, $m_{i}$ and $h_{i}$ and the vector-valued function
$\mathbf{f}(\cdot,\cdot)$, which describes the evolution of $\mathbf{x}(t)$ in
time, is formed by the right-hand sides of Eqs. 1 and 3. Notice that
$\mathbf{f}(\cdot,\cdot)$ also depends on a parameter vector $\theta$, which
for now is dropped from Eq. 4 for notational clarity. Components of $\theta$
are the maximal conductances $G_{i}$, the reversal potentials $E_{i}$ and the
various parameters that control the voltage-dependence of the steady states
and relaxation times in Eq. 3.
The above deterministic model does not capture the inherent variability in the
electrical activity of neurons, but rather some average behavior of
intrinsically stochastic events. In general, this variability originates from
various sources, such as the random opening and shutting of transmembrane ion
channels or the random bombardment of the neuron with external (e.g. synaptic)
stimuli[32]. Here, we model the inherent variability in single-neuron activity
by augmenting Eq. 4 with a noisy term and re-writing as follows:
$d\mathbf{x}(t)=\mathbf{f}(\mathbf{x}(t),t)dt+\sqrt{\Sigma_{x}}d\mathbf{W}_{\mathbf{x}}(t)$
(5)
where $\Sigma_{\mathbf{x}}$ is a covariance matrix and $\mathbf{W_{x}}(t)$ is
a standard Wiener process over the state space of $\mathbf{x}(t)$.
$\Sigma_{\mathbf{x}}$ may be a diagonal matrix of variances ($\sigma_{V}^{2}$,
$\sigma_{m_{i}}^{2}$ and $\sigma_{h_{i}}^{2}$) corresponding to each component
of the state vector.
Typically, we assume that the above model is coupled to a measurement
“device”, which permits indirect observations of the hidden state
$\mathbf{x}(t)$:
$\mathbf{y}(t)=\mathbf{g}(\mathbf{x}(t),\zeta(t))$ (6)
where $\zeta(t)$ is an observation noise vector. In the simplest case, the
vector of observations $\mathbf{y}(t)$ is one-dimensional and it may consist
of noisy measurements of the membrane potential:
$y(t)=V(t)+\sigma_{y}\mathcal{N}(0,1)$ (7)
where $\sigma_{y}$ is the standard deviation of the observation noise and
$\mathcal{N}(0,1)$ a random number sampled from a Gaussian distribution with
zero mean and standard deviation equal to unity. More complicated non-linear,
non-Gaussian observation functions may be used when, for example, the
measurements are recordings of the intracellular calcium concentration,
simultaneous recordings of the membrane potential and the intracellular
calcium concentration or simultaneous recordings of the membrane potential
from multiple sites (e.g. soma and dendrites) of a neuron.
Assuming that time $t$ is partitioned in a very large number $K$ of time steps
$\Delta t$, such that $t\in\\{t_{0},t_{1}=t_{0}+\Delta t,t_{2}=t_{0}+2\;\Delta
t,\ldots,t_{K}=K\;\Delta t\\}$ and the corresponding states are
$\mathbf{x}\in\\{\mathbf{x}_{0},\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{K}\\}$,
we can approximate the solution to Eq. 5 using the following difference
equation:
$\mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{f}(\mathbf{x}_{k},t_{k})\Delta
t+\sqrt{\Sigma_{\mathbf{x}}}(\mathbf{W}_{\mathbf{x},k+1}-\mathbf{W}_{\mathbf{x},k})$
(8)
where $\mathbf{W}_{\mathbf{x},k+1}-\mathbf{W}_{\mathbf{x},k}=\sqrt{\Delta
t}\;\mathbf{\xi}_{k}$ and $\mathbf{\xi}_{k}$ is a random vector with
components sampled from a normal distribution with zero mean and unit
variance. The above expression implements a simple rule for computing the
membrane potential, activation and inactivation variables at each point
$t_{k+1}$ of the discretized time based on information at the previous time
point $t_{k}$ and it can be considered as a specific instantiation of the
Euler-Maruyama method for the numerical solution of Stochastic Differential
Equations[33].
Then, the observation model becomes:
$\mathbf{y}_{k+1}=\mathbf{g}(\mathbf{x}_{k+1},\zeta_{k+1})$ (9)
In general, measurements do not take place at every point $t_{k}$ of the
discretized time, but rather at intervals of $\Delta k$ time steps (depending
on the resolution of the measurement device), thus generating a total of
$K/\Delta k$ measurements. For simplicity in the above description, we have
assumed that $\Delta k=1$. However, all the models we consider in the Results
section assume $\Delta k>1$.
In terms of probability density functions, the non-linear state-space model
defined by Eqs. 8 and 9 (known as the dynamics model} and the observation
model, respectively) can be written as:
$\displaystyle\mathbf{x}_{k+1}$ $\displaystyle\sim$ $\displaystyle
p(\cdot|\mathbf{x}_{k})$ (10) $\displaystyle\mathbf{y}_{k+1}$
$\displaystyle\sim$ $\displaystyle p(\cdot|\mathbf{x}_{k+1})$ (11)
where the initial state $\mathbf{x}_{0}$ is distributed according to a prior
density $p(\mathbf{x}_{0})$. The above formulas are known as the state
transition and observation densities, respectively[34].
### Simulation-Based Filtering and Smoothing
In many inference problems involving state-space models, a primary concern is
the sequential estimation of the following two conditional probability
densities[29]: (a) $p(\mathbf{x}_{k}|\mathbf{y}_{1:k})$ and (b)
$p(\mathbf{x}_{k}|\mathbf{y}_{1:K})$, where
$\mathbf{y}_{1:k}=\\{\mathbf{y}_{1},...,\mathbf{y}_{k}\\}$, i.e. the set of
observations (for example, a sequence of measurements of the membrane
potential) up to the time point $t_{k}$. Density (a), known as the filter
density, models the distribution of state $\mathbf{x}_{k}$ given all
observations up to and including the time point $t_{k}$, while density (b),
known as the smoother density, models the distribution of state
$\mathbf{x}_{k}$ given the whole set of observations up to the final time
point $t_{K}$.
In principle, the filter density can be estimated recursively at each time
point $t_{k}$ using Bayes’ rule appropriately[29]:
$p(\mathbf{x}_{k}|\mathbf{y}_{1:k})=\frac{p(\mathbf{y}_{k}|\mathbf{x}_{k})}{p(\mathbf{y}_{k}|\mathbf{y}_{1:k-1})}\int
p(\mathbf{x}_{k}|\mathbf{x}_{k-1})p(\mathbf{x}_{k-1}|\mathbf{y}_{1:k-1})d\mathbf{x}_{k-1}$
(12)
where $p(\mathbf{x}_{k}|\mathbf{x}_{k-1})$ and
$p(\mathbf{y}_{k}|\mathbf{x}_{k})$ are the state transition and observation
densities, respectively, and $p(\mathbf{x}_{k-1}|\mathbf{y}_{1:k-1})$ is the
filter density at the previous time step $t_{k-1}$.
Then, the smoother density can be obtained by using the following general
recursive formula:
$p(\mathbf{x}_{k}|\mathbf{y}_{1:K})=p(\mathbf{x}_{k}|\mathbf{y}_{1:k})\int\frac{p(\mathbf{x}_{k+1}|\mathbf{x}_{k})p(\mathbf{x}_{k+1}|\mathbf{y}_{1:K})}{p(\mathbf{x}_{k+1}|\mathbf{y}_{1:k})}d\mathbf{x}_{k+1}$
(13)
which evolves backwards in time and makes use of the pre-calculated filter,
$p(\mathbf{x}_{k}|\mathbf{y}_{1:k})$. Given either of the above posterior
densities, we can compute the expectation of any useful function of the hidden
model state as:
$\bar{h}_{k}=\int h(\mathbf{x}_{k})p(\mathbf{x}_{k}|\cdot)d\mathbf{x}_{k}$
(14)
where $p(\mathbf{x}_{k}|\cdot)$ is either the filter or the smoother density.
Common examples of $h(\mathbf{x}_{k})$ are $\mathbf{x}_{k}$ itself (giving the
mean $\mathbf{\bar{x}}_{k}$) and the squared difference from the mean (giving
the covariance of $\mathbf{x}_{k}$).
In practice, the computations defined by the above formulas can be performed
analytically only for linear Gaussian models using the Kalman smoother/filter
and for finite state-space hidden Markov models. For non-linear models, the
extended Kalman filter is a popular approach, which however can fail when non-
Gaussian or multimodal density functions are involved[34]. A more generally
applicable, albeit computationally more intensive approach, approximates the
filter and smoother densities using Sequential Monte Carlo (SMC) methods, also
known as particle filters[34, 35]. Within the SMC framework, the filter
density at each time point is approximated by a large number $N$ of discrete
samples or particles,
$\\{\mathbf{x}_{k}^{(1)},\ldots,\mathbf{x}_{k}^{(N)}\\}$, and associated non-
negative importance weights, $\\{w_{k}^{(1)},\ldots,w_{k}^{(N)}\\}$:
$p(\mathbf{x}_{k}|\mathbf{y}_{1:k})\approx\sum_{j=1}^{N}w_{k}^{(j)}\delta(\mathbf{x}_{k},\mathbf{x}_{k}^{(j)})\qquad,\qquad\sum_{j=1}^{N}w_{k}^{(j)}=1$
(15)
where $\delta(\mathbf{x}_{k},\mathbf{x}_{k}^{(j)})$ is the Dirac delta
function centered at the $j^{th}$ particle, $\mathbf{x}_{k}^{(j)}$.
Given an initial set of particles sampled from a prior distribution and their
associated weights, a simple update rule involves the following steps[29]:
Step 1:
For $j=1,\ldots,N$, sample a new set of particles from the proposal transition
density function,
$q(\mathbf{x}_{k+1}^{(j)}|\mathbf{x}_{k}^{(j)},\mathbf{y}_{k+1})$. In general,
one has enormous freedom in choosing the form of this density and even
condition it on future observations, if these are available (see, for example,
[36]). However, the simplest (and a quite common) choice is to use the
transition density as the proposal, i.e.
$q(\mathbf{x}_{k}|\mathbf{x}_{k-1},\mathbf{y}_{k})=p(\mathbf{x}_{k}|\mathbf{x}_{k-1})$.
This is the approach we follow in this paper.
Step 2:
For each new particle $\mathbf{x}_{k+1}^{(j)}$, evaluate the importance
weight:
$W_{k+1}^{(j)}=w_{k}^{(j)}p(\mathbf{y}_{k+1}|\mathbf{x}_{k+1}^{(j)})\frac{p(\mathbf{x}_{k+1}^{(j)}|\mathbf{x}_{k}^{(j)})}{q(\mathbf{x}_{k+1}^{(j)}|\mathbf{x}_{k}^{(j)},\mathbf{y}_{k+1})}$
(16)
Notice that when
$q(\mathbf{x}_{k}^{(j)}|\mathbf{x}_{k-1}^{(j)},\mathbf{y}_{k})=p(\mathbf{x}_{k}^{(j)}|\mathbf{x}_{k-1}^{(j)})$,
then the computation of the importance weights is significantly simplified,
i.e. $W_{k+1}^{(j)}=w_{k}^{(j)}p(\mathbf{y}_{k+1}|\mathbf{x}_{k+1}^{(j)})$.
Step 3:
Normalize the computed importance weights, by dividing each of them with their
sum, i.e.
$w_{k+1}^{(j)}=\frac{W_{k+1}^{(j)}}{\sum_{j=1}^{N}W_{k+1}^{(j)}}$ (17)
The derived set of weighted samples
$\\{\mathbf{x}_{k+1}^{(j)},w_{k+1}^{(j)}\\}$ is considered an approximation of
the filter density $p(\mathbf{x}_{k+1}|\mathbf{y}_{k+1})$.
In practice, the above algorithm is augmented with a re-sampling step
(preceding Step 1), during which $N$ particles are sampled from the set of
weighted particles computed at the previous iteration with probabilities
proportional to their weights[34, 35]. All re-sampled particles are given
weights equal to $1/N$. This step results in discarding particles with small
weights and multiplying particles with large weights, thus compensating for
the gradual degeneration of the particle filter i.e. the situation where all
particles but one have weights equal to zero. For performance reasons, the
resampling step may be applied only when the effective number of particles
drops below a threshold value, e.g. $N_{thr}=N/2$. An estimation of the
effective number of particles is given by
$\hat{N}_{eff}=\frac{1}{\sum_{j=1}^{N}w_{k+1}^{(j)}{}^{2}}$ (18)
The above filter can be extended to a fixed-lag smoother, if instead of
resampling just the particles at the current time step, we store and resample
all particles up to $L$ time steps before the current time step, i.e.
$\\{\mathbf{x}_{k-L}^{(j)},\ldots,\mathbf{x}_{k-1}^{(j)},\mathbf{x}_{k}^{(j)}\\}$[29].
The resampled particles can be considered a realization from a posterior
density $p(\mathbf{x}_{k}|\mathbf{y}_{1:k+L})$, which is an approximation of
the smoother density $p(\mathbf{x}_{k}|\mathbf{y}_{1:K})$, for sufficiently
large values of $L$.
Within this Monte Carlo framework, the expectation in Eq. 14 can be
approximated as:
$\bar{h}_{k}\approx\sum_{j=1}^{N}w_{k}^{(j)}h(\mathbf{x}_{k}^{(j)})$ (19)
for a large number $N$ of weighted samples.
### Simultaneous Estimation of Hidden States and Parameters
It is possible to apply the above standard filtering and smoothing techniques
to parameter estimation problems involving state-space models. The key
idea[29] is to define an extended state vector $\mathbf{z}_{k}$ by augmenting
the state vector $\mathbf{x}_{k}$ with the model parameters, i.e.
$\mathbf{z}_{k}=(\mathbf{\theta}_{k},\mathbf{x}_{k})^{T}$. Then, the time
evolution of the extended state-space model becomes:
$\mathbf{z}_{k+1}=\left(\begin{array}[]{c}\theta_{k+1}\\\
\mathbf{x}_{k+1}\end{array}\right)=\left(\begin{array}[]{c}\theta_{k}\\\
\mathbf{x}_{k}+\mathbf{f}(\mathbf{x}_{k},t_{k})\Delta
t+\sqrt{\Sigma_{\mathbf{x}}\Delta t}\xi_{k}\end{array}\right)$ (20)
while the observational model remains unaltered:
$\mathbf{y}_{k+1}=\mathbf{G}(\mathbf{z}_{k+1},\zeta_{k+1})=\mathbf{g}(\mathbf{x}_{k+1},\zeta_{k+1})$
(21)
The marginal posterior density of the parameter vector $\theta_{k}$ is given
by:
$p(\theta_{k}|\mathbf{y}_{1:K})=\int
p(\mathbf{z}_{k}|\mathbf{y}_{1:K})d\mathbf{x}_{k}=\int
p(\theta_{k},\mathbf{x}_{k}|\mathbf{y}_{1:K})d\mathbf{x}_{k}$ (22)
and, subsequently, the expectation of any function of $\theta_{k}$ can be
computed as in Eq. 14:
$\bar{h}_{k}=\int h(\theta_{k})p(\theta_{k}|\mathbf{y}_{1:K})d\theta_{k}$ (23)
Furthermore, given a set of particles and associated weights, which
approximate the smoother density $p(\mathbf{z}_{k}|\mathbf{y}_{1:K})$ as
outlined in the previous section, i.e.
$\\{\mathbf{z}_{k}^{(j)},w_{k}^{(j)}\\}=\\{\mathbf{x}_{k}^{(j)},\theta_{k}^{(j)},w_{k}^{(j)}\\}$
for $j=1,\ldots,N$, the above expectation can be approximated as:
$\bar{h}_{k}\approx\sum_{j=1}^{N}w_{k}^{(j)}h(\theta_{k}^{(j)})$ (24)
for large $N$.
Under this formulation, parameter estimation, which is traditionally treated
as an optimization problem, is reduced to an integration problem, which can be
tackled using filtering and smoothing methodologies for state-space models, a
well-studied subject in the field of Computational Statistics.
### Connection to Evolutionary Algorithms
It should be emphasized that although in Eq. 20 the parameter vector
$\mathbf{\theta}_{k}$ was assumed constant, i.e.
$\mathbf{\theta}_{k+1}=\mathbf{\theta}_{k}$, the same methodology applies in
the case of parameters that are naturally evolving in time, such as a time-
varying externally injected current $I_{ext}(t)$. A particularly interesting
case arises when an artificial evolution rule is imposed on a parameter
vector, which is otherwise constant by definition. Such a rule allows sampling
new parameter vectors based on samples at the previous time step, i.e.
$\theta_{k+1}\sim\mathrm{p}(\cdot|\theta_{k})$, and generating a sequence
$\\{\theta_{0},\theta_{1},\ldots\\}$, which explores the parameter space and,
ideally converges in a small optimal subset of it, after a sufficiently large
number of iterations. It is at this point that the opportunity to use
techniques borrowed from the domain of Evolutionary Algorithms arises. Here,
we assume that the artificial evolution of the parameter vector
$\mathbf{\theta}_{k}$ is governed by a version of the Covariance Matrix
Adaptation algorithm[30], a well-known Evolution Strategy, although the
modeler is free to make other choices (e.g. Differential Evolution[31]). For
the $j^{th}$ particle, we write:
$\theta_{k+1}^{(j)}=\eta_{k+1}^{(j)}+s_{k+1}^{(j)}\sqrt{Q_{k+1}}\lambda_{k+1}^{(j)}$
(25)
where $\lambda_{k+1}^{(j)}$ is a random vector with elements sampled from a
normal distribution with zero mean and unit variance. $\eta_{k+1}^{(j)}$ and
$Q_{k+1}$ are a mean vector and covariance matrix respectively, which are
computed as follows:
$\displaystyle\eta_{k+1}^{(j)}$ $\displaystyle=$
$\displaystyle(1-a)\theta_{k}^{(j)}+a\hat{E}[\theta_{k}]$ (26) $\displaystyle
Q_{k+1}$ $\displaystyle=$ $\displaystyle(1-b)Q_{k}+b\hat{C}ov[\theta_{k}]$
(27)
In the above expressions, a and $b$ are small adaptation constants and
$\hat{E}[\cdot]$ and $\hat{C}ov[\cdot]$ are the expectation and covariance of
the weighted sample of $\theta_{k}$, respectively. $s_{k+1}^{(j)}$ is a scale
parameter that evolves according to a log-normal update rule:
$s_{k+1}^{(j)}=s_{k}^{(j)}\exp(c\phi_{k+1}^{(j)})$ (28)
where $c$ is a small adaptation constant and
$\phi_{k+1}^{(j)}\sim\mathcal{N}(0,1)$ is a normally distributed random number
with zero mean and unit variance.
According to Eq. 25, the parameter vector $\theta_{k+1}^{(j)}$ is sampled at
each iteration of the algorithm from a multivariate normal distribution, which
is centered at $\eta_{k+1}^{(j)}$ and has a covariance matrix equal to
$s_{k+1}^{(j)}{}^{2}Q_{k+1}$:
$\theta_{k+1}^{(j)}\sim\mathcal{N}(\eta_{k+1}^{(j)},s_{k+1}^{(j)}{}^{2}Q_{k+1})$
(29)
Both $\eta_{k+1}^{(j)}$ and $Q_{k+1}$ are slowly adapting to the sample mean
$\hat{E}[\theta_{k}]$ and covariance $\hat{C}ov[\theta_{k}]$, with an
adaptation rate determined by the constants $a$ and $b$. Notice that by
switching off the adaptation process (i.e. by setting $a=b=c=0$),
$\theta_{k+1}^{(j)}$ evolves according to a multivariate Gaussian
distribution, which is centered at the previous parameter vector and has a
covariance matrix equal to $s_{0}^{(j)}{}^{2}Q_{0}$:
$\theta_{k+1}^{(j)}\sim\mathcal{N}(\theta_{k}^{(j)},s_{0}^{(j)}{}^{2}Q_{0})$
(30)
Therefore, given an initial set of weighted particles
$\\{\mathbf{z}_{0}^{(j)},w_{0}^{(j)}\\}=\\{s_{0}^{(j)},\theta_{0}^{(j)},\mathbf{x}_{0}^{(j)},w_{0}^{(j)}\\}$
sampled from some prior density function and an initial covariance matrix
$Q_{0}$, which may be set equal to the identity matrix, the smoothing
algorithm presented earlier becomes:
Step 1a:
Compute the expectation $\hat{E}[\theta_{k}]$ and covariance
$\hat{C}ov[\theta_{k}]$ of the weighted sample of $\theta_{k}$
Step 1b:
For $j=1,\ldots,N$, compute the scale factor $s_{k+1}^{(j)}$ according to Eq.
28. Notice that this scale factor is now part of the extended state
$\mathbf{z}_{k+1}^{(j)}$ for each particle
Step 1c:
For $j=1,\ldots,N$, compute the mean vector $\eta_{k+1}^{(j)}$, as shown in
Eq. 26
Step 1d:
Compute the covariance matrix $Q_{k+1}$, as shown in Eq. 27
Step 1e:
For $j=1,\ldots,N$, sample $\theta_{k+1}^{(j)}$, as shown in Eq. 25
Step 1f:
For $j=1,\ldots,N$, sample a new set of state vectors from the proposal
density
$q(\mathbf{x}_{k+1}^{(j)}|\mathbf{x}_{k}^{(j)},\theta_{k+1}^{(j)},\mathbf{y}_{k+1})$,
thus completing sampling the extended vectors $\mathbf{z}_{k+1}^{(j)}$. Notice
that the proposal density $q(\cdot|\cdot)$ is conditioned on the updated
parameter vector $\theta_{k+1}^{(j)}$.
Steps 2-3:
Execute steps 2 and 3 as described previously
Notice that in the algorithm outlined above, the order in which the components
of $\mathbf{z}_{k+1}^{(j)}$ are sampled is important. First, we sample the
scaling factor $s_{k+1}^{(j)}$. Then, we sample the parameter vector
$\theta_{k+1}^{(j)}$ given the updated $s_{k+1}^{(j)}$. Finally, we sample the
state vector $\mathbf{x}_{k+1}^{(j)}$ from a proposal, which is conditioned on
the updated parameter vector $\theta_{k+1}^{(j)}$. When resampling occurs, the
state vectors $\mathbf{x}_{k+1}^{(j)}$ with large importance weights are
selected and multiplied with high probability along with their associated
parameter vectors and scaling factors, thus resulting in a gradual self-
adaptation process. This self-adaptation mechanism is very common in the
Evolution Strategies literature.
### Implementation
The algorithm described in the previous section was implemented in MATLAB and
C (source code available as Supplementary Material; unmaintained FORTRAN code
is also available upon request from the first author) and tested on parameter
inference problems using simulated or actual electrophysiological data and a
number of Hodgkin-Huxley-type models: (a) a single-compartment model (derived
from the classic Hodgkin-Huxley model of neural excitability) containing a
leakage, transient sodium and delayed rectifier potassium current, (b) a two-
compartment model of a cat spinal motoneuron[37] and (c) a model of a B4
motoneuron in the Central Nervous System of the pond snail Lymnaea
stagnalis[38], which was developed as part of this study. Each of these models
is described in detail in the Results section. Models (a) and (b) were used
for generating noisy voltage traces at a sampling rate of $10KHz$ (one sample
every $0.1ms$). The simulated data was subsequently used as input to the
algorithm in order to estimate a large number of parameters; typically,
maximal conductances of ionic currents, reversal potentials, the parameters
governing the activation and inactivation kinetics of ionic currents, as well
as the levels of intrinsic and observation noise. Estimated parameter values
were subsequently compared against the true parameter values in the model. The
MATLAB environment was used for visualization and analysis of simulation
results. For the estimation of the unknown parameters in model (c), actual
electrophysiological data were used, as described in the next section.
Prior information was incorporated in the smoother by assuming that parameter
values were not allowed to exceed well-defined upper or lower limits (see
Tables 1, 2 and 3). For example, maximal conductances never received negative
values, while time constants were always larger than zero. At the beginning of
each simulation, the initial population of particles was uniformly sampled
from within the acceptable range of parameter values and, during each
simulation, parameters were forced to remain within their pre-defined limits.
All simulations were performed on an Intel dual-core i5 processor with 4 GB of
memory running Ubuntu Linux. The number of particles used in each simulation
was typically $100\times D$, where $D$ was the dimensionality of the extended
state $\mathbf{z}$ (equal to the number of free parameters and dynamic states
in the model). The time step $\Delta t$ in the Euler-Maruyama method was set
equal to $0.01ms$. The parameter $L$ of the fixed-lag smoother was set equal
to $100$ (unless stated otherwise), which is equivalent to a time window
$10ms$ wide (since data were sampled every $0.1ms$). The adaptation constants
$a$, $b$ and $c$ in Eqs. 26, 27 and 28 were all set equal to $0.01$, unless
stated otherwise. Depending on the size of $D$, the complexity of the model
and the length of the (actual or simulated) electrophysiological recordings,
simulation times ranged from a few minutes up to more than $12$ hours.
### Electrophysiology
As part of this study, we developed a single-compartment Hodgkin-Huxley-type
model of a B4 neuron in the pond-snail Lymnaea stagnalis[38]. B4 neurons are
part of the neural circuit that controls the rhythmic movements of the feeding
muscles via which the animal captures and ingests its food. The Lymnaea
central nervous system was dissected from adult animals (shell length
$20-30mm$) that were bred at the University of Leicester as described
previously[39]. All dissections were carried out in $HEPES$-buffered saline
containing (in $mM$) $50\;NaCl$, $1.6\;KCl$, $2\;MgCl_{2}$, $3.5\;CaCl_{2}$,
and $10\;HEPES$, $pH\;7.9$, in distilled water. All chemicals were purchased
from Sigma. The buccal ganglia containing the B4 neurons were separated from
the rest of the nervous system by cutting the cerebral buccal connectives and
the buccal-buccal connective was crushed to eliminate electrical coupling
between B4 neurons in the left and right buccal ganglion. Prior to recording,
excess saline was removed from the dish and small crystals of protease type
XIV were placed directly on top of the buccal ganglia to soften the connective
tissue and aid the impalement of individual neurons. The protease crystals
were washed of after about $30s$ with multiple changes of $HEPES$-buffered
saline. The B4 neuron was visually identified based on its size and position
and impaled with two sharp intracellular electrodes filled with a mixture of
$3M$ potassium acetate and $10mM$ potassium chloride (resistance $\sim
20M\Omega$). During the recording, the preparation was bathed in
$HEPES$-buffered saline plus $1mM$ hexamethonium chloride to block cholinergic
synaptic inputs and suppress spontaneous fictive feeding activity.
The signals from the two intracellular electrodes were amplified using a
Multiclamp 900A amplifier (Molecular Devices), digitized at a sampling
frequency of $10kHz$ using a CED1401plus A/D converter (Cambridge Electronic
Devices) and recorded on a PC using Spike2 version 6 software (Cambridge
Electronic Devices). A custom set of instructions using the Spike2 scripting
language was used to generate sequences of current pulses consisting of
individual random steps ranging in amplitude from $-4nA$ to $+4nA$ and a
duration from $1$ to $256ms$. The current signal was injected through one of
the recording electrodes whilst the second electrode was used to measure the
resulting changes in membrane potential.
## Results
### Hidden States, Intrinsic and Observational Noise are Simultaneously
Estimated Using the Fixed-Lag Smoother
The applicability of the fixed-lag smoother presented above was demonstrated
on a range of Hodgkin-Huxley-type models using simulated or actual
electrophysiological data. The first model we examined consisted of a single
compartment containing leakage, sodium and potassium currents, as shown below:
$dV=\frac{I_{ext}-G_{L}(V-E_{L})-G_{Na}m_{Na}^{3}h_{Na}(V-E_{Na})-G_{K}m_{K}^{4}(V-E_{K})}{C_{m}}dt+\sigma_{V}dW_{V}$
(31) $dm_{Na}=\frac{m_{\infty,Na}-m_{Na}}{\tau_{m_{Na}}}dt\qquad,\qquad
dh_{Na}=\frac{h_{\infty,Na}-h_{Na}}{\tau_{h_{Na}}}dt\qquad,\qquad
dm_{K}=\frac{m_{\infty,K}-m_{K}}{\tau_{m_{K}}}dt$ (32)
where $C_{m}=1\mu F/cm^{2}$. Notice the absence of noise in the dynamics of
$m_{Na}$, $h_{Na}$ and $m_{K}$, which is valid if we assume a very large
number of channels (see Supplementary Material for the case were noise is
present in the dynamics of these variables). The steady states and relaxation
times of the activation and inactivation gating variables were voltage-
dependent, as shown below (e.g. [5]):
$x_{\infty,i}=\left(1+\exp\frac{V_{H,x_{i}}-V}{V_{S,x_{i}}}\right)^{-1}$ (33)
and
$\tau_{x_{i}}=\tau_{min,x_{i}}+(\tau_{max,x_{i}}-\tau_{min,x_{i}})x_{\infty,i}\exp\left(\delta_{x_{i}}\frac{V_{H,x_{i}}-V}{V_{S,x_{i}}}\right)$
(34)
where $x\in\\{m,h\\}$ and $i\in\\{Na,K\\}$. The parameters $V_{H,x_{i}}$,
$V_{S,x_{i}}$, $\delta_{x_{i}}$, $\tau_{min,x_{i}}$and $\tau_{max,x_{i}}$ in
Eqs. 33 and 34 were chosen such that $x_{\infty,i}$ and $\tau_{x_{i}}$ fit
closely the corresponding steady-states and relaxation times of the classic
Hodgkin-Huxley model of neural excitability in the giant squid axon[40].
Observations consisted of noisy measurements of the membrane potential, as
shown in Eq. 7. The full set of parameter values in the above model is given
in Table 1.
First, we used the fixed-lag smoother to simultaneously infer the hidden
states ($V$, $m_{Na}$, $h_{Na}$, $m_{K}$) and standard deviations of the
intrinsic ($\sigma_{V}$) and observation ($\sigma_{y}$) noise based on
simulated recordings of the membrane potential $V$. These recordings were
generated by assuming a time-dependent $I_{ext}$ in Eq. 31, which consisted of
a sequence of current steps with amplitude randomly distributed between $-5\mu
A/cm^{2}$ and $20\mu A/cm^{2}$ and random duration up to a maximum of $20ms$.
Two simulated voltage recordings were generated corresponding to two different
levels of observation noise, $\sigma_{y}=0.5mV$ and $\sigma_{y}=50mV$,
respectively. The second value ($50mV$) was rather extreme and it was chosen
in order to illustrate the applicability of the method even at very high
levels of observation noise. Simulated data points were sampled every $0.1ms$
($10KHz$). The standard deviation of the intrinsic noise was set at
$\sigma_{V}=5mV$. The injected current $I_{ext}$ and the induced voltage trace
(for either value of $\sigma_{y}$) were then used as input to the smoother,
during the inference phase. At this stage, all other parameters in the model
(conductances, reversal potentials, and ionic current kinetics) were assumed
known, thus the extended state vector took the form
$\mathbf{z}=(s,\sigma_{V},\sigma_{y},V,m_{Na},h_{Na},m_{K})^{T}$, where $s$
was a scale factor as in Eq. 25. New samples for $s$ were taken from a log-
normal distribution (Eq. 28), while new samples for $\sigma_{V}$ and
$\sigma_{y}$ were drawn from an adaptive bivariate Gaussian distribution at
each iteration of the algorithm (Eq. 25). For each data set, smoothing was
repeated for two different values of the smoothing lag, i.e. $L=0$ and
$L=100$. $L=0$ corresponds to filtering, while $L=100$ corresponds to
smoothing with a fixed lag equal to $10ms$. Our results from this set of
simulations are summarized in Fig. 1.
We observed that at low levels of observation noise (Fig. 1A), the inferred
expectation of the voltage (solid blue and red lines) closely matched the
underlying (true) signal (solid black line). This was true for both values of
the fixed lag $L$ used for smoothing. However, at high levels of observation
noise (Fig. 1Bi), the true voltage was inferred with high fidelity when a
large value of the fixed lag ($L=100$) was used (solid red line), but not when
$L=0$ (solid blue line). Furthermore, the inferred expectations of the
unobserved dynamic variables $m_{Na}$, $h_{Na}$ and $m_{K}$ (solid red lines
in Fig. 1Bii) also matched the true hidden time series (solid black lines in
the same figure) remarkably well, when $L=100$.
We repeat that during these simulations an artificial update rule was imposed
on the two free standard deviations $\sigma_{V}$ and $\sigma_{y}$, as shown in
Eq. 25. The artificial evolution of these parameters is illustrated in Fig.
1Ci, where the inferred expectations of $s_{V}$ and $s_{y}$ are presented as
functions of time. These expectations converged immediately, fluctuating
around the true values of $s_{V}$ and $s_{y}$ (dashed lines in Fig. 1Ci). This
is also illustrated by the histograms in Fig. 1Cii, which were constructed
from the data points in Fig. 1Ci. We observed that the peaks of these
histograms were located quite closely to the true values of $\sigma_{V}$ and
$\sigma_{y}$ (dashed lines in Fig. 1Cii).
In summary, the fixed-lag smoother was able to recover the hidden states and
standard deviations of the intrinsic and observation noise in the model based
on noisy observations of the membrane potential. This was true even at high
levels of observation noise, subject to the condition that a sufficiently
large smoothing lag $L$ was adopted during the simulation.
### Adaptive Sampling Reduces the Variance of Inferred Parameter Distributions
and Accelerates Convergence of the Algorithm
Next, we treated two more parameters in the model as unknown, i.e. the maximal
conductances of the transient sodium ($G_{Na}$) and delayed rectifier
potassium ($G_{K}$) currents. The extended state vector, thus, took the form
$\mathbf{z}=(s,\sigma_{V},\sigma_{y},G_{Na},G_{K},V,m_{Na},h_{Na},m_{K})^{T}$.
As in the previous section, new samples for $s$ were drawn from a log-normal
distribution (Eq. 28), while $\sigma_{V}$, $\sigma_{y}$, $G_{Na}$ and $G_{K}$
were sampled by default from an adaptive multivariate Gaussian distribution at
each iteration of the algorithm (Eq. 25).
In order to examine the effect of this adaptive sampling approach on the
variance of the inferred parameter distributions, we repeated fixed-lag
smoothing assuming each time that different aspects of this adaptive sampling
process were switched off, as illustrated in Fig. 2\. First, we assumed that
no adaptation was imposed on $s$ or the “unknown” noise parameters and maximal
conductances, i.e. the constants $a$, $b$ and $c$ in Eqs. 26-28 were all set
equal to zero. In this case, the multivariate Gaussian distribution from which
new samples of $\sigma_{V}$, $\sigma_{y}$, $G_{Na}$ and $G_{K}$ were drawn
from reduced to Eq. 30. In addition, we assumed that $s_{0}^{(j)}$ in the same
equation was equal to $1$, for all samples $j$. Under these conditions, the
true values of the free parameters were correctly estimated through
application of the fixed-lag smoother, as illustrated for the case of $G_{Na}$
and $G_{K}$ in Figs. 2Ai and 2Aii.
Subsequently, we repeated smoothing assuming that the scale factor $s$ evolved
according to the log-normal update rule given by Eq. 28 with $c=0.01$, while
$a$ and $b$ were again set equal to $0$. As illustrated in Figs. 2Bi and 2Bii
for parameters $G_{Na}$ and $G_{K}$, by imposing this simple adaptation rule
on the multivariate Gaussian distribution from which the free parameters in
the model were sampled, we managed again to estimate correctly their values,
but this time the variance of the inferred parameter distributions (the width
of the histograms in Fig. 2Bii) was drastically reduced.
By further letting the mean and covariance of the proposal Gaussian
distribution in Eq. 25 adapt (by setting $a=b=0.01$ in Eqs. 26 and 27), we
achieved a further decrease in the spread of the inferred parameter
distributions (Figs. 2C and 2D). Parameters $\sigma_{y}$ and $\sigma_{V}$ and
the hidden states $V$, $m_{Na}$, $h_{Na}$ and $m_{K}$ were also inferred with
very high fidelity in all cases (as in Fig. 1), but the variance of the
estimated posteriors for $\sigma_{y}$ and $\sigma_{V}$ followed the same
pattern as the variance of $G_{Na}$ and $G_{K}$.
It is worth observing that when all three adaptation processes were switched
on (i.e. $a=b=c=0.01$), the algorithm converged to a single point in parameter
space within the first $1s$ of simulation, which coincided with the true
parameter values in the model (see Fig. 2D for the case of $G_{Na}$ and
$G_{K}$). At this point, the covariance matrix $\bar{s}_{k}^{2}Q_{k}$ became
very small (i.e. all its elements were less than $10^{-8}$, although the
matrix itself remained non-singular) and the mean $\bar{\eta}_{k}$ was very
close to the true parameter vector $\theta$. We note that
$\bar{s}_{k}=\hat{E}[s_{k}]$ and $\bar{\eta}_{k}=\hat{E}[\eta_{k}]$, where
$\hat{E}[\cdot]$ stands for the expectation computed over the population of
particles. In this case, it is not strictly correct to claim that the chains
in Fig. 2Di approximate the posteriors of the unknown parameters $G_{Na}$ and
$G_{K}$; since repeating the simulation many times would result in convergence
at slightly different points clustered tightly around the true parameter
values, it would be more reasonable to claim that these optimal points are
random samples from the posterior parameter distribution and they can be
treated as estimates of its mode.
Depending on the situation, one may wish to estimate the full posteriors of
the unknown parameters or just an optimal set of parameter values, which can
be used in a subsequent predictive simulation. In Fig 3A, we examined in more
detail how the scale factor $s_{k}$ affects the variance of the final
estimates, assuming that $a=b=c=0.01$. We repeat that each particle $j$
contains $s_{k}$ as a component of its extended state. Each scaling factor
$s_{k}^{(j)}$ is updated at each iteration of the algorithm following a
lognormal rule (Eq. 28, Step 1b of the algorithm in the Methods section).
Sampling new parameter vectors is conditioned on these updated scaling factors
(Eq. 25, Step 1e of the algorithm). When at a later stage weighting (and
resampling) of the particles occurs, the scaling factors that are associated
with high-weight parameters and hidden states are likely to survive into
subsequent iterations (or “generations”) of the algorithm. During the course
of this adaptive process, the scaling factors $s_{k}^{(j)}$ are allowed to
fluctuate only within predefines limits, similarly to the other components of
the extended state vector.
In Fig. 3Ai, we demonstrate the case where the scaling factors $s_{k}^{(j)}$
were allowed to take values from the prior interval $[0,2]$. We observed that
during the course of the simulation, the average value of the scaling factor,
$\bar{s}_{k}$, decreased gradually towards $0$ and this was accompanied by a
dramatic decrease in the variance of the inferred parameters $G_{Na}$ and
$G_{K}$, which eventually “collapsed” to a point in parameter space located
very close to their true values. This situation was the same as the one
illustrated in Fig. 2D. Notice that although $\bar{s}_{k}$ decreased towards
zero, it never actually took this value; it merely became very small ($\sim
0.01$). When we used a prior interval for $s_{k}^{(j)}$ with non-zero lower
bound (i.e. $[0.15,2$]; see Fig. 3Aii), the final estimates had a larger
variance, providing an approximation of the full posteriors of the “unknown”
parameters $G_{Na}$ and $G_{K}$. Thus, controlling the lower bound of the
prior interval for the scaling factors $s_{k}^{(j)}$ provides a simple method
for controlling the variance of the final estimates. Notice that the variance
of the final estimates also depends on the number of particles (Fig. 3B). A
smaller number of particles resulted in a larger variance of the estimates
(compare Fig. 3Bi to Fig. 3Bii). However, when a large number of particles was
already in use, further increasing their number did not significantly affect
the variance of the estimates or rte of convergence (compare Fig. 3Bii to Fig.
3Aii), indicating the presence of a ceiling effect.
The adaptive sampling of the scaling factors $s_{k}^{(j)}$ further depends on
parameter $c$ in Eq. 28, which determines the width of the lognormal
distribution from which new samples are drawn. The value of this parameter
provides a simple way to control the rate of convergence of the algorithm;
larger values of $c$ resulted in faster convergence (compare Fig. 4A to Fig
4B). The rate of convergence also depends on the number of particles in use
(compare Fig. 4A to Fig. 4C), although it is more sensitive to changes in
parameter $c$; dividing the value of $c$ by $2$ (Fig. 4B) had a larger effect
on the rate of convergence than dividing the number of particles by $10$ (Fig.
4C).
In summary, by assuming an adaptive sampling process for the unknown
parameters in the model, we managed to achieve a significant reduction in the
spread of the inferred posterior distributions of these parameters.
Furthermore, adjusting the prior interval and adaptation rate $c$ of the
scaling factors $s_{k}^{(j)}$ provides a straightforward way to control the
variance of the estimated posteriors and the rate of convergence of the
algorithm. Alternatively, we could have set $s_{k}^{(j)}=constant$, i.e. set
it to the same constant value for all particles $j$ and time steps $k$ (as in
Fig. 2A). However, by permitting $s_{k}^{(j)}$ to adapt within a predefined
interval, we potentially allow this parameter and, thus, the covariance
matrices $s_{k}^{(j)}{}^{2}Q_{k}$ take large values, which in turn would
permit the algorithm to escape local optima in the parameter space. For
example, the time profiles of $\bar{s}_{k}$ in Figs. 3 and 4 indicate that,
early during the simulations, this quantity had relatively large values, which
were associated with large variances of the posterior parameter estimates.
During this initial period, the algorithm has the potential to “jump” away
from local optima and towards more optimal regions of the parameter space. One
may see, here, a distant analogy to simulated annealing, where a fictitious
“temperature” control variable is gradually decreased, thus allowing the
system to escape local minima and gradually settle to more optimal regions of
the energy landscape.
### Increasing Observation Noise Reduces the Accuracy and Precision of the
Fixed-Lag Smoother
In a subsequent stage, we treated as unknown two more parameters in the model,
i.e. the reversal potentials for the sodium and potassium currents, $E_{Na}$
and $E_{K}$, respectively. Thus, the extended state vector became
$\mathbf{z}=(s,\sigma_{V},\sigma_{y},G_{Na},G_{K},E_{Na},E_{K},V,m_{Na},h_{Na},m_{K})^{T}$.
This time, we wanted to examine how increasing levels of observation noise
(i.e. the value of parameter $\sigma_{y}$) affect the inference of unknown
quantities in the model based on the fixed-lag smoother. For this reason, we
repeated smoothing on four simulated data sets (i.e. recordings of membrane
potential and the associated $I_{ext}$) corresponding to increasing values of
the standard deviation of the observation noise $\sigma_{y}$, i.e. $0.5mV$,
$5mV$, $25mV$ and $50mV$.
The results from this set of simulations are summarized in Fig. 5. For
$\sigma_{y}=0.5mV$, the expectations of the four parameters $G_{Na}$, $G_{K}$,
$E_{Na}$ and $E_{K}$ (red solid lines in Figs. 5Ai-iv) eventually converged to
their true values (dashed lines in the aforementioned figures). For
$\sigma_{y}=50mV$, the expectations of these parameters (light red solid lines
in Figs. 5Ai-iv) also converged, although the expectations for $G_{Na}$ (Fig.
5Ai) and, to a lesser degree, $G_{K}$ (Fig. 5Aii) deviated noticeably from
their true values. As expected, at higher levels of noise, the variance of the
final estimates was larger, although the rate of convergence did not seem to
be affected, due to the large number of particles we used ($N=1100$; see
ceiling effect in Fig. 3Bii). The inferred parameters $\sigma_{V}$ and
$\sigma_{y}$ (not illustrated for clarity) followed a similar convergence
pattern.
In Fig. 5B, we show, for each tested value of $\sigma_{y}$, the box plots of
the above four parameters, which were computed from the data points (as in
Fig. 5A) corresponding to time $t\geq 1s$. For each parameter and each value
of $\sigma_{y}$, the data were first normalized as follows:
$\tilde{x}_{k}=\frac{\bar{x}{}_{k}-x_{true}}{\sum_{k=1}^{K}\bar{x}_{k}}$ (35)
where $x\in\\{G_{Na},G_{K},E_{Na},E_{K}\\}$. The box plots in Fig. 5B were
constructed from the normalized data points $\tilde{x}_{k}$. The above
normalization was necessary since it made possible the comparison between
different data sets, each characterized by its own mean, variance and unit of
measurement. In the box plots in Fig. 5B, zero (i.e. the dashed lines)
corresponds to the true parameter values, while discrepancies from the true
parameter values along the y-axis are given in relation to the average
$\sum_{k=1}^{K}\bar{x}_{k}$. We may observe that for very low levels of
observation noise ($\sigma_{y}=0.5mV$), the posteriors of the four examined
parameters were clustered tightly around their true values, but for larger
levels of noise ($\sigma_{y}=5$, $25$ and $50mV$), we observed larger
discrepancies from the true parameter values and broader inferred posteriors.
The parameters following more noticeably this trend were the conductances
$G_{Na}$ and $G_{K}$, while $E_{Na}$ and, particularly, $E_{K}$ were less
affected. This indicates that smoothing is more sensitive to changes in some
model parameters than others and this is why these parameters were tightly
controlled. In summary, increasing the levels of measurement noise (i.e. the
value of parameter $\sigma_{y}$) decreased the accuracy and precision of the
algorithm, but it did not significantly affect the rate of convergence due to
the large number of particles used during the simulations.
### High-Dimensional Inference Problems are Resolved Given Sufficiently
Informative Priors
At the next stage, we treated all parameters in the model (a total of 23
parameters; see Table 1) as unknown. Therefore, the extended state vector took
the following ($28$-dimensional) form:
$\mathbf{z}=(s,\sigma_{V},\sigma_{y},G_{L},G_{i},E_{L},E_{i},V_{H,x_{i}},V_{S,x_{i}},\tau_{min,x_{i}},\tau_{max,x_{i}},\delta_{x_{i}},V,m_{Na},h_{Na},m_{K})^{T}$
where $i\in\\{Na,K\\}$ and $x\in\\{m,h\\}$. These parameters included the
standard deviations of intrinsic and observation noise ($\sigma_{V}$ and
$\sigma_{y}$, respectively), the maximal conductances $G_{i}$ and reversal
potentials $E_{i}$ of all currents in the model and the parameters controlling
the steady-states and relaxation times of activation and inactivation for the
sodium and potassium currents ($V_{H,x_{i}}$, $V_{S,x_{i}}$,
$\tau_{min,x_{i}}$, $\tau_{max,x_{i}}$ and $\delta_{x_{i}}$). The results from
this simulation are illustrated in Fig. 6.
We observed that the true signal (membrane potential) was inferred with very
high fidelity (Fig. 6Ai). The sodium activation $m_{Na}$ was also recovered
with very high accuracy, while estimation of the hidden states $h_{Na}$ and
$m_{K}$ (sodium inactivation and potassium activation, respectively) was also
satisfactory (despite significant deviations, the general form of the true
hidden states was recovered without any observable impact on the dynamics of
the membrane potential), as shown in Fig. 6Aii. Among the $23$ estimated
parameters, we illustrate (in Figs. 6B and 6C) the estimated posteriors for
the reversal potential of sodium $E_{Na}$ (Fig. 6B) and for parameters
$\tau_{max,m_{Na}}$ (Figs. 6Ci,ii) and $\tau_{max,m_{K}}$ (Figs. 6Ciii,iv),
which control the activation of sodium and potassium currents, respectively.
We focus on these parameters, because they represent three different
characteristic cases. The posteriors of parameters $E_{Na}$ and
$\tau_{max,m_{Na}}$ are unimodal (see Figs. 6Bii and 6Cii) and they were
estimated with relatively high accuracy. Particularly, the posterior for
$\tau_{max,m_{Na}}$ was estimated with very high precision and accuracy,
despite its broad prior interval (the y-axis in Fig. 6Ci and the x-axis in
Fig. 6Cii). On the other hand, the estimated posterior of $\tau_{max,m_{K}}$
covered a large part of its prior interval (the y-axis in Fig. 6Ciii and the
x-axis in Fig. 6Civ), its main mode was located at a slightly larger value
than the true parameter value, while at least two minor modes seem to be
present near the upper bound of the prior interval (the arrow in Fig. 6Civ).
These results reiterate our previous conclusion that smoothing may be
particularly sensitive to some parameters, but not to others. The posteriors
of parameters in the former category are very precise and narrow (as in the
case of $E_{Na}$ and, especially, $\tau_{max,m_{Na}}$), while the parameters
in the latter category are characterized by broader posteriors. Also, we can
observe that the fixed-lag smoother has the capability to provide a global
approximation of the unknown posteriors, including their variance and the
location of major and minor modes (i.e. global and local optima). An overview
of all inferred posteriors is given by the box plot in Fig. 6D, which was
constructed after all data (as in Figs. 6Bi, 6Ci and 6Ciii) were normalized
according to Eq. 35. Again, it may be observed that while some of the
estimated parameter posteriors are quite precise and accurate, such as
$\sigma_{y}$ (parameter $\\#2$), $E_{K}$ (parameter $\\#8$) and $V_{H,m_{Na}}$
(parameter $\\#9$), others are less precise and accurate, such as the maximal
conductances (parameters $\\#3$ to $\\#5$), $\tau_{max,h_{Na}}$ (parameter
$\\#19$) and $\delta_{h_{Na}}$ (parameter $\\#22$).
The simulation results presented above were obtained by assuming a prior
interval for the scaling factors $s_{k}^{(j)}$ equal to $[0.15,10]$. When we
repeated the simulation using the prior interval $[0,10]$, the true underlying
membrane potential was again inferred with very high fidelity (Fig. 7Ai),
while the hidden states $m_{Na}$, $h_{Na}$ and $m_{K}$ were also estimated
with sufficient accuracy (Fig. 7Aii). In this case, however, the estimates of
the “unknown” parameters converged to single points in parameter space (as
illustrated, for example, for parameters $E_{Na}$, $\tau_{max,m_{Na}}$ and
$\tau_{max,m_{K}}$ in Figs. 7Bi-ii), which fall within the support of the
posteriors illustrated in Figs. 6B and 6C. The activation and inactivation
steady states (Fig. 7Ci, red solid lines) and relaxation times (Fig. 7Cii, red
solid lines) as functions of voltage, which were computed from these
estimates, were also similar to their corresponding true functions, with the
curves for $\bar{\tau}_{h_{Na}}$ and $\bar{\tau}_{m_{K}}$ manifesting the
largest deviation from truth (black solid lines in Figs. 7Ci,ii). An overview
of the estimated parameter values (after normalizing using Eq. 35) is given in
Fig. 7Di. As stated previously, some estimates were close to their true
counterparts, while others were not. For example, the activation of the sodium
current $m_{Na}$ (Fig. 7Aii) and its steady state $m_{\infty,Na}$ (Fig. 7Ci),
which are important for the correct onset of the action potentials, were
inferred with relatively high accuracy. On the other hand, larger errors were
observed, for example, in the inference of sodium inactivation ($h_{Na}$; Fig.
7Aii) or in the estimation of $G_{Na}$ (parameter $\\#4$; Fig. 7Di), the
maximal conductance for the sodium current.
Given the fact that the data on which inference was based (a single noisy
recording of the membrane potential) was of much lower dimensionality than the
extended state we aimed to infer, the observed discrepancies between inferred
and true model quantities were unlikely to vanish unless we imposed more
strict constraints on the model. When we repeated the previous simulation
using more narrow prior intervals for some of the parameters controlling the
kinetics of the sodium and potassium currents in the model (see red dashed
boxes in Fig. 7Dii and bold intervals in Table 1), the estimated parameters
settled closer to their true values (Fig. 7Dii). This was true even for
parameters on which more narrow intervals were not directly applied, such as
the maximal conductances (i.e. parameters $\\#3$ to $\\#5$ in Fig. 7Dii), and
even when data with higher levels of observation noise wee used (Fig. 7Dii,
data points indicated with crosses; see also Fig. S3). It is important to
mention that using more narrow prior constraints only affected the accuracy of
the final estimates, not the quality of fitting the experimental data, which
in all cases was of very high fidelity. Alternatively, we could have
constrained the model by increasing the dimensionality of the observed signal,
e.g. by using simultaneously more that one unique voltage traces (each
generated under different conditions of injected current) during smoothing. We
examine the use of multiple data sets simultaneously as input to the fixed-lag
smoother later in the Results section.
In summary, the smoothing algorithm can be used to resolve high-dimensional
inference problems. In combination with sufficient prior information (in the
form of bounded regions within which parameters are allowed to fluctuate; see
Table 1), the fixed-lag smoother can provide estimates of the intrinsic and
observation noise, maximal conductances, reversal potentials and kinetics of
ionic currents in a single-compartment Hodgkin-Huxley-type neuron model, based
on low-dimensional noisy experimental data.
### Parameter Estimation in Compartmental Models is Straightforward Using the
Fixed-Lag Smoother
Next, we tested whether the fixed-lag smoother could be successfully applied
on inference problems involving more complex models than the one we used in
the previous sections. For this reason, we focused on a two-compartment model
of a vertebrate motoneuron containing sodium, potassium and calcium currents
and intracellular calcium dynamics, which were differentially distributed
among a soma and a dendritic compartment[37]. The model (modified
appropriately to include intrinsic noise terms) is summarized below:
$\displaystyle dV_{S}$ $\displaystyle=$
$\displaystyle\frac{I_{ext,S}-G_{L}(V_{S}-E_{L})-\frac{G_{C}}{p}(V_{S}-V_{D})-I_{Na}-I_{K}-I_{K(Ca),S}-I_{CaN,S}}{C_{m}}dt+\sigma_{V_{S}}dW_{V_{S}}$
(36) $\displaystyle dV_{D}$ $\displaystyle=$
$\displaystyle\frac{I_{ext,D}-G_{L}(V_{D}-E_{L})-\frac{G_{C}}{1-p}(V_{D}-V_{S})-I_{K(Ca),D}-I_{CaN,D}-I_{CaL}}{C_{m}}dt+\sigma_{V_{D}}dW_{V_{D}}$
(37)
where $V_{S}$ and $V_{D}$ is the membrane potential at the soma and dendritic
compartments, respectively, and $C_{m}=1\mu F/cm^{2}$. The leakage conductance
and reversal potential were $G_{L}=0.51mS/cm^{2}$ and $E_{L}=-60mV$,
respectively. The coupling conductance was $G_{C}=0.1mS/cm^{2}$ and the ratio
of the soma area to the total surface area of the cell was $p=0.1$. The
various ionic currents in the above model were as follows: (a) a transient
sodium current, $I_{Na}=G_{Na}m_{\infty,Na}^{3}h_{Na}(V_{S}-E_{Na})$, (b) a
delayed rectifier potassium current, $I_{K}=G_{K}m_{K}^{4}(V_{S}-E_{K})$, (c)
a calcium-activated potassium current,
$I_{K(Ca),X}=G_{K(Ca),X}\frac{[Ca^{2+}]_{X}}{[Ca^{2+}]_{X}+K_{d}}(V_{X}-E_{K})$,
where $X\in\\{S,D\\}$ and $K_{d}=0.2\mu M$ (the half-saturation constant), (d)
an N-type calcium current,
$I_{CaN,X}=G_{CaN,X}m_{CaN,X}^{2}h_{CaN,X}(V_{X}-E_{Ca})$, where
$X\in\\{S,D\\}$ and (e) an L-type calcium current,
$I_{CaL}=G_{CaL}m_{CaL}(V_{D}-E_{Ca})$. The various activation and
inactivation dynamic variables in the above model were modeled using first-
order relaxation kinetics (as in Eq. 32), where the various steady states were
assumed to be sigmoid functions of voltage (Eq. 33). Notice, that the
activation of $I_{Na}$ was assumed instantaneous and therefore, it was given
at any time by the voltage-dependent steady state $m_{\infty,Na}$. The
relaxation times for sodium inactivation and potassium activation were also
functions of voltage as in Eq. 34:
$\tau_{h_{Na}}=\tau_{max,h_{Na}}h_{\infty,Na}\exp\left(\delta_{h_{Na}}\frac{V_{H,h_{Na}}-V}{V_{S,h_{Na}}}\right)$
(38)
$\tau_{m_{K}}=\tau_{min,m_{K}}+(\tau_{max,m_{K}}-\tau_{min,m_{K}})m_{\infty,K}\exp\left(\delta_{m_{K}}\frac{V_{H,m_{K}}-V}{V_{S,m_{K}}}\right)$
(39)
where the parameters $\tau_{min,x_{i}}$, $\tau_{max,x_{i}}$ and
$\delta_{x_{i}}$ (with $x\in\\{m,h\\}$ and $i\in\\{Na,K\\}$) were chosen by
fitting the above expressions to the original model in [37]. The relaxation
times for the remaining activation and inactivation variables were constant.
All parameters values in the model are given in Table 2.
The intracellular calcium concentration at either the soma or the dendritic
compartment was also modeled by a first-order differential equation, as
follows:
$\frac{d[Ca^{2+}]_{X}}{dt}=f(aI_{Ca,X}-k[Ca^{2+}]_{X})\qquad,\qquad
X\in\\{S,D\\}$ (40)
where $f=0.01$, $a=0.009mol(C\mu m)^{-1}$ and $k=2ms^{-1}$. The total calcium
current is $I_{Ca,S}=I_{CaN}$ at the soma ($X=S$) and
$I_{Ca,D}=I_{CaN}+I_{CaL}$ at the dendritic compartment ($X=D$).
The observation model assumed simultaneous noisy recordings of the membrane
potential from both the soma and dendritic compartments, as follows:
$\left(\begin{array}[]{c}y_{S}\\\
y_{D}\end{array}\right)=\left(\begin{array}[]{c}V_{S}\\\
V_{D}\end{array}\right)+\left(\begin{array}[]{cc}\sigma_{y}&0\\\
0&\sigma_{y}\end{array}\right)\left(\begin{array}[]{c}\zeta_{S}\\\
\zeta_{D}\end{array}\right)$ (41)
where $\zeta_{X}\sim\mathcal{N}(0,1)$ with $X\in\\{S,D\\}$. Notice that
$\sigma_{y}$ is the same for both compartments.
In the above model, the externally injected currents $I_{ext,S}$ and
$I_{ext,D}$ were sequences of random current steps with duration up to $50ms$
(instead of $20ms$ as in the single-compartment model, due to the presence of
slower currents in the two-compartment model) and magnitude between $-5\mu
A/cm^{2}$ and $20\mu A/cm^{2}$. Current was injected in both the dendritic
compartment and the soma (instead of just in the soma), because preliminary
simulations indicated that this experimental setting facilitated parameter
estimation, presumably due to the generation of a more variable (and, thus,
information-rich) data set111It should be mentioned that the two-compartment
model allows for the physical separation of currents and as such it is a
slightly better approximation of a real neuron with differential expression of
individual currents in different cellular compartments. However, in no way
does it capture the full morphological complexity of a real neuron. As such,
current injection into the dendritic compartment can not be replicated
accurately in a real neuron as current injection in the model will have a
uniform effect on all currents in that compartment, whilst current injection
into the dendrite of a neuron would have far more complex effects on dendritic
currents, which potentially would be dependent on the distance from the
injection site. Thus, whilst it would be possible, albeit challenging, to
carry out dual recordings from the soma and dendrites in a real neuron this
would not be the same as the dual current injection in the model. In this
case, application of the fixed-lag smoother on a more spatially detailed model
would be necessary (and feasible). In principle, the method can also
assimilate other types of spatial data, such as calcium imaging data, in case
recordings from multiple neuron locations are not available (although we do
not examine this case in detail in this paper).. The injected currents and the
induced noisy voltage traces $y_{S}$ and $y_{D}$ comprised the simulated data
on which parameter estimation was based.
First, we aimed to infer the noise parameters and maximal conductances of all
voltage- and calcium-gated currents in the model, assuming that the kinetics
of these currents were known. This implied an extended-state vector with $22$
components as shown below
$\mathbf{z}=(s,\sigma_{X},\sigma_{y},G_{Na},G_{K},G_{K(Ca),X},G_{CaN,X},G_{CaL},V_{X},[Ca^{2+}]_{X},h_{Na},m_{K},m_{CaN,X},h_{CaN,X},m_{CaL})^{T}$
where $X\in\\{S,D\\}$. The results from this simulation are illustrated in
Figs. 8 and 9. The fixed-lag smoother managed to recover the hidden dynamic
states (including the time-evolution of the intracellular calcium; Fig. 8),
the standard deviations of the intrinsic and observation noise (Figs. 9Ai,ii)
and the true values of all the gated maximal conductances (Figs. 9Bi-iv) in
the model using approximately $2s$ of simulated data and $2200$ particles.
Notice that, in Figs. 8Ci-iv, the inferred hidden gating states (dashed red
lines) coincide extremely well with the true ones (solid black lines), which
is not surprising, since the voltage-dependent kinetics of these states were
assumed known and the true membrane potential at the soma and dendritic
compartment was recovered with very high fidelity (Figs. 8Ai,ii). Also, notice
that, in Figs. 9Aii, 9Biii and 9Biv, the estimation of the standard deviation
of the intrinsic noise, $\sigma_{V_{D}}$, and the maximal conductances of
calcium and calcium-dependent currents in the dendritic compartment
($G_{K(Ca),D}$, $G_{CaN,D}$ and $G_{CaL}$) was improved after injecting
current in both the soma and the dendritic compartment (compare the grey solid
lines, which correspond to injection in the soma only, to the color ones in
the aforementioned figures).
In a second stage, we assumed that the kinetics of all voltage-gated ionic
currents were also unknown, implying an extended state vector with $41$
components, as follows:
$\mathbf{z}=(\ldots,G_{CaL},V_{H,x_{i}},V_{S,x_{i}},\tau_{min,x_{i}},\tau_{max,x_{i}},\delta_{x_{i}},\tau_{o,x_{i}},V_{X},\ldots)^{T}$
where $X\in\\{S,D\\}$, $x\in\\{m,h\\}$ and $i\in\\{Na,K,CaN,CaL\\}$. Our
results from this simulation are summarized in Figs. 10 and 11. Again, the
membrane potential at the soma and the dendrite were inferred with very high
fidelity (Fig. 10Ai,ii). However, the estimated hidden dynamics of most ionic
currents and intracellular calcium concentrations in the model deviated
significantly from their true counterparts (Fig. 10B,C). The expectations of
all estimated parameters are illustrated in Fig. 11Ai. As in the case of the
single-compartment model, by imposing tighter prior constraints on some of the
parameters controlling the kinetics of ionic currents in the model (see red
dashed box in Fig. 11Aii and Table 2), we managed to reduce the discrepancies
of the estimates from their true values (Fig. 11Aii and Supplementary Fig.
S4). This was true even for parameters on which stricter priors were not
directly applied. The inference was completed after processing almost $3s$ of
data, as shown in Fig. 11B for the maximal conductances of sodium and
potassium currents at the soma. Interestingly, the algorithm seems to
temporarily settle at local optima (see arrows in Fig. 11B) before “jumping”
away and, eventually, converge at the final estimates. The inferred voltage-
dependent steady-states of the sodium, potassium and calcium currents (Figs.
11Ci,ii) and the relaxation times for the sodium inactivation and potassium
activation (Fig. 11Ciii) were also very similar to their true corresponding
functions. The algorithm remained operational when more noisy data were used,
as illustrated in Fig. 11Aii and in Supplementary Fig. S5.
An interesting fact regarding the simulation results presented in Figs. 10 and
11Ai was that, in order to obtain high-fidelity estimates of the true membrane
potential at the soma and dendritic compartment (as shown in Figs. 10Ai,ii) we
had to use more than $4100$ particles, the number calculated by the
$N=100\times\text{size of the extended state}$ rule (see Methods). In
particular, we used $8200$ particles, although we cannot exclude that a
smaller number may have sufficed. After applying more narrow prior constraints
(Figs. 11Aii, B, C, S4 and S5), using the number of particles calculated by
the above simple heuristic ($4100$ in this case) was again sufficient for
accurately inferring the true membrane potential (see Fig. S4Ai,ii and
S5Ai,ii). This implies that as the complexity (and dimensionality) of the
estimation problem increases, a non-linearly growing number of particles may
be required in order to obtain acceptable results, but this situation may be
compensated for by providing highly informative priors.
Given the large number of unknown parameters and hidden states in combination
with the low dimensionality of the data (notice that the intracellular calcium
concentration was assumed unobserved), it was truly remarkable that the
algorithm managed to recover much of the extended state vector with relatively
satisfactory accuracy. However, it should be noted that in our simulations we
assumed knowledge of important information, such as the passive conductances
$G_{L}$ and $G_{C}$ and the reversal potentials of sodium, potassium and
calcium currents. This and the fact that the availability of prior information
in the form of more narrow parameter boundaries improved significantly the
accuracy of the final estimates emphasizes our previous conclusion that prior
information is important for the successful inference of unknown model
parameters and hidden model states using the fixed-lag smoother. Given such
information, inference in complex compartmental models based on simultaneous
recordings from several neuron locations and, possibly, measurements of
intracellular calcium, can be naturally achieved via appropriate formulation
of the extended state vector and application of the fixed-lag smoother.
### Parameters in a Model of an Invertebrate Motoneuron were Inferred from
Actual Electrophysiological Data Using the Fixed-Lag Smoother
In a final set of simulations, we applied the smoother on actual
electrophysiological data in order to estimate the unknown parameters in a
single-compartment model of the B4 motoneuron from the nervous system of the
pond snail, Lymnaea stagnalis[38]. This neuron is part of a population of
motoneurons, which receive rhythmic electrical input from upstream Central
Pattern Generator interneurons and in turn innervate and control the movements
of the feeding muscles via which the animal captures and ingests its food.
Previous studies in these neurons have demonstrated the presence of a
transient inward sodium current $I_{Na}$, a delayed outward potassium current
$I_{K}$ and a transient outward potassium current $I_{A}$[41]. A
hyperpolarization-activated current $I_{h}$ was conditional on the presence of
serotonin in the solution [38] and, therefore, this current was not included
in this instance of the B4 model. Thus, the current conservation equation for
a single-compartment model of the B4 motoneuron (appropriately modified to
include an intrinsic noise term) took the following form:
$dV=\frac{I_{ext}-G_{L}(V-E_{L})-I_{Na}-I_{K}-I_{A}}{C_{m}}dt+\sigma_{V}dW_{V}$
(42)
where the leakage conductance, leakage reversal potential and membrane
capacitance in the above model were estimated a priori based on neuron
responses to negative current pulses ($G_{L}=0.11\mu S$, $E_{L}=-65mV$ and
$C_{m}=2.89nF$, respectively). The voltage-activated currents that appear in
the above expression were modeled as follows: (a)
$I_{Na}=G_{Na}m_{\infty,Na}^{3}h_{Na}(V-E_{Na})$, (b)
$I_{K}=G_{K}m_{K}^{4}(V-E_{K})$ and (c) $I_{A}=G_{A}m_{A}^{4}h_{A}(V-E_{K})$,
where $E_{Na}=35mV$ and $E_{K}=-67mV$ as in [41]. The dynamic activation and
inactivation variables of these currents ($h_{Na}$, $m_{K}$, $m_{A}$ and
$h_{A}$) obeyed first-order relaxation kinetics (as in Eq. 32) with voltage-
dependent steady-states (Eq. 33) and relaxation times (Eq. 34 with
$\tau_{min,x_{i}}=0$ and $\delta_{x_{i}}=0.5$), similarly to previously
published neuron models in the central nervous system of Lymnaea[42]. The
observation model was as in Eq. 7.
The raw data we used for inferring the parameters in the above model took the
form of four independent $3.5s$-long recordings of the membrane potential from
the same B4 motoneuron. Each recording was taken while injecting an external
current in the neuron consisting of a sequence of random steps ranging in
amplitude between $-4nA$ and $+4nA$ and with duration between $1$ and $256ms$.
A particular characteristic of the data generated under these conditions was
the presence of brief bursts of spikes, which were interrupted by relatively
long intervals of non-activity (corresponding to sub-threshold excitatory and
inhibitory current injections, respectively; see Figs. 12Ai-iv). These long
intervals of inactivity were not informative and they negatively affected the
performance of the smoother by permitting the random drift of particles
towards non-optimal regions of the parameter space (see Supplementary Fig.
S6). However, when the four recordings are considered together, the intervals
of inactivity at any single voltage trace overlap with intervals of activity
at the remaining three voltage traces, resulting in a four-dimensional data
set, where the overall intervals of inactivity were minimized. This four-
dimensional data set was used as input to the smoother during the inference
phase.
Thus, the $42$-dimensional extended state vector became:
$\mathbf{\mathbf{z}=}(s,G_{i},V_{H,x_{i}},V_{S,x_{i}},\tau_{max,x_{i}},V_{k},m_{Na,k},h_{Na,k},m_{K,k},m_{A,k},h_{A,k})^{T}$
where $x\in\\{m,h\\}$, $i\in\\{Na,K,A\\}$ and $k\in\\{1,2,3,4\\}$. Notice the
presence of four groups of hidden dynamic states, {$V_{k}$, $m_{Na,k}$,
$h_{Na,k}$, $m_{K,k}$, $m_{A,k}$, $h_{A,k}$}, where each group corresponds to
a different voltage trace (and associated externally injected current,
$I_{ext,k}$). The evolution of all four groups of dynamic variables was
governed by a common (shared) set of parameters. In total, we had to estimate
$17$ unknown parameters. The boundaries within which the values of these
parameters were allowed to fluctuate are given in Table 3 (indicated in bold)
and they were chosen from within the support of the posteriors in
Supplementary Fig. S7 (after a few trial-and-error simulations), which were
obtained by using the broader prior intervals given in Table 3. Notice that
the marginal distributions illustrated in Fig. S7 have large variance and
multiple modes and, although they provide a global view of the structure of
the parameter space, they cannot be used to identify a single combination of
optimal parameters values, since they do not include any information regarding
correlations between parameters. Using the major modes of the inferred
posteriors did not lead to an accurate (or even spiking) predictive model.
Thus, the estimation was based on using more narrow prior intervals, which
helped us estimate unimodal posteriors with small variance (see Fig. 12C) and,
thus, identify a single combination of optimal parameters that could be used
in predictive simulations. We cannot prove that other optimal combinations of
parameters do not exist, but we were not able to find any (i.e. by choosing
different narrow prior intervals) after a reasonable amount of time. Also,
notice that the standard deviations of the intrinsic and observation noise
were not subject to estimation, but instead they were given (through trial and
error) the minimal fixed values $\sigma_{V}=0.3mV$ and $\sigma_{y}=1mV$,
respectively. If left free during smoothing, the values of these parameters
fluctuated uncontrollably, masking the contribution of the remaining
parameters in the model and, thus, achieving an almost perfect (but
meaningless) smoothing of the experimental data. This is an indication that
the B4 model we used may be missing one or more relevant components, such as
additional currents and compartments (see below for further analysis of this
point). We did not observe this effect in the cases examined in the previous
sections, where simulated data was used, because the models responsible for
the generation of this data were, by definition, precisely known.
Our results from this set of simulations are illustrated in Fig. 12.
Simultaneous smoothing of all four data sets was again accomplished with high
fidelity, as illustrated in Figs. 12Ai-iv. The artificial evolution of the
expectations of the conductances for the transient sodium, persistent
potassium and transient potassium currents, as well as of some of the kinetic
parameters that were estimated in the model is illustrated in Figs. 12Bi-iii.
The distributions of all inferred parameters (normalized after replacing
$x_{true}$ in Eq. 35 with $\sum_{k=1}^{K}\bar{x}_{k}$, for each tested
parameter) are also illustrated in Fig. 12C. The inferred expectations of all
parameters are given in Table 3.
In order to examine the predictive value of the model given the estimated
parameter expectations in Table 3, we compared its activity to that of the
biological B4 neuron, when both were injected with a $30s$-long random current
consisting of a sequence of current pulses with amplitude ranging from $-4nA$
to $+4nA$ and duration from $1ms$ to $256ms$. Our results from this simulation
are illustrated in Fig. 13. We observed that the overall pattern of activity
of the model was similar to that of the biological neuron (Fig. 13A). Whilst
the model overall generated more action potentials, some individual spikes
were absent in the simulated data. A more detailed examination of our data
revealed specific differences between the biological and model neurons, which
explain the differences in the overall activity between the two (Fig. 13B, C).
The spike shape of the model neuron was quite similar to that of its
biological counterpart (Fig. 13Bi), including spike threshold, peak, trough
and height (i.e. trough-to-peak amplitude; Fig. 13Biii), but the simulated
spike had a slightly longer duration than the biological one (half-width:
$1.9ms$ vs $1.5ms$; Fig. 13Bii).
In a second set of experiments, both the biological and model neurons were
injected with $1s$-long current pulses ranging from $-4nA$ to $+4nA$ and their
current-voltage (IV) and current-frequency (IF) relations were constructed
(Fig. 13C). The IV plot showed some non-linear behavior in response to
negative current pulses in the experimental data (probably due to the presence
of a residual $I_{h}$ current), which was not present in the simulations (Fig.
13Ci). As a result, the slope of the part of the IV curve corresponding to
$0mV$ was more shallow in the simulations than in the experimental data.
Moreover, the rheobase was lower in the experimental data than in the model,
but the slope of the IF curve was steeper in the simulated data, which
resulted in higher firing rates for the model at injected currents larger than
approximately $3nA$ (Fig. 13Cii). This feature can account for the overall
level of spiking in the model neuron when compared to the biological one (Fig.
13A).
Overall, this analysis illustrates that the assumed B4 model did not capture
all the aspects of the real neuron. However, this does not mean that our
estimation method is flawed. It just shows that the model is actually missing
some relevant components, such as additional ionic currents or compartments,
which would be necessary for approximating more accurately the spatial
structure and biophysical properties of the biological neuron. In the first
part of the manuscript we have demonstrated that if the underlying model is
complete, then our method produced accurate estimates of the true parameter
values, given sufficient informative priors. Thus, it is safe to assume that
the observed differences between the biological and model neurons can be
minimized, if the fixed-lag smoother is applied on a more complex model of the
B4 motoneuron.
In summary, we used the fixed-lag smoother to estimate the unknown parameters
in a single-compartment model of an invertebrate motoneuron based on actual
electrophysiological data. The model, although a simplification of the actual
biological system, was still quite complex containing a number of non-linearly
interacting components and a total of $17$ unknown parameters. By using the
methodologies outlined in the previous sections, we managed to estimate the
values of these parameters, such that the resulting model mimicked with
satisfactory accuracy the overall activity of its biological counterpart.
Furthermore, we demonstrated the flexibility of the fixed-lag smoother by
showing how it can be used to process simultaneously multiple data sets, given
an appropriate formulation of the extended state vector.
## Discussion
Parameter estimation in conductance-based neuron models traditionally involves
a global optimization algorithm (for example, an evolutionary algorithm),
usually in combination with a local search method (such as gradient descent),
in order to find combinations of model parameters that minimize a pre-defined
cost function. In this paper, we have addressed the problem of parameter
estimation in Hodgkin-Huxley-type models of single neurons from a different
perspective. By adopting a hidden-dynamical-systems formalism and expressing
parameter estimation as an inference problem in these systems, we made
possible the application of a range of well-established inference methods from
the field of Computational Statistics. Although it is usually assumed that the
kinetics of ionic currents in a conductance-based model are known a priori,
here we assumed that this was not the case and, typically, we estimated
kinetic parameters, along with the maximal conductances and reversal
potentials of ionic currents in the models we examined.
The particular method we used was Kitagawa’s self-organizing state-space
model, which was implemented as a fixed-lag smoother. The smoother was
combined with an adaptive algorithm for sampling new sets of parameters akin
to the Covariance Matrix Adaptation Evolution Strategy. Alternatively, we
could have approximated the smoother distribution (Eq. 13) with a two-pass
algorithm, consisting of a forward filter followed by a backward smoothing
phase, which would make use of the precomputed filter[34]. This would require
storing the filter for the whole duration of the smoothed data, which in turn
would have very high memory requirements when large numbers of particles or
high-dimensional problems are considered. In contrast, the fixed-lag smoother
has the advantage that only the particles up to $L$ time steps in the past
need to be stored, which is less demanding in memory size and computationally
more efficient. Moreover, the fixed-lag smoother, being a single-pass
algorithm, was more natural to use in the context of on-line parameter
estimation.
The applicability of the algorithm was demonstrated on a number of
conductance-based models using noisy simulated or actual electrophysiological
data. In a recent study, it was found that increasing observation noise led to
an increase in the variance of parameter estimates and a decrease in the rate
of convergence of the algorithm[28]. Similarly, we observed that at high
levels of observation noise, although the algorithm remained functional, its
accuracy and precision were reduced (Fig. 5). It is emphasized that, at a
particular level of observation noise, the outcome of the algorithm is an
approximation of the posterior distributions of hidden states and unknown
parameters in the model, given the available experimental data and prior
information. In general, these approximate posteriors provide an overview of
the structure of the parameter space and they potentially have multiple modes
(or local optima). By taking advantage of the adaptive nature of the fixed-lag
smoother (and, in particular, by controlling the scaling factor that
determines the width of the proposal distribution in Eq. 25), we managed to
reduce the variance of these posteriors and, in the limit case, we could force
the algorithm to converge to a single optimal point (belonging to the support
of the parameter posteriors), which could subsequently be used in predictive
simulations (e.g. see Figs. 7D and 11A). Unlike the study in [28], we did not
observe any significant reduction in the rate of convergence of the algorithm
at high levels of observation noise, which was attributed to a ceiling effect
due to the large number of particles we used in our simulations (typically,
$100\times D$, where $D$ was the dimensionality of the estimation problem; see
Figs. 3B and 4C). Thus, we cannot exclude observing such a reduction in the
rate of convergence, if a smaller number of particles is used and/or problems
of higher dimension are examined. Furthermore, the proposed method requires
only a single forward pass of the experimental data, instead of multiple
passes, as in the case of off-line estimation methods, including the
Expectation Maximization (EM) algorithm. On the other hand, this means that,
in general, the proposed algorithm requires processing longer data time series
in order to converge. In addition, unlike off-line estimation methods, it does
not take into account the complete data trace at each iteration, but at most
$L$ past data points (but, also, see [36] for a partial “remedy” of this
situation). In principle, it would be possible to combine previous work on
parameter estimation (e.g. [43, 44]) within an EM inference framework in order
to estimate various types of parameters (including maximal conductances and
channel kinetics) in conductance-based neuron models. This could be an
interesting topic for further research.
Our main conclusion was that, using this algorithm and a set of low-
dimensional experimental data (typically, one or more traces of membrane
potential activity), it was possible to fit complex compartmental models to
this data with high fidelity and, simultaneously, estimate the hidden dynamic
states and optimal values of a large number of parameters in these models.
Based on simulation experiments using simulated data, we found that the
estimated optimal parameter values and hidden states were close to their true
counterparts, as long as sufficient prior information was made available to
the algorithm. This information took the form of knowledge of the values of
particular parameters (for example, the passive properties of the membrane) or
of relatively narrow ranges of permissible parameter values. Such prior
information could have included the kinetics of the ion currents that flow
through the membrane or the spatial distribution of various parameter values
along different neuron compartments (e.g. the ratio of maximal conductance A
between compartment 1 and compartment 2). In real-life situations, such
information may become available through current- or voltage-clamp
experiments. For example, the passive properties in the B4 model (membrane
capacitance, leakage maximal conductance and reversal potential) were inferred
from current-clamp data and, thus, they were fixed during the subsequent
smoothing phase.
It has been demonstrated that this requirement for prior information may be
relaxed, if the data set used as input was sufficiently variable to tease
apart the relative contribution of different parameters in a model[15]. A
well-established result in conductance-based modeling is that the same pattern
of electrical activity may be produced by different parameter configurations
of the same model[6, 7, 8, 9]. This implies that it is impossible to identify,
during the course of an optimization procedure, a unique set of parameters
using just this single pattern of activity as input to the method. For
example, as we observed in the case of the B4 model, the posteriors of the
estimated parameters may be characterized by multiple modes (i.e. local
optima) or quite large variances, which makes identification of a unique set
of optimal parameter values for use in predictive simulations rather difficult
(Supplementary Fig. S7). A more variable data set would be necessary in order
to constrain the model under study, thus forcing the optimization process to
converge towards a unique solution. It should be noted that this conclusion
was reached by treating as unknown only the maximal conductances in a
conductance-based model[15]. Although it is reasonable to assume that this
holds true when the kinetics of ion channels are also treated as unknown, it
still needs to be demonstrated whether the generation of a data set
sufficiently variable to constrain both the maximal conductances and kinetics
of ion channels in a complex conductance-based model is practical or even
feasible. A more pragmatic approach would be to rely on a mixture of prior
information and one or more sufficiently variable electrophysiological
recordings as input to the optimization algorithm. It was shown in this study
that both the injection of prior information (in the form mentioned above) and
the simultaneous assimilation of multiple data sets is straightforward using
the proposed algorithm.
It is important to notice that, unlike more traditional approaches, explicitly
defining a cost or fitness function was not required by the fixed-lag
smoother. Given the fact that the efficiency of any optimizer can be seriously
impeded by a poorly designed cost function, bypassing the need to define such
a function may be viewed as an advantage of the proposed method. As in
previous studies[43, 44], here lies the implicit assumption that by fitting
(or smoothing) with high fidelity the raw experimental data (for example, one
or more recordings of the membrane potential), the estimated model would
capture a whole range of features embedded in this data, such as the current-
frequency response of the neuron. Although this is a reasonable assumption, we
found that it did not hold completely true, when our knowledge of the form of
the underlying model was not exact, as in the case of the B4 neuron. In this
case, although we could achieve a very good smoothing of the experimental
data, subsequent predictive simulations using the inferred model parameters
revealed discrepancies between simulation output and experimental data. It is
likely that these discrepancies will be minimized, if important missing
components are added to the model, such as additional ionic currents or,
importantly, an approximation of the spatial structure of the biological
neuron.
An important outcome of this study was to demonstrate the intimate relation
between the self-organizing state-space model and evolutionary algorithms.
When used for parameter estimation, the self-organizing state-space model
undergoes at each iteration a process of new particle (individual) generation
(mutation/recombination) and resampling (selection and multiplication), which
parallels similar processes in evolutionary algorithms. At the root of this
parallelism is the fact that we need to impose an artificial evolution on
model parameters as part of the formulation of the self-organizing state-space
model (see Methods), thus providing a unique opportunity to merge the two
classes of algorithms. Here, we decided to combine the self-organizing state-
space model with an adaptive algorithm similar to the Covariance Matrix
Adaptation Evolution Strategy[30] and by following this adaptive strategy, we
managed to achieve a dramatic reduction in the variance of parameter
estimates. However, this choice is by no means exclusive and other
evolutionary algorithms may be chosen instead, e.g. the Differential Evolution
algorithm[31]. This is a topic open to further exploration. Notice that,
similarly to Evolutionary Algorithms, the proposed method has, in principle,
the ability to estimate the possibly multi-modal posterior distribution of the
unknown parameters in the examined model, i.e. it is a global estimation
method (for example, see Fig. 6C, 11B and S7). At each iteration, the
algorithm retains a population of particles, which are characterized by a
degree of variability and, thus, give the algorithm the opportunity to
randomly explore a wide range of the parameter space, spending on average more
time in the vicinity of optimal regions. By imposing narrow prior constraints
on some of the unknown parameters, we are effectively reducing the
dimensionality of the problem and we force the algorithm to converge towards a
particular optimum, which can be later used in predictive simulations.
A point of potential improvement concerns our choice of the proposal density,
$q(\mathbf{z}_{k}|\mathbf{z}_{k-1},\mathbf{y}_{k})$. Here, we made the common
and straightforward choice to use the transition density
$p(\mathbf{z}_{k}|\mathbf{z}_{k-1})$ as our proposal. However, the modeler is
free to make other choices. For example, a recent study demonstrated that the
efficiency of particle filters can be significantly increased by conditioning
the proposal density on future observations[36].
An important practical aspect of the proposed algorithm was its high
computational cost. This cost increased as a function of the number $N$ of
particles used during smoothing, the length of the fixed smoothing lag $L$,
the complexity of the model and the number of unknown parameters in the model.
Our simulations on an Intel dual-core i5 processor with four gigabytes of
memory took from a few minutes to more than 12 hours to complete. An emerging
trend in Scientific Computing is the use of modern massively parallel Graphics
Processing Units (GPUs) in order to accelerate general purpose computations,
as those presented in this paper. The utility of this approach in achieving
significant accelerations of Monte Carlo simulations has been recently
demonstrated[45] and it has even been applied recently on parameter estimation
problems in conductance-based models of single neurons[46]. Preliminary
results using a GPU-accelerated version of the fixed-lag smoother (data not
shown) have indeed demonstrated reduced simulation times, but the
accelerations we observed were not as dramatic as those reported in the
literature[45, 46]. This can always be attributed to the fact that our
implementation of the algorithm was not optimized. On the other hand, we
observed significant accelerations in our simulations involving the serial
implementation of the fixed-lag smoother, just by switching from an open-
source compiler (GNU) to a commercial one (Intel), which presumably emitted
better optimized machine code for the underlying hardware. Nevertheless, the
use of GPUs for general purpose computing is becoming common and it is likely
to become quite popular with the advent of cheaper hardware and, importantly,
more flexible and programmer-friendly Application Programming Interfaces
(APIs).
Overall, our results point towards a generic four-stage heuristic for
parameter estimation in conductance-based models of single neurons: (a) First,
the general structure of the model is decided, such as the number of ionic
currents and compartments it should include. (b) Second, prior information is
exploited in order to fix as many parameters as possible in the model and
tightly constrain the remaining ones. For example, the capacitance, reversal
potentials and leakage conductance in the model may be fixed to values
estimated from current-clamp data. By further exploiting current– and voltage-
clamp data, narrow constraints may be imposed on the remaining free (e.g.
kinetic) parameters in the model. (c) At a third stage, more precise parameter
value distributions are estimated by applying the fixed-lag smoother on
current-clamp data, such as one or more recordings of the electrical activity
of the membrane induced by random current injections. (d) Finally, the
predictive value of the model is assessed through comparison to independent
data sets and the model is modified, if necessary. It is important to notice
that the techniques outlined in this paper are applicable on a wide range of
research domains and that they provide a disciplined way to merge complex
stochastic dynamic models, noisy data and prior information under a common
inference framework.
In conclusion, the class of statistical estimation methods, which the
algorithm presented in this paper belongs to, in combination with Monte Carlo
approximation techniques are particularly suitable to address high-dimensional
inference problems in a disciplined manner. This makes them potentially useful
tools at the disposal of biophysical modelers of neurons and neural networks
and it is predicted that these methodologies will become more popular in the
future among this research community.
## Acknowledgments
This study was supported by the Future and Emerging Technologies (FET)
European Commission program BION (number 213219).
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## Figures
Figure 1: Simultaneous estimation of hidden states, intrinsic and observation
noise. Estimation was based on a simulated recording of membrane potential
with duration $1s$. For clarity, only $30ms$ of activity are shown in A and
Bi,ii. (A) Smoothing of the membrane potential (the observed variable), when
observation noise was low ($\sigma_{y}=0.5mV$). High-fidelity smoothing was
achieved for either small ($L=0$) or large ($L=100$) values of the fixed
smoothing lag $L$. Simulated and smoothed data are difficult to distinguish
due to their overlap. (Bi) Smoothing of the membrane potential at high levels
of observation noise ($\sigma_{y}=50mV$). A large value of the smoothing lag
($L=100$) was required for high-fidelity smoothing. (Bii) Inference of the
unobserved activation ($m_{Na}$, $m_{K}$) and inactivation ($h_{Na}$)
variables for sodium and potassium currents as functions of time, during
smoothing of the data shown in Bi for $L=100$. (Ci) Inference of the standard
deviations for the intrinsic and observation noise ($\sigma_{V}$ and
$\sigma_{y}$, respectively) during smoothing of the data shown in Bi for
$L=100$. Dashed lines indicate the true values of $\sigma_{V}$ and
$\sigma_{y}$. (Cii) Histograms of the time series for $\sigma_{V}$ and
$\sigma_{y}$ in Ci. Again, dashed lines indicate the true values of the
corresponding parameters. At this stage, maximal conductances, reversal
potentials and kinetic parameters in the model were assumed known. The number
of particles was $N=700$. Also, $a=b=c=0$. The scaling factors in Eq. 25 were
all considered equal to $1$. Figure 2: The effect of adaptive parameter
sampling on the variance of parameter estimates. Merging the fixed-lag
smoother with an adaptive sampling algorithm akin to the Covariance Matrix
Adaptation Evolution Strategy reduced significantly the variance of parameter
estimates. At this stage, the maximal conductances for the sodium ($G_{Na}$)
and potassium ($G_{K}$) currents were assumed unknown. Estimation was based on
a simulated recording of membrane potential with duration $1s$ and
$\sigma_{V}=\sigma_{y}=1mV$. (A) Inference of $G_{Na}$ and $G_{K}$ during
smoothing, when new parameter samples were drawn from a non-adaptive multi-
variate normal distribution (Eq. 30). Dashed lines indicate the true parameter
values. (B) Inference of $G_{Na}$ and $G_{K}$ during smoothing, when new
samples were drawn from a multi-variate normal distribution (Eq. 25) with an
adaptive scaling factor $s$ ($c=0.01$ in Eq. 28). (C) Inference of $G_{Na}$
and $G_{K}$ during smoothing, when new samples were drawn from a multi-variate
normal distribution (Eq. 25) with adaptive scaling (as in B) and mean
($a=0.01$ in Eq. 26). (D) Inference of $G_{Na}$ and $G_{K}$ during smoothing,
when new samples were drawn from a multi-variate normal distribution with
adaptive scaling (as in B), mean (as in C) and covariance ($b=0.01$ in Eq.
27). The histograms in the right plots were constructed from the time series
in the left plots. Membrane potential, activation and inactivation variables,
intrinsic and observation noise were also subject to estimation, as in Fig. 1.
Smoothing lag and number of particles were $L=100$ and $N=900$, respectively.
The prior interval of the scaling factors $s_{k}^{(j)}$ was $[0,10]$. Figure
3: The effect of the size of the scaling factor $s$ and the number of
particles $N$ on the variance of the estimates. Large minimal values of $s$
and small values of $N$ imply large variance of the estimates. (A) Resampling
of particles (see Methods) implies adaptation of (among others) the scaling
factors $s_{k}^{(j)}$, which gradually approach the lower bound of their prior
interval (red lines in Ai,ii). A prior interval with zero lower bound (i.e.
$[0,2]$) leads to estimates with negligible variance (Ai). A prior interval
with relatively large lower bound (e.g. $[0.15,2]$) leads to estimates with
non-zero variance (Aii). Notice that the expectation $\bar{s}$ in Ai does not
actually take the value $0$ (instead it becomes approximately equal to
$0.01$). (B) A small number of particles (Bi, $N=90$) implies estimates with
large variance (compare to Bii, $N=1800$). Notice that the difference between
Aii ($N=900$) and Bii ($N=1800$) is negligible, implying the presence of a
ceiling effect, when the number of particles becomes very large. In these
simulations, $L=100$ and $a=b=c=0.01$. Figure 4: The effect of adaptation of
the scaling factor $s$ and the number of particles $N$ on the speed of
convergence. A slow rate of adaptation for $s$ and a small number of particles
$N$ imply slow convergence of the algorithm. The rate at which $s_{k}^{(j)}$
adapts depends on the parameter $c$ in Eq. 28. Reducing $c$ in half results in
a significant decrease in the rate of convergence (compare A to B). Also,
reducing the number of particles by a factor of $10$ slows down the speed of
convergence (compare A to C), but not as much as when parameter $c$ was
adjusted. The plots on the right illustrate the profile of $\bar{s}$
associated with the estimation of the parameters on the left plots. In these
simulations, $L=100$, $a=b=0.01$ and the prior interval for the scaling
factors $s_{k}^{(j)}$ was $[0.15,2]$. Figure 5: The effect of observation
noise on the accuracy and precision of parameter estimates. Increasing
observation noise decreases the accuracy and precision of the fixed-lag
smoother. At this stage, the reversal potentials for the sodium and potassium
currents ($E_{Na}$ and $E_{K}$, respectively) were also considered unknown.
Estimation was based on a simulated recording of membrane potential with
duration 2s. The noise parameters were $\sigma_{V}=1mV$ and
$\sigma_{y}=0.5mV$, $5mV$, $25mV$ or $50mV$. (A) Inference of $G_{Na}$,
$G_{K}$, $E_{Na}$ and $E_{K}$ during smoothing. The accuracy of the estimates
decreases and their variance increases with increasing observation noise. (B)
The box plot of the time series in A for $t\geq 1s$. Data were first
normalized according to Eq. 35. The reduction in the accuracy and precision at
higher levels of observation noise were more prominent in the case of the
maximal conductances ($G_{Na}$ and $G_{K}$) and less prominent in the case of
reversal potentials ($E_{Na}$ and, particularly $E_{K}$). The membrane
potential, activation and inactivation variables, intrinsic and observation
noise were also subject to estimation, as in Fig. 1. In these simulations,
$L=100$, $N=1100$, $a=b=c=0.01$ and the prior interval of $s_{k}^{(j)}$ was
$[0.15,10]$. Figure 6: Estimation of all parameters in a single-compartment
conductance-based model using the fixed-lag smoother. Estimation was based on
a simulated recording of the membrane potential with duration $20s$. Noise
parameters were $\sigma_{V}=\sigma_{y}=1mV$. For clarity, only $35ms$ of
activity are illustrated in Ai,ii. (A) Smoothing of the membrane potential
(Ai) and the unobserved activation and inactivation variables for the sodium
and potassium currents (Aii). (B, C) Estimated posteriors for $E_{Na}$ (B),
$\tau_{max,m_{Na}}$ (Ci,ii) and $\tau_{max,m_{K}}$ (Ciii,iv). The histograms
on the right were constructed form the data on left. (D) Box plot of the $23$
estimated parameter posteriors in the model. These included the standard
deviations of intrinsic and observation noise, maximal conductances, reversal
potentials and kinetics of all currents in the model (see Table 1). The
estimates were first normalized according to Eq. 28. Parameter identification
numbers are as in Table 1. In these simulations, $L=100$, $N=2800$,
$a=b=c=0.01$ and the prior interval for $s_{k}^{(j)}$ was $[0.15,10]$. Figure
7: The effect of prior parameter intervals on the accuracy of the fixed-lag
smoother. Estimation was based on a simulated recording of the membrane
potential with duration $2s$. Noise parameters were
$\sigma_{V}=\sigma_{y}=1mV$. For clarity, only $35ms$ of activity are
illustrated in Ai,ii. Unlike Fig. 6, the prior interval for the scaling
factors $s_{k}^{(j)}$ was now assumed equal to $[0,10]$. (A) Smoothing of the
membrane potential (Ai) and the unobserved activation and inactivation
variables for the sodium and potassium currents (Aii). (B) Estimates for
parameters $E_{Na}$ (Bi), $\tau_{max,m_{Na}}$ and $\tau_{max,m_{K}}$ (Bii).
Convergence to an optimal parameter vector was achieved after approximately
$1.5s$ of activity. Notice that this optimal parameter vector falls within the
support of the corresponding parameter posteriors (see Figs. 6Bii, 6Cii and
6Civ). (C) Inferred steady states (Ci) and relaxation times (Cii) for the
activation and inactivation variables of sodium and potassium currents (red
lines) against their true counterparts (black lines). (D) Inferred parameter
values when broad (Di) or narrow (Dii) prior intervals were used for the
parameters controlling the kinetics of sodium and potassium ionic currents
(see Table I). Plots A, B and C correspond to plot Di. In Dii, we also
illustrate the estimated parameter values when very noisy data were used (see
also Supplementary Fig. S3). In these simulations, $L=100$, $N=2800$ and
$a=b=c=0.01$. Figure 8: Simultaneous estimation of hidden model states
(including intracellular calcium concentrations) and maximal conductances in a
two-compartment model of a vertebrate motoneuron (I). Estimation was based on
two $3s$-long simulated recordings of the membrane potential, each recorded
simultaneously from the soma and the dendritic compartment. Only part of the
recorded activity is illustrated in A, B and C for clarity. Notice the
different time scales between the right and left panels. (A) High-fidelity
smoothing of the membrane potential at the soma (Ai) and the dendritic
compartment (Aii). (B) Inference of the unobserved calcium concentrations at
the soma (Bi) and the dendrite (Bii). (C) Inference of the unobserved
activation and inactivation variables for the sodium and potassium currents
(Ci) and the N-type calcium current (Ciii) at the soma and the N-type (Cii)
and L-type (Civ) calcium currents at the dendritic compartment. Notice the
almost complete overlap between true (black lines) and inferred (red lines)
dynamic variables in Ci-iv. This was not surprising since we assumed, at this
stage, that the kinetics of all gated currents were known. In these
simulations, $L=100$, $N=2200$, $a=b=c=0.01$ and the prior interval for
$s_{k}^{(j)}$ was $[0,10]$. Figure 9: Simultaneous estimation of hidden model
states (including intracellular calcium concentrations) and maximal
conductances in a two-compartment model of a vertebrate motoneuron (II).
Inference of maximal conductances and noise parameters during fixed-lag
smoothing. (A) The standard deviations of the observation (Ai) and the
intrinsic (Aii) noise at the soma and the dendrite. (B) Inferred maximal
conductances of the sodium and potassium currents at the soma (Bi), of the
N-type calcium current and the calcium-activated potassium current at the soma
(Bii), of the calcium-activated potassium current at the dendrite (Biii) and
of the N-type and L-type calcium currents at the dendrite (Biv). In all cases,
parameter expectations gradually converged towards the true parameter values
(dashed lines) after less than $2s$. The grey lines in Aii, Biii and Biv
correspond to estimated parameters, when current was injected in the soma
only. In these simulations, $L=100$, $N=2200$, $a=b=c=0.01$ and the prior
interval for $s_{k}^{(j)}$ was $[0,10]$. Figure 10: Simultaneous estimation
of hidden model states, maximal conductances and kinetic parameters in a two-
compartment model of a vertebrate motoneuron (I). Estimation was based on two
simulated $4s$-long simultaneous recordings of the membrane potential from the
soma and dendritic compartment. Only part of this data is illustrated for
clarity. Notice the different time scales between the left and right panels.
(A) High-fidelity smoothing of the observed voltage at the soma (Ai) and the
dendrite (Aii). (B) Inference of unobserved calcium concentrations at the soma
(Bi) and dendritic compartment (Bii). (C) Inference of the unobserved
activation and inactivation variables for all voltage-gated currents at the
soma and the dendrite. Since the kinetics of voltage-gated currents were
assumed unknown, the difference between true (black lines) and inferred (red
lines) dynamic variables was significant (compare to Fig. 8). The inferred
parameters are shown in Fig. 7Ai. In these simulations, $L=100$, $N=8200$,
$a=b=c=0.01$ and the prior interval for $s_{k}^{(j)}$ was $[0,10]$. Figure 11:
Simultaneous estimation of hidden model states, maximal conductances and
kinetic parameters in a two-compartment model of a vertebrate motoneuron (II).
Inference of maximal conductances, noise and kinetic parameters during
smoothing. (A) Inferred parameters in the model using broad or narrow prior
intervals and high or low levels of observation noise. Estimates were
normalized according to Eq. 35. Parameter identification numbers are as in
Table 2. The estimates in Ai were obtained using broad prior intervals (see
Table 2). The maximal conductance $G_{CaN,S}$ (parameter #7) converged to zero
and, for this reason, it is indicated with a red square. These estimates
correspond to the results shown in Fig. 10. Estimates in Aii were obtained
using narrow prior intervals for some of the parameters controlling the
kinetics of ionic currents (see red dashed boxes) at either low
($\sigma_{y}=1mV$) or high ($\sigma_{y}=50mV$) levels of observation noise
(see also Supplementary Figs. S4 and S5). (B) Inferred maximal conductances
for sodium ($G_{Na}$) and potassium ($G_{K}$) when narrow prior intervals and
low levels of observation noise were used (circles in Aii). Notice the
temporary convergence of the estimates (arrows) before jumping away towards
their final values. (C) True (black lines) and inferred (red lines) activation
and inactivation steady-states for the sodium and potassium currents (Ci) and
the N-type and L-type calcium currents (Cii) and for the relaxation times for
sodium inactivation and potassium activation (Ciii), when narrow prior
intervals and low levels of observation noise were used (circles in Aii). In
these simulations, $L=100$, $a=b=c=0.01$ and the prior interval for
$s_{k}^{(j)}$ was $[0,10]$. The number of particles was $N=8200$ in Ai and
$N=4100$ in Aii, B and C (see main text for further comments). Figure 12:
Parameter estimation in a model invertebrate motoneuron based on actual
electrophysiological data. Estimation was based on four independent
$3.5s$-long recordings of the membrane potential from the same B4 motoneuron.
(A) Simultaneous, high-fidelity smoothing of the four membrane potential
recordings. (B) A total of $17$ free parameters in the model were inferred
during smoothing (see Table 3), including the maximal conductances of the
transient sodium and potassium and persistent potassium currents (Bi), the
half steady-state activation values (Bii) and the relaxation times for the
activation of the potassium currents (Biii). The remaining inferred parameters
are not illustrated for clarity, but they follow a similar convergence
pattern. (C) Box plot of all inferred parameters in the model. Parameter
identification numbers are as in Table 3. Estimates were normalized as
explained in the main text (the non-normalized mean parameter values are given
in Table 3). In this simulation, $L=100$, $N=3800$, $a=b=c=0.01$ and the prior
interval for $s_{k}^{(j)}$ was $[0.2,0.5]$. Figure 13: Comparison between B4
model activity and the biological neuron. (A) Response of the model and the
biological B4 motoneuron to a sequence of current steps with random amplitude
and duration. Current step amplitudes were from $-4nA$ to $+4nA$ and current
step durations from $1ms$ to $256ms$. Intrinsic and observation noise in the
model were $\sigma_{V}=0.3mV$ and $\sigma_{y}=1mV$, respectively. (B)
Comparison between model and biological B4 action potentials. The width of the
spikes was measured at half their peak amplitude. (C) Current-Voltage (IV) and
Current-Frequency (IF) relations for the model and biological B4 neurons. In
order to construct these relations both the model and biological neurons were
injected with $1s$-long current pulses with amplitude between $-4nA$ and
$+4nA$.
## Tables
Table 1: True and estimated values and prior intervals used during smoothing
for all parameters in the single-compartment conductance-based model
# | Parameter | Unit | True Value | Estimated Value1 | Lower Bound | Upper Bound
---|---|---|---|---|---|---
1 | $\sigma_{V}$ | $mV$ | $1.0$ | $1.0$ | $0.0$ | $10.0$
2 | $\sigma_{y}$ | $mV$ | $1.0$ | $1.0$ | $0.0$1 | $10.0$
3 | $G_{L}$ | $mS/cm^{2}$ | $0.3$ | $0.17$ | $0.0$ | $150.0$
4 | $G_{Na}$ | $mS/cm^{2}$ | $120$.0 | 34.3 | $0.0$ | $150.0$
5 | $G_{K}$ | $mS/cm^{2}$ | $36$.0 | $125.9$ | $0.0$ | $150.0$
6 | $E_{L}$ | $mV$ | $-54.4$ | $-32.49$ | $-100.0$ | $0.0$
7 | $E_{Na}$ | $mV$ | $55$.0 | $66.35$ | $0.0$ | $100.0$
8 | $E_{K}$ | $mV$ | $-77$.0 | $-77.8$ | $-100.0$ | $0.0$
9 | $V_{H,m_{Na}}$ | $mV$ | $-39.6$ | $-42.9$ | $-70.0$ (-45.0) | $-30.0$ (-35.0)
10 | $V_{H,h_{Na}}$ | $mV$ | $-62.2$ | $-58.2$ | $-70.0$ (-65.0) | $-30.0$ (-55.0)
11 | $V_{H,m_{K}}$ | $mV$ | $-51.5$ | $-43.0$ | $-70.0$ (-55.0) | $-30.0$ (-45.0)
12 | $V_{S,m_{Na}}$ | $mV$ | $9.5$ | $9.0$ | $5.0$ (5.0) | $25.0$ (10.0)
13 | $V_{S,h_{Na}}$ | $mV$ | $-7.1$ | $-9.7$ | $-25.0$ (-10.0) | $-5.0$ (-5.0)
14 | $V_{S,m_{K}}$ | $mV$ | $16.4$ | $19.6$ | $5.0$ (10.0) | $25.0$ (20.0)
15 | $\tau_{min,m_{Na}}$ | $ms$ | $0.0093$ | $0.009$ | $0.008$ | $1.0$
16 | $\tau_{min,h_{Na}}$ | $ms$ | $0.4$ | $0.6$ | $0.01$ | $1.0$
17 | $\tau_{min,m_{K}}$ | $ms$ | $0.5$ | $0.24$ | $0.01$ | $1.0$
18 | $\tau_{max,m_{Na}}$ | $ms$ | $1.0$ | $0.7$ | $0.01$ | $20.0$
19 | $\tau_{max,h_{Na}}$ | $ms$ | $16.1$ | $6.6$ | $0.01$ | $20.0$
20 | $\tau_{max,m_{K}}$ | $ms$ | $8.9$ | $12.2$ | $0.01$ | $20.0$
21 | $\delta_{m_{Na}}$ | - | $0.4$ | $0.4$ | $0.0$ (0.0) | $1.0$ (0.5)
22 | $\delta_{h_{Na}}$ | - | $0.4$ | $0.2$ | $0.0$ (0.0) | $1.0$ (0.5)
23 | $\delta_{m_{K}}$ | - | $0.8$ | $0.6$ | $0.0$ (0.5) | $1.0$ (1.0)
1These parameter values were estimated when we used the broad prior intervals
(see Fig. 7Di)
2Values in bold indicate the narrow prior intervals we used for generating
Fig. 7Dii (and Supplementary Fig. S3)
Table 2: True and estimated values and prior intervals used during smoothing
for all parameters in the two-compartment conductance-based model
# | Parameter | Unit | True Value | Estimated Value1 | Lower Bound | Upper Bound
---|---|---|---|---|---|---
1 | $\sigma_{V_{S}}$ | $mV$ | $1.0$ | $2.1$ | $0.0$ | $10.0$
2 | $\sigma_{V_{D}}$ | $mV$ | $1.0$ | $1.0$ | $0.0$ | $10.0$
3 | $\sigma_{y}$ | $mV$ | $1.0$ | $0.9$ | $0.01$ | $10.0$
4 | $G_{Na}$ | $mS/cm^{2}$ | $120$ | $88.8$ | $0.0$ | $150.0$
5 | $G_{K}$ | $mS/cm^{2}$ | $100$ | $48.1$ | $0.0$ | $150.0$
6 | $G_{K(Ca),S}$ | $mS/cm^{2}$ | $5.0$ | $3.2$ | $0.0$ | $20.0$
7 | $G_{CaN,S}$ | $mS/cm^{2}$ | $14.0$ | $0.0$ | $0.0$ | $20.0$
8 | $G_{K(Ca),D}$ | $mS/cm^{2}$ | $1.1$ | $0.72$ | $0.0$ | $5.0$
9 | $G_{CaN,D}$ | $mS/cm^{2}$ | $0.3$ | $0.64$ | $0.0$ | $1.0$
10 | $G_{CaL}$ | $mS/cm^{2}$ | $0.33$ | $0.2$ | $0.0$ | $1.0$
11 | $V_{H,m_{Na}}$ | $mV$ | -35.0 | $-29.7$ | -60.0 (-45.0) | $-20.0$ (-25.0)
12 | $V_{H,h_{Na}}$ | $mV$ | $-55.0$ | $-48.5$ | -60.0 (-65.0) | $-20.0$ (-45.0)
13 | $V_{H,m_{K}}$ | $mV$ | $-28.0$ | $-24.1$ | -60.0 (-40.0) | $-20.0$ (-20.0)
14 | $V_{H,m_{CaN}}$ | $mV$ | $-30.0$ | $-33.2$ | -60.0 (-40.0) | $-20.0$ (-20.0)
15 | $V_{H,h_{CaN}}$ | $mV$ | $-45.0$ | $-41.4$ | -60.0 (-55.0) | $-20.0$ (-35.0)
16 | $V_{H,m_{CaL}}$ | $mV$ | $-40.0$ | $-45.4$ | -60.0 (-50.0) | $-20.0$ (-30.0)
17 | $V_{S,m_{Na}}$ | $mV$ | $7.8$ | $8.9$ | $5.0$ (5.0) | $25.0$ (10.0)
18 | $V_{S,h_{Na}}$ | $mV$ | $-7.0$ | $-12.7$ | $-25.0$ (-10.0) | $-5.0$ (-5.0)
19 | $V_{S,m_{K}}$ | $mV$ | $15.0$ | $21.7$ | $5.0$ (10.0) | $25.0$ (20.0)
20 | $V_{S,m_{CaN}}$ | $mV$ | $5.0$ | $23.0$ | $3.0$ (3.0) | $23.0$ (8.0)
21 | $V_{S,h_{CaN}}$ | $mV$ | $-5.$0 | $-5.4$ | $-23.0$ (-8.0) | $-3.0$ (-3.0)
22 | $V_{S,m_{CaL}}$ | $mV$ | $7.0$ | $19.8$ | $5.0$ (5.0) | $25.0$ (10.0)
23 | $\tau_{min,m_{K}}$ | $ms$ | $0.65$ | $0.2$ | $0.01$ | $1.0$
24 | $\tau_{max,h_{Na}}$ | $ms$ | $30.3$ | $11.6$ | $0.01$ | $70.0$
25 | $\tau_{max,m_{K}}$ | $ms$ | $6.3$ | $7.3$ | $0.01$ | $10.0$
26 | $\delta_{h_{Na}}$ | - | $0.6$ | $0.2$ | $0.0$ (0.5) | $1.0$ (1.0)
27 | $\delta_{m_{K}}$ | - | $0.7$ | $0.7$ | $0.0$ (0.5) | $1.0$ (1.0)
28 | $\tau_{m_{CaN}}$ | $ms$ | $4.0$ | $9.8$ | $0.01$ | $10.0$
29 | $\tau_{h_{CaN}}$ | $ms$ | $40.0$ | $17.0$ | $0.01$ | $70.0$
30 | $\tau_{m_{CaL}}$ | $ms$ | $40.0$ | $48.1$ | $0.01$ | $70.0$
1These parameter values were estimated when we used the broad prior intervals
(see Fig. 11Ai)
2Values in bold indicate the narrow prior intervals we used for generating
Figs. 11Aii, 11B, 11C (and Supplementary Figs. S4 and S5)
Table 3: Estimated mean values and prior limits used during smoothing for all
parameters in the B4 model
# | Parameter | Unit | Estimated Mean Value1 | Lower Bound | Upper Bound
---|---|---|---|---|---
1 | $G_{Na}$ | $mS/cm^{2}$ | $24.9$ | $0.0$ | $60.0$
2 | $G_{K}$ | $mS/cm^{2}$ | $21.5$ | $0.0$ | $60.0$
3 | $G_{A}$ | $mS/cm^{2}$ | $23.3$ | $0.0$ | $60.0$
4 | $V_{H,m_{Na}}$ | $mV$ | $-25.1$ | $-70.0$ (-40.0) | $0.0$ (-20.0)
5 | $V_{H,h_{Na}}$ | $mV$ | $-24.1$ | $-70.0$ (-40.0) | $0.0$ (-20.0)
6 | $V_{H,m_{K}}$ | $mV$ | $-23.1$ | $-70.0$ (-40.0) | $0.0$ (-20.0)
7 | $V_{H,m_{A}}$ | $mV$ | $-10.2$ | $-70.0$ (-20.0) | $0.0$ (0.0)
8 | $V_{H,h_{A}}$ | $mV$ | $-53.6$ | $-70.0$ (-70.0) | $0.0$ (-40.0)
9 | $V_{S,m_{Na}}$ | $mV$ | $6.6$ | $5.0$ (5.0) | $25.0$ (10.0)
10 | $V_{S,h_{Na}}$ | $mV$ | $-6.5$ | $-25.0$ (-10.0) | $-5.0$ (-5.0)
11 | $V_{S,m_{K}}$ | $mV$ | $11.0$ | $5.0$ (10.0) | $25.0$ (15.0)
12 | $V_{S,m_{A}}$ | $mV$ | $6.8$ | $5.0$ (5.0) | $25.0$ (10.0)
13 | $V_{S,h_{A}}$ | $mV$ | $-20.1$ | $-25.0$ (-25.0) | $-5.0$ (-15.0)
14 | $\tau_{max,h_{Na}}$ | $ms$ | $22.9$ | $0.01$ (15.0) | $60.0$ (25.0)
15 | $\tau_{max,m_{K}}$ | $ms$ | $32.0$ | $0.01$ (25.0) | $60.0$ (35.0)
16 | $\tau_{max,m_{A}}$ | $ms$ | $29.5$ | $0.01$ (25.0) | $60.0$ (35.0)
17 | $\tau_{max,h_{A}}$ | $ms$ | $49.9$ | $0.01$ (35.0) | $60.0$ (60.0)
1These parameter values were estimated when we used the narrow prior intervals
(in bold; see Fig. 12)
2The parameter posteriors estimated when we used the broad prior intervals are
illustrated in Supplementary Fig. S7
|
arxiv-papers
| 2011-06-21T20:04:52 |
2024-09-04T02:49:19.965656
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dimitrios V. Vavoulis, Volko A. Straub, John A.D. Aston, Jianfeng Feng",
"submitter": "Dimitris Vavoulis",
"url": "https://arxiv.org/abs/1106.4317"
}
|
1106.4435
|
# Determination of $\boldsymbol{f_{s}/f_{d}}$ for 7 TeV $\boldsymbol{pp}$
collisions and a measurement of the branching fraction of the decay
$\boldsymbol{B^{0}\rightarrow D^{-}K^{+}}$
R. Aaij et al., (The LHCb Collaboration)
###### Abstract
The relative abundance of the three decay modes $B^{0}\rightarrow
D^{-}K^{+}\,$, $B^{0}\rightarrow D^{-}\pi^{+}\,$and $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$produced in 7 TeV $pp$ collisions at the LHC is determined
from data corresponding to an integrated luminosity of $35\mbox{\,pb}^{-1}$.
The branching fraction of $B^{0}\rightarrow D^{-}K^{+}\,$is found to be ${\cal
B}\left(B^{0}\rightarrow D^{-}K^{+}\,\right)=(2.01\pm 0.18^{\textrm{stat}}\pm
0.14^{\textrm{syst}})\times 10^{-4}$. The ratio of fragmentation fractions
$f_{s}/f_{d}$ is determined through the relative abundance of
$B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$to $B^{0}\rightarrow D^{-}K^{+}\,$and
$B^{0}\rightarrow D^{-}\pi^{+}\,$, leading to $f_{s}/f_{d}~{}=0.253\pm
0.017\pm 0.017\pm 0.020$, where the uncertainties are statistical, systematic,
and theoretical respectively.
###### pacs:
12.38.Qk, 13.60.Le, 13.87.Fh
Knowledge of the production rate of $B^{0}_{s}$ mesons is required to
determine any $B^{0}_{s}$ branching fraction. This rate is determined by the
$b\bar{b}$ production cross-section and the fragmentation probability $f_{s}$,
which is the fraction of $B^{0}_{s}$ mesons amongst all weakly-decaying bottom
hadrons. Similarly the production rate of $B^{0}$ mesons is driven by the
fragmentation probability $f_{d}$. The measurement of the branching fraction
of the rare decay $B^{0}_{s}\\!\rightarrow\mu^{+}\mu^{-}$ is a prime example
where improved knowledge of $f_{s}/f_{d}$ is needed to reach the highest
sensitivity in the search for physics beyond the Standard Model
Aaij:2011Bsmm_notitle . The ratio $f_{s}/f_{d}$ is in principle dependent on
collision energy and type as well as the acceptance region of the detector.
This is the first measurement of this quantity at the LHC.
The ratio $f_{s}/f_{d}$ can be extracted if the ratio of branching fractions
of $B^{0}$ and $B_{s}^{0}$ mesons decaying to particular final states $X_{1}$
and $X_{2}$, respectively, is known:
$\frac{f_{s}}{f_{d}}=\frac{N_{X_{2}}}{N_{X_{1}}}\frac{{\cal
B}(B^{0}\rightarrow X_{1})}{{\cal B}(B_{s}^{0}\rightarrow
X_{2})}\frac{\epsilon(B^{0}\rightarrow X_{1})}{\epsilon(B_{s}^{0}\rightarrow
X_{2})}.$ (1)
The ratio of the branching fraction of the $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$and $B^{0}\rightarrow D^{-}K^{+}\,$decays is dominated by
contributions from colour-allowed tree-diagram amplitudes and is therefore
theoretically well understood. In contrast, the ratio of the branching ratios
of the two decays $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$and
$B^{0}\rightarrow D^{-}\pi^{+}\,$can be measured with a smaller statistical
uncertainty due to the greater yield of the $B^{0}$ mode, but suffers from an
additional theoretical uncertainty due to the contribution from a $W$-exchange
diagram. Both ratios are exploited here to measure $f_{s}/f_{d}$ according to
the equations Fleischer:2010-2_notitle ; Fleischer:2010ay_notitle
$\frac{f_{s}}{f_{d}}=0.971\cdot\left|\frac{V_{us}}{V_{ud}}\right|^{2}\left(\frac{f_{K}}{f_{\pi}}\right)^{2}\frac{\tau_{B_{d}}}{\tau_{B_{s}}}\frac{1}{{\cal
N}_{a}{\cal
N}_{F}}\frac{\epsilon_{D^{-}K^{+}}}{\epsilon_{D^{-}_{s}\pi^{+}}}\frac{N_{D^{-}_{s}\pi^{+}}}{N_{D^{-}K^{+}}},$
(2)
and
$\frac{f_{s}}{f_{d}}=0.982\cdot\frac{\tau_{B_{d}}}{\tau_{B_{s}}}\frac{1}{{\cal
N}_{a}{\cal N}_{F}{\cal
N}_{E}}\frac{\epsilon_{D^{-}\pi^{+}}}{\epsilon_{D^{-}_{s}\pi^{+}}}\frac{N_{D^{-}_{s}\pi^{+}}}{N_{D^{-}\pi^{+}}}.$
(3)
Here $\epsilon_{X}$ is the selection efficiency of decay $X$ (including the
branching fraction of the $D$ decay mode used to reconstruct it), $N_{X}$ is
the observed number of decays of this type, the $V_{ij}$ are elements of the
CKM matrix, $f_{i}$ are the meson decay constants and the numerical factors
take into account the phase space difference for the ratio of the two decay
modes. Inclusion of charge conjugate modes is implied throughout. The term
${\cal N}_{a}$ parametrizes non-factorizable SU(3)-breaking effects; ${\cal
N}_{F}$ is the ratio of the form factors; ${\cal N}_{E}$ is an additional
correction term to account for the $W$-exchange diagram in the
$B^{0}\rightarrow D^{-}\pi^{+}\,$decay. Their values Fleischer:2010ay_notitle
; Fleischer:2010-2_notitle are ${\cal N}_{a}=1.00\pm 0.02$, ${\cal
N}_{F}=1.24\pm 0.08$, and ${\cal N}_{E}=0.966\pm 0.075$. The latest world
average HFAG_notitle is used for the $B$ meson lifetime ratio
$\tau_{B_{s}}/\tau_{B_{d}}=0.973\pm 0.015$. The numerical values used for the
other factors are: $|V_{us}|=0.2252$, $|V_{ud}|=0.97425$, $f_{\pi}=130.41$ and
$f_{K}=156.1$, with negligible associated uncertainties PDG_2010_notitle .
The observed yields of these three decay modes in $35\mbox{\,pb}^{-1}$ of data
collected with the LHCb detector in the 2010 running period are used to
measure $f_{s}/f_{d}$ averaged over the LHCb acceptance and to improve the
current measurement of the branching fraction of the $B^{0}\rightarrow
D^{-}K^{+}\,$decay mode Abe:2001waa_notitle .
The LHCb experiment DetectPaper_notitle is a single-arm spectrometer,
designed to study $B$ decays at the LHC, with a pseudorapidity acceptance of
$2<\eta<5$ for charged tracks. The first trigger level allows the selection of
events with $B$ hadronic decays using the transverse energy of hadrons
measured in the calorimeter system. The event information is subsequently sent
to a software trigger, implemented in a dedicated processor farm, which
performs a final online selection of events for later offline analysis. The
tracking system determines the momenta of $B$ decay products with a precision
of $\delta p/p=0.35$–$0.5\%$. Two Ring Imaging Cherenkov (RICH) detectors
allow charged kaons and pions to be distinguished in the momentum range 2–100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ DetectPaper .
The three decay modes, $B^{0}\rightarrow D^{-}(K^{+}\pi^{-}\pi^{-})\pi^{+}$,
$B^{0}\rightarrow D^{-}(K^{+}\pi^{-}\pi^{-})K^{+}$ and $B^{0}_{s}\rightarrow
D_{s}^{-}(K^{+}K^{-}\pi^{-})\pi^{+}$, are topologically identical and can
therefore be selected using identical geometric and kinematic criteria, thus
minimizing efficiency differences between them. Events are selected at the
first trigger stage by requiring a hadron with transverse energy greater than
$3.6$ GeV in the calorimeter. The second, software, stage
Gligorov:1300771_notitle ; hlt2toponote_notitle requires a two, three, or
four track secondary vertex with a high sum $p_{T}$ of the tracks, significant
displacement from the primary interaction, and at least one track with
exceptionally high $p_{T}$, large displacement from the primary interaction,
and small fit $\chi^{2}$.
The decays of $B$ mesons can be distinguished from background using variables
such as the $p_{T}$ and impact parameter $\chi^{2}$ of the $B$, $D$, and the
final state particles with respect to the primary interaction. In addition the
vertex quality of the $B$ and $D$ candidates, the $B$ lifetime, and the angle
between the $B$ momentum vector and the vector joining the $B$ production and
decay vertices are used in the selection. The $D$ lifetime and flight distance
are not used in the selection because the lifetimes of the $D^{-}_{s}$ and
$D^{-}$ differ by about a factor of two.
The event sample is first selected using the gradient boosted decision tree
technique TMVA_notitle , which combines the geometrical and kinematic
variables listed above. The selection is trained on a mixture of simulated
$B^{0}\rightarrow D^{-}\pi^{+}\,$ decays and combinatorial background selected
from the sidebands of the data mass distributions. The distributions of the
input variables for data and simulated signal events show excellent agreement,
justifying the use of simulated events in the training procedure.
Subsequently, $D^{-}$ ($D^{-}_{s}$) candidates are identified by requiring the
invariant mass under the $K\pi\pi$ ($KK\pi$) hypothesis to fall within the
selection window $1870^{+24}_{-40}$ ($1969^{+24}_{-40}$)
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where the mass resolution is
approximately $10$ ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The final
$B^{0}\rightarrow D^{-}\pi^{+}\,$and $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$subsamples consist of events that pass a particle
identification (PID) criterion on the bachelor particle, based on the
difference in log-likelihood between the charged pion and kaon hypotheses
(DLL) of $\textrm{DLL}(K-\pi)<0$, with an efficiency of $83.0\%$. The
$B^{0}\rightarrow D^{-}K^{+}\,$subsample consists of events with
$\textrm{DLL}(K-\pi)>5$, with an efficiency of $70.2\%$. Events not satisfying
either condition are not used.
The relative efficiency of the selection procedure is evaluated for all decay
modes using simulated events, where the appropriate resonances in the charm
decays are taken into account. As the analysis is only sensitive to relative
efficiencies, the impact of differences between data and simulation is small.
The relative efficiencies are
$\epsilon_{D^{-}\pi^{+}}/\epsilon_{D^{-}K^{+}}=1.221\pm 0.021$,
$\epsilon_{D^{-}K^{+}}/\epsilon_{D^{-}_{s}\pi^{+}}=0.917\pm 0.020$, and
$\epsilon_{D^{-}\pi^{+}}/\epsilon_{D^{-}_{s}\pi^{+}}=1.120\pm 0.025$, where
the errors are due to the limited size of the simulated event samples.
The relative yields of the three decay modes are extracted from unbinned
extended maximum likelihood fits to the mass distributions shown in Fig. 2.
The signal mass shape is described by an empirical model derived from
simulated events. The mass distribution in the simulation exhibits non-
Gaussian tails on either side of the signal. The tail on the right-hand side
is due to non-Gaussian detector effects and modeled with a Crystal Ball (CB)
function Skwarnicki:1986_notitle . A similar tail is present on the left-hand
side of the peak. In addition, the low mass tail contains a second
contribution due to events where hadrons have radiated photons that are not
reconstructed. The sum of these contributions is modeled with a second CB
function. The peak values of these two CB functions are constrained to be
identical.
Various backgrounds have to be considered, in particular the crossfeed between
the $D^{-}$ and $D^{-}_{s}$ channels, and the contamination in both samples
from $\Lambda_{b}\rightarrow\Lambda_{c}^{+}\pi^{-}\,$decays, where
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$. The $D^{-}_{s}$ contamination in
the $D^{-}$ data sample is reduced by loose PID requirements,
$\textrm{DLL}(K-\pi)<10$ (with an efficiency of $98.6\%$) and
$\textrm{DLL}(K-\pi)>0$ (with an efficiency of $95.6\%$), for the pions and
kaons from $D$ decays, respectively. The resulting efficiency to reconstruct
$B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$as background is evaluated, using
simulated events, to be 30 times smaller than for $B^{0}\rightarrow
D^{-}\pi^{+}\,$and 150 times smaller than for $B^{0}\rightarrow
D^{-}K^{+}\,$within the $B^{0}$ and $D^{-}$ signal mass windows. Taking into
account the lower production fraction of $B^{0}_{s}$ mesons, this background
is negligible.
The contamination from $\Lambda_{c}$ decays is estimated in a similar way.
However, different approaches are used for the $B^{0}$ and $B^{0}_{s}$ decays.
A contamination of approximately $2\%$ under the $B^{0}\rightarrow
D^{-}\pi^{+}\,$mass peak and below $1\%$ under the $B^{0}\rightarrow
D^{-}K^{+}\,$peak is found, and therefore no explicit $\textrm{DLL}(p-\pi)$
criterion is needed. The $\Lambda_{c}$ background in the $B^{0}_{s}$ sample
is, on the other hand, large enough that it can be fitted for directly.
A prominent peaking background to $B^{0}\rightarrow D^{-}K^{+}\,$is
$B^{0}\rightarrow D^{-}\pi^{+}\,$, with the pion misidentified as a kaon. The
small $\pi\rightarrow K$ misidentification rate (of about $4\%$) is
compensated by the larger branching fraction, resulting in similar event
yields. This background is modeled by obtaining a clean $B^{0}\rightarrow
D^{-}\pi^{+}\,$sample from the data and reconstructing it under the
$B^{0}\rightarrow D^{-}K^{+}\,$mass hypothesis. The resulting mass shape
depends on the momentum distribution of the bachelor particle. The momentum
distribution after the $\textrm{DLL}(K-\pi)>5$ requirement can be found by
considering the PID performance as a function of momentum. This is obtained
using a sample of $D^{*+}\rightarrow D^{0}\pi^{+}$decays, and is illustrated
in Fig. 1. The mass distribution is reweighted using this momentum
distribution to reproduce the $B^{0}\rightarrow D^{-}\pi^{+}\,$mass shape
following the DLL cut.
Figure 1: Probability, as a function of momentum, to correctly identify (full
symbols) a kaon or a pion when requiring $\textrm{DLL}(K-\pi)>5$ or
$\textrm{DLL}(K-\pi)<0$, respectively. The correspondent probability to
wrongly identify (open symbols) a pion as a kaon, or a kaon as a pion is also
shown. The data are taken from a calibration sample of $D^{*}\rightarrow
D(K\pi)\pi$ decays; the statistical uncertainties are too small to display.
The combinatorial background consists of events with random pions and kaons,
forming a fake $D^{-}$ or $D^{-}_{s}$ candidate, as well as real, $D^{-}$ or
$D^{-}_{s}$ mesons that combine with a random pion or kaon. The combinatorial
background is modeled with an exponential shape.
Other background components originate from partially reconstructed $B^{0}$ and
$B^{0}_{s}$ decays. In $B^{0}\rightarrow D^{-}\pi^{+}\,$these originate from
$B^{0}\rightarrow D^{*-}\pi^{+}\,$and $B^{0}\rightarrow
D^{-}\rho^{+}\,$decays, which can also be backgrounds for $B^{0}\rightarrow
D^{-}K^{+}\,$in the case of a misidentified bachelor pion. In
$B^{0}\rightarrow D^{-}K^{+}\,$there is additionally background from
$B^{0}\rightarrow D^{*-}K^{+}\,$decays. The invariant mass distributions for
the partially reconstructed and misidentified backgrounds are taken from large
samples of simulated events, reweighted according to the mass hypothesis of
the signal being fitted and the DLL cuts.
For $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$, the $B^{0}\rightarrow
D^{-}\pi^{+}\,$background peaks under the signal with a similar shape. In
order to suppress this peaking background, PID requirements are placed on both
kaon tracks. The kaon which has the same sign in the $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$and $B^{0}\rightarrow D^{-}\pi^{+}\,$decays is required to
satisfy $\textrm{DLL}(K-\pi)>0$, while the other kaon in the $D^{+}_{s}$ decay
is required to satisfy $\textrm{DLL}(K-\pi)>5$. Because of the similar shape,
a Gaussian constraint is applied to the yield of this background. The central
value of this constraint is computed from the $\pi\rightarrow K$
misidentification rate. The
$\Lambda_{b}\rightarrow\Lambda_{c}^{+}\pi^{-}\,$background shape is obtained
from simulated events, reweighted according to the PID efficiency, and the
yield allowed to float in the fit. Finally, the relative size of the
$B^{0}_{s}\rightarrow D^{-}_{s}\rho^{+}\,$and $B^{0}_{s}\rightarrow
D^{*-}_{s}\pi^{+}\,$backgrounds is constrained to the ratio of the
$B^{0}\rightarrow D^{-}\rho^{+}\,$and $B^{0}\rightarrow
D^{*-}\pi^{+}\,$backgrounds in the $B^{0}\rightarrow D^{-}\pi^{+}\,$fit, with
an uncertainty of $20\%$ to account for potential SU(3) symmetry breaking
effects.
The free parameters in the likelihood fits to the mass distributions are the
event yields for the different event types, i.e. the combinatorial background,
partially reconstructed background, misidentified contributions, the signal,
as well as the peak value of the signal shape. In addition the combinatoric
background shape is left free in the $B^{0}\rightarrow D^{-}\pi^{+}\,$and
$B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$fits, and the signal width is left
free in the $B^{0}\rightarrow D^{-}\pi^{+}\,$fit. In the $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$and $B^{0}\rightarrow D^{-}K^{+}\,$fits the signal width is
fixed to the value from the $B^{0}\rightarrow D^{-}\pi^{+}\,$fit, corrected by
the ratio of the signal widths for these modes in simulated events.
The fits to the full $B^{0}\rightarrow D^{-}\pi^{+}\,$, $B^{0}\rightarrow
D^{-}K^{+}\,$, and $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$data samples are
shown in Fig. 2. The resulting $B^{0}\rightarrow D^{-}\pi^{+}\,$and
$B^{0}\rightarrow D^{-}K^{+}\,$event yields are $4103\pm 75$ and $252\pm 21$,
respectively. The number of misidentified $B^{0}\rightarrow
D^{-}\pi^{+}\,$events under the $B^{0}\rightarrow D^{-}K^{+}\,$signal as
obtained from the fit is $131\pm 19$. This agrees with the number expected
from the total number of $B^{0}\rightarrow D^{-}\pi^{+}\,$events, corrected
for the misidentification rate determined from the PID calibration sample, of
$145\pm 5$. The $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$event yield is
$670\pm 34$.
Figure 2: Mass distributions of the $B^{0}\rightarrow D^{-}\pi^{+}\,$,
$B^{0}\rightarrow D^{-}K^{+}\,$, and $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$candidates (top to bottom). The indicated components are
described in the text.
The stability of the fit results has been investigated using different cut
values for both the PID requirement on the bachelor particle and for the
multivariate selection variable. In all cases variations are found to be small
in comparison to the statistical uncertainty.
The relative branching fractions are obtained by correcting the event yields
by the corresponding efficiency factors; the dominant correction comes from
the PID efficiency. The dominant source of systematic uncertainty is the
knowledge on the $B^{0}\rightarrow D^{-}\pi^{+}\,$branching fraction (for the
$B^{0}\rightarrow D^{-}K^{+}\,$branching fraction measurement) and the
knowledge of the $D^{-}$ and $D^{-}_{s}$ branching fractions (for the
$f_{s}/f_{d}$ measurement). An important source of systematic uncertainty is
the knowledge of the PID efficiency as a function of momentum, which is needed
to reweight the mass distribution of the $B^{0}\rightarrow
D^{-}\pi^{+}\,$decay under the kaon hypothesis for the bachelor track. This
enters in two ways: firstly as an uncertainty on the correction factors, and
secondly as part of the systematic uncertainty, since the shape for the
misidentified backgrounds relies on correct knowledge of the PID efficiency as
a function of momentum.
The performance of the PID calibration is evaluated by applying the same
method from data to simulated events, and the maximum discrepancy found
between the calibration method and the true mis-identification is attributed
as a systematic uncertainty. The $f_{s}/f_{d}$ measurement using
$B^{0}\rightarrow D^{-}K^{+}\,$and $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$is
more robust against PID uncertainties, since the final states have the same
number of kaons and pions.
Other systematic uncertainties are due to limited simulated event samples
(affecting the relative selection efficiencies), neglecting the
$\Lambda_{b}\rightarrow\Lambda_{c}^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$backgrounds in the $B^{0}\rightarrow D^{-}\pi^{+}\,$fits,
and the limited accuracy of the trigger simulation. Even though the ratio of
efficiencies is statistically consistent with unity, the maximum deviation is
conservatively assigned as a systematic uncertainty. The difference in
interaction probability between kaons and pions is estimated using MC
simulation. The systematic uncertainty due to possible discrepancies between
data and simulation is expected to be negligible and it is not taken into
account. The efficiency of the non-resonant $D_{s}$ decays varies across the
Dalitz plane, but has a negligible effect on the total $B_{s}^{0}\rightarrow
D_{s}^{-}\pi^{+}$ efficiency. The sources of systematic uncertainty are
summarized in Tab. 1.
The efficiency corrected ratio of $B^{0}\rightarrow D^{-}\pi^{+}\,$and
$B^{0}\rightarrow D^{-}K^{+}\,$yields is combined with the world average of
the $B^{0}\rightarrow D^{-}\pi^{+}\,$ PDG_2010_notitle branching ratio to
give
${\cal B}\left(B^{0}\rightarrow D^{-}K^{+}\,\right)=(2.01\pm 0.18\pm
0.14)\times 10^{-4}.$ (4)
The first uncertainty is statistical and the second systematic.
Table 1: Experimental systematic uncertainties for the ${\cal B}\left(B^{0}\rightarrow D^{-}K^{+}\,\right)$ and the two $f_{s}/f_{d}$ measurements. | ${\cal B}\left(B^{0}\rightarrow D^{-}K^{+}\,\right)$ | $f_{s}/f_{d}$
---|---|---
PID calibration | $2.5\%$ | $1.0\%$/$2.5\%$
Fit model | $2.8\%$ | $2.8\%$
Trigger simulation | $2.0\%$ | $2.0\%$
${\cal B}(B^{0}\rightarrow D^{-}\pi^{+}\,)$ | $4.9\%$ |
${\cal B}(D^{+}_{s}\rightarrow K^{+}K^{-}\pi^{+})$ | | $4.9\%$
${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})$ | | $2.2\%$
$\tau_{B_{s}}/\tau_{B_{d}}$ | | $1.5\%$
The theoretically cleaner measurement of $f_{s}/f_{d}$ uses $B^{0}\rightarrow
D^{-}K^{+}\,$and $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}\,$and is made
according to Eq. 2. Accounting for the exclusive $D$ branching fractions
${\cal B}(D^{+}\rightarrow K^{-}\pi^{+}\pi^{+})=(9.14\pm 0.20)\%$
CleoBRD_notitle and ${\cal B}(D_{s}^{+}\rightarrow
K^{-}K^{+}\pi^{+})=(5.50\pm 0.27)\%$ CLEO:2008cqa_notitle , the value of
$f_{s}/f_{d}$ is found to be
$f_{s}/f_{d}~{}=(0.310\pm 0.030^{\textrm{stat}}\pm
0.021^{\textrm{syst}})\times\frac{1}{{\cal N}_{a}{\cal N}_{F}},$ (5)
where the first uncertainty is statistical and the second is systematic. The
statistical uncertainty is dominated by the yield of the $B^{0}\rightarrow
D^{-}K^{+}\,$mode.
The statistically more precise but theoretically less clean measurement of
$f_{s}/f_{d}$ uses $B^{0}\rightarrow D^{-}\pi^{+}\,$and $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$and is, from Eq. 3,
$f_{s}/f_{d}~{}=(0.307\pm 0.017^{\textrm{stat}}\pm
0.023^{\textrm{syst}})\times\frac{1}{{\cal N}_{a}{\cal N}_{F}{\cal N}_{E}}.$
(6)
The two values for $f_{s}/f_{d}$ can be combined into a single value, taking
all correlated uncertainties into account and using the theoretical inputs
accounting for the $SU(3)$ breaking part of the form factor ratio, the non-
factorizable and W-exchange diagram:
$f_{s}/f_{d}~{}=0.253\pm 0.017^{\textrm{stat}}\pm 0.017^{\textrm{syst}}\pm
0.020^{\textrm{theor}}.$ (7)
In summary, with $35$ pb-1 of data collected using the LHCb detector during
the 2010 LHC operation at a centre-of-mass energy of 7 TeV, the branching
fraction of the Cabibbo-suppressed $B^{0}$ decay mode $B^{0}\rightarrow
D^{-}K^{+}\,$has been measured with better precision than the current world
average. Additionally, two measurements of the $f_{s}/f_{d}$ production
fraction are performed from the relative yields of $B^{0}_{s}\rightarrow
D^{-}_{s}\pi^{+}\,$with respect to $B^{0}\rightarrow D^{-}K^{+}\,$and
$B^{0}\rightarrow D^{-}\pi^{+}\,$. These values of $f_{s}/f_{d}$ are
numerically close to the values determined at LEP and at the Tevatron
HFAG_notitle .
## Acknowledgments
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Région Auvergne.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avan$c$cats (ICREA), Barcelona,
Spain
pHanoi University of Science, Hanoi, Viet Nam
(The LHCb Collaboration)
|
arxiv-papers
| 2011-06-22T12:32:25 |
2024-09-04T02:49:19.985019
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Amoraal, J.\n Anderson, R.B. Appleby, O. Aquines Gutierrez, L. Arrabito, A. Artamonov, M.\n Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad,\n T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, S. Brisbane, M. Britsch,\n T. Britton, N.H. Brook, A. B\\\"uchler-Germann, A. Bursche, J. Buytaert, S.\n Cadeddu, J.M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, X. Cid Vidal, P.E.L. Clarke, M. Clemencic, H.V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, F. Constantin, G.\n Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G.A. Cowan, R. Currie, B.\n D'Almagne, C. D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, S. De Capua,\n M. De Cian, F. De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del Buono, C.\n Deplano, O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista,\n D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L.\n Est\\`eve, A. Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli,\n S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M.\n Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J-C. Garnier, J. Garofoli, L. Garrido, C. Gaspar, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, S.\n Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, S.C. Haines, T.\n Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F.\n Harrison, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, W. Hofmann, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt,\n T. Huse, R.S. Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J.\n Imong, R. Jacobsson, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei,\n T.M. Karbach, J. Keaveney, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M.\n Kim, M. Knecht, S. Koblitz, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, S.\n Kukulak, R. Kumar, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty,\n A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.Y. Li, L. Li\n Gioi, M. Lieng, R. Lindner, C. Linn, B. Liu, G. Liu, J.H. Lopes, E. Lopez\n Asamar, N. Lopez-March, J. Luisier, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, J. Magnin, A. Maier, S. Malde, R.M.D. Mamunur, G. Manca, G.\n Mancinelli, N. Mangiafave, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti,\n A. Martens, L. Martin, A. Mart\\'in S\\'anchez, D. Martinez Santos, A.\n Massafferri, Z. Mathe, C. Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A.\n Mazurov, G. McGregor, R. McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel,\n R. Messi, S. Miglioranzi, D.A. Milanes, M.-N. Minard, S. Monteil, D. Moran,\n P. Morawski, J.V. Morris, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R.\n Muresan, B. Muryn, M. Musy, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli,\n M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, V. Niess,\n N. Nikitin, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, B. Pal, J.\n Palacios, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes,\n C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N.\n Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A.\n P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A.\n Petrella, A. Petrolini, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R.\n Plackett, S. Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo,\n D. Popov, B. Popovici, C. Potterat, A. Powell, T. du Pree, V. Pugatch, A.\n Puig Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, I. Raniuk, G.\n Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, K. Rinnert, D.A.\n Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, C. Rodriguez Cobo, P.\n Rodriguez Perez, G.J. Rogers, V. Romanovsky, J. Rouvinet, T. Ruf, H. Ruiz, G.\n Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann,\n M. Sannino, R. Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A.\n Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller,\n S. Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H.\n Schune, R. Schwemmer, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, N.\n Serra, J. Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov,\n Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A.\n Shires, R. Silva Coutinho, H.P. Skottowe, T. Skwarnicki, A.C. Smith, N.A.\n Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B.\n Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica,\n S. Stone, B. Storaci, U. Straumann, N. Styles, S. Swientek, M. Szczekowski,\n P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas,\n E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, M.T.\n Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja, P. Urquijo, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, K. Vervink, B. Viaud, I. Videau, X. Vilasis-Cardona, J.\n Visniakov, A. Vollhardt, D. Voong, A. Vorobyev, H. Voss, K. Wacker, S.\n Wandernoth, J. Wang, D.R. Ward, A.D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Yang, R. Young, O. Yushchenko, M. Zavertyaev, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Vladimir Vava Gligorov",
"url": "https://arxiv.org/abs/1106.4435"
}
|
1106.4458
|
# Fast Neutron Detection with 6Li-loaded Liquid Scintillator
B. M. Fisher brian.fisher@jhuapl.edu National Institute of Standards and
Technology, Gaithersburg, MD 20899 USA J. N. Abdurashitov Institute for
Nuclear Research, Russian Academy of Sciences, Moscow, 117312 Russia K. J.
Coakley National Institute of Standards and Technology, Boulder, CO 80305 USA
V. N. Gavrin D. M. Gilliam J. S. Nico A. A. Shikhin A. K. Thompson D. F.
Vecchia V. E. Yants
###### Abstract
We report on the development of a fast neutron detector using a liquid
scintillator doped with enriched 6Li. The lithium was introduced in the form
of an aqueous LiCl micro-emulsion with a di-isopropylnaphthalene-based liquid
scintillator. A 6Li concentration of 0.15 % by weight was obtained. A 125 mL
glass cell was filled with the scintillator and irradiated with fission-source
neutrons. Fast neutrons may produce recoil protons in the scintillator, and
those neutrons that thermalize within the detector volume can be captured on
the 6Li. The energy of the neutron may be determined by the light output from
recoiling protons, and the capture of the delayed thermal neutron reduces
background events. In this paper, we discuss the development of this 6Li-
loaded liquid scintillator, demonstrate the operation of it in a detector, and
compare its efficiency and capture lifetime with Monte Carlo simulations. Data
from a boron-loaded plastic scintillator were acquired for comparison. We also
present a pulse-shape discrimination method for differentiating between
electronic and nuclear recoil events based on the Matusita distance between a
normalized observed waveform and nuclear and electronic recoil template
waveforms. The details of the measurements are discussed along with specifics
of the data analysis and its comparison with the Monte Carlo simulation.
###### keywords:
capture-gated detection , fast neutron , lithium-6 , neutron detection
††journal: Nuclear Instruments and Methods A111Present address: Johns Hopkins
Applied Physics Laboratory, Laurel, MD, USA
## 1 Overview
Fast neutrons may be produced through several mechanisms. Naturally occurring
isotopes in the 238U and 232Th decay chains generate fast neutrons via
spontaneous fission, and ($\alpha$,n) reactions create neutrons in the 1 MeV
to 15 MeV range. Particle accelerators produce higher energy fast neutrons
through a variety of reactions, and neutrons of very high energies are
generated from cosmic-ray muon-induced spallation reactions [1]. Improvements
in the ability to detect and characterize these neutrons are of interest to
diverse research interests such as fundamental physics, neutron dosimetry, and
detection of low-level neutron emissions.
In the area of basic physics research, many classes of nuclear, particle, and
astrophysics experiments must be performed in underground laboratories to
reduce the backgrounds generated by naturally occurring radioactivity and
cosmic rays [2]. Although experimenters operate their detectors deep
underground and go to great lengths to optimize radiation shielding, the
experiments have become so sensitive that even these efforts may not be
sufficient. The characterization of the fast neutron fluence has become a
critical issue for experiments that require these extreme low-background
environments, such as neutrinoless double-beta decay [3, 4, 5], dark matter
searches [6, 7, 8, 9], and solar neutrino experiments [10, 11, 12, 13, 14,
15]. In some experiments, fast neutrons may be the dominant and potentially
irreducible background, thus necessitating precise information about the fast
neutron fluence and energy spectrum. The most reasonable approach to
addressing the problem is through the complete characterization of the
neutrons through both site-specific measurement [16] and benchmarking of
simulation codes [17, 18].
The health physics community is another area where improved fast neutron
detection and spectroscopy are needed [19]. Existing spectrometers allow the
determination of fluence spectra (and dose) for low energy neutron sources
such as 252Cf and D2O moderated 252Cf, but begin to have difficulty with
higher energy sources such as 241Am-Be($\alpha$, n) or 241Am-B($\alpha$, n).
Current spectrometers fail almost completely for determining neutron fields
with energies of tens of MeV and may require multiple measurements with
different detectors and complicated unfolding procedures. This need has only
grown due to the increased use of 14 MeV neutron generators in interdiction
and inspection technologies. Without good knowledge of these neutron spectra,
health physicists cannot accurately calibrate radiation protection instruments
and dosimeters, which may result in incorrect determination of dose received
by radiation workers.
An improved fast neutron detector has direct application to the detection of
low fluence rates of fast neutrons, such as from fissile material. The
technological challenges and requirements for measuring the fast neutron
fluence in the underground environment are very similar to this problem. A
highly efficient neutron spectrometer with reasonable resolution would be
capable of detecting low-level neutron signals from a number of sources.
In this paper, we present the developmental work performed using a small
detector filled with an enriched 6Li liquid scintillator. General methods of
neutron detection and spectroscopy may be found elsewhere [20, 21]. Section 2
discusses capture-gated neutron detection and the fabrication of the 6Li-
loaded liquid scintillator. In Section 3 the results of measurements with a
test detector of 6Li-loaded liquid scintillator and a B-loaded plastic
scintillator are presented along the analysis methods. Section 4 presents a
pulse-shape discrimination method for differentiating between electronic and
nuclear recoil events.
## 2 Neutron Detection with 6Li Scintillator
### 2.1 Capture-Gated Neutron Detection
The detection method used in this work is known as capture-gated neutron
spectroscopy [22, 23, 24, 25]. As illustrated in Figure 1, an incident fast
neutron preferentially scatters from a proton in a scintillation medium. The
proton recoils with approximately half (on average) of the neutron’s energy,
producing scintillation light. The neutron then scatters off another proton,
producing another proton recoil, and continues to scatter until it leaves the
detector volume or is captured, typically at thermal energies. A large
fraction of the neutrons that lose all their energy within the scintillator
volume and then capture will have lost most of their energy through neutron-
proton scattering, and the majority of the detected light will come from those
neutron-proton events. There are other mechanisms that may generate light, in
particular scattering from carbon [26], but these are small effects for the
measurements performed here.
An organic scintillator provides a high density of protons that will
efficiently moderate the neutron; a neutron with an energy of a few MeV loses
90 % of its energy in this manner during the first 10 ns in the scintillator.
The thermalized neutron diffuses within the bulk of the scintillator and may
capture on an isotope that was introduced because of its large neutron capture
cross section and unambiguous signature of the neutron capture. The time to
capture occurs within approximately 100 ${\rm{\mu}}$s and depends upon upon
the size and geometry of the detector and the concentration and neutron
capture cross section of the isotope used as the dopant. The delayed-
coincidence signature indicates that the fast neutron gave up essentially all
of its energy within the scintillator and provides a method of rejecting of
uncorrelated backgrounds.
Figure 1: An illustration of the principle of capture-gated detection. A fast
neutron impinges on the detector. It rapidly gives up its energy through
nuclear collisions, primarily with protons, in the moderation process. The
thermalized neutron diffuses in the medium until it is captured on a material
with a high capture cross section.
One must consider several factors when selecting which capture isotope is
appropriate for the application. Typical elements are gadolinium, boron, and
lithium. Gadolinium is commercially available and has a very large neutron
capture cross section, but it produces gamma rays of various energies (up to 9
MeV) that may be difficult to detect with small volume detectors. In addition,
the Gd gamma rays cannot be distinguished from background gamma rays by means
of pulse shape discrimination. With neutron capture on 10B, the 7Li can be
formed in a ground or excited state, leading to two reaction branches:
${}^{10}{\rm B}+n\rightarrow{{}^{7}{\rm Li}}^{*}\,(0.84\,{\rm
MeV})+\alpha\,(1.47\,{\rm MeV})$ (1) ${}^{7}{\rm Li}^{*}\rightarrow^{7}{\rm
Li}+\gamma\,(0.477\,{\rm MeV})$ ${}^{10}{\rm B}+n\rightarrow{{}^{7}{\rm
Li}}\,(1.01\,{\rm MeV})+\alpha\,(1.78\,{\rm MeV}).$ (2)
The branching ratio for the first reaction is 93.7 % and 6.3 % for the second.
The cross section is also large, and the alpha particle makes efficient
thermal detection possible with a small-volume detector. Boron-loaded
detectors are expensive, which tends to make scaling to larger-volume
detectors cost-prohibitive.
A good alternative to 10B is 6Li
${}^{6}{\rm Li}+n\rightarrow t\,(2.05\,{\rm MeV})+\alpha\,(2.73\,{\rm MeV}).$
(3)
The use of enriched 6Li as the dopant has the advantage of a large Q-value and
the production of two energetic charged-particles. The light yield of the
2-MeV triton is nearly a factor of 10 higher than that of the 1.5-MeV alpha
from neutron capture on 10B. The neutron capture signature from 6Li is well-
separated from the noise and most background sources. There is no concern
about the energy leaving the scintillator as there is with dopants that
produce gamma rays, and pulse shape discrimination between gammas and the
capture products is possible. The capture cross section is high, but it less
than for Gd and 10B.
The capture-gated detection method using 6Li has been implemented in different
ways. A 5-L detector was constructed for measurements in the Gran Sasso
Laboratory [27], and an 8-L detector was used to measure the fast neutron
background at the Modane Underground Laboratory in Modane, France [28]. Both
detectors used the commercial 6Li-doped liquid scintillator NE320 [29]. In a
different approach, a plastic scintillator impregnated with lithium gadolinium
borate crystals was developed and was demonstrated to detect neutrons over a
wide range of energies [24]. Studies of the response of B-loaded liquid
scintillator to monoenergetic neutrons have also been carried out [30].
### 2.2 6Li-loaded Liquid Scintillator
Isotope | Scintillator | $\sigma_{n}$ | $f$(%) | Volume | H/C
---|---|---|---|---|---
6Li | liquid | 941 b | 0.15 | 125 cm3 | 1.5
B | plastic | 3842 b | 1.1 | 103 cm3 | 1.1
Table 1: Properties of the two scintillator materials used in testing.
$\sigma_{n}$ is the 2200 m/s neutron capture cross section; $f$ is the
fractional weight of the isotope in the scintillator; and H/C is the hydrogen
to carbon ratio of the scintillator.
Natural lithium for use as a neutron capture material was introduced into
organic liquid scintillators more than 50 years ago [31, 32]. More recently a
solution made using 6Li-salicylate dissolved in a toluene-methanol solvent
[33] was developed. Scintillators with enriched 6Li were developed later, and
in the 1980s Nuclear Enterprises manufactured a lithium-loaded pseudocumene-
based scintillator (NE320 [34]), which was used to make several measurements
of neutron backgrounds in underground laboratories [29, 27, 28]. This
scintillator is no longer being manufactured, and to the authors’ knowledge
there are currently no vendors of scintillator doped with enriched 6Li. Hence,
detectors incorporating 6Li have not been made for many years due to the lack
of commercial availability of a scintillator.
Given the possible applications for a 6Li-doped scintillator, we have
investigated approaches for introducing lithium compounds into liquid
scintillators. After exploring methods for incorporating organolithium
compounds directly into an organic scintillator, we investigated technology
used in liquid-scintillation counting in the life sciences. Radioactive
samples used in biochemistry are often low-energy beta emitters in an aqueous
solution, such as tritium in urine. The very short range of these low-energy
$\beta$-particles make traditional counting very difficult. The liquid
scintillators used in this field have additional surfactants that allow for
the creation of micro-emulsions between the aqueous solution and the organic
liquid scintillator. This permits the counting sample to be uniformly
distributed throughout the scintillator, greatly increasing its counting
efficiency for low energy $\beta$-particles.
A critical property of the desired organic scintillator was its ability to
distribute and suspend a lithium-containing compound uniformly throughout its
volume. We found a scintillator manufactured by Zinsser Analytic [34] composed
of a solvent (very high-purity di-isopropylnaphthalene) with added surfactants
allowing introduction of aqueous solutions up to 40 % water by volume. Di-
isopropylnaphthalene is considered a safer solvent than the more common
pseudocumene-based liquid scintillators. It has a higher flashpoint
(approximately 150 ∘C) and is biodegradable, allowing for safer handling and
operation.
Figure 2: Block diagram of the data acquisition electronics. The two inputs to
the digitizer allow for different range settings.
A 10-molar aqueous solution of 6Li-enriched LiCl was created by reacting
enriched Li2CO3 (95 % 6Li by weight) with concentrated hydrochloric acid,
boiling off the excess acid, and dissolving the remaining dried LiCl salt in
pure de-ionized water. Mixing this LiCl (aqueous) solution with the liquid
scintillator, we achieved 6Li concentrations of approximately 0.15 % by weight
[35]. This concentration is comparable to what had been commercially available
and used previously [28, 27]. This scintillator mixture was found to have good
light properties, and no precipitation of LiCl was seen during the duration of
testing.
## 3 Detector Characterization
### 3.1 6Li-loaded Liquid Scintillator Detector
A 125-mL cell was filled with this scintillator and inserted into fast neutron
fields to demonstrate its effectiveness as a neutron detector. Some
characteristics of the cell and scintillator are given in Table 1. A 5.08-cm
diameter photomultiplier tube (PMT, Burle 8850) was mounted on one face of the
5.08-cm diameter by 5.08-cm long cylindrical cell, and the remaining walls
were coated with a diffuse, white reflector. The logic of the data acquisition
electronics is shown in the block diagram of Fig. 2. The PMT signal was
amplified and split into two signals. One output was delayed 64 ns to avoid
self-triggering, and it became the start signal in a time-to-amplitude
converter. To further reduce the probability of self-triggering, the
discriminator was vetoed for 500 ns after an event. The trigger was formed on
any event above a threshold followed by a second event that occurred within 40
$\mu$s; the waveform containing both events was digitized and recorded to
disk. The data acquisition system was a single 2-channel 2 GHz digital
oscilloscope card with 8-bit resolution.
Figure 3: Plot of the digitized PMT pulse of the detector response to a fast
neutron within the scintillator. A neutron enters the test cell and initially
scatters off the protons in the scintillator. After thermalizing within the
scintillator, it captures on a 6Li, in this example approximately 1.4 $\mu$s
later.
Figure 3 gives a sample trace showing the initial proton recoil of a fast-
neutron off hydrogen in the scintillator followed by the 6Li-capture pulse.
All of the analysis was performed on these digitized waveforms. Each waveform
was analyzed to determine the energy of each pulse and the relative timing of
the two events. The pulse offset was determined by averaging 200 ns of the
baseline near the stop event. The pulse height corresponds to the maximum
voltage of the pulse, and the time of the pulse was obtained from the timing
channel corresponding to the position of the maximum pulse height. Histograms
of these parameters from the measured data were constructed from the analysis
of all events.
The light yield of the scintillator is nonlinear as a function of proton
recoil energy. Thus, the sum of all the light from individual recoil events is
not proportional to the incident neutron energy. This results in the
degradation of the energy resolution. By segmenting the detector, one can
ensure that the probability of having more than one energy deposit in any
segment is very small and thus reconstruct the initial neutron energy by
summing the light output of the individual segments [36]. This produces a
detection scheme that can still be efficient while achieving good energy
resolution and suppression of uncorrelated backgrounds. There is an effort in
progress to construct a 16-channel pilot spectrometer based on this idea [37].
Figure 4: Results from a demonstration of the 6Li-loaded liquid scintillator
cell irradiated by a 252Cf source. The top plot shows the energy distribution
of the start events (primarily recoil events) and stop events (primarily
capture on 6Li). The peak in the spectrum of stop events corresponds to the
total energy of the 6Li reaction products. The bottom plot shows the delay
time (between start and stop events) distribution for event sequences where
the stop event produces light consistent with neutron capture on 6Li. The
error bars represent plus and minus the combined standard uncertainty.
The test cell was exposed to fission neutrons from 252Cf sources and 2.5-MeV
monoenergetic neutrons from a commercial neutron generator. Both the sources
and generator are maintained at the NIST Californium Neutron Irradiation
Facility (CNIF) [38]. The top plot of Figure 4 shows the energy spectrum for
both the start and stop pulses of the trigger when the detector was irradiated
with neutrons from a 252Cf source. The start events correspond primarily to
proton recoil events while the stop events primarily correspond to the capture
of the thermalized neutron on 6Li. The peak in the capture spectrum is due to
the reaction products from the 6Li(n,$\alpha$)3H reaction; most of the
scintillation light comes from the triton. The peak is clearly separated from
the noise, and the signal-to-background is more than 10:1. The energy spectra
were calibrated using the Compton edges of gamma sources such as 133Ba and
137Cs, as there is little photopeak detection in the liquid scintillator. Note
that the energy scale for both plots is given in the electron-equivalent
energy. The timing data were fit to a two-component exponential with a
constant background, as shown in the bottom plot of Figure 4. The error bars
represent plus and minus the combined standard uncertainty. A long component
arises from the capture of thermalized neutrons as they diffuse within the
bulk of the scintillator. The simulation indicates that at times below
$1\,\mu$s there is a population of partially thermalized neutrons that can
still capture on 6Li or 10B. These events give rise to a second, faster
component in the timing spectrum.
Because the data were digitized event-by-event, recoil events that do not
correspond to a valid neutron capture can be rejected in analysis. Two
important cuts are the energy of neutron capture and the timing of the
capture. An energy cut can be made around the 6Li capture peak (shown in Fig.
4) to reject with high probability recoil events that do not have a valid
capture associated with them. The capture peak was fit to a gaussian function,
and the energy window was chosen to be $\pm 3$ sigma around the peak channel
of the distribution. The bottom plot of Fig. 4 shows the time distribution of
capture times for the thermalizing neutron after the energy cut was made to
keep only events with neutron captures on 6Li. One can use the timing spectrum
to perform another cut. At long times, the correlated capture events are gone,
and one is left with the uncorrelated background. The spectrum of these
background events can be subtracted from energy spectrum of the recoil events.
Typically, the signal window was $0.5\,\mu$s to $10\,\mu$s and the background
window was $10\,\mu$s to $20\,\mu$s. The relative size of the windows could
vary slightly from run to run depending on the signal-to-background in the
particular time spectrum.
Data were also acquired in a similar data manner using plastic scintillator
loaded with 5 % natural boron (BC-454), corresponding to about 1 % 10B. Some
its characteristics are given in Table 1. The scintillator was a 5.08 cm by
5.08 cm right cylinder coupled to a 5.08 cm diameter PMT (Burle 8850). The
setup, data acquisition, and analysis were carried out in the same manner as
that done for the 6Li-loaded cell. These data were useful for comparison with
the 6Li data and also as another check on the computer simulations.
The top plot of Figure 5 shows the energy spectrum for both the start and stop
events when the detector was irradiated with neutrons from a 252Cf source. The
energy scale was calibrated using a 133Ba source and the Compton edge of the
477 keV photon, which is seen in the stop spectrum. The number of events for
the gamma is less than for the alphas due to the lower efficiency for stopping
the gammas in the small volume of plastic. The bottom plot of Figure 5 shows
the time distribution of neutron captures on the 10B. As with the 6Li cell,
the distribution was made after an energy cut on alpha peak in the stop
spectrum, thus requiring that there was a valid neutron capture. The
distribution of delayed-capture times fits well to the 2-component exponential
function.
Figure 5: The top plot shows the energy distribution of the start and stop events for the irradiation of plastic scintillator by neutron from 252Cf. The large peak in the spectrum of stop events comes from the alpha particle and the smaller peak to the right is the Compton edge of the 477 keV gamma. The bottom plot shows the time distribution of all events that have a valid neutron capture on 10B. | Pb shield | Activity | Number of | Range | $R_{Tot}$ | $R_{bkgd}$ | Threshold
---|---|---|---|---|---|---|---
| | (Bq) | Positions | (cm) | (s-1) | (s-1) | (keV)
6Li liquid | off | $1.2\times 10^{6}$ | 7 | 107 – 230 | 3.90 | 3.77 | 75
scintillator | on | $1.2\times 10^{6}$ | 6 | 86 – 230 | 3.72 | 3.40 | 76
| off | $1.0\times 10^{3}$ | 6 | 8 – 33 | 0.308 | 0.258 | 86
| on | $1.0\times 10^{3}$ | 6 | 9 – 30 | 0.0530 | 0.023 | 93
B-plastic | off | $1.2\times 10^{6}$ | 7 | 81 – 231 | 5.07 | 4.71 | 29
scintillator | on | $1.2\times 10^{6}$ | 7 | 74 – 221 | 3.32 | 2.01 | 31
| off | $1.0\times 10^{3}$ | 5 | 8 – 30 | 0.729 | 0.577 | 30
| on | $1.0\times 10^{3}$ | 5 | 8 – 30 | 0.149 | 0.021 | 32
Table 2: Table of some of the run parameters for the 6Li liquid scintillator
and the B-plastic scintillator. The first column states whether the
scintillator had a lead shield; the second column indicates the activity of
the 252Cf source; the third column gives the number of runs that were done at
different positions; the fourth column gives the range of source-detector
distances $r$; the fifth column gives the measured signal rate (at the closest
position only); the sixth column gives the background rate (at the closest
position only); and the seventh column list the low-energy threshold that was
used for both the data and simulation.
### 3.2 Efficiency Measurements
Both the 6Li cell and the B-plastic scintillator were irradiated by 252Cf
sources of known activity in the CNIF, and measurements of their efficiencies
for fast neutron detection were performed. Irradiations were carried out using
two source activities and two gamma shielding configurations, and the
measurements were done at several source-detector distances. For the gamma
shielding, there was either a thin (6 mm) annulus of lead around the detector
or there was no shielding at all. The purpose in using several configurations
was to test the robustness of the analysis and simulation under varying
conditions, such as the detected rate, detector composition, background
subtraction, and dead time. Table 2 gives some of the rates and run parameters
associated with the data set.
When irradiated with a source, the neutron rate $R_{n}$ is the difference
between the total measured rate $R_{tot}$ and the uncorrelated background rate
$R_{bkgd}$ that is associated with the source. The numerical values are
obtained after applying both the energy and timing cuts of Section 3.1. The
neutron rate can also be expressed as
$\displaystyle R_{n}=R_{tot}-R_{bkgd}=\epsilon
A_{\mathrm{Cf}}\Omega+R_{rr}+R_{b},$ (4)
where $\epsilon$ is the detector efficiency, $A_{\mathrm{Cf}}$ is the neutron
source activity, and $\Omega$ is the solid angle subtended by the detector
from the source. $R_{rr}$ is the rate in the detector due to room-return
neutrons (i.e., those source neutrons that scatter from the surrounding
environment into the detector), and $R_{b}$ is the background rate in the
detector when there is no source present. The measured rates must be obtained
above a low-energy threshold. The values were chosen to be safely above the
hardware threshold and the detector noise, and they are listed for each
configuration in Table 2.
The background rate $R_{b}$ is negligible compared with the room return and
can be ignored for these measurements. The room return contribution, however,
can be a significant fraction of the measured rate and must be taken into
account. The room return will be very dependent upon the geometry and material
composition of the room. In the CNIF the rate is largely constant over the
range of measurement positions for a given source activity [39, 40]. To
determine $R_{rr}$ for each detector and each source that were used,
measurements were taken with the source placed at several distances, $r$, from
the detector. When appropriately corrected for deadtime, the intercept of a
linear fit to the total rate in the detector versus $1/r^{2}$ gives the value
of $R_{rr}$. Figure 6 shows the fit for data acquired using the 6Li
scintillator with the lead shield in place and using the higher-activity 252Cf
source. With the measured values of the room return, the known source
activity, and the calculated solid angle, one obtains the efficiency of the
detector using Eq. 4. Table 3 shows the results for all eight configuration
under which data were acquired.
For the lifetimes, both the data and simulation were fit to a two-component
exponential function, as discussed in Section 3.1. The fast component was
obtained from the fit to the simulation and held fixed in the fit to the data.
For the B-plastic scintillator, the lifetime of the fast component was
$0.47\,\mu$s, and for the 6Li-liquid scintillator it was equal to $1.6\,\mu$s.
All the other parameters were allowed to float. This approach produced better
fits to the data, particularly where the small amplitude of the component
caused difficulty for the fit. The value of the long component for each
configuration is given in Table 3. The stated uncertainties arise from
statistical treatment of the data alone.
Figure 6: Plot of the total neutron rate $R_{n}$ versus $1/r^{2}$. The intercept of the linear fit gives $R_{rr}$, the contribution of neutrons that scatter from the room and are detected. These data are from the 6Li liquid scintillator with the lead annulus in place and using the higher-activity 252Cf source. The error bars represent plus and minus the combine standard uncertainty. | Pb shield | Activity | Efficiency $\epsilon\ (\times 10^{-3})$ | Lifetime ($\mu$s)
---|---|---|---|---
| | (Bq) | Data | Simulation | Data | Simulation
6Li liquid | off | $1.2\times 10^{6}$ | $1.3\pm 0.1$ | $1.4\pm 0.1$ | $12.0\pm 2.2$ | $10.2\pm 0.3$
scintillator | on | $1.2\times 10^{6}$ | $1.6\pm 0.1$ | $1.7\pm 0.1$ | $11.0\pm 1.9$ | $10.7\pm 0.2$
| off | $1.0\times 10^{3}$ | $1.3\pm 0.1$ | $1.2\pm 0.1$ | $14.2\pm 3.4$ | $10.5\pm 0.3$
| on | $1.0\times 10^{3}$ | $1.4\pm 0.1$ | $1.4\pm 0.1$ | $10.0\pm 2.2$ | $10.8\pm 0.4$
B-plastic | off | $1.2\times 10^{6}$ | $3.0\pm 0.2$ | $4.1\pm 0.5$ | $1.66\pm 0.20$ | $1.54\pm 0.17$
scintillator | on | $1.2\times 10^{6}$ | $4.0\pm 0.2$ | $5.0\pm 0.5$ | $1.80\pm 0.07$ | $1.69\pm 0.06$
| off | $1.0\times 10^{3}$ | $3.6\pm 0.2$ | $3.5\pm 0.5$ | $1.78\pm 0.08$ | $1.62\pm 0.06$
| on | $1.0\times 10^{3}$ | $3.9\pm 0.1$ | $4.4\pm 0.5$ | $1.84\pm 0.07$ | $1.70\pm 0.08$
Table 3: Comparison of the measured efficiency and capture lifetime from the
MCNP model to the eight configurations of the 6Li liquid scintillator and
B-plastic detectors. The uncertainty in the efficiency values is the standard
deviation of all the individual efficiencies determined at each value of the
detector-source distance $r$. The error bars for the data results come from
fits to the data and represent plus and minus the combined standard
uncertainty. The error bars for the simulation results represent the standard
uncertainty.
### 3.3 Comparison with Simulation
A Monte Carlo simulation was constructed using MCNP5 [41] to compare with the
measurements described in Section 3.2. Accurate benchmarking of this Monte
Carlo model against the laboratory measurements is important for understanding
the response of the detector and scintillator. The model of the test cell
consisted of a right cylinder with the same dimensions as the actual cell and
filled with the 0.15 % 6Li-loaded liquid scintillator. The 6 mm thick lead
annulus surrounding the cylinder was also modeled to compare with the shielded
measurements taken in the CNIF. The 252Cf source energy spectrum was assumed
to be Maxwellian, and the stainless steel source encapsulation was modeled to
take into account any attenuation arising from neutrons scattering in the
source holder. No attempt was made to model the CNIF room-return or
backgrounds as they were taken into account in the measurements.
A neutron slowing down in a liquid scintillator loses its energy primarily
through scattering with protons, and it is the interaction of each recoiling
proton that produces the scintillation light. If the number of scintillation
photons produced is a linear function of proton recoil energy, the response of
a detector can be modeled by converting the total energy deposition of a given
neutron’s slowing down history, a quantity easily obtainable from the standard
output tally of MCNP5. For incident gamma-rays the light-output function is
linear over a wide range of energies. For nearly all organic scintillators,
however, the light-response function for heavy charged particles is non-
linear. Thus, the light output from a neutron that deposits all of its energy
in a single proton scatter is not the same for one that deposits all of its
energy through multiple scatterings. To accurately model the detector response
to fast neutrons, light output from each individual proton recoil in a given
neutron history has to be summed to accurately model the scintillator response
to neutrons. We assumed that the light output of proton recoils in both the
liquid 6Li-loaded scintillator and the B-loaded plastic scintillator was the
same as that for the pseudocumene-based liquid scintillator NE-213 and the
plastic scintillator NE-102, which is given by
$E_{\mathrm{ee}}=0.95E_{\mathrm{p}}-8.0\left[1-\exp(-0.1\,E_{\mathrm{p}}^{0.9})\right],$
(5)
where $E_{\mathrm{p}}$ is the recoiling proton’s energy, and $E_{ee}$ is the
electron-equivalent light output in units of MeV [42, 43].
A custom post-processing routine was developed to parse the detailed neutron
history output from MCNP5 (called a PTRAC file) and to extract the pertinent
quantities from each of the individual scattering events for a given neutron
history. Only neutron histories in which the neutron scattered from at least
one proton and was captured on a 6Li or 10B nucleus were examined. For each
history in this subset, the electron-equivalent energy was calculated for each
proton recoil and then summed to give the total light output for that
particular incident neutron. The light output was summed from the start of the
neutron until the neutron was captured, and in addition, the time between the
first proton scatter and the capture was recorded and histogrammed.
The comparisons between the measurements and the model results are listed in
Table 3. The two parameters on which we primarily focused were the efficiency
and the capture lifetime. The same energy and timing cuts applied in the
analysis of the data were applied to the Monte Carlo results. Ten MCNP5 runs
were performed for each configuration (of source strength, shielding, and
detector type), each with a different initial seed to the random number
generator. The results for detector efficiencies and lifetime were averaged
across these 10 runs, and the reported error bars are the standard deviation
of the mean of those values. We did not quantify the systematic effects but
believe that the two more significant effects arise as a result of uncertainty
in the energy calibration of the measurements, which determines the value of
the energy threshold, and knowledge of the appropriate light output function.
The agreement between the simulation and measured data is good. For the test
cell containing 6Li liquid scintillator, an average efficiency for detecting
fast 252Cf neutrons was on the order of $1\times 10^{-3}$, which agreed well
with the MCNP5 Monte-Carlo models. The efficiency for the B-plastic was a few
times higher, which we attribute to the larger cross section and higher
concentration of the 10B in the scintillator. The simulation indicates that
the efficiencies for the shielded configurations were slightly higher than
those of the unshielded configurations. This can be attributed to in-
scattering of neutrons from the very dense lead shielding surrounding the
scintillator cells. Slight variations between the weak and strong source
efficiencies can be attributed to the differences in source encapsulation. The
time distribution of the capture events also agrees well with the measured
lifetimes.
## 4 Pulse-Shape Discrimination
The background rejection of the scintillator was very good due primarily to
the delayed capture on 6Li, as discussed in Section 2.1. In addition to timing
information, one can further discriminate between gamma rays (electronic
recoil scattering) and neutrons (nuclear recoil scattering or neutron capture)
based on the pulse waveform shape information. Due to the different energy
deposition mechanisms, the temporal probability density functions for
scintillation light creation times produced by gamma rays and neutrons are not
the same [44]. Hence, one can further discriminate background gamma events
from nuclear recoil and neutron capture events [45, 46, 47] based on the
observed pulse produced by each event.
We describe a pulse-shape discrimination (PSD) method based on the “Matusita
distance” [48, 49, 50] between an observed pulse and nuclear and electronic
recoil templates for normalized event waveforms. We also consider a standard
prompt ratio method. For both methods, we estimated a discrimination threshold
for accepting approximately 50 % of the nuclear recoil events.
### 4.1 Calibration Data
We acquired digitized pulses in two calibration measurements: one with 137Cs
that produced essentially only gamma ray waveforms and another with a 2.5-MeV
neutron generator that produced neutrons with some small admixture of gamma
rays. The data acquisition system that captured the waveforms is the same as
was used for the previous measurements. The purpose was to estimate the
expected value of a normalized and background-corrected pulse (that sums to 1)
for both the nuclear recoil and electronic recoil events. We estimated the
background level as a trimmed mean of all values of the pulse. The values that
contributed to the trimmed mean range from 0.125 quantile to 0.875 quantile of
the distribution of observed pulse values. We registered each pulse according
to the time when the pulses steeply rises from its baseline (background)
plateau to 0.3 times its maximum value. For each pulse, we determined the mean
value of the pulse for a 10 ns interval before this rise time. If this mean
value exceeded 0.003 times the pulse amplitude, we rejected the pulse.
### 4.2 Template Estimation
We estimated both the nuclear and electronic recoil templates with a k-means
cluster analysis [51] from calibration data produced by a 2.5-MeV generator,
shown in Fig. 7. Joint estimation of the nuclear and electronic recoil
templates is possible because this calibration data is contaminated by gamma
rays. Figure 7 shows the waveform templates determined from the 2.5-MeV
source. We determined the templates by minimizing the squared Euclidean
distance ($L^{2}$) of the normalized pulses within each of the two clusters.
We also estimated templates with a robust version of cluster analysis based on
an $L^{1}$ distance metric. In this approach, within each cluster, the median
value rather than the mean value is computed. The robust and non-robust
cluster analysis methods yield similar template estimates.
Figure 7: Waveform templates for nuclear recoil and electronic recoil events
determined by cluster analysis from calibration data from a 2.5-MeV neutron
source (contaminated by gamma rays)
From the 137Cs gamma ray source, we determined an electronic recoil template
by a robust signal averaging method. Each baseline-corrected pulse was
normalized so that its maximum value was 1. At each time sample, the trimmed
mean of all the processed pulses was computed, and the resulting pulse was
divided by its integral value. Values of the trimmed mean at each relative
time of interest between the 0.1 and 0.9 quantiles of the distribution were
averaged. For the 2.5-MeV source, we estimated a nuclear recoil template with
the same robust signal averaging method described above. The estimated nuclear
recoil and electronic recoil templates from the cluster analysis agree well
with the corresponding robust signal averaging estimates. Moreover, the
estimated nuclear recoil templates determined from start and stop pulses for
the 2.5 MeV case were in very close agreement for the range of amplitudes that
we attribute to neutron capture on 6Li.
### 4.3 Discrimination statistics
The Matusita distances between a normalized pulse of interest, $p_{m}$, and
the template pulses for the electronic recoil $\hat{p}_{e}$ and nuclear recoil
events $\hat{p}_{n}$ are
$\displaystyle
d_{e}=\sum_{i}(~{}~{}\sqrt{p_{m}(i)}-\sqrt{\hat{p}_{e}(i)}~{}~{})^{2}$ (6)
and
$\displaystyle
d_{n}=\sum_{i}(~{}~{}\sqrt{p_{m}(i)}-\sqrt{\hat{p}_{n}(i)}~{}~{})^{2},$ (7)
where $i$ is the time increment for the digitized pulse. The normalized pulses
sum to 1. Negative values are set to 0 before taking square roots in the above
equations. Our primary PSD statistic is
$\displaystyle\log R=\log\frac{d_{n}}{d_{e}}.$ (8)
For comparison, we also computed a prompt ratio statistic
$\displaystyle f_{p}=\frac{X_{p}}{X_{T}},$ (9)
where ${X_{p}}$ is the integrated pulse from $t=0$ to $t_{o}$, and $X_{T}$ is
the integrated pulse over all times. Here, we set $t_{o}$ to be the time where
the nuclear and electronic recoil pulses cross.
For both discrimination statistics, Figure 8 and Figure 9, we estimate an
amplitude dependent discrimination threshold based on events that produce
$\log R$ values less than 0. We then formed a curve in (amplitude, $\log R$)
or (amplitude, $f_{p}$) space. For each method, we sorted the corresponding
curve data according to amplitude bins and determine the median amplitude and
median discrimination statistic within each bin. In sequence, we fit a
monotonic regression model [52] and then a smoothing spline to each curve. The
degrees of freedom of the smoothing spline were determined by cross validation
[53] We determined a threshold for each particular amplitude by evaluating the
smoothing spline model at that amplitude.
Figure 8: Empirical distribution of $\log R$ statistics.
The separation between the $\log R$ statistics appears more dramatic than the
separation between the $f_{p}$ statistics for the 137Cs and 2.5-MeV sources.
Theoretically, we expect that the $\log R$ statistic conveys more information
because it is based on a 201-bin representation of the observed pulse whereas
the prompt ratio is based on a 2-bin representation of the observed pulse. A
careful quantification of the relative performance of PSD algorithms based on
these two statistics is a topic for further study. One could also form larger
bins to smooth out noise before computing a $\log R$ statistic for any pulse
as discussed in Ref. [54, 55]. In future experiments, our digital acquisition
system will have a higher (10-bit or 12-bit) resolution compared to the 8-bit
resolution of the data shown in this study. This should facilitate refinement
of our PSD techniques. In this work, we neglected to account for the energy
dependence of the templates. In future work, we may account for this
dependence.
Figure 9: Empirical distribution of prompt ratio statistics. The width of the
prompt time window is determined by where the nuclear and electronic recoil
templates cross.
## 5 Summary and Conclusions
A liquid scintillator doped with 0.15 % 6Li by weight was fabricated and made
into a test cell. The process of making the scintillator does not require
complicated chemical techniques, and the data acquisition system and analysis
are straightforward. The cost of the raw materials was not high. The cell was
tested in a known field of fast neutrons and the capture-gated detection
method worked as expected. The data acquired for both the 6Li-liquid and the
B-plastic scintillator were compared with Monte Carlo simulations yielding
good agreement. The detection efficiency is not large (on the order of
$10^{-3}$), but that is primarily due to the small size of the test cell.
Starting with a detector of this volume, the efficiency scales more rapidly
than the volume, so a high efficiency detector need not be prohibitively large
or expensive. The spectroscopic performance of the detector was not
demonstrated here, but in future work, we will investigate the response of the
detector to 2.5-MeV and 14-MeV monoenergetic neutrons.
One sample has been used in this testing over the course of about one year and
did not shown noticeable degradation. There is some loss of light yield due to
the introduction of the LiCl, but it is not prohibitively large. Future work
will focus on quantifying optical properties and increasing the concentration
of the 6Li in the scintillator. In principle, higher concentrations would be
better in terms of increasing the efficiency and decreasing the capture time,
but one also has to consider the potential degradation of light transmission
in the scintillator. We note that the variety of possible 6Li-loaded
scintillators has not been explored. LiCl was chosen initially due to its
relatively high solubility in water and for the ease with which LiCl can be
produced from isotopically-enriched lithium carbonate. There may be other Li-
containing compounds where one can form stable emulsions with high
concentrations of 6Li. Scintillators from other manufacturers should be
investigated for durability and optical properties. While this paper gives
data on the characteristics of detecting fast neutrons, the scintillator is
naturally a very efficient detector of thermal neutrons. Future work will
examine those properties.
Timing information based on the delay time between a neutron energy deposit
and its capture on 6Li enables one to reject backgrounds with high
probability. To improve background rejection based on this timing information,
we presented a PSD method based on the log-ratio of the Matusita distances
between a normalized waveform and template waveforms for nuclear recoil and
electronic recoil events. In the future, we plan to quantify the relative
performance of this multidimensional method (Figure 8) with respect to a
simpler prompt ratio discrimination method (Figure 9). We presented a method
to determine the width of the prompt time window based on when the template
waveforms cross. We also presented a nonparametric method to determine an
approximate 50 percent nuclear recoil acceptance threshold for the prompt
ratio and Matusita distance PSD methods.
We thank C. Bass and C. Heimbach of NIST and E. J. Beise, H. Breuer and T.
Langford of the University of Maryland for useful discussions. This work was
supported in part by the American Civil Research and Development Foundation,
grant no. RP2-2277.
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|
arxiv-papers
| 2011-06-22T14:19:45 |
2024-09-04T02:49:19.994192
|
{
"license": "Public Domain",
"authors": "B. M. Fisher, J. N. Abdurashitov, K. J. Coakley, V. N. Gavrin, D. M.\n Gilliam, J. S. Nico, A. A. Shikhin, A. K. Thompson, D. F. Vecchia, and V. E.\n Yants",
"submitter": "Jeffrey Nico",
"url": "https://arxiv.org/abs/1106.4458"
}
|
1106.4622
|
# Existence of positive solutions to quasi-linear elliptic equations with
exponential growth in the whole Euclidean space
Yunyan Yang yunyanyang@ruc.edu.cn Department of Mathematics, Renmin
University of China, Beijing 100872, P. R. China
###### Abstract
In this paper a quasi-linear elliptic equation in the whole Euclidean space is
considered. The nonlinearity of the equation is assumed to have exponential
growth or have critical growth in view of Trudinger-Moser type inequality.
Under some assumptions on the potential and the nonlinearity, it is proved
that there is a nontrivial positive weak solution to this equation. Also it is
shown that there are two distinct positive weak solutions to a perturbation of
the equation. The method of proving these results is combining Trudinger-Moser
type inequality, Mountain-pass theorem and Ekeland’s variational principle.
###### keywords:
Trudinger-Moser inequality, singular Trudinger-Moser inequality, $N$-Laplace
equation, exponential growth
###### MSC:
46E30, 46E35, 35J20, 35J60, 35B33
††journal: ***
## 1 Introduction
Let $\Omega\subset\mathbb{R}^{N}$ be a bounded smooth domain. There are
fruitful results on the following problem
$\left\\{\begin{array}[]{lll}-\Delta_{p}u=f(x,u)\quad{\rm
in}\quad\Omega\\\\[6.45831pt] u\in W_{0}^{1,p}(\Omega),\end{array}\right.$
(1.1)
where $\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)$. When $p=2$ and
$|f(x,u)|\leq c(|u|+|u|^{q-1})$, $1<q\leq 2^{*}=2N/(N-2)$, $N\geq 3$. Among
pioneer works we mention Brézis [8], Brézis-Nirenberg [10], Bartsh-Willem [11]
and Capozzi-Fortunato-Palmieri [13]. For $p\leq N$ and $p^{2}\leq N$, Garcia-
Alonso [23] generalized Brézis-Nirenberg’s existence and nonexistence results
to $p$-Laplace equation. When $\Omega=\mathbb{R}^{N}$ and $p=2$, one may
consider the semilinear Schrödinger equation instead of (1.1):
$\left\\{\begin{array}[]{lll}-\Delta u+V(x)u=f(x,u)\quad{\rm
in}\quad\mathbb{R}^{N}\\\\[6.45831pt] u\in
W^{1,N}(\mathbb{R}^{N}),\end{array}\right.$ (1.2)
where again $|f(x,u)|\leq c(|u|+|u|^{q-1})$, $1<q\leq 2^{*}=2N/(N-2)$. Many
papers are devoted to (1.2), we refer the reader to Kryszewski-Szulkin [25],
Alama-Li [6], Ding-Ni [17] and Jeanjean [24]. Sobolev embedding theorem and
the critical point theory, particularly moumtain-pass theorem would play an
important role in studying problems (1.1) and (1.2) since both of them have
variational structure. When $p=N$ and $f(x,u)$ behaves like
$e^{\alpha|u|^{{N}/{(N-1)}}}$ as $|u|\rightarrow\infty$, problem (1.1) was
studied by Adimurthi [2], Adimurthi-Yadava[4], Ruf et al [15, 16], J. M. do Ó
[19], Panda [30] and the references therein. To the author’s knowledge, all
theses results are based on Trudinger-Moser inequality [28, 31, 34] and
critical point theory.
In this paper we consider the existence of positive solutions of the quasi-
linear equation
$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^{\beta}},\,\,\,x\in\mathbb{R}^{N}\,\,(N\geq
2),$ (1.3)
where $\Delta_{N}u={\rm div}(|\nabla u|^{N-2}\nabla u)$,
$V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a continuous function, $f(x,s)$ is
continuous in $\mathbb{R}^{N}\times\mathbb{R}$ and behaves like $e^{\alpha
s^{{N}/{(N-1)}}}$ as $s\rightarrow+\infty$, and $0\leq\beta<N$. Problem (1.3)
can be compared with (1.2) in this way: Sobolev embedding theorem can be
applied to (1.2), while Trudinger-Moser type embedding theorem can be applied
to (1.3). When $\beta=0$, problem (1.3) was studied by D. Cao [12] in the case
$N=2$, by Panda [29], J. M. do Ó [18] and Alves-Figueiredo [7] in general
dimensional case. When $0<\beta<N$, problem (1.3) is closely related to a
singular Trudinger-Moser type inequality, namely
Theorem A ([5]). For all $\alpha>0$, $0\leq\beta<N$, and $u\in
W^{1,N}(\mathbb{R}^{N})$ $(N\geq 2)$, there holds
$\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$
(1.4)
Furthermore, we have for all
$\alpha\leq\left(1-\frac{\beta}{N}\right)\alpha_{N}$ and $\tau>0$,
$\sup_{\int_{\mathbb{R}^{N}}(|\nabla u|^{N}+\tau|u|^{N})dx\leq
1}\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$
(1.5)
This inequality is sharp : for any
$\alpha>\left(1-\frac{\beta}{N}\right)\alpha_{N}$, the supremum is infinity.
This theorem extends a result of Adimurthi-Sandeep [3] on a bounded smooth
domain. When $\beta=0$ and $\tau=1$, (1.5) was proved by B. Ruf in the case
$N=2$ via symmetrization method and by Li-Ruf [26] in general dimensional case
via the method of blow-up analysis. When $\beta=0$ and $\alpha<\alpha_{N}$,
(1.5) was first proved by Cao [12] in the case $N=2$, and then by Panda [29],
J. M. do Ó [18] in general dimensional case. A similar but different type
inequality was obtained by Adachi-Tanaka [1].
We assume the following two conditions on the potential $V(x)$:
$(V_{1})$ $V(x)\geq V_{0}>0$ in $\mathbb{R}^{N}$ for some $V_{0}>0$;
$(V_{2})$ The function $\frac{1}{V(x)}$ belongs to
$L^{\frac{1}{N-1}}(\mathbb{R}^{N})$.
As for the nonlinearity $f(x,s)$ we suppose the following:
$(H_{1})$ There exist constants $\alpha_{0}$, $b_{1}$, $b_{2}>0$ such that for
all $(x,s)\in\mathbb{R}^{N}\times\mathbb{R}^{+}$,
$|f(x,s)|\leq
b_{1}s^{N-1}+b_{2}\left\\{e^{\alpha_{0}|s|^{N/(N-1)}}-S_{N-2}(\alpha_{0},s)\right\\};$
$(H_{2})$ There exists $\mu>N$ such that for all $x\in\mathbb{R}^{N}$ and
$s>0$,
$0<\mu F(x,s)\equiv\mu\int_{0}^{s}f(x,t)dt\leq sf(x,s);$
$(H_{3})$ There exist constant $R_{0}$, $M_{0}>0$ such that for all
$x\in\mathbb{R}^{N}$ and $s\geq R_{0}$,
$F(x,s)\leq M_{0}f(x,s).$
Define a function space
$E=\left\\{u\in
W^{1,N}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(x)|u|^{N}dx<\infty\right\\}.$
(1.6)
We say that $u\in E$ is a weak solution of problem (1.3) if for all
$\varphi\in E$ we have
$\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\varphi+V(x)|u|^{N-2}u\varphi\right)dx=\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\varphi
dx.$
The assumption $(V_{1})$ implies that $E$ is a reflexive Banach space when
equipped with the norm
$\|u\|_{E}\equiv\left\\{\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{N}+V(x)|u|^{N}\right)dx\right\\}^{\frac{1}{N}}$ (1.7)
and for any $q\geq N$, the embedding
$E\hookrightarrow W^{1,N}(\mathbb{R}^{N})\hookrightarrow
L^{q}(\mathbb{R}^{N})$
is continuous. However $(V_{2})$ together with $(V_{1})$ implies that
$E\hookrightarrow L^{q}(\mathbb{R}^{N})$ is compact for all $q\geq 1$ (see
Lemma 2.4 below). Surprisingly the assumption $(V_{2})$ is much better than
$(V_{2}^{\prime})$ $V(x)\rightarrow+\infty$ as $|x|\rightarrow+\infty$,
since $(V_{1})$ together with $(V_{2}^{\prime})$ only leads to the compact
embedding $E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq N$ (see for
example Costa [14] for details). This is the case in [5, 21]. However in this
paper our argument of proving main results seriously depends on the compact
embedding $E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.
For any $\beta:0\leq\beta<N$, we define a singular eigenvalue for the
$N$-Laplace operator by
$\lambda_{\beta}=\inf_{u\in E,\,u\not\equiv
0}\frac{\|u\|_{E}^{N}}{\int_{\mathbb{R}^{N}}\frac{|u|^{N}}{|x|^{\beta}}dx}.$
(1.8)
It is easy to see that $\lambda_{\beta}>0$. Write $m(r)=\sup_{|x|\leq r}V(x)$
and
$\mathcal{M}=\inf_{r>0}\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}},$
(1.9)
where $\alpha_{0}$ is given by $(H_{1})$. If $V(x)$ is continuous and
$(V_{1})$ is satisfied, then $m(r)$ is a positive continuous function and
$\mathcal{M}$ can be attained by some $r>0$.
One of our main results can be stated as follows:
Theorem 1.1. Assume that $V(x)$ is a continuous function satisfying $(V_{1})$
and $(V_{2})$. $f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is a
continuous function and the hypothesis $(H_{1})$, $(H_{2})$ and $(H_{3})$
hold. Furthermore we assume
$None$ $None$
Then the equation (1.3) has a nontrivial positive mountain-pass type weak
solution.
Here and throughout this paper, we say that a weak solution $u$ is positive if
$u(x)\geq 0$ for almost every $x\in\mathbb{R}^{N}$. It should be pointed out
that $\mathcal{M}$ is not the best constant in $(H_{5})$. It would be
interesting if one can find an explicit smaller number replacing
$\mathcal{M}$.
In [5], Theorem A has been employed to study a perturbation of the equation
(1.3), namely
$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^{\beta}}+\epsilon
h,\,\,\,x\in\mathbb{R}^{N}\,\,(N\geq 2),$ (1.10)
where $\epsilon>0$ is a constant and $h:\mathbb{R}^{N}\rightarrow\mathbb{R}$
is a function belonging to $E^{*}$, the dual space of $E$. If $V(x)$ satisfies
$(V_{1})$, $(V_{2}^{\prime})$, and $f(x,s)$ satisfies $(H_{1})-(H_{4})$, then
it was shown in [5] that when $\epsilon>0$ is sufficiently small and
$h\not\equiv 0$, the problem (1.10) has two weak solutions: one is of
mountain-pass type and the other is of negative energy. But we can not
conclude that the two solutions are distinct. In this paper, replacing
$(V_{2}^{\prime})$ by $(V_{2})$ and imposing additional condition $(H_{5})$,
we can prove that the above two solutions are distinct, namely
Theorem 1.2. Suppose that $f(x,s)$ is continuous in
$\mathbb{R}^{N}\times\mathbb{R}$ and $(H_{1})-(H_{5})$ hold. $V(x)$ is
continuous in $\mathbb{R}^{N}$ satisfying $(V_{1})$ and $(V_{2})$, $h$ belongs
to $E^{*}$, the dual space of $E$, with $h\geq 0$ and $h\not\equiv 0$. Then
there exists $\epsilon_{0}>0$ such that if $0<\epsilon<\epsilon_{0}$, then the
problem (1.10) has two distinct positive weak solutions.
The proof of Theorem 1.1 and Theorem 1.2 is based on Theorem A, the mountain-
pass theorem without the Palais-Smale condition [32] and the Ekeland’s
variational principle [35], which were also used in [5, 21]. Let us make some
reduction on problems (1.3) and (1.10). Set
$\widetilde{f}(x,s)=\left\\{\begin{array}[]{lll}0,&f(x,s)<0\\\\[6.45831pt]
f(x,s),&f(x,s)\geq 0.\end{array}\right.$
Assume $u\in E$ is a weak solution of
$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{\widetilde{f}(x,u)}{|x|^{\beta}}+\epsilon
h,$ (1.11)
where $h\geq 0$ and $\epsilon>0$, then the negative part of $u$, namely
$u_{-}(x)=\left\\{\begin{array}[]{lll}0,&u(x)>0\\\\[6.45831pt] u(x),&u(x)\leq
0\end{array}\right.$
belongs to the function space $E$ and satisfies
$\displaystyle\int_{\mathbb{R}^{N}}(|\nabla u_{-}|^{N}+V(x)|u_{-}|^{N})dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\widetilde{f}(x,u)}{|x|^{\beta}}u_{-}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{-}dx$
$\displaystyle=$ $\displaystyle\epsilon\int_{\mathbb{R}^{N}}hu_{-}dx\leq 0.$
Hence $u_{-}(x)=0$ for almost every $x\in\mathbb{R}^{N}$ and thus $u$ is a
positive weak solution of (1.11). This together with $(H_{2})$ implies
$f(x,u)\geq 0$. It follows that $\widetilde{f}(x,u)=f(x,u)$. Therefore $u$ is
also a positive weak solution of (1.10). When $h=0$, (1.10) becomes (1.3).
Based on this, to prove Theorems 1.1 and 1.2, it suffices to find weak
solutions of (1.3) and (1.10) with $f$ replaced by $\widetilde{f}$
respectively. So throughout this paper, we can assume without loss of
generality
$f(x,s)\equiv 0,\quad\forall s<0.$ (1.12)
Before ending this introduction, we would like to mention that results similar
to Theorem 1.2 in two dimensional case, i.e. $N=2$, was obtained by J. M. do Ó
[21]. Similar problems for bi-Laplace equation in $\mathbb{R}^{4}$ was
considered by the author in [36]. For compact Riemannian manifold case, we
refer the reader to [22, 37]. Also it should be remarked that results obtained
in [5] and in the present paper still hold if there is only the subcritical
case of (1.5), namely for any $\alpha<(1-\beta/N)\alpha_{N}$ and $\tau>0$,
$\sup_{\int_{\mathbb{R}^{N}}(|\nabla u|^{N}+\tau|u|^{N})dx\leq
1}\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$
In fact, in [7, 12, 18, 29], all the contributors only used the above
subcritical inequality.
The remaining part of this paper is organized as follows: In section 2, we
display several key estimates in later compactness analysis. In section 3, we
consider the functionals related to problems (1.3) and (1.10). Finally Theorem
1.1 is proved in section 4 and Theorem 1.2 is proved in section 5.
## 2 Key estimates
In this section we will derive several technical lemmas for our use later. For
any integer $N\geq 2$ and real number $s$, we define a function
$\zeta:\mathbb{N}\times\mathbb{R}\rightarrow\mathbb{R}$ by
$\zeta(N,s)=e^{s}-\sum_{k=0}^{N-2}\frac{s^{k}}{k!}=\sum_{k=N-1}^{\infty}\frac{s^{k}}{k!}.$
(2.1)
Lemma 2.1. Let $s\geq 0$, $p\geq 1$ be real numbers and $N\geq 2$ be an
integer. Then there holds
$\left(\zeta(N,s)\right)^{p}\leq\zeta(N,ps).$ (2.2)
Proof. We prove (2.2) by induction with respect to $N$. Define a function
$\phi(s)=(e^{s}-1)^{p}-(e^{ps}-1).$
It is easy to see that for $s\geq 0$ and $p\geq 1$,
$\phi^{\prime}(s)=p(e^{s}-1)^{p-1}-pe^{ps}\leq 0.$
Hence $\phi(s)\leq\phi(0)=0$ and thus (2.2) holds for $N=2$. Suppose (2.2)
holds for $N\geq 2$, we only need to prove that
$\left(\zeta(N+1,s)\right)^{p}\leq\zeta(N+1,ps).$ (2.3)
For this purpose we set
$\psi(s)=\left(e^{s}-\sum_{k=0}^{N-1}\frac{s^{k}}{k!}\right)^{p}-\left(e^{ps}-\sum_{k=0}^{N-1}\frac{{(ps)}^{k}}{k!}\right).$
A straightforward calculation shows
$\displaystyle\psi^{\prime}(s)$ $\displaystyle=$ $\displaystyle
p\left(e^{s}-\sum_{k=0}^{N-1}\frac{s^{k}}{k!}\right)^{p-1}\left(e^{s}-\sum_{k=1}^{N-1}\frac{s^{k-1}}{(k-1)!}\right)$
$\displaystyle\quad-\left(pe^{ps}-p\sum_{k=1}^{N-1}\frac{(ps)^{k-1}}{(k-1)!}\right)$
$\displaystyle\leq$ $\displaystyle
p\left\\{\left(e^{s}-\sum_{k=1}^{N-1}\frac{s^{k-1}}{(k-1)!}\right)^{p}-\left(e^{ps}-\sum_{k=1}^{N-1}\frac{(ps)^{k-1}}{(k-1)!}\right)\right\\}$
$\displaystyle=$ $\displaystyle
p\left\\{\left(e^{s}-\sum_{k=0}^{N-2}\frac{s^{k}}{k!}\right)^{p}-\left(e^{ps}-\sum_{k=0}^{N-2}\frac{(ps)^{k}}{k!}\right)\right\\}\leq
0.$
Here we have used the induction assumption $(\zeta(N,s))^{p}\leq\zeta(N,ps)$.
Thus $\psi(s)\leq\psi(0)=0$ for $s\geq 0$, and whence (2.3) holds. Therefore
(2.2) holds for any integer $N\geq 2$. $\hfill\Box$
Lemma 2.2. For all $N\geq 2$, $s\geq 0$, $t\geq 0$, $\mu>1$ and $\nu>1$ with
$1/\mu+1/\nu=1$, there holds
$\zeta(N,s+t)\leq\frac{1}{\mu}\zeta(N,\mu s)+\frac{1}{\nu}\zeta(N,\nu t).$
Proof. Observing that
$\frac{\partial^{2}}{\partial s^{2}}\zeta(2,s)=e^{s}\geq
0,\quad\frac{\partial^{2}}{\partial s^{2}}\zeta(3,s)=e^{s}\geq 0$
and when $N\geq 4$,
$\frac{\partial^{2}}{\partial
s^{2}}\zeta(N,s)=e^{s}-\sum_{k=2}^{N-2}\frac{s^{k-2}}{(k-2)!}=e^{s}-\sum_{k=0}^{N-4}\frac{s^{k}}{k!}\geq
0,$
we conclude that $\zeta(N,s)$ is convex with respect to $s$ for all $N\geq 2$.
Hence
$\zeta(N,s+t)=\zeta\left(N,\frac{1}{\mu}\mu s+\frac{1}{\nu}\nu
t\right)\leq\frac{1}{\mu}\zeta(N,\mu s)+\frac{1}{\nu}\zeta(N,\nu t).$
This concludes the lemma. $\hfill\Box$
Lemma 2.3. Let $(w_{n})$ be a sequence in $E$. Suppose $\|w_{n}\|_{E}=1$,
$w_{n}\rightharpoonup w_{0}$ weakly in $E$, $w_{n}(x)\rightarrow w_{0}(x)$ and
$\nabla w_{n}(x)\rightarrow\nabla w_{0}(x)$ for almost every
$x\in\mathbb{R}^{N}$. Then for any
$p:0<p<\frac{1}{(1-\|w_{0}\|_{E}^{N})^{{1}/{(N-1)}}}$
$\sup_{n}\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{N}(1-\beta/N)p|w_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx<\infty.$
(2.4)
Proof. Noticing that
$|w_{n}|^{\frac{N}{N-1}}=|w_{n}-w_{0}+w_{0}|^{\frac{N}{N-1}}\leq(1+\epsilon)|w_{n}-w_{0}|^{\frac{N}{N-1}}+c(\epsilon)|w_{0}|^{\frac{N}{N-1}},$
we have by using Lemma 2.2
$\displaystyle\zeta\left(N,\alpha_{N}(1-\beta/N)p|w_{n}|^{\frac{N}{N-1}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\zeta\left(N,\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p|w_{n}-w_{0}|^{\frac{N}{N-1}}\right)$
$\displaystyle+\frac{1}{\nu}\zeta\left(N,\nu
c(\epsilon)\alpha_{N}(1-\beta/N)p|w_{0}|^{\frac{N}{N-1}}\right)$
$\displaystyle\leq$
$\displaystyle\zeta\left(N,\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p\|w_{n}-w_{0}\|_{E}^{\frac{N}{N-1}}\left(\frac{|w_{n}-w_{0}|}{\|w_{n}-w_{0}\|_{E}}\right)^{\frac{N}{N-1}}\right)$
$\displaystyle+\zeta\left(N,\nu
c(\epsilon)\alpha_{N}(1-\beta/N)p|w_{0}|^{\frac{N}{N-1}}\right),$
where $\mu>1$, $\nu>1$ and $1/\mu+1/\nu=1$. By Brézis-Lieb’s Lemma [9],
$\|w_{n}-w_{0}\|_{E}^{N}=1-\|w_{0}\|_{E}^{N}+o_{n}(1),$
where $o_{n}(1)\rightarrow 0$ as $n\rightarrow\infty$. Hence for any
$p:0<p<\frac{1}{(1-\|w_{0}\|_{E}^{N})^{{1}/{(N-1)}}}$, one can choose
$\epsilon>0$ sufficiently small and $\mu>1$ sufficiently close to $1$ such
that
$\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p\|w_{n}-w_{0}\|_{E}^{\frac{N}{N-1}}<\alpha_{N}(1-\beta/N).$
Now (2.4) follows from Theorem A immediately. $\hfill\Box$
Lemma 2.4. Assume $V:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is
continuous and $(V_{1})$, $(V_{2})$ hold. Then $E$ is compactly embedded in
$L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.
Proof. By $(V_{1})$, the standard Sobolev embedding theorem implies that the
following embedding is continuous
$E\hookrightarrow W^{1,N}(\mathbb{R}^{N})\hookrightarrow
L^{q}(\mathbb{R}^{N})\quad{\rm for\,\,all}\quad N\leq q<\infty.$
It follows from the Hölder inequality and $(V_{2})$ that
$\int_{\mathbb{R}^{N}}|u|dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\left(\int_{\mathbb{R}^{N}}V|u|^{N}dx\right)^{1/N}\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\|u\|_{E}.$
For any $\gamma:1<\gamma<N$, there holds
$\int_{\mathbb{R}^{N}}|u|^{\gamma}dx\leq\int_{\mathbb{R}^{N}}(|u|+|u|^{N})dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\|u\|_{E}+\frac{1}{V_{0}}\|u\|_{E}^{N},$
where $V_{0}$ is given by $(V_{1})$. Thus we get continuous embedding
$E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.
To prove that the above embedding is also compact, take a sequence of
functions $(u_{k})\subset E$ such that $\|u_{k}\|_{E}\leq C$ for all $k$, we
must prove that up to a subsequence there exists some $u\in E$ such that
$u_{k}$ convergent to $u$ strongly in $L^{q}(\mathbb{R}^{N})$ for all $q\geq
1$. Without loss of generality we may assume
$\left\\{\begin{array}[]{lll}u_{k}\rightharpoonup u&{\rm weakly\,\,in}\quad
E\\\\[6.45831pt] u_{k}\rightarrow u&{\rm strongly\,\,in}\quad L^{q}_{\rm
loc}(\mathbb{R}^{N}),\,\,\forall q\geq 1\\\\[6.45831pt] u_{k}\rightarrow
u&{\rm almost\,\,everywhere\,\,in}\quad\mathbb{R}^{N}.\end{array}\right.$
(2.5)
In view of $(V_{2})$, for any $\epsilon>0$, there exists $R>0$ such that
$\left(\int_{|x|>R}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}<\epsilon.$
Hence
$\int_{|x|>R}|u_{k}-u|dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\left(\int_{\mathbb{R}^{N}}V|u|^{N}dx\right)^{1/N}\leq\epsilon\|u_{k}-u\|_{E}\leq
C\epsilon.$ (2.6)
Here and in the sequel we often denote various constants by the same $C$. On
the other hand, it follows from (2.5) that $u_{k}\rightarrow u$ strongly in
$L^{1}(\mathbb{B}_{R}(0))$, where $\mathbb{B}_{R}(0)\subset\mathbb{R}^{N}$ is
the ball centered at $0$ with radius $R$. This together with (2.6) leads to
$\limsup_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{k}-u|dx\leq C\epsilon.$
Since $\epsilon$ is arbitrary, we obtain
$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{k}-u|dx=0.$
For $q>1$, it follows from the continuous embedding $E\hookrightarrow
L^{s}(\mathbb{R}^{N})$ ($s\geq 1$) that
$\displaystyle\int_{\mathbb{R}^{N}}|u_{k}-u|^{q}dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|u_{k}-u|^{\frac{1}{2}}|u_{k}-u|^{(q-\frac{1}{2})}dx$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{N}}|u_{k}-u|dx\right)^{1/2}\left(\int_{\mathbb{R}^{N}}|u_{k}-u|^{2q-1}dx\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\mathbb{R}^{N}}|u_{k}-u|dx\right)^{1/2}\rightarrow 0$
as $k\rightarrow\infty$. This concludes the lemma. $\hfill\Box$
## 3 Functionals and compactness analysis
### 3.1 The functionals and their profiles
As we mentioned in the introduction, problems (1.3) and (1.10) have
variational structure. To apply the critical point theory, we define the
functional $J_{\beta,\,\epsilon}:E\rightarrow\mathbb{R}$ by
$J_{\beta,\,\epsilon}(u)=\frac{1}{N}\|u\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx-\epsilon\int_{\mathbb{R}^{N}}hudx,$
where $\epsilon\geq 0$, $0\leq\beta<N$, $\|u\|_{E}$ is the norm of $u\in E$
defined by (1.7) and $F(x,s)=\int_{0}^{s}f(x,t)dt$ is the primitive of
$f(x,s)$. Assume $f$ satisfies the hypothesis $(H_{1})$. Then there exist some
positive constants $\alpha_{1}>\alpha_{0}$ and $b_{3}$ such that for all
$(x,s)\in\mathbb{R}^{N}\times\mathbb{R}$, $F(x,s)\leq
b_{3}\zeta(N,\alpha_{1}|s|^{N/(N-1)})$. Thus $J_{\beta,\,\epsilon}$ is well
defined thanks to Theorem A. In the case $\epsilon=0$, we denote $J_{\beta,0}$
for simplicity by
$J(u)=\frac{1}{N}\|u\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx.$
The profiles of the functionals $J_{\beta,\epsilon}$ and $J(u)$ are well
described in the following lemma.
Lemma 3.1. Assume $(V_{1})$, $(H_{1})$, $(H_{2})$, $(H_{3})$ and $(H_{4})$ are
satisfied. Then $(i)$ for any nonnegative, compactly supported function $u\in
W^{1,N}(\mathbb{R}^{N})\setminus\\{0\\}$, there holds
$J_{\beta,\,\epsilon}(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$; $(ii)$
there exists $\epsilon_{1}>0$ such that when $0<\epsilon<\epsilon_{1}$, one
can find $r_{\epsilon}$, $\vartheta_{\epsilon}>0$ such that
$J_{\beta,\,\epsilon}(u)\geq\vartheta_{\epsilon}$ for all $u$ with
$\|u\|_{E}=r_{\epsilon}$, where $r_{\epsilon}$ can be further chosen such that
$r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. When $\epsilon=0$,
there exists $\delta>0$ and $r>0$ such that $J(u)\geq\delta$ for all
$\|u\|_{E}=r$; $(iii)$ assume $\epsilon>0$ and $h\not\equiv 0$, there exists a
constant $\tau>0$ such that if $0<t<\tau$, then $\inf_{\|u\|_{E}\leq
t}J_{\beta,\epsilon}(u)<0$.
Proof. We refer the reader to ([5], Lemmas 4.1, 4.2 and 4.3) for details. It
is remarkable that we can also apply Lemma 2.1 and Lemma 2.4 instead of
decreasing rearrangement argument in the proof of ([5], Lemma 4.2) and thus
simplify it. $\hfill\Box$
To use the critical point theory, we need some regularity of the functionals
$J_{\beta,\epsilon}$ and $J$. In fact, by Proposition 1 in [21] and standard
arguments (see for example [32]), one can see that both $J_{\beta,\epsilon}$
and $J$ belong to $\mathcal{C}^{1}(E,\mathbb{R})$. A straightforward
calculation shows
$\displaystyle\langle
J^{\prime}(u),\phi\rangle=\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\phi+V|u|^{N-2}u\phi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\phi
dx,$ (3.1) $\displaystyle\langle
J_{\beta,\,\epsilon}^{\prime}(u),\phi\rangle=\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{N-2}\nabla
u\nabla\phi+V|u|^{N-2}u\phi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\phi
dx-\epsilon\int_{\mathbb{R}^{N}}h\phi dx\qquad$ (3.2)
for all $\phi\in E$. Hence weak solutions of (1.3) and (1.10) are critical
points of $J$ and $J_{\beta,\,\epsilon}$ respectively.
### 3.2 Min-Max value
In this subsection, we prepare for estimating the min-max value of the
functionals $J$ and $J_{\beta,\epsilon}$. The idea is to construct a sequence
of functions $M_{n}\in E$ and estimate $\max\limits_{t\geq 0}J(tM_{n})$ and
$\max\limits_{t\geq 0}J_{\beta,\epsilon}(tM_{n})$. Recall Moser’s function
sequence
$\widetilde{M}_{n}(x,r)=\frac{1}{\omega_{N-1}^{1/N}}\left\\{\begin{array}[]{lll}(\log
n)^{1-1/N},&|x|\leq r/n\\\\[6.45831pt] \frac{\log\frac{r}{|x\,|}}{(\log
n)^{1/N}},&r/n<|x|\leq r\\\\[6.45831pt] 0,&|x|>r.\end{array}\right.$
Let $M_{n}(x,r)=\frac{1}{\|\widetilde{M}_{n}\|_{E}}\widetilde{M}_{n}(x,r)$.
Then $M_{n}$ belongs to $E$ with its support in $\mathbb{B}_{r}(0)$ and
$\|M_{n}\|_{E}=1$.
Lemma 3.2. Assume $V(x)$ is continuous and $(V_{1})$ is satisfied. Then there
holds
$\|\widetilde{M}_{n}\|_{E}^{N}\leq 1+\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right),$ (3.3)
where $m(r)=\max\limits_{|x|\leq r}V(x)$ and $o_{n}(1)\rightarrow 0$ as
$n\rightarrow\infty$.
Proof. It is easy to calculate
$\int_{\mathbb{R}^{N}}|\nabla\widetilde{M}_{n}|^{N}dx=\frac{1}{\omega_{N-1}}\int_{\frac{r}{n}\leq|x|\leq
r}\frac{1}{|x|^{N}\log n}dx=1.$
Integration by parts gives
$\displaystyle\int_{\frac{r}{n}\leq|x|\leq
r}\left(\log\frac{r}{|x|}\right)^{N}dx$ $\displaystyle=$
$\displaystyle\omega_{N-1}\int_{\frac{r}{n}}^{r}s^{N-1}\left(\log\frac{r}{s}\right)^{N}ds$
$\displaystyle=$
$\displaystyle-\frac{\omega_{N-1}}{N}\left(\frac{r}{n}\right)^{N}\left(\log
n\right)^{N}+\omega_{N-1}\int_{\frac{r}{n}}^{r}s^{N-1}\left(\log\frac{r}{s}\right)^{N-1}ds$
$\displaystyle=$
$\displaystyle-{\omega_{N-1}}\left(\frac{r}{n}\right)^{N}\left\\{\frac{1}{N}(\log
n)^{N}+\frac{1}{N}(\log n)^{N-1}+\frac{N-1}{N^{2}}(\log n)^{N-2}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.+\cdots+\frac{(N-1)(N-2)\cdots
3}{N^{N-2}}(\log n)^{2}\right\\}$
$\displaystyle+\omega_{N-1}\frac{(N-1)!}{N^{N-2}}\int_{\frac{r}{n}}^{r}s^{N-1}\log\frac{r}{s}ds$
$\displaystyle=$
$\displaystyle\omega_{N-1}\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1).$
Hence
$\displaystyle\int_{\mathbb{R}^{N}}|\widetilde{M}_{n}|^{N}dx$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{N-1}}\int_{|x|\leq r/n}(\log
n)^{N-1}dx+\frac{1}{\omega_{N-1}}\int_{\frac{r}{n}\leq|x|\leq
r}\frac{\left(\log\frac{r}{|x|}\right)^{N}}{\log n}dx$ $\displaystyle=$
$\displaystyle\left(\frac{r}{n}\right)^{N}\frac{(\log
n)^{N-1}}{N}+\frac{1}{\omega_{N-1}\log n}\int_{\frac{r}{n}\leq|x|\leq
r}\left(\log\frac{r}{|x|}\right)^{N}dx$ $\displaystyle=$
$\displaystyle\frac{1}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right),$
and thus
$\displaystyle\|\widetilde{M}_{n}\|_{E}^{N}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\widetilde{M}_{n}|^{N}dx+\int_{\mathbb{R}^{N}}V(x)|\widetilde{M}_{n}|^{N}dx$
$\displaystyle\leq$ $\displaystyle
1+m(r)\int_{\mathbb{R}^{N}}|\widetilde{M}_{n}|^{N}dx$ $\displaystyle=$
$\displaystyle 1+\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right).$
This is exactly (3.3). $\hfill\Box$
Lemma 3.3. Assume $(V_{1})$, $(H_{1})$, $(H_{2})$, $(H_{3})$ and $(H_{5})$.
There exists some $n\in\mathbb{N}$ such that
$\max_{t\geq
0}J(tM_{n})<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.4)
Furthermore for the above $n$ there exists some $\epsilon^{*}>0$ and
$\delta^{*}>0$ such that if $0\leq\epsilon<\epsilon^{*}$, then
$\max_{t\geq
0}J_{\beta,\epsilon}(tM_{n})<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}-\delta^{*}.$
(3.5)
Proof. We first prove (3.4). By $(H_{5})$ and (1.9) (the definition of
$\mathcal{M}$), there exists some $r>0$ such that
$\beta_{0}>\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$
(3.6)
Suppose by contradiction that for all $n\in\mathbb{N}$
$\max_{t\geq
0}J(tM_{n})\geq\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.7)
By $(i)$ of Lemma 3.1, $\forall n\in\mathbb{N}$, there exists $t_{n}>0$ such
that
$J(t_{n}M_{n})=\max_{t\geq 0}J(tM_{n}).$
Thus (3.7) gives
$J(t_{n}M_{n})=\frac{t_{n}^{N}}{N}-\int_{\mathbb{R}^{N}}\frac{F(x,t_{n}M_{n})}{|x|^{\beta}}dx\geq\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
Noticing that $F(x,\cdot)\geq 0$, we have
$t_{n}^{N}\geq\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.8)
It is easy to see that at $t=t_{n}$,
$\frac{d}{dt}\left(\frac{t^{N}}{N}-\int_{\mathbb{R}^{N}}\frac{F(x,tM_{n})}{|x|^{\beta}}dx\right)=0,$
or equivalently
$t_{n}^{N}=\int_{\mathbb{R}^{N}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx.$
(3.9)
By $(H_{5})$, $\forall\eta>0$, $\exists R_{\eta}>0$ such that for all
$x\in\mathbb{R}^{N}$ and $u\geq R_{\eta}$
$uf(x,u)\geq(\beta_{0}-\eta)e^{\alpha_{0}|u|^{\frac{N}{N-1}}}.$ (3.10)
By Lemma 3.2, when $|x|\leq\frac{r}{n}$, we have
$\displaystyle M_{n}^{\frac{N}{N-1}}(x,r)$ $\displaystyle\geq$
$\displaystyle\frac{1}{\omega_{N-1}^{\frac{1}{N-1}}}\frac{\log
n}{1+\frac{1}{N-1}\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right)}$ (3.11) $\displaystyle=$
$\displaystyle\omega_{N-1}^{-\frac{1}{N-1}}\log
n-\omega_{N-1}^{-\frac{1}{N-1}}{m(r)}\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1).$
Hence we have by combining (3.9) and (3.10) that
$\displaystyle{}t_{n}^{N}$ $\displaystyle\geq$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
(3.12) $\displaystyle=$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\left(\log
n-m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)\right)}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle(\beta_{0}-\eta)\frac{\omega_{N-1}}{N-\beta}\left(\frac{r}{n}\right)^{N-\beta}e^{\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\left(\log
n-m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)\right)}.$
This yields that $t_{n}$ is a bounded sequence. In view of (3.8), we can also
see from (3.12) that
$\lim_{n\rightarrow\infty}t_{n}^{N}=\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.13)
For otherwise there exists some $\delta>0$ such that for sufficiently large
$n$
$t_{n}^{N}\geq\left(\delta+\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
Thus
$\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\geq
N-\beta+\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}\delta$
and whence the right hand of (3.12) tends to infinity which contradicts the
bounded-ness of $t_{n}$.
Now we estimate $\beta_{0}$. It follows from (3.9) and (3.10) that
$\displaystyle t_{n}^{N}$ $\displaystyle\geq$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx+\int_{t_{n}M_{n}<R_{\eta}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx{}$
(3.14)
$\displaystyle\qquad-(\beta_{0}-\eta)\int_{t_{n}M_{n}<R_{\eta}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx.$
Since $M_{n}\rightarrow 0$ almost everywhere in $\mathbb{R}^{N}$, we have by
using the Lebesgue’s dominated convergence theorem
$\displaystyle\lim_{n\rightarrow\infty}\int_{t_{n}M_{n}<R_{\eta}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx=0,$
(3.15)
$\displaystyle\lim_{n\rightarrow\infty}\int_{t_{n}M_{n}<R_{\eta}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx=\int_{|x|\leq
r}\frac{1}{|x|^{\beta}}dx=\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}.$ (3.16)
Using (3.8),
$\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx\geq\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx+\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx.$
(3.17)
On one hand we have by (3.11)
$\displaystyle\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle\geq$ $\displaystyle
e^{\alpha_{N}(1-\beta/N)\omega_{N-1}^{-\frac{1}{N-1}}\log
n-\omega_{N-1}^{-\frac{1}{N-1}}{m(r)}\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}\int_{|x|\leq\frac{r}{n}}\frac{1}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{N-\beta}\left(\frac{r}{n}\right)^{N-\beta}e^{(N-\beta)\log
n-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}.$
On the other hand, by definition of $M_{n}$,
$\displaystyle\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{(N-\beta)\left((\log
n)^{-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}\log\frac{r}{|x|}\right)^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\omega_{N-1}\int_{\frac{r}{n}}^{r}t^{N-\beta-1}e^{(N-\beta)\left((\log
n)^{-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}\log\frac{r}{t}\right)^{\frac{N}{N-1}}}dt$
$\displaystyle=$ $\displaystyle\omega_{N-1}r^{N-\beta}\int_{0}^{{(\log
n)^{1-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}}}(\log
n)^{1/N}\|\widetilde{M}_{n}\|_{E}$ $\displaystyle\qquad
e^{(N-\beta)s^{\frac{N}{N-1}}-(N-\beta)\|\widetilde{M}_{n}\|_{E}(\log
n)^{1/N}s}ds$ $\displaystyle\geq$
$\displaystyle\omega_{N-1}r^{N-\beta}\int_{0}^{{(\log
n)^{1-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}}}(\log
n)^{1/N}\|\widetilde{M}_{n}\|_{E}$ $\displaystyle\qquad
e^{-(N-\beta)\|\widetilde{M}_{n}\|_{E}(\log n)^{1/N}s}ds$ $\displaystyle=$
$\displaystyle\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}\left(1-e^{-(N-\beta)\log
n}\right).$
Here we have used the change of variable
$t=re^{-\|\widetilde{M}_{n}\|_{E}(\log n)^{1/N}s}$ in the third equality.
Hence we obtain by passing to the limit $n\rightarrow\infty$ in (3.17)
$\liminf_{n\rightarrow\infty}\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx\geq\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}\left(1+e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}\right).$
This together with (3.13)-(3.16) implies
$\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}\geq(\beta_{0}-\eta)\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$
Since $\eta>0$ is arbitrary, we have
$\beta_{0}\leq\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$
This contradicts (3.6) and ends the proof of (3.4).
Secondly it follows from (3.4) and the definition of $J_{\beta,\epsilon}$ that
(3.5) holds. $\hfill\Box$
### 3.3 Palais-Smale sequence
In this subsection, we will show that the weak limit of a Palais-Smale
sequence for $J_{\beta,\epsilon}$ is the weak solution of $(\ref{prob-1})$.
(Respectively the weak limit of a Palais-Smale sequence for $J$ is also the
weak solution of $(\ref{prob-0})$.)
Lemma 3.4. Assume that $(V_{1})$, $(V_{2})$, $(H_{1})$, $(H_{2})$ and
$(H_{3})$ are satisfied. Let $(u_{n})\subset E$ be an arbitrary Palais-Smale
sequence of $J_{\beta,\epsilon}$, i.e.,
$J_{\beta,\epsilon}(u_{n})\rightarrow
c,\,\,J^{\prime}_{\beta,\epsilon}(u_{n})\rightarrow 0\,\,{\rm
in}\,\,E^{*}\,\,{\rm as}\,\,n\rightarrow\infty,$ (3.18)
where $E^{*}$ denotes the dual space of $E$. Then there exist a subsequence of
$(u_{n})$ (still denoted by $(u_{n})$) and $u\in E$ such that
$u_{n}\rightharpoonup u$ weakly in $E$, $u_{n}\rightarrow u$ strongly in
$L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$, and
$\displaystyle\left\\{\begin{array}[]{lll}\nabla u_{n}(x)\rightarrow\nabla
u(x)\quad{\rm a.\,\,e.\,\,\,in}\quad\mathbb{R}^{N}\\\\[6.45831pt]
\frac{f(x,\,u_{n})}{|x|^{\beta}}\rightarrow\frac{f(x,\,u)}{|x|^{\beta}}\,\,{\rm
strongly\,\,in}\,\,L^{1}(\mathbb{R}^{N})\\\\[6.45831pt]
\frac{F(x,\,u_{n})}{|x|^{\beta}}\rightarrow\frac{F(x,\,u)}{|x|^{\beta}}\,\,{\rm
strongly\,\,in}\,\,L^{1}(\mathbb{R}^{N}).\end{array}\right.$
Furthermore $u$ is a weak solution of (1.10). The same conclusion holds when
$\epsilon=0$.
Proof. Assume $(u_{n})$ is a Palais-Smale sequence of $J_{\beta,\epsilon}$. By
$(\ref{PS})$, we have
$\displaystyle\frac{1}{N}\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx-\epsilon\int_{\mathbb{R}^{N}}hu_{n}dx\rightarrow
c\,\,{\rm as}\,\,n\rightarrow\infty,$ (3.20)
$\displaystyle\left|\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\psi+V|u_{n}|^{N-2}u_{n}\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}\psi
dx-\epsilon\int_{\mathbb{R}^{N}}h\psi
dx\right|\leq\tau_{n}\|\psi\|_{E}\qquad\quad$ (3.21)
for all $\psi\in E$, where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$.
Noticing that (1.12), we have by $(H_{2})$ that $0\leq\mu F(x,u_{n})\leq
u_{n}f(x,u_{n})$ for some $\mu>N$. Taking $\psi=u_{n}$ in (3.21) and
multiplying (3.20) by $\mu$, we have
$\displaystyle\left(\frac{\mu}{N}-1\right)\|u_{n}\|_{E}^{N}$
$\displaystyle\leq$
$\displaystyle\left(\frac{\mu}{N}-1\right)\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{\mu
F(x,u_{n})-f(x,u_{n})u_{n}}{|x|^{\beta}}dx$ $\displaystyle\leq$
$\displaystyle\mu|c|+\tau_{n}\|u_{n}\|_{E}+(\mu+1)\epsilon\|h\|_{E^{*}}\|u_{n}\|_{E}$
Therefore $\|u_{n}\|_{E}$ is bounded. It then follows from (3.20), (3.21) that
$\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u_{n}}{|x|^{\beta}}dx\leq
C,\quad\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx\leq C$ (3.22)
for some constant $C$ depending only on $\mu$, $N$ and $\|h\|_{E^{\ast}}$. By
Lemma 2.4, up to a subsequence, $u_{n}\rightarrow u$ strongly in
$L^{q}(\mathbb{R}^{N})$ for some $u\in E$, $\forall q\geq 1$. This immediately
leads to $u_{n}\rightarrow u$ almost everywhere in $\mathbb{R}^{N}$. Now we
claim that up to a subsequence
$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{|f(x,\,u_{n})-f(x,\,u)|}{|x|^{\beta}}dx=0.$
(3.23)
In fact, since $f(x,\cdot)\geq 0$, it suffices to prove that up to a
subsequence
$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{f(x,\,u_{n})}{|x|^{\beta}}dx=\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{f(x,\,u)}{|x|^{\beta}}dx.$
(3.24)
Since $u,\,\frac{f(x,u)}{|x|^{\beta}}\in L^{1}(\mathbb{R}^{N})$, we have
$\lim_{\eta\rightarrow+\infty}\int_{|u|\geq\eta}\frac{f(x,u)}{|x|^{\beta}}dx=0.$
Let $C$ be the constant in (3.22). Given any $\delta>0$, one can select some
$M>{C}/{\delta}$ such that
$\int_{|u|\geq M}\frac{f(x,u)}{|x|^{\beta}}dx<\delta.$ (3.25)
It follows from (3.22) that
$\int_{|u_{n}|\geq
M}\frac{f(x,u_{n})}{|x|^{\beta}}dx\leq\frac{1}{M}\int_{|u_{n}|\geq
M}\frac{f(x,u_{n})u_{n}}{|x|^{\beta}}dx<\delta.$ (3.26)
For all $x\in\\{x\in\mathbb{R}^{N}:|u_{n}|<M\\}$, by our assumption $(H_{1})$,
there exists a constant $C_{1}$ depending only on $M$ such that
$|f(x,u_{n}(x))|\leq C_{1}|u_{n}(x)|^{N-1}$. Notice that
$|x|^{-\beta}{|u_{n}|^{N-1}}\rightarrow|x|^{-\beta}|u|^{N-1}$ strongly in
$L^{1}(\mathbb{R}^{N})$ and $u_{n}\rightarrow u$ almost everywhere in
$\mathbb{R}^{N}$. By the generalized Lebesgue’s dominated convergence theorem,
we obtain
$\lim_{n\rightarrow\infty}\int_{|u_{n}|<M}\frac{f(x,u_{n})}{|x|^{\beta}}dx=\int_{|u|<M}\frac{f(x,u)}{|x|^{\beta}}dx.$
(3.27)
Combining (3.25), (3.26) and (3.27), we can find some $K>0$ such that when
$n>K$,
$\left|\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}dx\right|<3\delta.$
Hence (3.24) holds and thus our claim (3.23) holds. By $(H_{1})$ and
$(H_{3})$, there exist constants $c_{1}$, $c_{2}>0$ such that
$F(x,u_{n})\leq c_{1}|u_{n}|^{N}+c_{2}f(x,u_{n}).$
In view of (3.23) and Lemma 2.4, it follows from the generalized Lebesgue’s
dominated convergence theorem
$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{|F(x,\,u_{n})-F(x,\,u)|}{|x|^{\beta}}dx=0.$
Using the argument of proving (4.26) in [5], we have $\nabla
u_{n}(x)\rightarrow\nabla u(x)$ a. e. in $\mathbb{R}^{N}$ and
$|\nabla u_{n}|^{N-2}\nabla u_{n}\rightharpoonup|\nabla u|^{N-2}\nabla
u\quad{\rm
weakly\,\,in}\quad\left(L^{\frac{N}{N-1}}(\mathbb{R}^{N})\right)^{N}.$
Finally passing to the limit $n\rightarrow\infty$ in $(\ref{2})$, we have
$\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\psi+V|u|^{N-2}u\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\psi
dx-\epsilon\int_{\mathbb{R}^{4}}h\psi dx=0$
for all $\psi\in C_{0}^{\infty}(\mathbb{R}^{N})$, which is dense in $E$. Hence
$u$ is a weak solution of $(\ref{prob-1})$. After checking the above argument,
$\epsilon$ need not to be nonzero, i.e. the same conclusion holds for $J$.
$\hfill\Box$
Remark 3.5. Similar results of Lemma 3.4 was also established by J. M. do Ó in
two dimensional case [20] and by the author for bi-Laplace equation in four
dimensional Euclidean space [36].
## 4 Nontrivial positive solution
In this section, we will prove Theorem 1.1. It suffices to look for nontrivial
critical points of the functional $J$ in the function space $E$.
Proof of Theorem 1.1. By $(i)$ and $(ii)$ of Lemma 3.1, $J$ satisfies all the
hypothesis of the mountain-pass theorem except for the Palais-Smale condition:
$J\in\mathcal{C}^{1}(E,\mathbb{R})$; $J(0)=0$; $J(u)\geq\delta>0$ when
$\|u\|_{E}=r$; $J(e)<0$ for some $e\in E$ with $\|e\|_{E}>r$. Then using the
mountain-pass theorem without the Palais-Smale condition [32], we can find a
sequence $(u_{n})$ in $E$ such that
$J(u_{n})\rightarrow c>0,\quad J^{\prime}(u_{n})\rightarrow 0\,\,{\rm
in}\,\,E^{*},$
where
$c=\min_{\gamma\in\Gamma}\max_{u\in\gamma}J(u)\geq\delta$
is the min-max value of $J$, where
$\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. By
(3.1), this is equivalent to saying
$\displaystyle\frac{1}{N}\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx\rightarrow
c\,\,{\rm as}\,\,n\rightarrow\infty,$ (4.1)
$\displaystyle\left|\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\psi+V|u_{n}|^{N-2}u_{n}\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}\psi
dx\right|\leq\tau_{n}\|\psi\|_{E}\qquad\quad$ (4.2)
for all $\psi\in E$, where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$. By
Lemma 3.4, up to a subsequence, there holds
$\left\\{\begin{array}[]{lll}u_{n}\rightharpoonup u\,\,{\rm
weakly\,\,in}\,\,E\\\\[6.45831pt] u_{n}\rightarrow u\,\,{\rm
strongly\,\,in}\,\,L^{q}(\mathbb{R}^{N}),\,\,\forall q\geq 1\\\\[6.45831pt]
\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx=\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx\\\\[6.45831pt]
u\,\,{\rm
is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{prob-0}).\end{array}\right.$ (4.3)
Now suppose by contradiction $u\equiv 0$. Since $F(x,0)=0$ for all
$x\in\mathbb{R}^{N}$, it follows from (4.1) and (4.3) that
$\lim_{n\rightarrow\infty}\|u_{n}\|_{E}^{N}=Nc>0.$ (4.4)
Thanks to the hypothesis $(H_{5})$, we have
$0<c<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}$
by applying Lemma 3.3. Thus there exists some $\eta_{0}>0$ and $K>0$ such that
$\|u_{n}\|_{E}^{N}\leq\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}-\eta_{0}\right)^{N-1}$
for all $n>K$. Choose $q>1$ sufficiently close to $1$ such that
$q\alpha_{0}\|u_{n}\|_{E}^{\frac{N}{N-1}}\leq(1-\beta/N)\alpha_{N}-\alpha_{0}\eta_{0}/2$
for all $n>N$. By $(H_{1})$,
$|f(x,u_{n})u_{n}|\leq
b_{1}|u_{n}|^{N}+b_{2}|u_{n}|\zeta\left(N,\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\right),$
where the function $\zeta(\cdot,\cdot)$ is defined by (2.1). It follows from
the Hölder inequality, Lemma 2.1 and Theorem A that
$\displaystyle\int_{\mathbb{R}^{N}}\frac{|f(x,u_{n})u_{n}|}{|x|^{\beta}}dx$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+b_{2}\int_{\mathbb{R}^{N}}\frac{|u_{n}|\zeta\left(N,\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+b_{2}\left(\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{q^{\prime}}}{|x|^{\beta}}dx\right)^{1/{q^{\prime}}}\left(\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,q\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx\right)^{1/{q}}$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+C\left(\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{q^{\prime}}}{|x|^{\beta}}dx\right)^{1/{q^{\prime}}}\rightarrow
0\quad{\rm as}\quad n\rightarrow\infty.$
Here we used (4.3) again (precisely $u_{n}\rightarrow u$ in
$L^{s}(\mathbb{R}^{N})$ for all $s\geq 1$) in the last step of the above
estimates. Inserting this into (4.2) with $\psi=u_{n}$, we have
$\|u_{n}\|_{E}\rightarrow 0\quad{\rm as}\quad n\rightarrow\infty,$
which contradicts (4.4). Therefore $u\not\equiv 0$ and we obtain a nontrivial
weak solution of (1.3). $\hfill\Box$
## 5 Multiplicity results
In this section we will prove Theorem 1.2. The proof is divided into three
steps, namely
Step 1. Let $\epsilon_{1}$ be given by $(ii)$ of Lemma 3.1, and
$\epsilon^{*}$, $\delta^{*}$ be given by Lemma 3.3. Then when
$0<\epsilon<\epsilon_{1}$, there exists a sequence $(v_{n})\subset E$ such
that
$J_{\beta,\epsilon}(v_{n})\rightarrow c_{M},\quad
J_{\beta,\epsilon}^{\prime}(v_{n})\rightarrow 0,$ (5.1)
where $c_{M}$ is a min-max value of $J_{\beta,\epsilon}$. Let
$\epsilon_{2}=\min\\{\epsilon_{1},\epsilon^{*}\\}$. Then when
$0<\epsilon<\epsilon_{2}$, we can take $c_{M}$ such that
$0<c_{M}<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}-\delta^{*}.$
(5.2)
In addition, up to a subsequence, there holds $v_{n}\rightharpoonup u_{M}$
weakly in $E$, and $u_{M}$ is a weak solution of (1.10).
Proof. By $(i)$ and $(ii)$ of Lemma 3.1, when $0<\epsilon<\epsilon_{1}$,
$J_{\beta,\epsilon}$ satisfies the following condition:
$J_{\beta,\epsilon}\in\mathcal{C}^{1}(E,\mathbb{R})$;
$J_{\beta,\epsilon}(0)=0$; $J_{\beta,\epsilon}(u)\geq\vartheta_{\epsilon}>0$
when $\|u\|_{E}=r_{\epsilon}$; $J_{\beta,\epsilon}(e)<0$ for some $e\in E$
with $\|e\|>\max\\{r_{\epsilon},1\\}$. Then using the mountain-pass theorem
without the Palais-Smale condition [32], we can find a sequence $(v_{n})$ in
$E$ such that
$J_{\beta,\,\epsilon}(v_{n})\rightarrow c_{M}>0,\quad
J^{\prime}_{\beta,\,\epsilon}(v_{n})\rightarrow 0\,\,{\rm in}\,\,E^{*},$
where
$c_{M}=\min_{\gamma\in\Gamma}\max_{u\in\gamma}J_{\beta,\epsilon}(u)\geq\vartheta_{\epsilon}$
is a min-max value of $J_{\beta,\,\epsilon}$, where
$\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. Clearly
(5.2) follows from Lemma 3.3. The last assertion follows from Lemma 3.4
immediately. $\hfill\Box$
Step 2. Let $r_{\epsilon}$ be given by $(ii)$ of Lemma 3.1 such that
$r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. There exists
$\epsilon_{3}:0<\epsilon_{3}<\epsilon_{2}$ such that if
$0<\epsilon<\epsilon_{3}$, then there exists a sequence $(u_{n})\subset E$
such that
$J_{\beta,\epsilon}(u_{n})\rightarrow c_{\epsilon}:=\inf_{\|u\|_{E}\leq
r_{\epsilon}}J_{\beta,\epsilon}(u)$ (5.3)
and
$J_{\beta,\epsilon}^{\prime}(u_{n})\rightarrow 0\quad{\rm in}\quad
E^{*}\quad{\rm as}\quad n\rightarrow\infty,$ (5.4)
where $c_{\epsilon}<0$ and $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow
0$. In addition, up to a subsequence, there holds $u_{n}\rightarrow u_{0}$
strongly in $E$, and $u_{0}$ is a weak solution of (1.10) with
$J_{\beta,\epsilon}(u_{0})=c_{\epsilon}$.
Proof. Let $r_{\epsilon}$ be given by $(ii)$ of Lemma 3.1, i.e.
$J_{\beta,\epsilon}(u)>\vartheta_{\epsilon}>0$ for all $u$ with
$\|u\|_{E}=r_{\epsilon}$. Since $r_{\epsilon}\rightarrow 0$ as
$\epsilon\rightarrow 0$, one can choose
$\epsilon_{3}:0<\epsilon_{3}<\epsilon_{2}$ such that when
$0<\epsilon<\epsilon_{3}$,
$r_{\epsilon}<\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{\frac{N-1}{N}}.$
(5.5)
By $(H_{1})$ and $(H_{2})$, we have
$F(x,u)\leq
b_{1}|u|^{N}+b_{2}|u|\zeta\left(N,\alpha_{0}\|u\|_{E}^{{N}/{(N-1)}}\left({|u|}/{\|u\|_{E}}\right)^{N/(N-1)}\right).$
(5.6)
Here again $\zeta(\cdot,\cdot)$ is defined by (2.1). When $\|u\|_{E}\leq
r_{\epsilon}$, we have $\alpha_{0}\|u\|_{E}^{N/(N-1)}<(1-\beta/N)\alpha_{N}$.
It then follows from Lemma 2.1 and Theorem A that $F(x,u)/|x|^{\beta}$ is
bounded in $L^{p}(\mathbb{R}^{N})\cap L^{1}(\mathbb{R}^{N})$ for some $p>1$
when $\|u\|_{E}\leq r_{\epsilon}$. Hence $J_{\beta,\epsilon}$ has lower bound
on the ball $B_{r_{\epsilon}}=\\{u\in E:\|u\|_{E}\leq r_{\epsilon}\\}$.
Since the closure of $B_{r_{\epsilon}}$, $\overline{B}_{r_{\epsilon}}\subset
E$ is a complete metric space with the metric given by the norm of $E$, convex
and $J_{\beta,\epsilon}$ is of class $\mathcal{C}^{1}$ and has lower bound on
$\overline{B}_{r_{\epsilon}}$. By the Ekeland’s variational principle [35],
there exists a sequence $(u_{n})\subset\overline{B}_{r_{\epsilon}}$ such that
(5.3) and (5.4) hold.
By $(iii)$ of Lemma 3.1, $c_{\epsilon}<0$. Since $r_{\epsilon}\rightarrow 0$
as $\epsilon\rightarrow 0$, noticing (5.6), we have by using the Hölder
inequality and Lemma 2.4
$\sup_{\|u\|_{E}\leq
r_{\epsilon}}\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx\rightarrow
0,\quad\sup_{\|u\|_{E}\leq r_{\epsilon}}\int_{\mathbb{R}^{N}}hudx\rightarrow
0$
as $\epsilon\rightarrow 0$. This implies $c_{\epsilon}\rightarrow 0$ as
$\epsilon\rightarrow 0$.
Now we are proving the last assertion. Assume $u_{n}\rightharpoonup u_{0}$
weakly in $E$. (5.4) is equivalent to
$|\langle
J_{\beta,\epsilon}^{\prime}(u_{n}),\phi\rangle|\leq\tau_{n}\|\phi\|_{E},\quad\forall\phi\in
E,$ (5.7)
where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$. Recalling (3.2) and
choosing $\phi=u_{n}-u_{0}$ in (5.7), we have
$\displaystyle\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u_{0})+V(x)|u_{n}|^{N-2}u_{n}(u_{n}-u_{0})\right)dx$
$\displaystyle-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}(u_{n}-u_{0})dx-\epsilon\int_{\mathbb{R}^{N}}h(u_{n}-u_{0})dx=o_{n}(1),$
where $o_{n}(1)\rightarrow 0$ as $n\rightarrow\infty$. Hölder inequality
together with (5.5), Theorem A and Lemma 2.4 implies that
$\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}(u_{n}-u_{0})dx=o_{n}(1),\quad\epsilon\int_{\mathbb{R}^{N}}h(u_{n}-u_{0})dx=o_{n}(1).$
Hence
$\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u_{0})+V(x)|u_{n}|^{N-2}u_{n}(u_{n}-u_{0})\right)dx=o_{n}(1).$
(5.8)
On the other hand, since $u_{n}\rightharpoonup u_{0}$ weakly in $E$, we obtain
$\int_{\mathbb{R}^{N}}\left(|\nabla u_{0}|^{N-2}\nabla
u_{0}\nabla(u_{n}-u_{0})+V(x)|u_{0}|^{N-2}u_{0}(u_{n}-u_{0})\right)dx=o_{n}(1).$
(5.9)
Subtracting (5.9) from (5.8), using a well known inequality (see for example
Chapter 10 of [27])
$2^{2-N}|b-a|^{N}\leq\langle|b|^{N-2}b-|a|^{N-2}a,b-a\rangle,\quad\forall
a,\,b\in\mathbb{R}^{N},$ (5.10)
we obtain $\|u_{n}-u_{0}\|_{E}^{N}\rightarrow 0$ and thus $u_{n}\rightarrow
u_{0}$ strongly in $E$ as $n\rightarrow\infty$. Since
$J_{\beta,\epsilon}\in\mathcal{C}^{1}(E,\mathbb{R})$, there hold
$J_{\beta,\epsilon}(u_{0})=c_{\epsilon}$ and
$J_{\beta,\epsilon}^{\prime}(u_{0})=0$, i.e. $u_{0}$ is a weak solution of
(1.10). $\hfill\Box$
Step 3. There exists $\epsilon_{0}:0<\epsilon_{0}<\epsilon_{3}$ such that if
$0<\epsilon<\epsilon_{0}$, then $u_{M}\not\equiv u_{0}$.
Proof. Suppose by contradiction that $u_{M}\equiv u_{0}$. Then
$v_{n}\rightharpoonup u_{0}$ weakly in $E$. By (5.1),
$J_{\beta,\epsilon}(v_{n})\rightarrow c_{M}>0,\quad|\langle
J_{\beta,\epsilon}^{\prime}(v_{n}),\phi\rangle|\leq\gamma_{n}\|\phi\|_{E}$
(5.11)
with $\gamma_{n}\rightarrow 0$ as $n\rightarrow\infty$. On one hand, by Lemma
3.4, we have
$\int_{\mathbb{R}^{N}}\frac{F(x,v_{n})}{|x|^{\beta}}dx\rightarrow\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx\quad{\rm
as}\quad n\rightarrow\infty.$ (5.12)
Here and in the sequel, we do not distinguish sequence and subsequence. On the
other hand, since $v_{n}\rightharpoonup u_{0}$ weakly in $E$, it follows from
the Hölder inequality and Lemma 2.4 that
$\int_{\mathbb{R}^{N}}hv_{n}dx\rightarrow\int_{\mathbb{R}^{N}}hu_{0}dx.\quad{\rm
as}\quad n\rightarrow\infty.$ (5.13)
Inserting (5.12) and (5.13) into (5.11), we obtain
$\frac{1}{N}\|v_{n}\|_{E}^{N}=c_{M}+\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{0}dx+o_{n}(1).$
(5.14)
In the same way, one can derive
$\frac{1}{N}\|u_{n}\|_{E}^{N}=c_{\epsilon}+\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{0}dx+o_{n}(1).$
(5.15)
Combining (5.14) and (5.15), we have
$\|v_{n}\|_{E}^{N}-\|u_{0}\|_{E}^{N}=N\left(c_{M}-c_{\epsilon}+o_{n}(1)\right).$
(5.16)
From Step 2, we know that $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow
0$. This together with (5.2) leads to the existence of
$\epsilon_{0}:0<\epsilon_{0}<\epsilon_{3}$ such that if
$0<\epsilon<\epsilon_{0}$, then
$0<c_{M}-c_{\epsilon}<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(5.17)
Write
$w_{n}=\frac{v_{n}}{\|v_{n}\|_{E}},\quad
w_{0}=\frac{u_{0}}{\left(\|u_{0}\|_{E}^{N}+N(c_{M}-c_{\epsilon})\right)^{1/N}}.$
It follows from (5.16) and $v_{n}\rightharpoonup u_{0}$ weakly in $E$ that
$w_{n}\rightharpoonup w_{0}$ weakly in $E$. Notice that
$\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{0}|v_{n}|^{N/(N-1)}\right)}{|x|^{\beta}}dx=\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{0}\|v_{n}\|_{E}^{{N}/{(N-1)}}|w_{n}|^{N/(N-1)}\right)}{|x|^{\beta}}dx.$
By (5.16) and (5.17), a straightforward calculation shows
$\lim_{n\rightarrow\infty}\alpha_{0}\|v_{n}\|_{E}^{\frac{N}{N-1}}\left(1-\|w_{0}\|_{E}^{N}\right)^{\frac{1}{N-1}}<\left(1-\frac{\beta}{N}\right)\alpha_{N}.$
Whence Lemma 2.3 together with Lemma 2.1 implies that
${\zeta\left(N,\alpha_{0}|v_{n}|^{N/(N-1)}\right)}/{|x|^{\beta}}$ is bounded
in $L^{q}(\mathbb{R}^{N})$ for some $q>1$. By $(H_{1})$,
$|f(x,v_{n})|\leq
b_{1}|v_{n}|^{N-1}+b_{2}\zeta(N,\alpha_{0}|v_{n}|^{\frac{N}{N-1}}).$
Then it follows from the continuous embedding $E\hookrightarrow
L^{p}(\mathbb{R}^{N})$ for all $p\geq 1$ that $f(x,v_{n})/|x|^{\beta}$ is
bounded in $L^{q_{1}}(\mathbb{R}^{N})$ for some $q_{1}$: $1<q_{1}<q$. This
together with Lemma 2.4 and the Hölder inequality gives
$\left|\int_{\mathbb{R}^{N}}\frac{f(x,v_{n})(v_{n}-u_{0})}{|x|^{\beta}}dx\right|\leq\left\|\frac{f(x,v_{n})}{|x|^{\beta}}\right\|_{L^{q_{1}}(\mathbb{R}^{N})}\left\|v_{n}-u_{0}\right\|_{L^{{q_{1}^{\prime}}}(\mathbb{R}^{N})}\rightarrow
0,$ (5.18)
where $1/q_{1}+1/q_{1}^{\prime}=1$.
Taking $\phi=v_{n}-u_{0}$ in (5.11), we have by using (5.13) and (5.18) that
$\int_{\mathbb{R}^{N}}\left(|\nabla v_{n}|^{N-2}\nabla
v_{n}\nabla(v_{n}-u_{0})+V(x)|v_{n}|^{N-2}v_{n}(v_{n}-u_{0})\right)dx\rightarrow
0.$ (5.19)
However the fact $v_{n}\rightharpoonup u_{0}$ weakly in $E$ leads to
$\int_{\mathbb{R}^{N}}\left(|\nabla u_{0}|^{N-2}\nabla
u_{0}\nabla(v_{n}-u_{0})+V(x)|u_{0}|^{N-2}u_{0}(v_{n}-u_{0})\right)dx\rightarrow
0.$ (5.20)
Subtracting (5.20) from (5.19), using the inequality (5.10), we have
$\|v_{n}-u_{0}\|_{E}^{N}\rightarrow 0.$
This together with (5.16) implies that
$c_{M}=c_{\epsilon},$
which is absurd since $c_{M}>0$ and $c_{\epsilon}<0$. Therefore we end Step 3
and complete the proof of Theorem 1.2. $\hfill\Box$
Acknowledgement. The author is supported by the program for NCET.
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|
arxiv-papers
| 2011-06-23T04:30:21 |
2024-09-04T02:49:20.007576
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yunyan Yang",
"submitter": "Yunyan Yang",
"url": "https://arxiv.org/abs/1106.4622"
}
|
1106.4745
|
# Pattern polynomial graphs
A. Satyanarayana Reddy111Department of Mathematics and Statistics, Indian
Institute of Technology, Kanpur, India 208016; (e-mail: satya@iitk.ac.in).
Shashank K Mehta 222Department of Computer Science and Engineering, Indian
Institute of Technology, Kanpur, India 208016; (e-mail:
skmehta@cse.iitk.ac.in). This work was partly supported by Research-I
Foundation, IIT-Kanpur.
###### Abstract
A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra
is a coherent algebra. In this study we will find a necessary and sufficient
condition for a graph to be a pattern polynomial graph. Some of the properties
of the graphs which are polynomials in the pattern polynomial graph have been
studied. We also identify known graph classes which are pattern polynomial
graphs.
Keywords: Adjacency algebra of a graph, coherent algebra, automorphisms of a
graph, distance regular graphs.
Mathematics Subject Classification(2010): 05C50, 05E40,05E18
## 1 Introduction and preliminaries
Let $M_{n}({\mathbb{C}})$ denote the set of all $n\times n$ matrices over the
field of complex numbers ${\mathbb{C}}$. Let ${\mathbb{C}}[A]$ denote the set
of all matrices which are polynomials in $A$ with coefficients from
${\mathbb{C}}$. Clearly ${\mathbb{C}}[A]$ is an algebra over ${\mathbb{C}}$
for any $A\in M_{n}({\mathbb{C}})$. The dimension of ${\mathbb{C}}[A]$ over
${\mathbb{C}}$ as a vector space, is the degree of the minimal polynomial of
$A$. If $A$ is diagonalizable, then from the following lemma, it’s dimension
is equal to the number of distinct eigenvalues of $A$.
###### Lemma 1.1 (Hoffman $\&$ Kunze [10]).
A matrix is diagonalizable if and only if its minimal polynomial has all
distinct linear factors over ${\mathbb{C}}$.
###### Definition 1.1.
Hadamard product of two $n\times n$ matrices $A$ and $B$ is denoted by $A\odot
B$ and is defined as $(A\odot B)_{xy}=A_{xy}B_{xy}$.
Two $n\times n$ matrices $A$ and $B$ are said to be disjoint if their Hadamard
product is the zero matrix.
###### Definition 1.2.
A subalgebra of $M_{n}(\mathbb{C})$ is called coherent if it contains the
matrices $I$ and $J$ and if it is closed under conjugate-transposition and
Hadamard multiplication. Here $J$ denotes the matrix with every entry being
$1$.
###### Theorem 1.2.
[4] Every coherent algebra contains unique basis of mutually disjoint $0,1$\-
matrices (matrices with entries either $0$ or $1$).
We call this unique basis of mutually disjoint 0-1 matrices as a _standard
basis_.
###### Corollary 1.3.
Every $0,1$-matrix in a coherent algebra is the sum of one or more matrices in
its standard basis.
###### Proof.
Let $\mathcal{M}$ be a coherent algebra over ${\mathbb{C}}$ with its standard
basis $\\{M_{1},\ldots M_{t}\\}$. Let $B\in\mathcal{M}$ be a $0,1$-matrix,
then $B=\sum_{i=1}^{t}a_{i}M_{i}$ where $a_{i}\in{\mathbb{C}}$. $B=B\odot
B=\sum_{i=1}^{t}a_{i}^{2}M_{i}\Rightarrow a_{i}^{2}=a_{i}$. Hence the result
follows. ∎
###### Corollary 1.4.
If $\mathcal{M}$ is a commutative coherent algebra over ${\mathbb{C}}$, then
$\dim(\mathcal{M})\leq n$.
###### Proof.
Since $J$ commutes with every element in $\mathcal{M}$. Hence all the
row(column) sums of every matrix in $\mathcal{M}$ are equal. Consequently the
number of elements in the standard basis of $\mathcal{M}$ is atmost $n$, as
the matrices in the standard basis of $\mathcal{M}$ are disjoint and whose sum
is $J$. ∎
###### Observation 1.5.
The intersection of coherent algebras is again a coherent algebra.
###### Definition 1.3.
Let $A\in M_{n}({\mathbb{C}})$, then coherent closure of $A$, denoted by
$\langle\langle A\rangle\rangle$ or ${\mathcal{CC}}(A)$, is the smallest
coherent algebra containing $A$.
In Section 2 we will see a necessary and sufficient condition for any matrix
$A\in M_{n}({\mathbb{C}})$ such that ${\mathbb{C}}[A]={\mathcal{CC}}(A)$. In
the remaining sections we consider $A$ to be the adjacency matrix of a graph
$X$. If $A$ is the adjacency matrix of a graph $X$ and
${\mathbb{C}}[A]={\mathcal{CC}}(A)$, then $X$ will be called a pattern
polynomial graph. Some of the properties of pattern polynomial graphs are
given in Section 3. In Section 4 we will see a few graph classes which are
pattern polynomial graphs. Few partially balanced incomplete block designs
from pattern polynomial graphs are constructed in Section 5. Properties of
graphs which are polynomials in the pattern polynomial graph are provided in
the Section 6.
## 2 ${\mathbb{C}}[A]={\mathcal{CC}}(A)$
In this section we will see a necessary and sufficient condition for a matrix
$A\in M_{n}({\mathbb{C}})$ such that ${\mathbb{C}}[A]={\mathcal{CC}}(A)$. For
that we construct a vector space which lies in between
${\mathbb{C}}[A]\;and\;{\mathcal{CC}}(A)$ as follows.
Let $\ell$ be the degree of the minimal polynomial of $A$. Then
$\\{I,A,\ldots,A^{\ell-1}\\}$ is a basis for ${\mathbb{C}}[A]$ over
${\mathbb{C}}$. Let
$\textbf{y}=(y_{0},y_{1}\ldots,y_{\ell-1})\in{\mathbb{C}}^{\ell}$ be a
variable and
$B(\textbf{y})=y_{0}I+y_{1}A+\dots
y_{l-1}A^{\ell-1}=\begin{bmatrix}p_{11}(\textbf{y})&p_{12}(\textbf{y})&\dots&p_{1n}(\textbf{y})\\\
p_{21}(\textbf{y})&p_{22}(\textbf{y})&\dots&p_{2n}(\textbf{y})\\\
\vdots&\vdots&\ddots&\vdots\\\
p_{n1}(\textbf{y})&p_{n2}(\textbf{y})&\dots&p_{nn}(\textbf{y})\end{bmatrix}.$
Where $p_{ij}(\textbf{y})=p_{ij}(y_{0},y_{1}\ldots,y_{\ell-1})$ is a
polynomial in the variables $y_{0},y_{1},\ldots,y_{\ell-1}$. Let us assume
$\\{q_{1}(\textbf{y}),q_{2}(\textbf{y}),\ldots,q_{r}(\textbf{y})\\}$ be the
set of distinct polynomials in the matrix $B(\textbf{y})$. We define the
matrices called pattern matrices of $A$, as $P_{j}\;(1\leq j\leq r)$ as
$(P_{j})_{s,t}=\begin{cases}1,&{\mbox{ if
}}B(\textbf{y})_{s,t}=q_{j}(\textbf{y}),\\\ 0,&{\mbox{otherwise.
}}\end{cases}$
Let ${\mathcal{L}}(A)=L\\{P_{1},P_{2},\ldots,P_{r}\\}$ denote the linear span
of matrices $P_{1},\ldots,P_{r}$. Then ${\mathcal{L}}(A)$ is a subspace of
$M_{n}({\mathbb{C}})$. From the definition of ${\mathcal{L}}(A)$ we have the
following observation.
###### Observation 2.1.
1. 1.
$P_{i}\odot P_{j}=0\;\forall i,j\;1\leq i,j\leq r$,
$\sum_{i=1}^{r}P_{i}=J\in{\mathcal{L}}(X)$, $I\in{\mathcal{L}}(A)$.
2. 2.
${\mathcal{L}}(A)$ is closed under Hadamard product.
###### Proof.
Let $M,N\in{\mathcal{L}}(A)$, so
$M=\sum_{i=1}^{r}a_{i}P_{i}\;,N=\sum_{i=1}^{r}b_{i}P_{i}$ where
$a_{i},b_{i}\in{\mathbb{C}}$. Then $M\odot
N=\sum_{i=1}^{r}a_{i}b_{i}P_{i}\in{\mathcal{L}}(A)$. ∎
3. 3.
${\mathcal{L}}(A)$ is the smallest subspace of $M_{n}({\mathbb{C}})$ closed
under Hadamard product and contains all powers of $A$. Consequently
${\mathbb{C}}[A]\subseteq{\mathcal{L}}(A)\subseteq{\mathcal{CC}}(A)$ and
$l\leq r$.
4. 4.
If $P_{i}^{T}\in\\{P_{1},P_{2},\ldots,P_{r}\\}\;for\;all\;1\leq i\leq r$, then
${\mathcal{L}}(A)$ is also closed under conjugate transposition. In
particular, if $A$ is symmetric, then all pattern matrices are symmetric hence
${\mathcal{L}}(A)$ is closed under conjugate transposition.
With the above observation, we are providing the main result of this section.
###### Theorem 2.2.
Let $A\in M_{n}({\mathbb{C}})$ be a symmetric matrix. Then
${\mathbb{C}}[A]={\mathcal{CC}}(A)$ if and only if $\ell=r$.
###### Proof.
If ${\mathbb{C}}[A]={\mathcal{CC}}(A)$, then
${\mathbb{C}}[A]={\mathcal{L}}(A)$ hence $\ell=r$. Conversely suppose $\ell=r$
then ${\mathbb{C}}[A]={\mathcal{L}}(A)$. In particular ${\mathcal{L}}(A)$ is
closed under ordinary multiplication. Hence from the above obseravtion
${\mathcal{L}}(A)$ is a coherent algebra but ${\mathcal{CC}}(A)$ is the
smallest coherent algebra containing ${\mathbb{C}}[A]$. Hence the result
follows. ∎
## 3 Pattern polynomial graphs
From now onwards we suppose that $A$ (or A(X)) is the adjacency matrix of a
graph $X$, hence $A$ is a symmetric $0,1$ matrix. We denote ${\mathbb{C}}[A]$
by ${\mathcal{A}}(X)$, ${\mathcal{CC}}(A)$ by ${\mathcal{CC}}(X)$ and
${\mathcal{L}}(A)$ by ${\mathcal{L}}(X)$. We call ${\mathcal{A}}(X)$ as the
adjacency algebra of the graph $X$, ${\mathcal{CC}}(X)$ as coherent closure of
$X$. First we will provide a few results on adjacency algebra of a graph. Then
we will see some additional properties of the vector space ${\mathcal{L}}(X)$.
For two vertices $u$ and $v$ of a connected graph $X$, let $d(u,v)$ denote the
length of the shortest path from $u$ to $v$. Then the diameter of a connected
graph $X=(V,E)$ is $\max\\{d(u,v):\;u,v\in V\\}$. It is shown in Biggs [3]
that if $X$ is a connected graph with diameter $d$, then
$d+1\leq\dim({\mathcal{A}}(X))\leq n$ (1)
where $\dim({\mathcal{A}}(X))$ is the dimension of ${\mathcal{A}}(X)$ as a
vector space over ${\mathbb{C}}$.
A graph $X_{1}=(V(X_{1}),E(X_{1}))$ is said to be isomorphic to the graph
$X_{2}=(V(X_{2}),E(X_{2}))$, written $X_{1}\cong X_{2}$, if there is a one-to-
one correspondence $\rho:V(X_{1})\rightarrow V(X_{2})$ such that
$\\{v_{1},v_{2}\\}\in E(X_{1})$ if and only if
$\\{\rho(v_{1}),\rho(v_{2})\\}\in E(X_{2})$. In such a case, $\rho$ is called
an isomorphism of $X_{1}$ and $X_{2}$. An isomorphism of a graph $X$ onto
itself is called an automorphism. The collection of all automorphisms of a
graph $X$ is denoted by $\text{Aut}(X)$. It is well known that $\text{Aut}(X)$
is a group under composition of two maps. It is easy to see that
$\text{Aut}(X)=\text{Aut}(X^{c})$, where $X^{c}$ is the complement of the
graph X. If $X$ is a graph with $n$ vertices we can think of $\text{Aut}(X)$
as a subgroup of ${\mathcal{S}}_{n}$. Under this correspondence, if a graph
$X$ has $n$ vertices then $\text{Aut}(X)$ consists of $n\times n$ permutation
matrices and for each $g\in\text{Aut}(X)$, $P(g)$ will denote the
corresponding permutation matrix. The next result gives a method to check
whether a given permutation matrix is an element of $\text{Aut}(X)$ or not.
###### Lemma 3.1 (Biggs [3]).
Let A be the adjacency matrix of a graph $X$. Then $g\in\text{Aut}(X)$ is an
automorphism of $X$ if and only if $P(g)A=AP(g)$.
The following result is very useful in this work, which provides the necessary
and sufficient condition for a graph to be connected and regular.
###### Lemma 3.2 (Biggs [3]).
A graph $X$ is connected regular if and only if $J\in{\mathcal{A}}(X)$.
###### Corollary 3.3.
If $X$ is a regular graph then $J$ is polynomial in either $A$ or $A^{c}$.
###### Proof.
For every graph $X$, either $X$ or $X^{c}$ is connected. Hence the result
follow from the above Lemma. ∎
Let $X$ be a connected regular graph. Then from the above lemma we have
$A(X^{c})\in{\mathcal{A}}(X)$. Hence we have the following corollary.
###### Corollary 3.4.
Let $X$ be a connected regular graph. Then $X^{c}$ is connected if and only if
${\mathcal{A}}(X)={\mathcal{A}}(X^{c})$.
###### Definition 3.1.
A graph $X$ is said to be a _pattern polynomial graph_ if its pattern matrices
are polynomials in the adjacency matrix of $X$.
###### Lemma 3.5.
A graph $X$ is a pattern polynomial graph if and only if
${\mathcal{A}}(X)={\mathcal{CC}}(X)$.
Consequently if $X$ is a pattern polynomial graph, then
${\mathcal{A}}(X)={\mathcal{L}}(X)={\mathcal{CC}}(X)$. For any graph $X$, we
have ${\mathcal{L}}(X)={\mathcal{L}}(X^{c})$ and
${\mathcal{CC}}(X)={\mathcal{CC}}(X^{c})$. Hence we have the following result.
###### Corollary 3.6.
Let $X$ be a pattern polynomial graph and $X^{c}$ is also connected. Then
${\mathcal{A}}(X)={\mathcal{L}}(X)={\mathcal{CC}}(X)={\mathcal{A}}(X^{c})={\mathcal{L}}(X^{c})={\mathcal{CC}}(X^{c})$.
### 3.1 Properties of pattern polynomial graphs
In this subsection we prove that if $X$ is a pattern polynomial graph, then
$X$ is necessarily a
a) connected regular graph b) distance polynomial graph c) walk regular graph
d) strongly distance-balanced graph. e) edge regular graph whenever $A$ is
also a pattern matrix. We also show that every pattern polynomial graph except
$K_{2}$ (complete graph with 2 vertices) has at least one multiple eigenvalue.
In particular, if $X$ is a pattern polynomial graph with odd number of
vertices, then we show that $\dim({\mathcal{A}}(X))\leq\frac{n-1}{2}$.
Throughout this section we suppose that $X$ is a pattern polynomial graph that
is ${\mathcal{A}}(X)={\mathcal{L}}(X)={\mathcal{CC}}(X)$ or $\ell=r$. From
Lemma 3.2 we have the following result.
###### Lemma 3.7.
Every pattern polynomial graph is a connected regular graph.
It is easy to see that the converse of the above lemma is not true. The above
lemma provides a necessary condition to have
${\mathcal{A}}(X)={\mathcal{L}}(X)$. That is if the graph $X$ is either not
connected or not regular, then ${\mathcal{A}}(X)\subsetneq{\mathcal{L}}(X)$.
Now we will state another necessary condition stronger than this.
Distance polynomial graph
Let $X=(V,E)$ be a connected graph with diameter $d$. The _$k$ -th distance
matrix_ $A_{k}(0\leq k\leq d)$ of $X$, is defined as
$(A_{k})_{rs}=\left\\{\begin{array}[]{cl}1,&{\mbox{ if }}d(v_{r},v_{s})=k\\\
0,&{\mbox{ otherwise.}}\end{array}\right.$
It follows that
$A_{0}=I\;\mbox{(Identity matrix)},\;A_{1}=A,\;A_{0}+A_{1}+\dots+A_{d}=J.$
A connected graph $X$ of diameter $d$ is said to be a distance polynomial
graph if $A_{k}\in{\mathcal{A}}(X)\;for\;0\leq k\leq d$. From the Lemma 3.2,
the following observation is evident.
###### Observation 3.8.
Every distance polynomial graph is a regular connected graph. The converse is
generally not true but every regular connected graph of diameter $2$ is
distance polynomial.
###### Lemma 3.9.
If $X$ be a connected graph of diameter $d$, then
$A_{k}\in{\mathcal{L}}(X)\;(0\leq k\leq d)$, where $A_{k}$ is the $k$-th
distance matrix of $X$.
###### Proof.
We prove the result by induction on $d$.
$A_{0}(=I),A_{1}(=A)\in{\mathcal{A}}(X)\subseteq{\mathcal{L}}(X)$. So the
result is true for $d=1$. Suppose that
$A_{0},A_{1},\ldots,A_{s-1}\in{\mathcal{L}}(X)$. Then $J-I-A_{0}-A_{1}-\dots-
A_{s-1}\in{\mathcal{L}}(X)$. Let $M=A^{s}\odot(J-I-A_{1}-\dots-
A_{s-1})\in{\mathcal{L}}(X)$. Observe that $M_{ij}\neq 0$ if and only if
$(A_{s})_{ij}=1$. As $M\in{\mathcal{L}}(X)$ we have
$M=\sum_{i=1}^{r}a_{i}P_{i}\;\mbox{where}\;a_{i}\in{\mathbb{C}}.$ (2)
Hence $A_{s}=\sum_{i:a_{i}\neq 0}P_{i}\in{\mathcal{L}}(X)$. ∎
Now from Corollary 1.3, every distance matrix is the sum of one or more
pattern matrices.
###### Corollary 3.10.
Every pattern polynomial graph is a distance polynomial graph.
From Observation 3.8, every connected regular graph of diameter $2$ is
distance polynomial. But all connected regular graphs of diameter $2$ are not
pattern polynomial graphs.
###### Definition 3.2 (Paul M.Weichsel [15]).
Let $v$ be a vertex in the graph $X$ of diameter $d$. The generalized degree
of $v$ is the $d$-tuple $(k_{1},k_{2}\ldots,k_{d})$, where $k_{i}$ is the
number of vertices whose distance from $v$ is $i$. The graph G is called
_super-regular_ if each vertex has the same generalized degree.
###### Theorem 3.11 (Paul M.Weichsel [15]).
Let $X$ be a connected graph of diameter $d$. $X$ is super-regular graph if
and only if $(A_{i}A_{j})_{rs}=(A_{j}A_{i})_{rs}$ for all $r,s$.
From the Corollary 3.10 every pattern polynomial graph is a super regular
graph. In the present literature super-regular graphs are also called
distance-degree regular or strongly distance-balanced graphs for details refer
[14].
Walk-regular graph
A graph $X$ is said to be walk-regular if for each $s$, the number of closed
walks of length $s$ starting at a vertex $v$ is independent of the choice of
$v$.
###### Theorem 3.12.
[9] Let $A$ be the adjacency matrix of a graph $X$. Then $X$ is walk-regular
if and only if the diagonal entries of $A^{s}\;\forall s$ are all equal.
The following lemma and corollary can be obtained from the Corollary 1.4 and
above theorem.
###### Lemma 3.13.
Every pattern polynomial graph is a walk-regular graph.
###### Corollary 3.14.
If $X$ is pattern polynomial graph then every pattern matrix other than the
identity matrix is the adjacency matrix of a regular graph.
###### Remark 3.15.
Let $X$ be a pattern polynomial graph and $P\in{\mathcal{A}}(X)$ be a
permutation matrix. Then from Corollary 1.3 and from the above result $P$ is
an element in the standard basis of ${\mathcal{A}}(X)$. Further it is easy to
see that the set of all permutation matrices in ${\mathcal{A}}(X)$ forms an
elementary abelian 2-group since matrices in ${\mathcal{A}}(X)$ are symmetric.
Let $X$ be a pattern polynomial graph and $\\{I=P_{1},P_{2},\ldots,P_{r}\\}$
be the set of its pattern matrices. Let us call the graph $X_{P_{i}}\;2\leq
i\leq r$ as pattern graph of $X$ with adjacency matrix $P_{i}$. Then form
Lemma 3.1, we have $\text{Aut}(X)\subseteq\text{Aut}(X_{P_{i}})$. In fact in
the next section we will show that
$\text{Aut}(X)\subseteq\text{Aut}(X_{P_{i}})$ is true even if $X$ is not a
pattern polynomial graph. Now we will show that every pattern polynomial graph
except $K_{2}$ has at least one multiple eigenvalue. In order to prove this,
we need the following definition and the result.
###### Definition 3.3.
A graph is said to be vertex transitive if its automorphism group acts
transitively on $V$. That is for any two vertices $x,y\in V,\exists g\in G$
such that $g(x)=y$.
###### Lemma 3.16.
[Biggs [3]] Let $X$ be a $k$-regular vertex transitive graph, and $\lambda$ be
a simple eigenvalue of $X$. Then
$\lambda=\left\\{\begin{array}[]{cl}k,&{\mbox{ if }}\;|V|\;\mbox{is odd},\\\
\mbox{one of the integers}\;2\alpha-k\;(\;0\leq\alpha\leq k),&{\mbox{
if}}\;|V|\;\mbox{is even.}\end{array}\right.$
###### Corollary 3.17.
If $X$ is a vertex transitive graph and $X\neq K_{2}$, then $X$ has at least
one multiple eigenvalue.
###### Corollary 3.18.
If $X$ is a pattern polynomial graph and $X\neq K_{2}$, then $X$ has a
multiple eigenvalue.
###### Proof.
If all eigenvalues of $X$ are simple, then $\dim({\mathcal{A}}(X))=n$ so every
pattern matrix is a symmetric permutation matrix whose sum is $J$.
Consequently $X$ is a vertex transitive graph. ∎
We can extend the above result to arbitrary graphs in the following manner.
###### Corollary 3.19.
Let $X$ be a graph with $n>2$ vertices and ${\mathcal{CC}}(X)$ is a
commutative algebra. Then $\dim({\mathcal{A}}(X))\leq n-1$ and
$\dim({\mathcal{CC}}(X))\leq n$.
###### Proof.
First observe that $X$ has at least one multiple eigenvalue if and only if
$\dim({\mathcal{A}}(X))\leq n-1$. Now from the Corollary 1.4, we have
$\dim({\mathcal{CC}}(X))\leq n$. If $\dim({\mathcal{A}}(X))=n$, then we will
get a contradiction from above corollary. ∎
If $X$ is a pattern polynomial graph with odd number of vertices, then from
Corollary 3.14 we have stronger result than above.
###### Lemma 3.20.
If $X$ is a pattern polynomial graph with odd number of vertices, then
$\dim({\mathcal{A}}(X))\leq\frac{n+1}{2}$.
###### Proof.
First observe that $\dim({\mathcal{A}}(X))$ is the number of pattern matrices.
From Corollary 3.14, all pattern graphs of $X$ are regular with odd number of
vertices. Consequently each is an even regular graph with regularity $\geq 2$.
Hence $X$ has atmost $\frac{n-1}{2}$ pattern graphs. ∎
A graph is said to be an edge-regular graph if any two of its adjacent
vertices have the same number of common neighbours. The following result is
easy to see.
###### Lemma 3.21.
If $X$ is a pattern polynomial graph and its adjacency matrix itself is a
pattern matrix then $X$ is an edge-regular graph.
###### Proof.
Let $X$ be a pattern polynomial graph with adjacency matrix $A$. Let
$\\{P_{0}=I,P_{1}=A,P_{2},\ldots,P_{r}\\}$ be the standard basis of
${\mathcal{A}}(X)$. Hence $A^{2}=a_{0}I+a_{1}A+a_{2}P_{2}+\dots+a_{r}P_{r}$
where $a_{i}\in{\mathbb{C}}$. Consequently any two adjacent vertices have
exactly $a_{1}$ common neighbours. ∎
In the following section, we see few classes of graphs which satisfy the
condition $\ell=r$. Consequently they are pattern polynomial graphs
## 4 Some graph classes which are pattern polynomial
In this section we will prove that the following classes of graphs are pattern
polynomial graphs a)orbit polynomial graphs b) distance regular graphs hence
distance transitive graphs c) connected compact regular graphs.
###### Definition 4.1.
Let $G$ be a subset of $n\times n$ permutation matrices forming a group. Then
${\mathcal{V}}_{{\mathbb{C}}}(G)=\\{A\in M_{n}({\mathbb{C}}):PA=AP\;\forall
P\in G\\}$ forms an algebra over ${\mathbb{C}}$ called the centralizer algebra
of the group $G$.
###### Definition 4.2.
If G is a group acting on a set $V$, then $G$ also acts on $V\times V$ by
$g(x,y)=(g(x),g(y))$. The orbits of G on $V\times V$ are called orbitals. In
the context of graphs, the orbitals of graph $X$ are orbitals of its
automorphism group $\text{Aut}(X)$ acting on the vertex set of $X$. That is,
the orbitals are the orbits of the arcs/non-arcs of the graph $X=(V,E)$. The
number of orbitals is called the rank of $X$.
An orbital can be represented by a $0,1$-matrix $M$ where $M_{ij}$ is $1$ if
$(i,j)$ belongs to the orbital. We can associate directed graphs to these
matrices. If the matrices are symmetric, then these can be treated as
undirected graphs.
###### Observation 4.1.
* •
The ‘1’ entries of any orbital matrix are either all on the diagonal or all
are off diagonal.
* •
The orbitals containing $1$’s on the diagonal will be called diagonal
orbitals.
###### Definition 4.3.
The centralizer algebra of a graph $X$ denoted by ${\mathcal{V}}(X)$ is the
centralizer algebra of its automorphism group acting on the vertex set of $X$.
###### Theorem 4.2.
[11] ${\mathcal{V}}_{(}X)$ is a coherent algebra and orbitals of
$\text{Aut}(X)$ acting on the vertex set of $X$ form its unique 0-1 matrix
basis.
Since $\text{Aut}(X)=\text{Aut}(X^{c})$, we have ${\mathcal{V}}(X)=V(X^{c})$.
Also ${\mathcal{CC}}(X)$ is the smallest coherent algebra containing $A(X)$
and ${\mathcal{V}}(X)$ is a coherent algebra of $X$ containing $A(X)$ so
${\mathcal{CC}}(X)\subseteq{\mathcal{V}}(X)$.
#### Orbit polynomial graphs
###### Definition 4.4.
A graph $X=(V,E)$ is orbit polynomial graph if each orbital matrix is a member
of ${\mathcal{A}}(X)$. That is, each orbital matrix is a polynomial in $A$.
###### Lemma 4.3.
If $X$ is orbit polynomial graph if and only if
${\mathcal{A}}(X)={\mathcal{V}}(X)$
If $X$ is a orbit polynomial graph, then
${\mathcal{A}}(X)={\mathcal{CC}}(X)={\mathcal{V}}(X)$. Hence every orbit
polynomial graph is a pattern polynomial graph.
If $X$ is an orbit polynomial graph and $X^{c}$ is connected, then from above
lemma we have
${\mathcal{A}}(X)={\mathcal{A}}(X^{c})={\mathcal{CC}}(X)=CC(X^{c})={\mathcal{V}}(X)=V(X^{c})$.
So we have the following result.
###### Corollary 4.4.
If $X$ is an orbit polynomial graph and $X^{c}$ is connected then $X^{c}$ is
also a orbit polynomial graph.
For any graph $X$, we have
${\mathcal{A}}(X)\subseteq{\mathcal{L}}(X)\subseteq{\mathcal{CC}}(X)\subseteq{\mathcal{V}}(X)$.
Consequently from Corollary 1.3 every pattern matrix is the sum of one or more
orbital matrices. Further by definition, orbital matrices commute with all
automorphisms of $X$ hence we have the following result.
###### Lemma 4.5.
Let $X$ be any graph and $\\{P_{1},P_{2},\ldots,P_{r}\\}$ be the set of all
pattern matrices of adjacency matrix of $X$. Then
$\text{Aut}(X)\subseteq\text{Aut}(X_{P_{i}})\;1\leq i\leq r$.
Every connected vertex transitive graph of prime order is an orbit polynomial
graph, see [Beezer [1]]. Following lemma gives a stronger result.
###### Lemma 4.6.
If $X$ is a connected graph of prime order, then $X$ is orbit polynomial graph
if and only if ${\mathcal{V}}(X)$ is commutative.
###### Proof.
If $X$ is an orbit polynomial graph then clearly ${\mathcal{V}}(X)$ is
commutative. Conversely suppose that ${\mathcal{V}}(X)$ is commutative, then
the identity matrix is in the standard basis of ${\mathcal{V}}(X)$.
Consequently $X$ is a vertex transitive graph, hence the result follows. ∎
An easy consequence of this lemma is that if $X$ is a connected graph of prime
order and ${\mathcal{V}}(X)$ is commutative, then $X$ is a pattern polynomial
graph.
#### Distance transitive graphs
###### Definition 4.5.
A graph $X$ is _distance transitive_ if for all vertices $u,v,x,y$ of $X$ such
that $d(u,v)=d(x,y)$ then there is a $g$ in $\text{Aut}(X)$ satisfying
$g(u)=x$ and $g(v)=y$.
###### Remark 4.7.
From the definition of distance transitivity following facts are immediate: a)
For a distance transitive graph with diameter $d$, distance matrices and
orbital matrices coincide, consequently its rank is $d+1$. b) From Equation 1
and the fact that ${\mathcal{A}}(X)\subseteq{\mathcal{V}}(X)$, if $X$ is a
distance transitive graph with diameter $d$, then dimension of
${\mathcal{A}}(X)$ is $d+1$. Further orbital matrices form a basis for
${\mathcal{A}}(X)$. This also implies the following lemma.
Now the following result is immediate from the Theorem 4.11 and the above
Remark.
###### Lemma 4.8.
Every distance transitive graph is an orbit polynomial graph.
Converse of the above lemma is not true, as every vertex transitive graph of
prime order is not a distance transitive graph. so we have
Distance transitive graph $\Rightarrow$ Orbit polynomial graph $\Rightarrow$
Pattern polynomial graph $\Rightarrow$ Distance polynomial graph.
#### Compact graphs
###### Definition 4.6.
A graph $X$ is said to be _compact_ if every doubly stochastic matrix which
commutes with $A(X)$ is a convex combination of matrices from $\text{Aut}(X)$.
A permutation group on a set X is generously transitive if, given any two
points, there is a permutation which interchanges them.
###### Theorem 4.9.
[8] Let $X$be a connected regular graph with $r$ distinct eigenvalues. If
$X$is compact, then $\text{Aut}(X)$ is a generously transitive permutation
group with rank $r$.
In a compact connected regular graph $X$ the number of distinct eigenvalues of
$A(X)$ is same as the number of orbitals. It is also the dimension of
${\mathcal{A}}(X)$. Hence we have the following corollary from the fact that
${\mathcal{A}}(X)\subseteq{\mathcal{V}}(X)$.
###### Corollary 4.10.
Every compact connected regular graph is orbit polynomial graph.
Godsil [8] showed that if $n\geq 7$, then the line graph of the complete graph
$K_{n}$, is a distance transitive graph but not compact graph. It is also easy
to check that if $X$ is compact, then so is its compliment $X^{c}$ . Hence if
$X$ is connected compact regular graph and $X^{c}$ is also connected, then
$X^{c}$ is compact connected regular graph but it need not be a distance
transitive graph $X=C_{6}$(the cycle graph) is such an example.
Now we will see class of graphs which are pattern polynomial graphs but need
not be orbit polynomials graphs.
#### Distance regular Graphs
###### Definition 4.7.
A connected graph is distance regular if for any two vertices $u$ and $v$, the
number of vertices at distance $i$ from $u$ and $j$ from $v$ depends only on
$i$, $j$, and the distance between $u$ and $v$. These graphs are necessarily
regular, since $u$ may be equal to $v$.
It is easy to see that every distance transitive graph is distance regular. In
fact, there are many distance regular graphs whose automorphism group is
trivial [13]. The following theorem establishes that every distance regular
graph is a pattern polynomial graph.
###### Theorem 4.11.
[Damerell [5]] Let $X$ be a distance regular graph with diameter $d$. Then
$\\{A_{0},A_{1},\dots,A_{d}\\}$ is a basis for the adjacency algebra
${\mathcal{A}}(X)$, and consequently the dimension of ${\mathcal{A}}(X)$ is
$d+1$.
###### Corollary 4.12.
Every distance regular graph is a pattern polynomial graph.
Observe that if $X$ is a distance regular graph then $A$ itself is a pattern
matrix. Consequently if $X$ is a distance regular graph with diameter $\geq
3$, then $X^{c}$ is not distance regular. Now if $X$ is a distance regular
graph and $X^{c}$ is connected then from Corollary 3.6
${\mathcal{A}}(X)={\mathcal{A}}(X^{c})={\mathcal{CC}}(X)={\mathcal{CC}}(X^{c})$.
Hence $X^{c}$ is also pattern polynomial graph but it need not be a distance
regular graph for example $C_{6}^{c}$. Further there are distance regular
graphs whose automorphism group is trivial [13]. So they can’t be orbit
polynomial graphs. Finally, if $X$ is a distance regular graph with diameter
$\geq 3$ with trivial automorphism group and $X^{c}$ is connected, then
$X^{c}$ is neither distance regular nor orbit polynomial graph, but it is a
pattern polynomial graph.
The following diagram gives the relationship among some of the graph classes
which we studied in this work.
Regular connected GraphsDistance polynomial graphsPattern polynomial
graphsOrbit polynomial graphsCompact connected regular graphsDistance
transitive graphsDistance regular graphs
## 5 PBIBD(t)s from pattern polynomial graphs
A design is an ordered pair $(V,\mathcal{B})$ with point set $V$ and set of
blocks $\mathcal{B}$ such that $\mathcal{B}$ is a collection of subsets of
$V$.
A design $(V,\mathcal{B})$ is called $t-(v,k,{\lambda})$ design(some times
only t-design) if $|V|=v,|B|=k\;\forall B\in\mathcal{B}$ and each subset of
$V$ of cardinality $t$ is contained in ${\lambda}$ blocks. One can show by
counting that a $t$-design is an $i$-design for each $0\leq i\leq t$. In fact
a $t-(v,k,{\lambda})$ design is a $i-(v,k,{\lambda}_{i})$ design with
${\lambda}_{i}={\lambda}{v-i\choose i-1}/{k-i\choose t-i}$ for each $0\leq
i\leq t$.
A balanced incomplete block design(BIBD) is a 2-design. The parameters
${\lambda}_{1}$ and ${\lambda}_{0}$ are usually denoted by $r_{1}$
(replication number) and $b$ (number of blocks).
###### Definition 5.1.
Given $v$symbols $1,2,\ldots,v$, a relation satisfying the following condition
is said to be an _symmetric association scheme_ with $m$ association classes:
1. 1.
Any two symbols $\alpha$ and $\beta$ are either first,second,.. or mth
associates and this relationship is symmetrical. We denote $(\alpha,\beta)=i$,
when $\alpha$ and $\beta$ ith associates.
2. 2.
Every symbol $\alpha$ has $n_{i}$, $i$th associates, the number $n_{i}$ being
independent of $\alpha$.
3. 3.
If $(\alpha,\beta)=i$ the number of symbols $\gamma$ that satisfy
simultaneously $(\alpha,\gamma)=j$ and $(\beta,\gamma)=j^{\prime}$ is
$p^{i}_{jj^{\prime}}$ and this number is independent of $\alpha$ and $\beta$.
Further $p^{i}_{jj^{\prime}}=p^{i}_{j^{\prime}j}$
The numbers $v,n_{i},p^{i}_{jj^{\prime}}$ are called the parameters of the
association scheme. If the relations are not symmetric, then it is called an
association scheme.
Let $R_{i}=\\{(\alpha,\beta)|(\alpha,\beta)=i\\}$ be the set of all $i$-th
associates. Then the relation $R_{i}$ of an association scheme can be
described by a $0,1$-matrices $A_{i}$. Hence above definition can be described
in terms of matrices as follows.
An association scheme with $d$ associate classes is a set
$\mathfrak{A}=\\{A_{0},\ldots,A_{d}\\}$ of 0,1-matrices such that
1. 1.
$A_{0}=I$.
2. 2.
$A_{0}+A_{1}+\dots+A_{d}=J$.
3. 3.
$A_{i}^{T}\in\mathfrak{A}$.
4. 4.
$A_{i}A_{j}=A_{j}A_{i}\in span(\mathfrak{A})$.
If $A_{i}^{T}=A_{i}(1\leq i\leq d)$, then $\mathfrak{A}$ is a symmetric
association scheme also called Bose-Mesner algebra. For example, if $X$ is a
pattern polynomial graph, then ${\mathcal{A}}(X)$ is a Bose-Mesner algebra.
But every Bose-Mesner algebra can not be obtained in this way. Now we will
give an example of a Bose-Mesner algebra $\mathfrak{A}$ which is not equal to
adjacency algebra of any graph.
Let $G$ be a finite abelian group of order $n(>2)$. Each element of $G$ gives
rise to a permutation of $G$, the permutation corresponding to ’$a$ maps g in
$G$ to $ga$’. Hence for each element $g$ in $G$ we have a permutation matrix
$P_{g}$; the map $g\rightarrow P_{g}$ is a group homomorphism. Therefore
$P_{g}P_{h}=P_{h}P_{g},P(g^{-1})=P_{g}^{T}$. We have $P(1)=I$ and $\sum_{g\in
G}P_{g}=J$. Hence the matrices $P_{g}$ forms an association scheme with $v-1$
classes. Note the association scheme obtained in this way is same as
centralizer algebra ${\mathcal{V}}_{\mathbb{C}}(G)$ which is further equal to
group algebra ${\mathbb{C}}[G]$. The restricted centralizer algebra
${\mathcal{V}}_{r}(G)$, consisting of all real, symmetric matrices in
${\mathcal{V}}_{\mathbb{C}}(G)$ is a real subalgebra of
${\mathcal{V}}_{\mathbb{C}}(G)$, which is closed under Hadamard product and
spanned by the matrices $P_{g}+P^{T}_{g}\;\forall g\in G$. That is
${\mathcal{V}}_{r}(G)$ is a symmetric association scheme. Further if we assume
that $G$ is an elementary abelian 2-group, then
${\mathbb{C}}[G]={\mathcal{V}}_{r}(G)={\mathcal{V}}_{\mathbb{C}}(G)$.
Consequently ${\mathbb{C}}[G]$ is a Bose-Mesner algebra and
$\dim({\mathbb{C}}[G])=n$. Now from Corollary 3.19 there exists no graph $X$
with $n>2$ vertices such that ${\mathcal{A}}(X)={\mathbb{C}}[G]$.
###### Definition 5.2.
Given an $m$-association scheme on $v$-symbols a PBIBD(m) with $m$ associate
classes is defined as follows. A PBIBD(m) with $m$ associate classes is an
arrangement of $v$ symbols in $b$ sets of size $k(<v)$ such that
1. 1.
Every symbol occurs at most once in a set.
2. 2.
Every symbol occurs in $r$ sets.
3. 3.
Two symbols $\alpha$ and $\beta$ occur in $\lambda_{i}$ sets, if
$(\alpha,\beta)=i$ and $\lambda_{i}$ is independent of symbols $\alpha$ and
$\beta$.
The numbers $v,b,r,k,\lambda_{i}$ are the parameters of the PBIBD. The PBIBD
is usually identified by the association scheme of the symbols. For more
information on design theory the reader is referred to Raghavarao [12]. For
any PBIBD(t) we will write the parameters as
$(v,b,r_{1},k_{1},\lambda_{1},\ldots,\lambda_{t})$ where $v$ is the number of
points , $b$ is the number of blocks, $r_{1}$ is called the replication
number, $k_{1}$ is the number of elements in any block of the design.
If $N$ is the incidence matrix of a design $\mathcal{D}$, then we say that the
design $\mathcal{D}$ is obtained from the graph $X$ if
$NN^{T}\in{\mathcal{A}}(X)$. For example, if $X$ is a pattern polynomial graph
with $n$ vertices , then
1. 1.
Every BIBD with $n$ points is obtained from $X$. In fact every t-design with
$n$ points and $t\geq 2$ is obtained from $X$.
2. 2.
Let a graph $Y$ be a polynomial in $X$ and $\mathcal{D}_{1}=(V(Y),E(Y))$ be a
design with points as vertices of graph $Y$ and blocks as edges of $Y$. Then
$NN^{T}=D+A(Y)\in{\mathcal{A}}(X)$ where $N$ is the incidence matrix of design
$\mathcal{D}_{1}$, which is also 0,1-incidence matrix of the graph $Y$ and $D$
is the diagonal matrix with diagonal entries are degree of vertices of $Y$.
Note that $\mathcal{D}_{1}$ is a PBIBD(r), where $r$ is the degree of the
minimal polynomial of $X$.
Let $v$ be any vertex in a graph $Z$, $N(v)$ be the set of vertices which are
adjacent to $v$ in $Z$, $\mathcal{B}_{2}=\\{N(v)|v\in V(Z)\\}$ and
$\mathcal{B}_{3}=\\{N(v)\cup\\{v\\}|v\in V(Z)\\}$. If
$\mathcal{D}_{2}=(V(Z),\mathcal{B}_{2})$ and
$\mathcal{D}_{3}=(V(Z),\mathcal{B}_{3})$, then $\mathcal{D}_{2}$ and
$\mathcal{D}_{3}$ are designs on $V(Z)$ with
$n=|V(Z)|=|\mathcal{B}_{2}|=|\mathcal{B}_{3}|=b$. Hence $A(Z)$ is the
incidence matrix of the designs $\mathcal{D}_{2}$ and $I+A(Z)$ is the
incidence matrix of $\mathcal{D}_{3}$. If $Z$ is a $k$-regular graph, then all
blocks in the design $\mathcal{D}_{2}(\mathcal{D}_{3})$ are $k$-subsets ($k+1$
subsets) of $V(Z)$. And each vertex belongs exactly $k$ ($k+1$) blocks. In
other words $\mathcal{D}_{2}(\mathcal{D}_{3})$ is a 1-design with
$r_{1}=k_{1}$. Further if we assume $Z$ is a polynomial in a pattern
polynomial graph $X$, then the designs $\mathcal{D}_{2}$ and $\mathcal{D}_{3}$
are PBIBD(r)s where $r$ is the degree of minimal polynomial of $X$. Hence we
have the following result.
###### Lemma 5.1.
Let a k-regular graph $Y$ be a polynomial in a pattern polynomial graph $X$
with $n$ vertices and $\mathcal{D}_{i},i=2,3$ are designs on $V(Y)$ defined as
above. Then $\mathcal{D}_{2}(\mathcal{D}_{3})$ is a PBIBD(r) obtained from $X$
with parameters
$(n,n,k,k,\lambda_{1},\ldots,\lambda_{r})((n,n,k+1,k+1,\lambda^{{}^{\prime}}_{1},\ldots,\lambda^{{}^{\prime}}_{r}))$
for some ${\lambda}_{i}$ and ${\lambda}_{i}^{{}^{\prime}}$.
## 6 On the polynomial of a pattern polynomial graph
A graph $Y$ is said to be a polynomial in a graph $X$ if
$A(Y)\in{\mathcal{A}}(X)$. For an arbitrary graph $X$ it seems difficult to
find whether a given graph is polynomial in $X$ or not. This question is
answered for orbit poynomial graph and distance regular graphs by [Robert
A.Beezer [2]] and [Paul M.Weichsel [15]] respectively. In the following lemma
we generalize those results to all pattern polynomial graphs.
###### Lemma 6.1.
If $X$ is a pattern polynomial graph with standard basis
$\\{P_{1},P_{2},\ldots,P_{r}\\}$ where $P_{1}=I$, then a graph $Y$ is a
polynomial in $X$ if and only if $A(Y)=\sum_{i=2}^{r-1}a_{i}P_{i}$ where
$a_{i}\in\\{0,1\\}$.
###### Proof.
Direct consequence of Corollary 1.3. ∎
###### Corollary 6.2.
There are $2^{r-1}$ graphs in the adjacency algebra of a pattern polynomial
graph $X$, where $r$ is the degree of the minimal polynomial of $A(X)$.
Another trivial fact is as follows.
###### Lemma 6.3.
Let a graph $Y$ be a polynomial in a pattern polynomial graph $X$, then
${\mathcal{CC}}(Y)\subseteq{\mathcal{CC}}(X)$.
If a graph $Y$ is a polynomial in a pattern polynomial graph $X$, then
${\mathcal{CC}}(Y)$ is a symmetric (every matrix in ${\mathcal{CC}}(Y)$ is
symmetric) commutative algebra. Hence
1. 1.
$Y$ is a walk regular graph,
2. 2.
$Y$ is a strongly distance-balanced graph, from Lemma 3.9,
3. 3.
$Y$ has a multiple eigenvalue, whenever $Y\neq K_{2}$, from Corollary 3.19,
4. 4.
$\dim({\mathcal{CC}}(Y))\leq n-1$, from Corollary 3.19. Further if the number
of vertices in $Y$ is odd, then $\dim({\mathcal{CC}}(Y))\leq\frac{n+1}{2}$.
Now it is interesting to answer the following question: If $Y$ is a graph such
that ${\mathcal{CC}}(Y)$ is symmetric commutative algebra, then “does there
exist a pattern polynomial graph $X$ such that $Y$ is a polynomial in $X$?”.
For example, if $Y$ is a circulant graph (Cayley graph on cyclic group) with
$n$ vertices, then clearly ${\mathcal{CC}}(Y)$ is symmetric commutative
algebra and it is also known that $Y$ is a polynomial in cycle graph $C_{n}$,
which is a pattern polynomial graph.
If a graph $Y$ is a polynomial in a graph $X$, then there exists a unique
polynomial $p_{Y}(x)\in{\mathbb{C}}[x]$, with degree less than the degree of
the minimal of $X$, such that $A(Y)=p_{Y}(A(X))$. It is called representor
polynomial of $Y$. If $X$ is a pattern polynomial graph and
$A(Y)=\sum_{i}a_{i}P_{i}$, then $p_{Y}=\sum_{i}a_{i}p_{X_{P_{i}}}$. If a graph
$Y$ is a polynomial in $X$ with representor polynomial $p_{Y}(x)$, then the
eigenvalues of $A(Y)$ are $p_{Y}(\lambda_{i})$, where $\lambda_{i}\;(0\leq
i\leq n-1)$ are eigenvalues of $A(X)$.
The following result gives whether a graph $Y$ which is a polynomial in a
graph $X$ is singular or not. Recall a graph is said to be singular if its
adjacency matrix is singular.
###### Lemma 6.4.
Let a graph $Y$ be a polynomial in a graph $X$. Then $Y$ is singular if and
only if $\deg(\gcd(p(x),p_{Y}(x)))\geq 1$, where $p(x)$ is the minimal
polynomial of $A(X)$ and $p_{Y}(x)$ is the representor polynomial of $Y$ with
respect to $X$.
## References
* [1] Robert A.Beezer, Orbit polynomial graphs of prime order, Discrete Mathematics 67 (1987) 139-147.
* [2] Robert A.Beezer, A disrespectful polynomial, Linear Algebra and its Applications, Volume 128, January 1990, Pages 139-146.
* [3] N.L. Biggs, Algebraic Graph Theory(second ed.), Cambridge University Press, Cambridge (1993).
* [4] A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance regular Graphs, Springer-Verlag, (1989).
* [5] Damerell. R.M, On Moore graphs, proc. Cambridge Philos. sec.74,227-236.
* [6] Philip J. Davis. Circulant matrices ”A Wiley-interscience publications,(1979).
* [7] Chris D. Godsil $\&$ Gordon Royle. Algebraic Graph Theory, Springer-Verlag, (2001).
* [8] Chris D. Godsil, Compact graphs and equitable partitions , Linear algebra and applications, 255:259-266 (1997).
* [9] C. D. Godsil and B. D. McKay, Feasibility conditions for the existence of walk-regular graphs, Linear algebra and its applications 30:51-61(1980).
* [10] Kenneth Hoffman and Ray Kunge, Linear Algebra (second edition), Prentice-Hall, (1971).
* [11] M.klin, C.R$\ddot{u}$cker,G.R$\ddot{u}$cker, G.Tinhofer, Algebraic Combinatorics in Mathematical Chemistry Methods and Algorithms.I. Permutation Groups and Coherent (cellular) Algebras. Technical report Technische universit$\ddot{a}$t M$\ddot{u}$nchen, TUM-M9510(1995).
* [12] Raghavarao, Constructions and combinatorial problems in Design of Experiments, Wiley, New York (1971).
* [13] E.Spence, Regular two-graphs on 36 vertices, Linear Alg. Appl. 226-228 (1995), 459-497.
* [14] Sergio Cabello and Primo$\check{z}$ Luk$\check{s}$i$\check{c}$, The complexity of obtaining a distance-balanced graph, The electronic journal of combinatorics 18 (2011), P49.
* [15] Paul M.Weichsel, On distance-regularity in graphs, Journal of combinatorial theory, Series B 32, 156-161 (1982).
|
arxiv-papers
| 2011-06-23T14:50:02 |
2024-09-04T02:49:20.019389
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A.Satyanarayana Reddy and Shashank K Mehta",
"submitter": "Satyanarayana Reddy Arikatla",
"url": "https://arxiv.org/abs/1106.4745"
}
|
1106.4853
|
# AUTOMORPHIC EQUIVALENCE OF LINEAR ALGEBRAS.
A.Tsurkov
Institute of Mathematics and Statistics.
University São Paulo.
Rua do Matão, 1010
Cidade Universitária
São Paulo - SP - Brasil - CEP 05508-090
arkady.tsurkov@gmail.com
###### Abstract
This research is motivated by universal algebraic geometry. We consider in
universal algebraic geometry the some variety of universal algebras $\Theta$
and algebras $H\in\Theta$ from this variety. One of the central question of
the theory is the following: When do two algebras have the same geometry? What
does it mean that the two algebras have the same geometry? The notion of
geometric equivalence of algebras gives a sort of answer to this question.
Algebras $H_{1}$ and $H_{2}$ are called geometrically equivalent if and only
if the $H_{1}$-closed sets coincide with the $H_{2}$-closed sets. The notion
of automorphic equivalence is a generalization of the first notion. Algebras
$H_{1}$ and $H_{2}$ are called automorphicaly equivalent if and only if the
$H_{1}$-closed sets coincide with the $H_{2}$-closed sets after some ”changing
of coordinates”.
We can detect the difference between geometric and automorphic equivalence of
algebras of the variety $\Theta$ by researching of the automorphisms of the
category $\Theta^{0}$ of the finitely generated free algebras of the variety
$\Theta$. By [3] the automorphic equivalence of algebras provided by inner
automorphism degenerated to the geometric equivalence. So the various
differences between geometric and automorphic equivalence of algebras can be
found in the variety $\Theta$ if the factor group $\mathfrak{A/Y}$ is big.
Hear $\mathfrak{A}$ is the group of all automorphisms of the category
$\Theta^{0}$, $\mathfrak{Y}$ is a normal subgroup of all inner automorphisms
of the category $\Theta^{0}$.
In [4] the variety of all Lie algebras and the variety of all associative
algebras over the infinite field $k$ were studied. If the field $k$ has not
nontrivial automorphisms then group $\mathfrak{A/Y}$ in the first case is
trivial and in the second case has order $2$. We consider in this paper the
variety of all linear algebras over the infinite field $k$. We prove that
group $\mathfrak{A/Y}$ is isomorphic to the group
$\left(U\left(k\mathbf{S}_{\mathbf{2}}\right)\mathfrak{/}U\left(k\left\\{e\right\\}\right)\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$,
where $\mathbf{S}_{\mathbf{2}}$ is the symmetric group of the set which has
$2$ elements, $U\left(k\mathbf{S}_{\mathbf{2}}\right)$ is the group of all
invertible elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$,
$e\in\mathbf{S}_{\mathbf{2}}$, $U\left(k\left\\{e\right\\}\right)$ is a group
of all invertible elements of the subalgebra $k\left\\{e\right\\}$,
$\mathrm{Aut}k$ is the group of all automorphisms of the field $k$.
So even the field $k$ has not nontrivial automorphisms the group
$\mathfrak{A/Y}$ is infinite. This kind of result is obtained for the first
time.
The example of two linear algebras which are automorphically equivalent but
not geometrically equivalent is presented in the last section of this paper.
This kind of example is also obtained for the first time.
## 1 Introduction.
In the first two sections we consider some variety $\Theta$ of one-sorted
algebras of the signature $\Omega$. Denote by
$X_{0}=\left\\{x_{1},x_{2},\ldots,x_{n},\ldots\right\\}$ a countable set of
symbols, and by $\mathfrak{F}\left(X_{0}\right)$ the set of all finite subsets
of $X_{0}$. We will consider the category $\Theta^{0}$, whose objects are all
free algebras $F\left(X\right)$ of the variety $\Theta$ generated by finite
subsets $X\in\mathfrak{F}\left(X_{0}\right)$. Morphisms of the category
$\Theta^{0}$ are homomorphisms of free algebras.
We denote some time $F\left(X\right)=F\left(x_{1},x_{2},\ldots,x_{n}\right)$
if $X=\left\\{x_{1},x_{2},\ldots,x_{n}\right\\}$ and even
$F\left(X\right)=F\left(x\right)$ if $X$ has only one element.
We assume that our variety $\Theta$ possesses the IBN property: for free
algebras $F\left(X\right),F\left(Y\right)\in\Theta$ we have
$F\left(X\right)\cong F\left(Y\right)$ if and only if
$\left|X\right|=\left|Y\right|$. In this case we have [4, Theorem 2] this
decomposition
$\mathfrak{A=YS}.$ (1.1)
of the group $\mathfrak{A}$ of all automorphisms of the category $\Theta^{0}$.
Hear $\mathfrak{Y}$ is a group of all inner automorphisms of the category
$\Theta^{0}$ and $\mathfrak{S}$ is a group of all strongly stable
automorphisms of the category $\Theta^{0}$.
###### Definition 1.1
An automorphism $\Upsilon$ of a category $\mathfrak{K}$ is inner, if it is
isomorphic as a functor to the identity automorphism of the category
$\mathfrak{K}$.
This means that for every $A\in\mathrm{Ob}\mathfrak{K}$ there exists an
isomorphism $s_{A}^{\Upsilon}:A\rightarrow\Upsilon\left(A\right)$ such that
for every $\alpha\in\mathrm{Mor}_{\mathfrak{K}}\left(A,B\right)$ the diagram
$\begin{array}[]{ccc}A&\overrightarrow{s_{A}^{\Upsilon}}&\Upsilon\left(A\right)\\\
\downarrow\alpha&&\Upsilon\left(\alpha\right)\downarrow\\\
B&\underrightarrow{s_{B}^{\Upsilon}}&\Upsilon\left(B\right)\end{array}$
commutes.
###### Definition 1.2
. An automorphism $\Phi$ of the category $\Theta^{0}$ is called strongly
stable if it satisfies the conditions:
1. A1)
$\Phi$ preserves all objects of $\Theta^{0}$,
2. A2)
there exists a system of bijections $\left\\{s_{F}^{\Phi}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ such that $\Phi$ acts on the morphisms
$\alpha:D\rightarrow F$ of $\Theta^{0}$ by this way:
$\Phi\left(\alpha\right)=s_{F}^{\Phi}\alpha\left(s_{D}^{\Phi}\right)^{-1},$
(1.2)
3. A3)
$s_{F}^{\Phi}\mid_{X}=id_{X},$ for every free algebra $F=F\left(X\right)$.
The subgroup $\mathfrak{Y}$ is a normal in $\mathfrak{A}$. We will calculate
the factor group $\mathfrak{A/Y\cong S/S\cap Y}$. This calculation is very
important for universal algebraic geometry.
All definitions of the basic notions of the universal algebraic geometry can
be found, for example, in [1], [2] and [3]. In universal algebraic geometry we
consider a ”set of equations” $T\subset F\times F$ in some finitely generated
free algebra $F$ of the arbitrary variety of universal algebras $\Theta$ and
we ”resolve” these equations in $\mathrm{Hom}\left(F,H\right)$, where
$H\in\Theta$. The set $\mathrm{Hom}\left(F,H\right)$ serves as an ”affine
space over the algebra $H$”. Denote by $T_{H}^{\prime}$ the set
$\left\\{\mu\in\mathrm{Hom}\left(F,H\right)\mid T\subset\ker\mu\right\\}$.
This is the set of all solutions of the set of equations $T$. For every set of
”points” $R$ of the affine space $\mathrm{Hom}\left(F,H\right)$ we consider a
congruence of equations defined by this set:
$R_{H}^{\prime}=\bigcap\limits_{\mu\in R}\ker\mu$. For every set of equations
$T$ we consider its algebraic closure $T_{H}^{\prime\prime}$ in respect to the
algebra $H$. A set $T\subset F\times F$ is called $H$-closed if
$T=T_{H}^{\prime\prime}$. An $H$-closed set is always a congruence.
###### Definition 1.3
Algebras $H_{1},H_{2}\in\Theta$ are geometrically equivalent if and only if
for every $X\in\mathfrak{F}\left(X_{0}\right)$ and every $T\subset
F\left(X\right)\times F\left(X\right)$ fulfills
$T_{H_{1}}^{\prime\prime}=T_{H_{2}}^{\prime\prime}$.
Denote the family of all $H$-closed congruences in $F$ by $Cl_{H}(F)$. We can
consider the category $C_{\Theta}\left(H\right)$ of the coordinate algebras
connected with the algebra $H\in\Theta$. Objects of this category are quotient
algebras $F\left(X\right)/T$, where $X\in\mathfrak{F}\left(X_{0}\right)$,
$T\in Cl_{H}(F\left(X\right))$. Morphisms of this category are homomorphisms
of algebras.
###### Definition 1.4
Let
$Id\left(H,X\right)=\bigcap\limits_{\varphi\in\mathrm{Hom}\left(F\left(X\right),H\right)}\ker\varphi$
be the minimal $H$-closed congruence in $\ F\left(X\right)$. Algebras
$H_{1},H_{2}\in\Theta$ are automorphically equivalent if and only if there
exists a pair $\left(\Phi,\Psi\right),$ where
$\Phi:\Theta^{0}\rightarrow\Theta^{0}$ is an automorphism,
$\Psi:C_{\Theta}\left(H_{1}\right)\rightarrow$ $C_{\Theta}\left(H_{2}\right)$
is an isomorphism subject to conditions:
1. A.
$\Psi\left(F\left(X\right)/Id\left(H_{1},X\right)\right)=F\left(Y\right)/Id\left(H_{2},Y\right)$,
where $\Phi\left(F\left(X\right)\right)=F\left(Y\right)$,
2. B.
$\Psi\left(F\left(X\right)/T\right)=F\left(Y\right)/\widetilde{T}$, where
$T\in Cl_{H_{1}}(F\left(X\right))$, $\widetilde{T}\in
Cl_{H_{2}}(F\left(Y\right))$,
3. C.
$\Psi$ takes the natural epimorphism
$\overline{\tau}:F\left(X\right)/Id\left(H_{1},X\right)\rightarrow
F\left(X\right)/T$ to the natural epimorphism
$\Psi\left(\overline{\tau}\right):F\left(Y\right)/Id\left(H_{2},Y\right)\rightarrow
F\left(Y\right)/\widetilde{T}$.
Note that if such a pair $\left(\Phi,\Psi\right)$ exists, then $\Psi$ is
uniquely defined by $\Phi$.
We can say, in certain sense, that automorphic equivalence of algebras is a
coinciding of the structure of closed sets after some ”changing of
coordinates” provided by automorphism $\Phi$.
Algebras $H_{1}$ and $H_{2}$ are geometrically equivalent if and only if an
inner automorphism $\Phi:\Theta^{0}\rightarrow\Theta^{0}$ provides the
automorphic equivalence of algebras $H_{1}$ and $H_{2}$. So, only strongly
stable automorphism $\Phi$ can provide us automorphic equivalence of algebras
which not coincides with geometric equivalence of algebras. Therefore, in some
sense, difference from the automorphic equivalence to the geometric
equivalence is measured by the factor group $\mathfrak{A/Y\cong S/S\cap Y}$.
## 2 Verbal operations and strongly stable automorphisms.
For every word $w=w\left(x_{1},\ldots,x_{k}\right)\in F\left(X\right)$, where
$F\left(X\right)\in\mathrm{Ob}\Theta^{0}$,
$X=\left\\{x_{1},\ldots,x_{k}\right\\}$ and for every algebra $H\in\Theta$ we
can define a $k$-ary operation $w_{H}^{\ast}$ on $H$ by
$w_{H}^{\ast}\left(h_{1},\ldots,h_{k}\right)=w\left(h_{1},\ldots,h_{k}\right)=\gamma_{h}\left(w\left(x_{1},\ldots,x_{k}\right)\right),$
where $\gamma_{h}$ is a homomorphism $F\left(X\right)\ni
x_{i}\rightarrow\gamma_{h}\left(x_{i}\right)=h_{i}\in H$, $1\leq i\leq k$.
This operation we call the verbal operation induced on the algebra $H$ by the
word $w\left(x_{1},\ldots,x_{k}\right)\in F\left(X\right)$. A system of words
$W=\left\\{w_{i}\mid i\in I\right\\}$ such that $w_{i}\in F\left(X_{i}\right)$
, $X_{i}=\left\\{x_{1},\ldots,x_{k_{i}}\right\\},$ determines a system of
$k_{i}$-ary operations $\left(w_{i}\right)_{H}^{\ast}$ on $H$. Denote the set
$H$ with the system of these operation by $H_{W}^{\ast}$.
We have a correspondence between strongly stable automorphisms and systems of
words which define the verbal operation and fulfill some conditions. This
correspondence explained in [4] and [5]: We denote the signature of our
variety $\Theta$ by $\Omega$, by $k_{\omega}$ we denote the arity of $\omega$
for every $\omega\in\Omega$. We suppose that we have the system of words
$W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$ satisfies the conditions:
1. Op1)
$w_{\omega}\left(x_{1},\ldots,x_{k_{\omega}}\right)\in
F\left(X_{\omega}\right)$, where
$X_{\omega}=\left\\{x_{1},\ldots,x_{k_{\omega}}\right\\}$;
2. Op2)
for every $F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ there exists an
isomorphism $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such that
$\sigma_{F}\mid_{X}=id_{X}$.
$F_{W}^{\ast}\in\Theta$ so isomorphisms $\sigma_{F}$ are defined uniquely by
the system of words $W$.
The set $S=\left\\{\sigma_{F}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ is a system of bijections which satisfies
the conditions:
1. B1)
for every homomorphism $\alpha:A\rightarrow B\in\mathrm{Mor}\Theta^{0}$ the
mappings $\sigma_{B}\alpha\sigma_{A}^{-1}$ and
$\sigma_{B}^{-1}\alpha\sigma_{A}$ are homomorphisms;
2. B2)
$\sigma_{F}\mid_{X}=id_{X}$ for every free algebra
$F\in\mathrm{Ob}\Theta^{0}$.
So we can define the strongly stable automorphism by this system of
bijections. This automorphism preserves all objects of $\Theta^{0}$ and acts
on morphism of $\Theta^{0}$ by formula (1.2), where $s_{F}^{\Phi}=$
$\sigma_{F}$.
Vice versa if we have a strongly stable automorphism $\Phi$ of the category
$\Theta^{0}$ then its system of bijections
$S=\left\\{s_{F}^{\Phi}:F\rightarrow F\mid F\in\mathrm{Ob}\Theta^{0}\right\\}$
defined uniquely. Really, if $F\in\mathrm{Ob}\Theta^{0}$ and $f\in F$ then
$s_{F}^{\Phi}\left(f\right)=s_{F}^{\Phi}\alpha\left(x\right)=\left(s_{F}^{\Phi}\alpha\left(s_{D}^{\Phi}\right)^{-1}\right)\left(x\right)=\left(\Phi\left(\alpha\right)\right)\left(x\right),$
(2.1)
where $D=F\left(x\right)$ \- $1$-generated free linear algebra - and
$\alpha:D\rightarrow F$ homomorphism such that $\alpha\left(x\right)=f$.
Obviously that this system of bijections $S=\left\\{s_{F}^{\Phi}:F\rightarrow
F\mid F\in\mathrm{Ob}\Theta^{0}\right\\}$ fulfills conditions B1 and B2 with
$\sigma_{F}=s_{F}^{\Phi}$.
If we have a system of bijections $S=\left\\{\sigma_{F}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$ which fulfills conditions B1 and B2 than
we can define the system of words
$W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$ satisfies the conditions
Op1 and Op2 by formula
$w_{\omega}\left(x_{1},\ldots,x_{k_{\omega}}\right)=\sigma_{F_{\omega}}\left(\omega\left(\left(x_{1},\ldots,x_{k_{\omega}}\right)\right)\right)\in
F_{\omega},$ (2.2)
where $F_{\omega}=F\left(X_{\omega}\right)$.
By formulas (2.1) and (2.2) we can check that there are
1. 1.
one to one and onto correspondence between strongly stable automorphisms of
the category $\Theta^{0}$ and systems of bijections satisfied the conditions
B1 and B2
2. 2.
one to one and onto correspondence between systems of bijections satisfied the
conditions B1 and B2 and systems of words satisfied the conditions Op1 and
Op2.
So we can find a strongly stable automorphism $\Phi$ of the category
$\Theta^{0}$ by finding a system of words which fulfills conditions Op1 and
Op2.
## 3 Verbal operations in linear algebras.
From now on, we consider the variety $\Theta$ of all linear algebras over
infinite field $k$. We consider linear algebras as one-sorted universal
algebras, i. e., multiplication by scalar we consider as $1$-ary operation for
every $\lambda\in k$: $H\ni h\rightarrow\lambda h\in H$ where $H\in\Theta$.
Hence the signature $\Omega$ of algebras of our variety contains these
operations: $0$-ary operation $0$; $\left|k\right|$ $1$-ary operations of
multiplications by scalars; $1$-ary operation $-:h\rightarrow-h$, where $h\in
H$, $H\in\Theta$; $2$-ary operation $\cdot$ and $2$-ary operation $+$. We will
finding the system of words $W=\left\\{w_{\omega}\mid\omega\in\Omega\right\\}$
satisfies the conditions Op1 and Op2. We denote the words corresponding to
these operations by $w_{0}$, $w_{\lambda}$ for all $\lambda\in k$, $w_{-}$,
$w_{\cdot}$, $w_{+}$.
For arbitrary $F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ we denote
$F\left(X\right)=\bigoplus\limits_{i=1}^{\infty}F_{i}$ the decomposition to
the linear spaces of elements which are homogeneous according the sum of
degrees of generators from the set $X$. We also denote the two-sides ideals
$\bigoplus\limits_{i=j}^{\infty}F_{i}=F^{j}$. From now on, the word ”ideal”
means two sided ideal of linear algebra.
We denote the group of all automorphisms of the field $k$ by $\mathrm{Aut}k$.
Our variety $\Theta$ possesses the IBN property, because $\left|X\right|=\dim
F/F^{2}$ fulfills for all free algebras $F=F\left(X\right)\in\Theta$. So we
have the decomposition (1.1) for group of all automorphisms of the category
$\Theta^{0}$.
Now we need to prove one technical fact about $1$-generated free linear
algebra $F\left(x\right)$.
###### Lemma 3.1
Let $\left\\{u_{1},\ldots,u_{r}\right\\}$ is the set of all monomials of
degree $n$ in $F\left(x\right)$ (basis of $F_{n}$),
$\left\\{v_{1},\ldots,v_{t}\right\\}$ is the set of all monomials of degree
$m$ in $F\left(x\right)$ (basis of $F_{m}$), $\varphi$ is an arbitrary
function from $\left\\{1,\ldots,n\right\\}$ to $\left\\{1,\ldots,t\right\\}$.
Denote by $\varphi\left(u_{l}\right)$ the monomial which is a results of
substitution into monomial $u_{l}$ ($1\leq l\leq r$) instead $j$-th from left
entry of $x$ the monomial $v_{\varphi\left(j\right)}$ ($1\leq j\leq n$). All
these monomials are distinct, i. e.,
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$ if and
only if $\varphi_{1}=\varphi_{2}$ and $u_{l_{1}}=u_{l_{2}}$, where
$\varphi_{1},\varphi_{2}:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$u_{l_{1}},u_{l_{2}}\in\left\\{u_{1},\ldots,u_{r}\right\\}$.
Proof. We will prove this lemma by induction by $n$ \- degree of monomials
from $\left\\{u_{1},\ldots,u_{r}\right\\}$. The claim of the lemma is trivial
for $n=1$. We assume that the claim of the lemma is proved for monomials which
have degree $<n$. We suppose that
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$, where
$\deg u_{l_{1}}=\deg u_{l_{2}}=n>1$,
$\varphi_{1},\varphi_{2}:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$.
$u_{l_{i}}=u_{l_{i}}^{(1)}\cdot u_{l_{i}}^{(2)}$, where $i=1,2$. We denote
$\deg u_{l_{i}}^{(1)}=c_{i}$. $1\leq c_{i}<n$ for $i=1,2$. For $i=1,2$ we have
$\varphi_{i}\left(u_{l_{i}}\right)=\varphi_{i}^{(1)}\left(u_{l_{i}}^{(1)}\right)\cdot\varphi_{i}^{(2)}\left(u_{l_{i}}^{(2)}\right)$,
where
$\varphi_{i}^{(1)}:\left\\{1,\ldots,c_{i}\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$\varphi_{i}^{(2)}:\left\\{1,\ldots,n-c_{i}\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$,
$\varphi_{i}^{(1)}\left(j\right)=\varphi_{i}\left(j\right)$ for $1\leq j\leq
c_{i}$, $\varphi_{i}^{(2)}\left(j\right)=\varphi_{i}\left(c_{i}+j\right)$ for
$1\leq j\leq n-c_{1}$.
$\varphi_{1}\left(u_{l_{1}}\right)=\varphi_{2}\left(u_{l_{2}}\right)$ if and
only if
$\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)=\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)$
and
$\varphi_{1}^{(2)}\left(u_{l_{1}}^{(2)}\right)=\varphi_{2}^{(2)}\left(u_{l_{2}}^{(2)}\right)$.
If $c_{1}\neq c_{2}$ then
$\deg\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)=c_{1}m\neq\deg\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)=c_{2}m$,
hence
$\varphi_{1}^{(1)}\left(u_{l_{1}}^{(1)}\right)\neq\varphi_{2}^{(1)}\left(u_{l_{2}}^{(1)}\right)$
and $\varphi_{1}\left(u_{l_{1}}\right)\neq\varphi_{2}\left(u_{l_{2}}\right)$.
So $c_{1}=c_{2}$ and, by our assumption,
$\varphi_{1}^{(1)}=\varphi_{2}^{(1)}$, $u_{l_{1}}^{(1)}=u_{l_{2}}^{(1)}$,
$\varphi_{1}^{(2)}=\varphi_{2}^{(2)}$, $u_{l_{1}}^{(2)}=u_{l_{2}}^{(2)}$.
Therefore $\varphi_{1}=\varphi_{2}$ and $u_{l_{1}}=u_{l_{2}}$.
###### Corollary 1
Let $f\left(x\right),g\left(x\right)\in F\left(X\right)$.
$f\left(g\left(x\right)\right)$ is a result of substitution of
$g\left(x\right)$ in $f\left(x\right)$ instead $x$.
$f\left(g\left(x\right)\right)\in F_{1}$ if and only if
$f\left(x\right),g\left(x\right)\in F_{1}$.
Proof. We write $f\left(x\right)$ and $g\left(x\right)$ as sum of its
homogeneous components:
$f\left(x\right)=f_{1}\left(x\right)+f_{2}\left(x\right)+\ldots+f_{n}\left(x\right)$,
$g\left(x\right)=g_{1}\left(x\right)+g_{2}\left(x\right)+\ldots+g_{m}\left(x\right)$,
$f_{i}\left(x\right),g_{i}\left(x\right)\in F_{i}$. We assume that $n>1$ or
$m>1$, $f_{n}\left(x\right)\neq 0$ and $g_{m}\left(x\right)\neq 0$.
$f\left(g\left(x\right)\right)=f_{1}\left(g\left(x\right)\right)+f_{2}\left(g\left(x\right)\right)+\ldots+f_{n}\left(g\left(x\right)\right)$.
Addenda of the maximal possible degree of $x$, which can appear in
$f\left(g\left(x\right)\right)$, i. e., addenda of degree $nm$ can appear in
$f_{n}\left(g\left(x\right)\right)$. They coincide with addenda of
$f_{n}\left(g_{m}\left(x\right)\right)$. Denote
$f_{n}\left(x\right)=\lambda_{1}u_{1}+\ldots+\lambda_{r}u_{r}$,
$g_{m}\left(x\right)=\mu_{1}v_{1}+\ldots+\mu_{t}v_{t}$, where
$\left\\{u_{1},\ldots,u_{r}\right\\}$ is the set of all monomials of degree
$n$ in $F\left(x\right)$, $\left\\{v_{1},\ldots,v_{t}\right\\}$ is the set of
all monomials of degree $m$ in $F\left(x\right)$, $\lambda_{i},\mu_{j}\in k$.
Not all $\left\\{\lambda_{1},\ldots,\lambda_{r}\right\\}$ and not all
$\left\\{\mu_{1},\ldots,\mu_{t}\right\\}$ are equal to $0$ by our assumption.
$f_{n}\left(g_{m}\left(x\right)\right)=\lambda_{1}u_{1}\left(g_{m}\left(x\right)\right)+\ldots+\lambda_{r}u_{r}\left(g_{m}\left(x\right)\right)$.
If we open the brackets in
$u_{l}\left(g_{m}\left(x\right)\right)=u_{l}\left(\mu_{1}v_{1}+\ldots+\mu_{t}v_{t}\right)$
($1\leq l\leq r$), we obtain addenda, which are results of substitution into
monomial $u_{l}$ instead all entry of $x$ some monomial from
$\mu_{1}v_{1},\ldots,\mu_{t}v_{t}$ in all possible options. We can say more
formal: for every function
$\varphi:\left\\{1,\ldots,n\right\\}\rightarrow\left\\{1,\ldots,t\right\\}$ we
obtain an addendum which is a results of substitution into monomial $u_{l}$
instead $j$-th from left entry of $x$ the monomial
$\mu_{\varphi\left(j\right)}v_{\varphi\left(j\right)}$ ($1\leq j\leq n$).
Therefore all addenda, which we obtain after the opening of the brackets in
$f_{n}\left(g_{m}\left(x\right)\right)$, distinct from the monomials discussed
in Lemma 3.1 only by coefficients. All these addenda have degree $nm>1$,
because $n>1$ or $m>1$. So addenda of $f_{n}\left(g_{m}\left(x\right)\right)$
can not cancel one another by Lemma 3.1. These addenda can not be canceled by
other addenda $f\left(g\left(x\right)\right)$, because all other addenda have
degree $<nm$. Therefore all these addenda equal to $0$, because
$f\left(g\left(x\right)\right)\in F_{1}$. For
$l\in\left\\{1,\ldots,r\right\\}$ and $j\in\left\\{1,\ldots,t\right\\}$ we
take the addendum which is a results of substitution into monomial
$\lambda_{l}u_{l}$ instead all entries of $x$ the monomial $\mu_{j}v_{j}$. The
coefficient of this addendum is $\lambda_{l}\mu_{j}^{n}=0$. So
$\lambda_{l}\mu_{j}=0$ for all $l\in\left\\{1,\ldots,r\right\\}$ and all
$j\in\left\\{1,\ldots,t\right\\}$. It contradicts the fact that
$f_{n}\left(x\right)\neq 0$ and $g_{m}\left(x\right)\neq 0$.
###### Theorem 3.1
The system of words
$W=\left\\{w_{0},w_{\lambda}\left(\lambda\in
k\right),w_{-},w_{+},w_{\cdot}\right\\}$ (3.1)
satisfies the conditions Op1 and Op2 if and only if $w_{0}=0$,
$w_{\lambda}=\varphi\left(\lambda\right)x_{1}$, $w_{-}=-x_{1}$,
$w_{+}=x_{1}+x_{2}$, $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi$ is
an automorphism of the field $k$, $a,b\in k$, $a\neq\pm b$.
Proof. Let $W$ (see (3.1) ) satisfies the conditions Op1 and Op2.
$w_{0}$ is an element of the $0$-generated free linear algebra. There is only
one element in this algebra: $0$. This is the only one opportunity for
$w_{0}$.
$w_{\lambda}\in F\left(x\right)$ for every $\lambda\in k$. Denote
multiplications by scalars in $\left(F\left(x\right)\right)_{W}^{\ast}$ by
$\ast$, i. e., $\lambda\ast f=w_{\lambda}\left(f\right)$ for every $f\in
F\left(x\right)$ and every $\lambda\in k$.
$\left(F\left(x\right)\right)_{W}^{\ast}\in\Theta$, therefore, if $\lambda=0$
then $0\ast x=w_{0}\left(x\right)=0$. If $\lambda\neq 0$ then
$1\ast x=\left(\lambda^{-1}\lambda\right)\ast
x=\lambda^{-1}\ast\left(\lambda\ast
x\right)=w_{\lambda^{-1}}\left(w_{\lambda}\left(x\right)\right)=x.$
Hence $w_{\lambda}=\varphi\left(\lambda\right)x$ by Corollary 1 from Lemma
3.1, where $\varphi\left(\lambda\right)\in k$. We can write
$\varphi\left(0\right)=0$. Also we have that for all
$\lambda_{1},\lambda_{2}\in k$ fulfills
$\left(\lambda_{1}\lambda_{2}\right)\ast
x=\varphi\left(\lambda_{1}\lambda_{2}\right)x$
and
$\left(\lambda_{1}\lambda_{2}\right)\ast
x=\lambda_{1}\ast\left(\lambda_{2}\ast
x\right)=\lambda_{1}\ast\left(\varphi\left(\lambda_{2}\right)x\right)=$
$\varphi\left(\lambda_{1}\right)\left(\varphi\left(\lambda_{2}\right)x\right)=\left(\varphi\left(\lambda_{1}\right)\varphi\left(\lambda_{2}\right)\right)x.$
So
$\varphi\left(\lambda_{1}\right)\varphi\left(\lambda_{2}\right)=\varphi\left(\lambda_{1}\lambda_{2}\right)$.
If $\mu\in k\setminus\left\\{0\right\\}$, then the $1$-ary operation of
multiplication by scalar $\mu$ is a verbal operation defined by some word
$w_{\mu}^{\ast}\left(x\right)\in\left(F\left(x\right)\right)_{W}^{\ast}$,
written be the operations defined by system of words $W$ \- see [5,
Proposition 4.2]. Hence, $\mu f=w_{\mu}^{\ast}\left(f\right)$ holds for every
$f\in F\left(x\right)$. Also there is
$w_{\mu^{-1}}^{\ast}\left(x\right)\in\left(F\left(x\right)\right)_{W}^{\ast}$
such that $\mu^{-1}f=w_{\mu^{-1}}^{\ast}\left(f\right)$ for every $f\in
F\left(x\right)$. $x=\mu^{-1}\left(\mu
x\right)=w_{\mu^{-1}}^{\ast}\left(w_{\mu}^{\ast}\left(x\right)\right)$. There
exists by Op2 an isomorphism
$\sigma_{F\left(x\right)}:F\left(x\right)\rightarrow\left(F\left(x\right)\right)_{W}^{\ast}$
such that $\sigma_{F\left(x\right)}\left(x\right)=x$. So
$\left(F\left(x\right)\right)_{W}^{\ast}$ is also $1$-generated free linear
algebra of $\Theta$ with the free generator $x$. Hence there exists a
decomposition
$\left(F\left(x\right)\right)_{W}^{\ast}=\bigoplus\limits_{i=1}^{\infty}F_{i}^{\ast}$,
where $F_{i}^{\ast}$ are linear spaces of elements which are homogeneous
according the degree of $x$ but in respect of operations defined by system of
words $W$. Therefore $w_{\mu}^{\ast}\left(x\right)=\lambda\ast x$, where
$\lambda\in k$, by Corollary 1 from Lemma 3.1. So $\mu x=\lambda\ast
x=\varphi\left(\lambda\right)x$ and $\mu=\varphi\left(\lambda\right)$, hence
$\varphi:k\rightarrow k$ is a surjection.
$w_{+}\in F\left(x_{1},x_{2}\right)=F$. There exists $n\in\mathbb{N}$, such
that
$w_{+}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)+p_{2}\left(x_{1},x_{2}\right)+\ldots+p_{n}\left(x_{1},x_{2}\right),$
where $p_{i}\left(x_{1},x_{2}\right)\in F_{i}$, $1\leq i\leq n$. We have for
every $\lambda\in k$ that
$w_{+}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=\lambda\ast
w_{+}\left(x_{1},x_{2}\right)=\varphi\left(\lambda\right)w_{+}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)p_{n}\left(x_{1},x_{2}\right)$
and
$w_{+}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=p_{1}\left(\lambda\ast
x_{1},\lambda\ast x_{2}\right)+p_{2}\left(\lambda\ast x_{1},\lambda\ast
x_{2}\right)+\ldots+p_{n}\left(\lambda\ast x_{1},\lambda\ast x_{2}\right)=$
$p_{1}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)+p_{2}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)+\ldots+p_{n}\left(\varphi\left(\lambda\right)x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\left(\varphi\left(\lambda\right)\right)^{2}p_{2}\left(x_{1},x_{2}\right)+\ldots+\left(\varphi\left(\lambda\right)\right)^{n}p_{n}\left(x_{1},x_{2}\right).$
We can take $\lambda\in k$ such that $\varphi\left(\lambda\right)$ is not a
solution of any equation $x^{i}=x$, where $2\leq i\leq n$. So,
$p_{i}\left(x_{1},x_{2}\right)=0$ for $2\leq i\leq n$ by equality of the
homogeneous components. Therefore $w_{+}=\alpha x_{1}+\beta x_{2}$, where
$\alpha,\beta\in k$. If we denote the operation defined by $w_{+}$ in
$\left(F\left(x_{1},x_{2}\right)\right)_{W}^{\ast}$ by $\bot$, then $x_{1}\bot
x_{2}=x_{2}\bot x_{1}$ holds, so $\alpha x_{1}+\beta x_{2}=\alpha x_{2}+\beta
x_{1}$ and $\alpha=\beta$. Also $x_{1}\bot 0=x_{1}$ holds and $\alpha
x_{1}=x_{1}$, so $\alpha=\beta=1$. Now, by consideration of $F\left(x\right)$,
we can conclude that for all $\lambda_{1},\lambda_{2}\in k$ fulfills
$\varphi\left(\lambda_{1}+\lambda_{2}\right)x=\left(\lambda_{1}+\lambda_{2}\right)\ast
x=\lambda_{1}\ast x\bot\lambda_{2}\ast x=$ $\lambda_{1}\ast x+\lambda_{2}\ast
x=\varphi\left(\lambda_{1}\right)x+\varphi\left(\lambda_{2}\right)x=\left(\varphi\left(\lambda_{1}\right)+\varphi\left(\lambda_{2}\right)\right)x,$
so $\varphi\left(\lambda_{1}+\lambda_{2}\right)=$
$\varphi\left(\lambda_{1}\right)+\varphi\left(\lambda_{2}\right)$ and
$\varphi$ is an automorphism of the field $k$.
Its clear now that $w_{-}=-x\in F\left(x\right)$, because
$w_{-}\left(x\right)=-1\ast x=\varphi\left(-1\right)x=\left(-1\right)x=-x.$
$w_{\cdot}\in F\left(x_{1},x_{2}\right)$. We write $w_{\cdot}$ as sum of its
homogeneous components according the degree of $x_{1}$:
$w_{\cdot}\left(x_{1},x_{2}\right)=p_{0}\left(x_{1},x_{2}\right)+p_{1}\left(x_{1},x_{2}\right)+p_{2}\left(x_{1},x_{2}\right)+\ldots+p_{n}\left(x_{1},x_{2}\right).$
We denote the operation defined by $w_{\cdot}$ in
$\left(F\left(x_{1},x_{2}\right)\right)_{W}^{\ast}$ by $\times$. So we have
for every $\lambda\in k$ that
$\left(\lambda\ast x_{1}\right)\times x_{2}=\lambda\ast\left(x_{1}\times
x_{2}\right)=\varphi\left(\lambda\right)w_{\cdot}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)p_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)p_{n}\left(x_{1},x_{2}\right).$
and
$\left(\lambda\ast x_{1}\right)\times
x_{2}=w_{\cdot}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)=$
$p_{0}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+p_{1}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+p_{2}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)+\ldots+p_{n}\left(\varphi\left(\lambda\right)x_{1},x_{2}\right)=$
$p_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)p_{1}\left(x_{1},x_{2}\right)+\left(\varphi\left(\lambda\right)\right)^{2}p_{2}\left(x_{1},x_{2}\right)+\ldots+\left(\varphi\left(\lambda\right)\right)^{n}p_{n}\left(x_{1},x_{2}\right).$
We can take, as above, $\lambda\in k$ such that by equality of the homogeneous
components we obtain that
$w_{\cdot}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)$. Now we
write $w_{\cdot}\left(x_{1},x_{2}\right)=p_{1}\left(x_{1},x_{2}\right)$ as sum
of its homogeneous components according the degree of $x_{2}$:
$w_{\cdot}\left(x_{1},x_{2}\right)=r_{0}\left(x_{1},x_{2}\right)+r_{1}\left(x_{1},x_{2}\right)+r_{2}\left(x_{1},x_{2}\right)+\ldots+r_{m}\left(x_{1},x_{2}\right).$
We have for every $\lambda\in k$ that
$x_{1}\times\left(\lambda\ast x_{2}\right)=\lambda\ast\left(x_{1}\times
x_{2}\right)=\varphi\left(\lambda\right)w_{\cdot}\left(x_{1},x_{2}\right)=$
$\varphi\left(\lambda\right)r_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)r_{m}\left(x_{1},x_{2}\right).$
and
$x_{1}\times\left(\lambda\ast
x_{2}\right)=w_{\cdot}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$r_{0}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+r_{1}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+r_{2}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)+\ldots+r_{m}\left(x_{1},\varphi\left(\lambda\right)x_{2}\right)=$
$r_{0}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)r_{1}\left(x_{1},x_{2}\right)+\varphi\left(\lambda\right)^{2}r_{2}\left(x_{1},x_{2}\right)+\ldots+\varphi\left(\lambda\right)^{m}r_{m}\left(x_{1},x_{2}\right).$
And, as above, we can conclude that
$w_{\cdot}\left(x_{1},x_{2}\right)=r_{1}\left(x_{1},x_{2}\right)$ where
$r_{1}\left(x_{1},x_{2}\right)$ is a homogeneous element of
$F\left(x_{1},x_{2}\right)$ such that
$\deg_{x_{1}}r_{1}\left(x_{1},x_{2}\right)=1$ and
$\deg_{x_{2}}r_{1}\left(x_{1},x_{2}\right)=1$. Therefore
$w_{\cdot}\left(x_{1},x_{2}\right)=ax_{1}x_{2}+bx_{2}x_{1}$, where $a,b\in k$.
If $a=b$ then the operation defined by $w_{\cdot}\left(x_{1},x_{2}\right)$ is
commutative. If $a=-b$ then the operation defined by
$w_{\cdot}\left(x_{1},x_{2}\right)$ is anticommutative. The isomorphisms
$\sigma_{F}:F\rightarrow F_{W}^{\ast}$, where $F\in\mathrm{Ob}\Theta^{0}$ can
not exists in both these cases if $F$ is not a $0$-generated free algebra.
Therefore we prove that if the system of words (3.1) satisfies the conditions
Op1 and Op2 then $w_{0}=0$, $w_{\lambda}=\varphi\left(\lambda\right)x_{1}$ for
all $\lambda\in k$, $w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$,
$w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$, where $\varphi$ is an automorphism of the
field $k$, $a,b\in k$, $a\neq\pm b$.
Now we must prove that for all $\varphi\in\mathrm{Aut}k$ and all $a,b\in k$
such that $a\neq\pm b$ the system of words (3.1) where $w_{0}=0$,
$w_{\lambda}=\varphi\left(\lambda\right)x_{1}$ for all $\lambda\in k$,
$w_{-}=-x_{1}$, $w_{+}=x_{1}+x_{2}$, $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$
fulfills condition Op2. It means that we must build for every
$F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ an isomorphism
$\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such that $\sigma_{F}\mid_{X}=id_{X}$.
We will prove, first of all, that $H_{W}^{\ast}\in\Theta$ for every
$H\in\Theta$. Operations defined by $w_{0}$, $w_{-}$, $w_{+}$ coincide with
$0$, $-$, $+$. So identities of the variety $\Theta$ (axioms of the linear
algebra) relating to these operations fulfill in $H_{W}^{\ast}$. Hence we only
need to check the axioms that involve the operations defined by $w_{\cdot}$
and $w_{\lambda}$ ($\lambda\in k$). As above we denote these operations by
$\times$ and by $\lambda\ast$.
$\lambda\ast\left(x+y\right)=\varphi\left(\lambda\right)\left(x+y\right)=\varphi\left(\lambda\right)x+\varphi\left(\lambda\right)y=\lambda\ast
x+\lambda\ast y,$ $\left(\lambda\mu\right)\ast
x=\varphi\left(\lambda\mu\right)x=\varphi\left(\lambda\right)\varphi\left(\mu\right)x=\varphi\left(\lambda\right)\left(\mu\ast
x\right)=\lambda\ast\left(\mu\ast x\right),$ $\left(\lambda+\mu\right)\ast
x=\varphi\left(\lambda+\mu\right)x=\left(\varphi\left(\lambda\right)+\varphi\left(\mu\right)\right)x=\varphi\left(\lambda\right)x+\varphi\left(\mu\right)x=\lambda\ast
x+\mu\ast x,$ $1\ast x=\varphi\left(1\right)x=1x=x,$
$x\times\left(y+z\right)=ax\left(y+z\right)+b\left(y+z\right)x=axy+axz+byx+bzx=x\times
y+x\times z,$ $\left(y+z\right)\times
x=a\left(y+z\right)x+bx\left(y+z\right)=ayx+azx+bxy+bxz=y\times x+z\times x,$
$\lambda\ast\left(x\times
y\right)=\varphi\left(\lambda\right)\left(axy+byx\right)=a\left(\varphi\left(\lambda\right)x\right)y+by\left(\varphi\left(\lambda\right)x\right)=\left(\varphi\left(\lambda\right)x\right)\times
y=$ $\left(\lambda\ast x\right)\times y=x\times\left(\lambda\ast y\right)$
fulfills for every $x,y,z\in H$, $\lambda,\mu\in k$.
Hence there exists a homomorphism $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ such
that $\sigma_{F}\mid_{X}=id_{X}$ for every
$F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$. Our goal is to prove that these
homomorphisms are isomorphisms. We will prove by induction by $i$ that
$\sigma_{F}\left(F_{i}\right)=F_{i}.$ (3.2)
for every $i\in\mathbb{N}$. If $X=\left\\{x_{1},\ldots,x_{n}\right\\}$ then
every element of $F_{1}$ has form $\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n}$,
where $\lambda_{1},\ldots,\lambda_{n}\in k$.
$\sigma_{F}\left(\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n}\right)=\lambda_{1}\ast\sigma_{F}\left(x_{1}\right)+\ldots+\lambda_{n}\ast\sigma_{F}\left(x_{n}\right)=\varphi\left(\lambda_{1}\right)x_{1}+\ldots+\varphi\left(\lambda_{n}\right)x_{n},$
so $\sigma_{F}\left(F_{1}\right)\subset F_{1}$.
$\sigma_{F}\left(\varphi^{-1}\left(\lambda_{1}\right)x_{1}+\ldots+\varphi^{-1}\left(\lambda_{n}\right)x_{n}\right)=\lambda_{1}x_{1}+\ldots+\lambda_{n}x_{n},$
so $\sigma_{F}\left(F_{1}\right)=F_{1}$.
Let (3.2) proved for $i$ such that $1\leq i<r$. Every element of $F_{r}$ is a
linear combination of the monomials of the form $uv$, where $u\in F_{i}$,
$v\in F_{j}$, $i+j=r$.
$\sigma_{F}\left(uv\right)=\sigma_{F}\left(u\right)\times\sigma_{F}\left(v\right)=a\sigma_{F}\left(u\right)\sigma_{F}\left(v\right)+b\sigma_{F}\left(v\right)\sigma_{F}\left(u\right),$
so $\sigma_{F}\left(F_{r}\right)\subset F_{r}$, because, by our assumption,
$\sigma_{F}\left(u\right)\in F_{i}$, $\sigma_{F}\left(v\right)\in F_{j}$.
Also, if $u=\sigma_{F}\left(\widetilde{u}\right)$,
$v=\sigma_{F}\left(\widetilde{v}\right)$, where $\widetilde{u}\in F_{r}$,
$\widetilde{v}\in F_{t}$, then
$\sigma_{F}\left(\widetilde{u}\widetilde{v}\right)=\sigma_{F}\left(\widetilde{u}\right)\times\sigma_{F}\left(\widetilde{v}\right)=u\times
v=auv+bvu$
$\sigma_{F}\left(\widetilde{v}\widetilde{u}\right)=\sigma_{F}\left(\widetilde{v}\right)\times\sigma_{F}\left(\widetilde{u}\right)=v\times
u=avu+buv=buv+avu,$
fulfills. $a\neq\pm b$, so the matrix $\left(\begin{array}[]{cc}a&b\\\
b&a\end{array}\right)$ is regular, hence there exist $\alpha,\beta\in k$ such
that
$uv=\alpha\sigma_{F}\left(\widetilde{u}\widetilde{v}\right)+\beta\sigma_{F}\left(\widetilde{v}\widetilde{u}\right)=\sigma_{F}\left(\varphi^{-1}\left(\alpha\right)\widetilde{u}\widetilde{v}+\varphi^{-1}\left(\beta\right)\widetilde{v}\widetilde{u}\right).$
Therefore $\sigma_{F}\left(F_{r}\right)=F_{r}$. We can conclude that
$\sigma_{F}$ is an epimorphism.
Now we will prove that $\ker\sigma_{F}=0$. Let $f\in\ker\sigma_{F}\subset
F\left(X\right)$. There exists $m\in\mathbb{N}$ such that
$f\in\bigoplus\limits_{i=1}^{m}F_{i}$.
$\sigma_{F}\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)=\bigoplus\limits_{i=1}^{m}F_{i}$
by (3.2). $\sigma_{F}$ is a linear mapping from the linear space
$\bigoplus\limits_{i=1}^{m}F_{i}$ with the original multiplication by scalars
in $F$ to the $\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}$ \- the
linear space $\bigoplus\limits_{i=1}^{m}F_{i}$ with the multiplication by
scalars which we denote by $\ast$. From formulas
$\sum\limits_{i=1}^{k}\left(\lambda_{i}\ast
e_{i}\right)=\sum\limits_{i=1}^{k}\varphi\left(\lambda_{i}\right)e_{i}$ and
$\sum\limits_{i=1}^{k}\lambda_{i}e_{i}=\sum\limits_{i=1}^{k}\left(\varphi^{-1}\left(\lambda_{i}\right)\ast
e_{i}\right)$ we can conclude that if $E$ is a basis of the linear space
$\bigoplus\limits_{i=1}^{m}F_{i}$ then $E$ is a basis of the linear space
$\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}$. So
$\dim\bigoplus\limits_{i=1}^{m}F_{i}=\dim\left(\bigoplus\limits_{i=1}^{m}F_{i}\right)_{W}^{\ast}<\infty$,
therefore $\ker\left(\sigma_{F}\mid\bigoplus\limits_{i=1}^{m}F_{i}\right)=0$
and $f=0$.
## 4 Group $\mathfrak{A/Y}$.
From now on, $W$ is a system of words (3.1) which fulfills conditions Op1 and
Op2.
The decomposition (1.1) is not split in general case, i. e. $\mathfrak{S\cap
Y\neq}\left\\{1\right\\}$ in general case. The strongly stable automorphism
$\Phi$ of the category $\Theta^{0}$ which corresponds to the system of words
$W$ is inner, by [4, Lemma 3], if and only if for every
$F\in\mathrm{Ob}\Theta^{0}$ there exists an isomorphism $c_{F}:F\rightarrow
F_{W}^{\ast}$ such that $c_{F}\alpha=\alpha c_{D}$ fulfills for every
$\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$ (by [5, Remark
3.1] $\alpha$ is also a homomorphism from $D_{W}^{\ast}$ to $F_{W}^{\ast}$).
Hear we need to prove one technical lemma.
###### Lemma 4.1
If $F=F\left(X\right)\in\mathrm{Ob}\Theta^{0}$ and $c_{F}:F\rightarrow
F_{W}^{\ast}$ is an isomorphism then there exists an isomorphism
$c_{i}:F/F^{i}\rightarrow F_{W}^{\ast}/F^{i}$ such that
$\chi_{i}^{\ast}c_{F}=c_{i}\chi_{i}$, where $\chi_{i}:F\rightarrow F/F^{i}$
and $\chi_{i}^{\ast}:F_{W}^{\ast}\rightarrow F_{W}^{\ast}/F^{i}$ are natural
homomorphisms, $i\in\mathbb{N}$.
Proof. If $H\in\Theta$ and $I$ is an ideal of $H$. If $\lambda\in k$, $y\in
I$, $h\in H$, then $\lambda\ast y=\varphi\left(\lambda\right)y\in I$, $y\times
h=ayh+bhy\in I$, analogously $h\times y\in I$. Therefore $I$ is an ideal of
$H_{W}^{\ast}$. Hence $F^{i}$ is an ideal of $F_{W}^{\ast}$.
If $\sigma_{F}:F\rightarrow F_{W}^{\ast}$ is an isomorphism such that
$\sigma_{F}\mid_{X}=id_{X}$, then by (3.2) we have
$c_{F}^{-1}\left(F^{i}\right)=c_{F}^{-1}\sigma_{F}\left(F^{i}\right)=F^{i}$
because $c_{F}^{-1}\sigma_{F}:F\rightarrow F$ is an isomorphism. So
$c_{F}\left(F^{i}\right)=F^{i}$. It finishes the proof.
###### Proposition 4.1
The strongly stable automorphism $\Phi$ which corresponds to the system of
words $W$ is inner if and only if $\varphi=id_{k}$ and $b=0$.
Proof. We suppose that strongly stable automorphism $\Phi$ which corresponds
to the system of words $W$ is inner. We assume that $\varphi\neq id_{k}$, i.,
e., there exists $\lambda\in k$ such that
$\varphi\left(\lambda\right)\neq\lambda$. We denote $F=F\left(x\right)$. We
take $\alpha\in\mathrm{End}F$, such that $\alpha\left(x\right)=\lambda x$. We
suppose that $c_{F}:F\rightarrow F_{W}^{\ast}$ is an isomorphism. $c_{2}$ is
defined as in the Lemma 4.1, and we by this Lemma we have:
$\chi_{2}^{\ast}c_{F}\left(x\right)=c_{2}\chi_{2}\left(x\right)=\mu\ast\chi_{2}^{\ast}\left(x\right)=\chi_{2}^{\ast}\left(\mu\ast
x\right)=\chi_{2}^{\ast}\left(\varphi\left(\mu\right)x\right),$
where operations in algebra $F_{W}^{\ast}/F^{2}$ we denote by same symbols as
operations in algebra $F_{W}^{\ast}$ and $\mu\in
k\setminus\left\\{0\right\\}$. Therefore
$c_{F}\left(x\right)\equiv\varphi\left(\mu\right)x\left(\mathop{\mathrm{m}od}F^{2}\right)$.
$\alpha\left(F^{2}\right)\subset F^{2}$ fulfils, so
$\alpha
c_{F}\left(x\right)=\alpha\left(\varphi\left(\mu\right)x+f_{2}\right)\equiv\alpha\left(\varphi\left(\mu\right)x\right)=\varphi\left(\mu\right)\alpha\left(x\right)=\varphi\left(\mu\right)\lambda
x\left(\mathop{\mathrm{m}od}F^{2}\right),$
where $f_{2}\in F^{2}$.
$c_{F}\alpha\left(x\right)=c_{F}\left(\lambda x\right)=\lambda\ast
c_{F}\left(x\right)=\varphi\left(\lambda\right)c_{F}\left(x\right)\equiv\varphi\left(\lambda\right)\varphi\left(\mu\right)x\left(\mathop{\mathrm{m}od}F^{2}\right).$
$\mu\neq 0$, so $\varphi\left(\mu\right)\neq 0$,
$\varphi\left(\lambda\right)\neq\lambda$ hence $\alpha c_{F}\neq c_{F}\alpha$.
This contradiction proves that $\varphi=id_{k}$.
Now we denote $F=F\left(x_{1},x_{2}\right)\in\mathrm{Ob}\Theta^{0}$. By our
assumption there exists an isomorphism $c_{F}:F\rightarrow F_{W}^{\ast}$ such
that $c_{F}\alpha=\alpha c_{F}$ fulfills for every $\alpha\in\mathrm{End}F$.
$c_{2}$ is defined as in the Lemma 4.1. $\alpha\left(F^{2}\right)\subset
F^{2}$ so we can define the homomorphism
$\widetilde{\alpha}:F/F^{2}\rightarrow F/F^{2}$ such that
$\widetilde{\alpha}\chi_{2}=\chi_{2}\alpha$. From $c_{F}\alpha=\alpha c_{F}$
we can conclude $c_{2}\widetilde{\alpha}=\widetilde{\alpha}c_{2}$ fulfills. By
Lemma 4.1 $c_{2}$ is a regular linear mapping. We can take the endomorphisms
$\alpha$ such that $\widetilde{\alpha}$ will be an arbitrary linear mapping
from $k^{2}$ to $k^{2}$. Therefore $c_{2}$ must be a regular linear mapping
from $k^{2}$ to $k^{2}$ which commutate with all linear mappings from $k^{2}$
to $k^{2}$. Hence $c_{2}$ must be a scalar mapping, i.e.,
$\chi_{2}^{\ast}c_{F}\left(x_{i}\right)=c_{2}\chi_{2}\left(x_{i}\right)=\lambda\chi_{2}^{\ast}\left(x_{i}\right)=\chi_{2}^{\ast}\left(\lambda
x_{i}\right),$
where $\lambda\in k\setminus\left\\{0\right\\}$, $i=1,2$. Therefore
$c_{F}\left(x_{i}\right)=\lambda x_{i}+f_{i}$, where $f_{i}\in F^{2}$,
$i=1,2$. We can remark that now we consider the case when $\varphi=id_{k}$,
hence we need not distinguish between multiplication by scalar in $F$ and
$F_{W}^{\ast}$.
Now we take $\alpha\in\mathrm{End}F$ such that
$\alpha\left(x_{1}\right)=x_{1}x_{2}$, $\alpha\left(x_{2}\right)=0$. If $u$ is
a monomial which contain only entries of $x_{1}$, then
$\deg_{x_{1}}\alpha\left(u\right)+\deg_{x_{2}}\alpha\left(u\right)=2\deg_{x_{1}}u$.
If a monomial $u$ contain at least one entry of $x_{2}$, then
$\alpha\left(u\right)=0$. Hence $\alpha\left(F^{2}\right)\subset F^{3}$. So we
have
$c_{F}\alpha\left(x_{1}\right)=c_{F}\left(x_{1}x_{2}\right)=c_{F}\left(x_{1}\right)\times
c_{F}\left(x_{2}\right)=ac_{F}\left(x_{1}\right)c_{F}\left(x_{2}\right)+bc_{F}\left(x_{2}\right)c_{F}\left(x_{1}\right)=$
$a\left(\lambda x_{1}+f_{1}\right)\left(\lambda
x_{2}+f_{2}\right)+b\left(\lambda x_{2}+f_{2}\right)\left(\lambda
x_{1}+f_{1}\right)\equiv
a\lambda^{2}x_{1}x_{2}+b\lambda^{2}x_{2}x_{1}\left(\mathop{\mathrm{m}od}F^{3}\right).$
$\alpha c_{F}\left(x_{1}\right)=\alpha\left(\lambda
x_{1}+f_{1}\right)\equiv\lambda
x_{1}x_{2}\left(\mathop{\mathrm{m}od}F^{3}\right).$
Hence we conclude $b=0$ from $c_{F}\alpha=\alpha c_{F}$.
If $b=0$, i. e., $w_{\cdot}=ax_{1}x_{2}$, $a\neq 0$, then we take
$c_{F}\left(f\right)=a^{-1}f$ for every $F\in\mathrm{Ob}\Theta^{0}$ and every
$f\in F$. It is obvious that $c_{F}$ is a regular linear mapping.
$c_{F}\left(f_{1}\right)\times
c_{F}\left(f_{2}\right)=ac_{F}\left(f_{1}\right)c_{F}\left(f_{2}\right)=a\left(a^{-1}f_{1}\right)\left(a^{-1}f_{2}\right)=a^{-1}f_{1}f_{2}=c_{F}\left(f_{1}f_{2}\right).$
for every $f_{1},f_{2}\in F$. So $c_{F}:F\rightarrow F_{W}^{\ast}$ is an
isomorphism. It fulfils
$c_{F}\alpha\left(d\right)=a^{-1}\alpha\left(d\right)=\alpha\left(a^{-1}d\right)=\alpha
c_{F}\left(d\right)$
for every $\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$ and
every $d\in D$.
###### Proposition 4.2
The group $\mathfrak{S\cong}G\mathfrak{\leftthreetimes}\mathrm{Aut}k$, where
$G$ is the group of all regular $2\times 2$ matrices over field $k$, which
have a form $\left(\begin{array}[]{cc}a&b\\\ b&a\end{array}\right)$ and every
$\varphi\in\mathrm{Aut}k$ acts on the group $G$ by this way:
$\varphi\left(\begin{array}[]{cc}a&b\\\
b&a\end{array}\right)=\left(\begin{array}[]{cc}\varphi\left(a\right)&\varphi\left(b\right)\\\
\varphi\left(b\right)&\varphi\left(a\right)\end{array}\right)$.
Proof. We will define the mapping
$\tau:G\mathfrak{\leftthreetimes}\mathrm{Aut}k\rightarrow\mathfrak{S}$. If
$g\varphi\in G\mathfrak{\leftthreetimes}\mathrm{Aut}k$, where
$g=\left(\begin{array}[]{cc}a&b\\\ b&a\end{array}\right)$, then we define
$\tau\left(g\varphi\right)=\Phi\in\mathfrak{S}$, where $\Phi$ corresponds to
the system of words $W$ with $w_{\lambda}=\varphi\left(\lambda\right)x_{1}$
for every $\lambda\in k$ and $w_{\cdot}=ax_{1}x_{2}+bx_{2}x_{1}$. By Section 2
and Theorem 3.1 $\tau$ is bijection.
We consider $\tau\left(g_{1}\varphi_{1}\right)=\Phi_{1}$ and
$\tau\left(g_{2}\varphi_{2}\right)=\Phi_{2}$, where
$g_{1}\varphi_{1},g_{2}\varphi_{2}\in
G\mathfrak{\leftthreetimes}\mathrm{Aut}k$ and
$g_{1}=\left(\begin{array}[]{cc}a_{1}&b_{1}\\\ b_{1}&a_{1}\end{array}\right)$,
$g_{2}=\left(\begin{array}[]{cc}a_{2}&b_{2}\\\ b_{2}&a_{2}\end{array}\right)$.
Both these strongly stable automorphisms preserves all objects of $\Theta^{0}$
and acts on morphisms of $\Theta^{0}$ by theirs systems of bijections
$\left\\{s_{F}^{\Phi_{i}}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}$, for $i=1,2$, according the formula (1.2).
We have
$\Phi_{2}\Phi_{1}\left(\alpha\right)=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\alpha\left(s_{D}^{\Phi_{1}}\right)^{-1}\left(s_{D}^{\Phi_{2}}\right)^{-1}$
for every $\left(\alpha:D\rightarrow F\right)\in\mathrm{Mor}\Theta^{0}$. So
strongly stable automorphism
$\Phi_{2}\Phi_{1}=\tau\left(g_{2}\varphi_{2}\right)\tau\left(g_{1}\varphi_{1}\right)$
preserves all objects of $\Theta^{0}$ and acts on morphisms of $\Theta^{0}$ by
system of bijections
$\left\\{s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}:F\rightarrow F\mid
F\in\mathrm{Ob}\Theta^{0}\right\\}.$
This system of bijections satisfies the conditions B1 and B2, so we can define
the words $w_{\lambda}^{\Phi_{2}\Phi_{1}}$ for every $\lambda\in k$ and
$w_{\cdot}^{\Phi_{2}\Phi_{1}}$ which correspond to the automorphism
$\Phi_{2}\Phi_{1}$ by formula (2.2). The words
$w_{\lambda}^{\Phi_{i}}(\lambda\in k)$ and $w_{\cdot}^{\Phi_{i}}$ which
correspond to the automorphism $\Phi_{i}$ have forms
$w_{\lambda}^{\Phi_{i}}=\varphi_{i}\left(\lambda\right)x_{1}(\lambda\in k)$
and $w_{\cdot}^{\Phi_{i}}=a_{i}x_{1}x_{2}+b_{i}x_{2}x_{1}$ for $i=1,2$. So
$w_{\lambda}^{\Phi_{2}\Phi_{1}}=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\left(\lambda
x_{1}\right)=s_{F}^{\Phi_{2}}\left(w_{\lambda}^{\Phi_{1}}\right)=s_{F}^{\Phi_{2}}\left(\varphi_{1}\left(\lambda\right)x_{1}\right)=\varphi_{2}\left(\varphi_{1}\left(\lambda\right)\right)x_{1}=\left(\varphi_{2}\varphi_{1}\right)\left(\lambda\right)x_{1}$
for every $\lambda\in k$ and
$w_{\cdot}^{\Phi_{2}\Phi_{1}}=s_{F}^{\Phi_{2}}s_{F}^{\Phi_{1}}\left(x_{1}x_{2}\right)=s_{F}^{\Phi_{2}}\left(w_{\cdot}^{\Phi_{1}}\right)=s_{F}^{\Phi_{2}}\left(a_{1}x_{1}x_{2}+b_{1}x_{2}x_{1}\right)=$
$\varphi_{2}\left(a_{1}\right)s_{F}^{\Phi_{2}}\left(x_{1}x_{2}\right)+\varphi_{2}\left(b_{1}\right)s_{F}^{\Phi_{2}}\left(x_{2}x_{1}\right)=$
$\varphi_{2}\left(a_{1}\right)\left(a_{2}x_{1}x_{2}+b_{2}x_{2}x_{1}\right)+\varphi_{2}\left(b_{1}\right)\left(a_{2}x_{2}x_{1}+b_{2}x_{1}x_{2}\right)=$
$\left(\varphi_{2}\left(a_{1}\right)a_{2}+\varphi_{2}\left(b_{1}\right)b_{2}\right)x_{1}x_{2}+\left(\varphi_{2}\left(a_{1}\right)b_{2}+\varphi_{2}\left(b_{1}\right)a_{2}\right)x_{2}x_{1}.$
because $s_{F}^{\Phi_{i}}:F\rightarrow F_{W_{i}}^{\ast}$ is an isomorphism,
$i=1,2$. Hence
$\Phi_{2}\Phi_{1}=\tau\left(g_{2}\varphi_{2}\right)\tau\left(g_{1}\varphi_{1}\right)=\tau\left(g_{2}\varphi_{2}\left(g_{1}\right)\varphi_{2}\varphi_{1}\right)=\tau\left(g_{2}\varphi_{2}\cdot
g_{1}\varphi_{1}\right).$
###### Corollary 1
Group $\mathfrak{S\cap Y}$ is isomorphic to the group $k^{\ast}I_{2}$ of the
regular $2\times 2$ scalar matrices over field $k$.
Proof. By Propositions 4.1 and 4.2.
###### Corollary 2
$\mathfrak{A/Y\cong}\left(G\mathfrak{/}k^{\ast}I_{2}\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$.
Proof. By Proposition 4.2 and Corollary 1 we have that
$\mathfrak{A/Y\cong}\left(G\mathfrak{\leftthreetimes}\mathrm{Aut}k\right)\mathfrak{/}k^{\ast}I_{2}$.
And we have
$\left(G\mathfrak{\leftthreetimes}\mathrm{Aut}k\right)\mathfrak{/}k^{\ast}I_{2}\mathfrak{\cong}\left(G\mathfrak{/}k^{\ast}I_{2}\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$
because $k^{\ast}I_{2}\vartriangleleft G$ and for every
$\varphi\in\mathrm{Aut}k$ $\varphi\left(k^{\ast}I_{2}\right)\subset
k^{\ast}I_{2}$ fulfills.
The symmetric group of the set which has $2$ elements -
$\mathbf{S}_{\mathbf{2}}$ can be embedded in the multiplicative structure of
the algebra $\mathbf{M}_{\mathbf{2}}\left(k\right)$ of the $2\times 2$
matrices over field $k$:
$\mathbf{S}_{\mathbf{2}}\ni\left(12\right)\rightarrow\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\in\mathbf{M}_{\mathbf{2}}\left(k\right)$, so $G\cong
U\left(k\mathbf{S}_{\mathbf{2}}\right)$, where
$U\left(k\mathbf{S}_{\mathbf{2}}\right)$ is the group of all invertible
elements of the group algebra $k\mathbf{S}_{\mathbf{2}}$. Also
$k^{\ast}I_{2}\cong U\left(k\left\\{e\right\\}\right)$, where
$e\in\mathbf{S}_{\mathbf{2}}$, $k\left\\{e\right\\}$ is a subalgebra of
$k\mathbf{S}_{\mathbf{2}}$, $U\left(k\left\\{e\right\\}\right)$ is a group of
all invertible elements of this subalgebra. Therefore
$\mathfrak{A/Y\cong}\left(U\left(k\mathbf{S}_{\mathbf{2}}\right)\mathfrak{/}U\left(k\left\\{e\right\\}\right)\right)\mathfrak{\leftthreetimes}\mathrm{Aut}k$,
where every $\varphi\in\mathrm{Aut}k$ acts on the algebra
$k\mathbf{S}_{\mathbf{2}}$ by natural way:
$\varphi\left(ae+b\left(12\right)\right)=\varphi\left(a\right)e+\varphi\left(b\right)\left(12\right)$.
## 5 Example of two linear algebras which are automorphically equivalent but
not geometrically equivalent.
We take $k=\mathbb{Q}$. $\Theta$ will be the variety of all linear algebras
over $k$. $H$ will be the $2$-generated linear algebra, which is free in the
variety corresponding to the identity $\left(x_{1}x_{1}\right)x_{2}=0$. We
consider the strongly stable automorphism $\Phi$ of the category $\Theta^{0}$
corresponding to the system of words $W$, where $b\neq 0$. Algebras $H$ and
$H_{W}^{\ast}$ are automorphically equivalent by [5, Theorem 5.1].
###### Proposition 5.1
Algebras $H$ and $H_{W}^{\ast}$ are not geometrically equivalent.
Proof. Let $F=F\left(x_{1},x_{2}\right)$. The ideal
$I=Id\left(H,\left\\{x_{1},x_{2}\right\\}\right)$ of the all two-variables
identities which are fulfill in the algebra $H$ will be the smallest
$H$-closed set in $F$, because $I=\left(0\right)_{H}^{\prime\prime}$, where
$0\in F$. If algebras $H$ and $H_{W}^{\ast}$ are geometrically equivalent then
the structures of the $H$-closed sets and of the $H_{W}^{\ast}$-closed sets in
$F$ coincide. Hence $I$ must be the smallest $H_{W}^{\ast}$-closed set in $F$.
By [5, Remark 5.1]
$T\rightarrow\sigma_{F}T$ (5.1)
is a bijection from the structure of the $H_{W}^{\ast}$-closed sets in $F$ to
the structure of the $H$-closed sets in $F$. Hear $\sigma_{F}:F\rightarrow
F_{W}^{\ast}$ is an isomorphism from condition Op2. It is clear that the
bijection (5.1) preserves inclusions of sets. So it transforms the smallest
$H_{W}^{\ast}$-closed set to the smallest $H$-closed set, i. e.
$I=\sigma_{F}I$ must fulfills.
It is obviously that $I\subset F^{3}$. By (3.2) $\sigma_{F}I\subset F^{3}$. We
will compare the linear subspaces $I/F^{4}$ and
$\left(\sigma_{F}I\right)/F^{4}$.
$I=\left\langle\alpha\left(\left(x_{1}x_{1}\right)x_{2}\right)\mid\alpha\in\mathrm{End}F\right\rangle$.
Let
$\alpha\left(x_{i}\right)\equiv\alpha_{1i}x_{1}+\alpha_{2i}x_{2}\left(\mathop{\mathrm{m}od}F^{2}\right)$,
where $i=1,2$, $\alpha_{ji}\in k$. Then
$\alpha\left(\left(x_{1}x_{1}\right)x_{2}\right)\equiv\left(\left(\alpha_{11}x_{1}+\alpha_{21}x_{2}\right)\left(\alpha_{11}x_{1}+\alpha_{21}x_{2}\right)\right)\left(\alpha_{12}x_{1}+\alpha_{22}x_{2}\right)\left(\mathop{\mathrm{m}od}F^{4}\right).$
We achieve after the extending of brackets that $I/F^{4}$ is a subspace of the
linear space spanned by the elements of $F^{3}/F^{4}$ which have form
$\left(x_{i}x_{j}\right)x_{k}+F^{4}$, where $i,j,k=1,2$. But
$\sigma_{F}I\ni\sigma_{F}\left(\left(x_{1}x_{1}\right)x_{2}\right)=a\sigma_{F}\left(x_{1}x_{1}\right)\sigma_{F}\left(x_{2}\right)+b\sigma_{F}\left(x_{2}\right)\sigma_{F}\left(x_{1}x_{1}\right)=$
$=a\left(a+b\right)\left(x_{1}x_{1}\right)x_{2}+b\left(a+b\right)x_{2}\left(x_{1}x_{1}\right).$
We have that $a+b\neq 0$, $b\neq 0$, so
$I/F^{4}\neq\left(\sigma_{F}I\right)/F^{4}$ and $I\neq\sigma_{F}I$. This
contradiction proves that algebras $H$ and $H_{W}^{\ast}$ are not
geometrically equivalent.
## References
* [1] B. Plotkin, Varieties of algebras and algebraic varieties. Categories of algebraic varieties. Siberian Advanced Mathematics, Allerton Press, 7:2 (1997), pp. 64 – 97.
* [2] B. Plotkin, Some notions of algebraic geometry in universal algebra, Algebra and Analysis, 9:4 (1997), pp. 224 – 248, St. Petersburg Math. J., 9:4, (1998) pp. 859 – 879.
* [3] B. Plotkin, Algebras with the same (algebraic) geometry, Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, v.242, (2003), pp. 17 – 207.
* [4] B. Plotkin, G. Zhitomirski On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 – 367.
* [5] A. Tsurkov, Automorphic equivalence of algebras. International Journal of Algebra and Computation. 17:5/6, (2007), 1263–1271.
|
arxiv-papers
| 2011-06-23T22:53:02 |
2024-09-04T02:49:20.030640
|
{
"license": "Public Domain",
"authors": "A. Tsurkov",
"submitter": "Arkady Tsurkov",
"url": "https://arxiv.org/abs/1106.4853"
}
|
1106.5005
|
††thanks: Present address: Joint Quantum Institute, NIST and University of
Maryland, Gaithersburg, MD 20899, USA††thanks: Present address: QUEST, Leibniz
Universität Hannover, 30167 Hannover and PTB, 38116 Braunschweig,
Germany††thanks: Present address: Halcyon Molecular, Redwood City, CA 94063,
USA††thanks: Present address: Centre for Quantum Technologies, National
University of Singapore, Singapore 117543, Singapore††thanks: Present address:
School of Physics, The University of Sydney, NSW 2006 Australia.
# Improved high-fidelity transport of trapped-ion qubits through a multi-
dimensional array
R.B. Blakestad C. Ospelkaus A.P. VanDevender J.H. Wesenberg M.J. Biercuk
D. Leibfried D.J. Wineland National Institute of Standards and Technology,
325 Broadway, Boulder, Colorado 80305, USA
###### Abstract
We have demonstrated transport of 9Be+ ions through a 2D Paul-trap array that
incorporates an X-junction, while maintaining the ions near the motional
ground-state of the confining potential well. We expand on the first report of
the experiment Blakestad et al. (2009), including a detailed discussion of how
the transport potentials were calculated. Two main mechanisms that caused
motional excitation during transport are explained, along with the methods
used to mitigate such excitation. We reduced the motional excitation below the
results in Ref. Blakestad et al. (2009) by a factor of approximately 50. The
effect of a mu-metal shield on qubit coherence is also reported. Finally, we
examined a method for exchanging energy between multiple motional modes on the
few-quanta level, which could be useful for cooling motional modes without
directly accessing the modes with lasers. These results establish how trapped
ions can be transported in a large-scale quantum processor with high fidelity.
###### pacs:
37.10.Ty, 37.10.Rs, 03.67.Lx
## I Introduction
The reliable transport of quantum information will enable operations between
any arbitrarily selected qubits in a quantum processor and is essential to
realize efficient, large-scale quantum information processing (QIP). Trapped
ions are a promising system in which to study QIP Blatt and Wineland (2008);
Monroe and Lukin (2008); Häffner et al. (2008), and several approaches to
achieving reliable information transport have been proposed Cirac and Zoller
(2000); Blatt and Wineland (2008); Monroe and Lukin (2008); Häffner et al.
(2008); Wineland et al. (1998); Kielpinski et al. (2002); Duan and Monroe
(2010); Lin et al. (2009). In most demonstrated entangling gate operations
that use ions, qubits stored in the internal atomic states of ions are
entangled by coupling the internal states with a single shared motional mode
through a laser-induced interaction Cirac and Zoller (1995); Blatt and
Wineland (2008); Monroe and Lukin (2008); Häffner et al. (2008). However, as
the number of ions grows large ($>10$), it becomes difficult to isolate a
single mode during gate operations Hughes et al. (1996); Wineland et al.
(1998). One way around this issue is to distribute the ions over an array of
harmonic potentials, where the number of ions in each trapping potential can
remain small. The potentials can be adjusted temporally to transport the ions
throughout the array and combine selected ions into a particular harmonic
potential. Once combined, gate operations can be performed on the selected
ions by use of a local shared mode of motion Wineland et al. (1998);
Kielpinski et al. (2002).
Initial demonstrations of such distributed architectures have incorporated
simple linear arrays Rowe et al. (2002); Barrett et al. (2004); Huber et al.
(2008); Home et al. (2009); Eble et al. (2009), where all ions are confined in
potential minima on a line along an axis of the trap. The order of ions within
the linear array can even be changed Splatt et al. (2009). However,
multidimensional arrays Kielpinski et al. (2002); Wineland et al. (1998)
provide the greatest flexibility in ion-trap processor architectures, and
permit more efficient reordering of ion strings for deterministic gate
operations. The key technical element that must be realized towards this end
is the two-dimensional junction, which consists of multiple intersecting
linear arrays. The potentials in a junction are more complicated than those in
a linear array, making transport through a junction challenging. Since the
fidelity of the gates is highest if the ions are near their motional ground
state, it is important that transport through such arrays be performed
reliably and with minimal excitation of the ion’s motion in its local trapping
potential. If multiple transports are needed, each transport should contribute
well under a single quantum of motional excitation, though sympathetic cooling
can be used to remove excess motional energy, at the cost of increased
experiment duration (and accompanying decoherence) Kielpinski et al. (2002).
For simple linear arrays, reliable transport with little motional excitation
has been demonstrated Rowe et al. (2002); Barrett et al. (2004); Home et al.
(2009).
To date, transport through a T-junction Hensinger et al. (2006), an X-junction
Blakestad et al. (2009) and surface-electrode Y-junctions Amini et al. (2010);
Moehring et al. (2011) have been demonstrated. However, such transport has not
yet been demonstrated with sufficiently-low motional excitation (at or below a
single quantum). Using the apparatus in Ref. Blakestad et al. (2009), we have
now realized highly reliable transport through an X-junction with excitation
of less than one quantum of motion per transport, a decrease of approximately
50 compared to the results in Ref. Blakestad et al. (2009). This has allowed
us to observe a process where energy can be exchanged between motional modes
in certain situations, and demonstrates motional control over the ions at the
single-quantum level. The paper is organized as follows: we begin in Sec. II
with a description of the X-junction trap array used for transport. Section
III lays out the procedure for calculating the time-dependent trapping
potentials that transport the ion. A description of the basic transport
experiment is given in Sec. IV. Section V covers the various mechanisms that
excite the ion’s motion during transport, as well as the filtering techniques
used to mitigate those excitations. This understanding of the noise sources,
and the subsequent improved filtering techniques, allowed the reduction in
motional excitation relative to Ref. Blakestad et al. (2009). To mitigate the
effects of magnetic field fluctuations on qubit decoherence, a mu-metal shield
and field-coil current stabilization were used, which is explained in Sec. VI.
Finally, in Sec. VII, we discuss a procedure for swapping motional energy
between motional modes at the center of the junction array. This swapping
process can potentially be used to laser-cool multiple modes of motion without
the need for a direct interaction between the cooling laser and every motional
mode.
## II X-junction Array
The X-junction array was based on the design of previous two-layer linear RF
Paul traps Rowe et al. (2002); Barrett et al. (2004); Jost (2010). The current
trap consisted of a stack of five high-purity alumina ($99.6\%$ Al2O3) wafers
clamped together (Fig. 1) with screws (visible in Fig. 2). The trap electrodes
resided in the ‘top’ and ‘bottom’ wafers. These wafers ($125~{}\mu$m thick)
were laser machined to cut out ‘main channels’ through the wafers, with
opposite sides of the channel forming rf and control electrodes. Slits,
nominally perpendicular to the main channel axes, separated the control
electrode side of the channel into a series of cantilevered structures to
produce separate control electrodes. Electrodes were formed onto the Al2O3 by
evaporating through a shadow-mask a $30$ nm titanium adhesion layer followed
by $0.5~{}\mu$m of gold, then overcoating with $3~{}\mu$m of electroplated
gold. Care was taken to coat all sides of each cantilevered structure to
minimize exposed dielectric that could otherwise charge and shift the
potential minima in an uncontrolled way.
Figure 1: A cross sectional view (not to scale) of the five-wafer stack, in
the $\hat{x},\hat{y}$ plane at the experiment zone ($\mathcal{E}$). Each wafer
had a channel cut through it to define the electrode structure and to provide
a path for laser beams to pass through the wafer stack. The top and bottom
wafers provided the confining potential; the ions were trapped between these
electrodes as indicated. The RC low-pass filters were surface-mounted to the
filter board with gold ribbon attached by use of resistive welds. The bias
wafer was a single electrode used to null stray electric fields along
$\hat{y}$. Gold (represented in yellow) was coated on the top side of the trap
wafers and wrapped around to the bottom side, and vice-versa for the bias
wafer. Gold wire bonds connected traces on the trap wafers to traces on the
filter board.
A spacer wafer provided a separation of $250~{}\mu$m between the two trap
electrode layers. These three wafers sat atop a $500~{}\mu$m thick ‘filter
board’, upon which in-vacuum RC filtering components were mounted. The ‘bias
wafer’ resembled the ‘top’ and ‘bottom’ wafers but with a single continuous
control electrode extending along all sides of the main channels. The bias
wafer sat below the filter board and was used to compensate stray electric
fields along $\hat{y}$.
Figure 2: Top view of the filter board and trap wafers. The filter board fills
the entire image, while the top wafer is the rotated square visible on the
right of the image. Cap screws, visible in two corners of the top wafer, held
the wafer stack together. Wire bonds connected the filter board traces to the
top and bottom trap wafers. Surface-mount resistive and capacitive elements on
the filter board provided filtering for the control potentials (see Fig. 10).
Gauge pins were used to help align the wafers during assembly. A misalignment
error of approximately $0.22^{\circ}$ was measured between the $\hat{z}$ axes
of the two electrode wafers, and this error was included in the computer model
of the trap used to determine the appropriate transport potentials.
Figure 3: (a) Cross-sectional view of the two layers of electrodes in the
X-junction array. (b) Top view of the electrode layout, with the rf electrodes
indicated, and all other (control) electrodes held at rf ground. The bottom
trap wafer, which sat below these electrodes, had a nearly identical set of
electrodes but with rf and control electrodes exchanged across the main
channel. Ions were trapped in the main channels between the rf and control
electrodes. Forty-six control electrodes (some of which are numbered for
reference) supported 18 different trapping zones. The load zone
($\mathcal{L}$), the main experiment zone ($\mathcal{E}$), the vertical zone
($\mathcal{V}$), the horizontal zone ($\mathcal{F}$) and the center of the
junction ($\mathcal{C}$) are labeled. (c) Schematic of the rf bridges from an
oblique angle (not to scale).
The electrode layout of the array is depicted in Fig. 3 and consisted of 46
control electrodes that produced 18 possible trapping zones. The experiment
zone, $\mathcal{E}$, was chosen as the zone where the ions interacted with
lasers for cooling and qubit operations. In addition to $\mathcal{E}$, zones
$\mathcal{F}$, $\mathcal{V}$, and $\mathcal{C}$ (at the center of the
junction) composed the four destinations of the transport protocols. The final
zone of interest was the load zone, $\mathcal{L}$, where the ions were
initially trapped.
The trap dimensions were similar to those in Refs. Barrett et al. (2004); Jost
(2010). The width of the channel between the rf and control electrodes was
$200~{}\mu$m, except near $\mathcal{L}$, where it increased to $300~{}\mu$m to
increase the volume of the loading zone and, with it, the loading probability.
Most control electrodes extended $200~{}\mu$m along the trap axis, but those
nearest to the junction were $100~{}\mu$m wide to ensure sufficient control
when ions were transported in this region.
At $\mathcal{C}$, two main channels crossed to form an X-junction, and two rf
bridges connected the rf electrodes on opposite sides of that junction (one on
the top trap wafer and one on the bottom). Without such bridges, the array
would not have provided harmonic three-dimensional confinement at the center
of the junction Chiaverini et al. (2005); Wesenberg (2009). The widths of the
bridges were $70~{}\mu$m, though the trapping potential was not strongly
dependent on this dimension.
These bridges introduced four axial pseudopotential barriers, one in each of
the entrances to the junction (along $\pm\hat{x}$ and $\pm\hat{z}$). Figure 4
shows the two simulated pseudopotential barriers along the $\hat{z}$ legs in
the X-junction array going toward $\mathcal{E}$ and $\mathcal{F}$ (the
asymmetry was due to the trap misalignment mentioned above). The height of
these barriers was a significant fraction of the transverse pseudopotential
trapping depth and was approximately 0.3 eV for 9Be+ , with rf potential of
$V_{\mathrm{rf}}\approx 200$ V (peak amplitude) and frequency
$\Omega_{\mathrm{rf}}\approx 2\pi\times 83$ MHz. At the apex of the barriers,
just outside the center of the junction, the pseudopotential was anti-
confining in the axial direction but still harmonically confining in the two
radial directions. It was possible to use the control electrodes to overwhelm
this anti-confinement and produce a 3D harmonic confining potential at all
points along the axis of the array.
Figure 4: Simulated pseudopotential barriers along the $z$ axis produced by
the rf bridges in the X-junction, with 0 being the junction’s center. Here, we
assumed $V_{\mathrm{rf}}\approx 200$ V and $\Omega_{\mathrm{rf}}\approx
2\pi\times 83$ MHz. The asymmetry between the two barriers was due to a slight
misalignment of the trap wafers.
Zone $\mathcal{E}$ was positioned far ($880~{}\mu$m) from the junction to
reduce the residual slope of the pseudopotential barrier in this zone. The
amplitude of the axial pseudopotential at $\mathcal{E}$ was estimated, by use
of computer models, to be $2.9\times 10^{-5}$ eV with a $8.7\times 10^{-8}$
eV$/\mu$m axial gradient, which would give rise to an axial ’micromotion’
amplitude of 47 nm at the drive amplitude specified above.
Figure 5: A Be oven was positioned out of the plane of this figure in the
positive $y$ direction (above the trap) and could be heated to produce a flux
of neutral Be. This Be would then travel down onto the trap, with a portion of
the flux passing into trap’s main channel at the load zone. Copropagating
photoionization and Doppler-cooling laser beams intersected the Be in the load
zone at 45∘ to the $xz$ plane of the page and parallel to $-\hat{y}+\hat{z}$.
An ‘L’-shaped oven barrier obscured the line-of-sight between the oven flux
and the zones used during the transport experiments to prevent neutral Be from
accumulating on the surfaces of the electrodes near the junction. This barrier
was positioned just above the trap electrodes, extending 1.6 cm along
$\hat{y}$ out of the plane of the page. Additional laser access was available
for beams passing through $\mathcal{E}$ (at 45∘ to the $xz$ plane) allowing
for cooling, detection, and gate operations at $\mathcal{E}$.
Ions were loaded into the array from a flux of neutral Be that passed through
$\mathcal{L}$ and was photoionized with a mode-locked laser that after two
stages of doubling produced 235 nm resonant with the S-to-P transition of
neutral Be. To help prevent buildup of neutral Be from the beam in other
regions of the array, $\mathcal{L}$ was located sufficiently far from
$\mathcal{E}$. In addition $\mathcal{L}$ was displaced along $x$ from
$\mathcal{E}$, by use of two $135^{\circ}$ bends in the main channel, to allow
an ‘L’-shaped stainless-steel shield to be placed 0.5 mm above the trap
wafers, preventing neutral Be from striking the experiment zone while allowing
laser access, as shown in Fig. 5. Transporting ions through such $135^{\circ}$
bends is relatively straightforward, and we were able to easily transport ions
between $\mathcal{L}$ and $\mathcal{E}$.
Whenever an ion was lost, a new ion was loaded into $\mathcal{L}$ and
immediately transferred to $\mathcal{E}$. It was also possible to use zone
$\mathcal{L}$ as a reservoir zone, where extra ions were loaded and held in
reserve until needed to replace ions lost in the experiment region. This
allowed the loading process to be performed less often, which avoided heating
the neutral Be oven and the concomitant degradation of the vacuum. Potentially
many ions could be simultaneously stored in such a zone, though we only stored
a small number and did not regularly make use of this feature of the trap. By
enabling a better vacuum, a reservoir can significantly increase ion lifetime.
In this scenario, it is desirable to maintain Doppler cooling in $\mathcal{L}$
to extend the ion lifetime.
## III Transport Potentials
The first demonstrations of ion transport in a multi-zone trap involved moving
an ion along a linear array Rowe et al. (2002). A protocol where two ions were
placed in a single trapping well and separated into two wells or combined from
two wells into a single well was also demonstrated Rowe et al. (2002); Barrett
et al. (2004). Since then, transport through linear arrays has been extended
to other contexts Splatt et al. (2009); Huber et al. (2008); Eble et al.
(2009), including transport through a junctions Hensinger et al. (2006);
Blakestad et al. (2009); Amini et al. (2010); Moehring et al. (2011) and
switching of ion order Hensinger et al. (2006); Splatt et al. (2009).
Here we outline the process used to calculate the time-series of control
potentials, or ‘waveforms’, used to transport ions through the X-junction.
This same basic procedure would be generally applicable to many ion-transport
situations. The goal was to move ions quickly, over long distances, while
maintaining low excitation of the ion’s secular motion in its local potential,
and traversing non-trivial potential landscapes such as those near junctions.
Ideally, the ion should move along the axial direction of the array while
remaining at the transverse pseudopotential minimum. The control electrodes
were used to create an overall harmonic trapping well whose minimum moved
along this desired trajectory. The procedure for determining waveforms can be
broken down into four steps: modeling the trap, determining the constraints,
solving for the appropriate potentials, and assigning the time dependence of
potentials.
An electrostatic model of the trap was constructed by use of boundary element
method (BEM) software Sadiku (2009); Singer et al. (2010). For each of $N$
electrodes, the model was run once, applying 1 V to the $n^{\textrm{th}}$
given electrode while grounding all other electrodes. The potential resulting
from each of these voltage configurations, $\tilde{\phi}_{n}(\mathbf{r})$, was
extracted (in the form of a $5~{}\mu$m grid) in the region through which the
ion would pass. These individual electrode potentials could then be weighted
by the actual voltage applied to the electrode, $V_{n}$, and summed to find
the total potential:
$\Phi(\mathbf{r})=\phi_{\mathrm{ps}}(\mathbf{r})+\displaystyle\sum_{n=1}^{N}V_{n}\tilde{\phi}_{n}(\mathbf{r}).$
(1)
Here we have included the contribution of the rf pseudopotential
$\phi_{\mathrm{ps}}$. The pseudopotential was found by first modeling the rf
potential $\tilde{\phi}_{\mathrm{rf}}$ as if it were a static potential at 1
V. Then an additional step was used to convert the rf potential into a time-
independent pseudopotential by use of
$\phi_{\mathrm{ps}}(\mathbf{r})=\frac{q}{4m\Omega_{\mathrm{rf}}^{2}}\left(V_{\mathrm{rf}}\nabla\tilde{\phi}_{\mathrm{rf}}(\mathbf{r})\right)^{2},$
(2)
where $V_{\mathrm{rf}}$ was the peak voltage applied to the rf electrode, and
$q$ and $m$ are the charge and mass of 9Be+ , respectively. (Throughout this
section, all $\phi$ potentials (including $\phi_{\mathrm{ps}}$) are reported
as electric potentials (in units of V) and not energy potentials (units of
eV); these are related by a factor of $q$.)
The waveform was built up from a string of individual solutions, where each
solution satisfied a set of constraints on the trapping potential centered at
a certain position. These constraints are defined below, but relate to
defining the secular frequencies and orientation of the principle axes of the
potential. By advancing that position by $5~{}\mu$m along the intended ion
trajectory for each subsequent solution, the series of potential steps was
created that moved the potential well along the sequence of positions. In
theory, the constraints can be set to completely define a harmonic potential
localized at the desired position, while also constraining the three secular
frequencies and the orientation of the principal axes. This would imply nine
constraints, which we assume for now, though below we will relax some of these
constraints when solving for the experiment waveforms.
To produce a trapping potential, $\Phi(\mathbf{r})$, with a minimum at
$\mathbf{r}_{0}=(x_{0},y_{0},z_{0})$, we enforce
$\nabla\Phi(\mathbf{r}_{0})\doteq 0,$ (3)
where $\doteq$ is used to mean ‘constrained to be true’.
The Hessian matrix,
$\mathcal{H}(\mathbf{r}_{0})\equiv
q\left[{\begin{array}[]{ccc}\frac{\partial^{2}}{\partial
x^{2}}&\frac{\partial^{2}}{\partial x\partial y}&\frac{\partial^{2}}{\partial
x\partial z}\\\ \frac{\partial^{2}}{\partial y\partial
x}&\frac{\partial^{2}}{\partial y^{2}}&\frac{\partial^{2}}{\partial y\partial
z}\\\ \frac{\partial^{2}}{\partial z\partial x}&\frac{\partial^{2}}{\partial
z\partial y}&\frac{\partial^{2}}{\partial
z^{2}}\end{array}}\right]\Phi(\mathbf{r}_{0}),$ (4)
can be used to extract the remaining six parameters of the harmonic potential:
the eigenvalues $\lambda_{i}$ of $\mathcal{H}(\mathbf{r}_{0})$ are related to
the secular frequencies, $\lambda_{i}=m\omega_{i}^{2}$ and the eigenvectors
point along the principal axes. By completely constraining the Hessian, we
constrain these quantities. Note that the Hessian is symmetric
($\mathcal{H}=\mathcal{H}^{\mathrm{T}}$), and has only six independent
entries.
It is most convenient to evaluate the Hessian in the frame of the desired
principal axes, $(x^{\prime},y^{\prime},z^{\prime})$, in which case the
Hessian constraint equation simplifies to
$\mathcal{H}(\mathbf{r}_{0})\doteq
m\left[{\begin{array}[]{ccc}\omega_{x^{\prime}}^{2}&0&0\\\
0&\omega_{y^{\prime}}^{2}&0\\\
0&0&\omega_{z^{\prime}}^{2}\end{array}}\right],$ (5)
where diagonal entries constrain the desired secular frequencies
$(\omega_{x^{\prime}},\omega_{y^{\prime}},\omega_{z^{\prime}})$, and the off-
diagonal entries constrain the principal axes to point along
$(x^{\prime},y^{\prime},z^{\prime})$. If the Hessian is evaluated in a
different basis, the right-hand side of Eq. (5) will not be diagonal, and the
frequency and axis constraints are mixed. Nonetheless, an appropriate choice
for the right-hand side can still be made in that case Blakestad (2010). From
here on, we assume $(x,y,z)=(x^{\prime},y^{\prime},z^{\prime})$.
In the interest of compact nomenclature, it is convenient to define several
column vectors:
$\mathbf{V}\equiv\left[{\begin{array}[]{ccccc}1&V_{1}&V_{2}&\dots&V_{N}\end{array}}\right]^{\mathrm{T}}$
(6)
and
$\Psi(\mathbf{r})\equiv\left[{\begin{array}[]{ccccc}\phi_{\mathrm{ps}}(\mathbf{r})&\tilde{\phi}_{1}(\mathbf{r})&\tilde{\phi}_{2}(\mathbf{r})&\dots&\tilde{\phi}_{N}(\mathbf{r})\end{array}}\right]^{\mathrm{T}},$
(7)
where $\mathbf{A}^{\mathrm{T}}$ denotes the transpose of $\mathbf{A}$ and
$\Phi(\mathbf{r}_{0})=\Psi^{\mathrm{T}}(\mathbf{r}_{0})\mathbf{V}$. Finally,
we define the 12-component operator
$\begin{split}\mathcal{P}\equiv&\bigg{[}\begin{array}[]{cccccc}\frac{\partial}{\partial
x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial
z}&\frac{\partial^{2}}{\partial x^{2}}&\frac{\partial^{2}}{\partial x\partial
y}&\frac{\partial^{2}}{\partial x\partial z}\end{array}\\\
&\quad\begin{array}[]{cccccc}\frac{\partial^{2}}{\partial y\partial
x}&\frac{\partial^{2}}{\partial y^{2}}&\frac{\partial^{2}}{\partial y\partial
z}&\frac{\partial^{2}}{\partial z\partial x}&\frac{\partial^{2}}{\partial
z\partial y}&\frac{\partial^{2}}{\partial
z^{2}}\end{array}\bigg{]}^{\mathrm{T}},\end{split}$ (8)
where the first three components are the gradient and the next nine components
are the Hessian operator.
The nine position, frequency, and axis constraints defined by Eqs. 3 and 5 can
be assembled into one equation:
$\mathbf{C}_{1}\bigl{(}\mathcal{P}\otimes\Psi^{\mathrm{T}}(\mathbf{r}_{0})\bigr{)}\mathbf{V}\doteq\mathbf{C}_{2},$
(9)
where $\mathbf{C}_{1}$ is a $j\times 12$ matrix and $\mathbf{C}_{2}$ is a
$j\times 1$ column vector, where $j=9$ for this example.
The position constraints in Eq. 3 can be reconstructed by using
$\mathbf{C}_{1}$ to pick out the three gradient components of $\mathcal{P}$
and $\mathbf{C}_{2}$ to set them to zero. The constraints in Eq. 5 can be
treated in a similar manner. Thus, to encode the nine desired constraints, we
use
$\mathbf{C}_{1}=\left[{\begin{array}[]{ccc@{\hspace{11pt}}ccccccccc}1&0&0\hfil\hskip
11.0&0&0&0&0&0&0&0&0&0\\\ 0&1&0\hfil\hskip 11.0&0&0&0&0&0&0&0&0&0\\\
0&0&1\hfil\hskip 11.0&0&0&0&0&0&0&0&0&0\\\\[6.0pt] 0&0&0\hfil\hskip
11.0&1&0&0&0&0&0&0&0&0\\\ 0&0&0\hfil\hskip 11.0&0&0&0&0&1&0&0&0&0\\\
0&0&0\hfil\hskip 11.0&0&0&0&0&0&0&0&0&1\\\\[6.0pt] 0&0&0\hfil\hskip
11.0&0&1&0&0&0&0&0&0&0\\\ 0&0&0\hfil\hskip 11.0&0&0&1&0&0&0&0&0&0\\\
0&0&0\hfil\hskip 11.0&0&0&0&0&0&1&0&0&0\end{array}}\right]$ (10)
and
$\mathbf{C}_{2}=\left[{\begin{array}[]{c}0\\\ 0\\\ 0\\\\[6.0pt]
(m/q)\omega_{x}^{2}\\\ (m/q)\omega_{y}^{2}\\\ (m/q)\omega_{z}^{2}\\\\[6.0pt]
0\\\ 0\\\ 0\end{array}}\right].$ (11)
Additional white space has been inserted in both equations to aid the reader
by separating the position, frequency, and principal axis constraints into
groups in the vertical direction, as well as separating the gradient and
Hessian components of Eq. 10 in the horizontal direction.
Once $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ are determined, Eq. (9) can be
solved for $\mathbf{V}$ by inverting
$\mathbf{C}_{1}\bigl{(}\mathcal{P}\otimes\Psi^{\mathrm{T}}(\mathbf{r}_{0})\bigr{)}$,
thus determining the control voltages that create the desired trapping
potential. This inversion may not be strictly possible, as is the case when
the number of constraints does not equal the number of control potentials,
leading to an over- or under-determined problem. Also, we are interested only
in solutions where the magnitudes of all control voltages are smaller than a
maximal voltage $V_{\mathrm{max}}$ (for our apparatus, $V_{\mathrm{max}}=10$
V). To achieve this, we use a constrained least-squares optimization
algorithm, as described in Ref. Gill et al. (1981), to calculate
$\min_{\lvert V_{i}\rvert\leq
V_{\mathrm{max}}}\left|\mathbf{C}_{1}\bigl{(}\mathcal{P}\otimes\Psi^{\mathrm{T}}(\mathbf{r}_{0})\bigr{)}\mathbf{V}-\mathbf{C}_{2}\right|^{2}.$
(12)
In cases where Eq. (9) is over-constrained, this method yields a “best-fit”
$\mathbf{V}$. When Eq. (9) is under-constrained, as is usually the case for
large trap arrays with many electrodes, it returns a null space in addition to
$\mathbf{V}$, which can be added to $\mathbf{V}$ to find multiple independent
solutions.
Nine constraints were used above, but many are unnecessary. For QIP in a
linear trap array, constraining the axial mode frequency and orientation is
often sufficient. Parameters for the other two modes are less important and
often achieve reasonable values without being constrained, in which case they
can be omitted from the constraint matrices.
In addition to explicitly defined user constraints, there are implicit
physical and geometric constraints that must be considered. As an example,
take the three secular frequencies of the ion, $\omega_{x}$, $\omega_{y}$, and
$\omega_{z}$. These frequencies result from a hybrid potential that includes
both pseudopotential and control potentials. The contributions from both
potentials can be separated mathematically into components,
$\omega_{\textrm{rf},i}$ and $\tilde{\omega}_{i}$ respectively, which add in
quadrature to give the overall frequency:
$\omega_{i}^{2}=\tilde{\omega_{i}}^{2}+\omega_{\textrm{rf},i}^{2}$. (An
imaginary frequency component would imply antitrapping, while a real component
yields trapping.) The control electrodes produce a quasi-static electric
field, which Laplace’s equation requires to be divergenceless. This places a
physical constraint on the frequencies components due to the control
potential, namely $\displaystyle\sum_{i=1}^{3}\tilde{\omega}_{i}^{2}=0$. Thus,
Laplace’s equation permits only certain combinations of the secular
frequencies. For a linear Paul trap, where $\omega_{\textrm{rf},z}=0$, the
secular frequencies must obey
$\omega_{x}^{2}+\omega_{y}^{2}+\omega_{z}^{2}=2\omega_{\mathrm{rf}}^{2},$ (13)
where $\omega_{\mathrm{rf}}$ is the pseudopotential radial trapping frequency.
The trap geometry can place constraints on the trapping potentials, as well.
For example, in traps where the geometry contains some symmetry, the
potentials must preserve that symmetry. Care must be exercised to ensure that
user-defined constraints do not contradict physical or geometry constraints,
as this will invalidate the solution.
Though we invoke only position, frequency, and orientation constraints here,
other varieties of user-defined constraints can be easily included with this
framework, and a more complete discussion of these constraints is presented in
Blakestad (2010). The constraints used to construct the waveforms in the
X-junction array were as follows:
1. 1.
The position of the potential minimum was constrained in three directions to
be at $\mathbf{r}_{0}$.
2. 2.
One of the principal axes was constrained to lie along the trap axis (which
involves two constraints on axes orientation).
3. 3.
The ion axial frequency was constrained (usually to 3.6 MHz).
4. 4.
The voltages were constrained to be between $\pm 10~{}$V (to conform to the
limits of the voltage supplies used in the experiment).
This relatively sparse set of constraints tended to give good solutions at
most locations considered. Item 4 is an inequality constraint that is easily
implemented by use of the constrained least-squares method.
When solving waveforms that transport across multiple zones, $\mathbf{V}$ can
become discontinuous from step to step, especially when transitioning between
sets of control electrodes. These jumps occur when an under-constrained
problem (with null space rank $>0$) has multiple linearly-independent
solutions and the algorithm returns a different solution from one step to the
next: during transport there will be some position at which it is suddenly
easier to produce the desired potential using a new combination of electrodes.
In principle, such jumps should not have adverse effects on the potential at
the ion, as the potentials on both side of the jump fulfill the same
constraints and should transition smoothly. However, since the potentials on
the electrodes are filtered, we would expect the potential at the ion to
experience a transitory jump during the transition.
These solution jumps can be handled by various means. We used the constrained
least-squares method to seed each new solution with the solution of the
previous step while introducing a cost for deviating from the previous
solution by replacing Eq. 12 with
$\min_{\lvert V_{i}\rvert\leq V_{\mathrm{max}},\ \lvert
V_{i}-V_{i,\mathrm{last}}\rvert\leq\alpha}\left|\mathbf{C}_{1}\bigl{(}\mathcal{P}\otimes\Psi^{\mathrm{T}}(\mathbf{r}_{0})\bigr{)}\mathbf{V}-\mathbf{C}_{2}\right|^{2},$
(14)
for a positive constant $\alpha$. This removes the need for iteratively
choosing weights to keep the voltages within bounds, as suggested in Ref.
Singer et al. (2010). This forced the jump transition to be extended over
multiple steps, rather than allowing a discontinuous jump. Another approach is
to average the two $\mathbf{V}$’s on each side of the discontinuity, taking
advantage of the linearity of the equations, to produce an intermediate
solution that still satisfies the constraints Uys . Performing several steps
of such averaging will smooth the jump. Alternately, trial and error can often
be used to determine a set of constraints that does not produce a jump, but
this can require significant effort.
#### III.0.1 Transport timing
If the spatial interval between steps in the waveform is small enough, the
potential, once applied to the electrodes, will move smoothly from step to
step Blakestad (2010). The velocity of the potential well (and, thus, the ion)
is controlled by the rate at which the waveform steps are updated on the
electrodes. In our case, the control potentials were supplied by digital-to-
analog converters (DACs) that had a constant update rate
$R_{\mathrm{DAC}}=480$ kHz and the number of update steps was adjusted to
change the velocity.
Different velocity profiles have been considered for minimizing excitation
while transporting Reichle et al. (2006); Hucul et al. (2008). In this report,
the ions were usually transported by use of a constant velocity with equally
spaced waveform steps. This could potentially lead to the ion being ‘kicked’
as the velocity jumps at the beginning and end of the transport, resulting in
motional excitation. However, these velocity jumps were smoothed by low-pass
filters placed on the control potentials (see Sec. V.2). A smoother
‘sinusoidal’ velocity profile was also tested but was abandoned after
observing no discernible difference in the amount of motional excitation by
use of the different profiles. This suggests that both transport protocols
were well within the adiabatic regime at the speeds used.
Low-pass filtering (160 kHz corner in our case) can also potentially distort
the waveforms when transporting quickly, placing an upper limit on the ion
speed. However, the practical speed limit was set by the combination of the
maximum update rate of the digital-to-analog converters and the number of
update points required to accurately produce a continuous harmonic potential
in the region of the pseudopotential barrier. This limit was experimentally
determined for each waveform by adjusting the number of update points until
minimum motional excitation was achieved. If faster DACs are available and
distortion of the waveforms due to low-pass filtering is of concern, the
waveform can be pre-compensated to account for these distortions and produce
the desired waveform at the ion.
Figure 6: (a) The waveform (as a function of position rather than time) used
when transporting an ion from the experiment zone, located at $z=-880~{}\mu$m,
to the center of the junction at $z=0~{}\mu$m. We plot voltage versus the $z$
position of the minimum of the trapping potential during the transport. The
locations of the electrodes near the junction are depicted, along with their
electrode number, by the rectangles in the bottom of the figure. The region
from $-100$ to $0~{}\mu$m is inside of the junction. The voltage traces are
numbered to show which electrode they correspond to. Electrodes 8 and 9 remain
near 0 V and are omitted for clarity. In addition, the potentials applied to
the control electrodes on the bottom wafer are not displayed, as they are
nearly identical to those applied on the top wafer. (b) A schematic of the
trap, showing the range over which this waveform transported.
The waveforms used to transport from $\mathcal{E}$ to $\mathcal{C}$ are
displayed in Fig. 6 as a function of the position of the minimum of the
trapping potential (the ion’s location). The potentials applied to the lower
trap-wafer control electrodes (on opposite sides of the main channel) were
nearly identical and are omitted for clarity. These waveforms could be run
left to right to transport an ion $880~{}\mu$m from $\mathcal{E}$ to
$\mathcal{C}$, or they could be run in reverse. The waveforms that transported
ions into the other two branches of the junction (to $\mathcal{F}$ and
$\mathcal{V}$) were similar to this waveform due to the approximate symmetry
of the trap.
In a typical transport, the potential minimum was moved at a constant
velocity, and there was a direct linear relationship between the location of
the minimum (horizontal axis of Fig. 6(a)) and the time elapsed since the
beginning of the transport. The typical transport duration for the waveforms
in Fig. 6(a) was approximately 165 $\mu$s, with 50 $\mu$s to cross the
pseudopotential barrier.
Some control potentials reached the $\pm 10$ V limit placed by use of the
constrained least-squares method while traversing the pseudopotential barrier
near the junction. Other control potentials had sharp and abrupt changes,
which resulted from the constraint in Eq. 14 that prevented ‘solution jumping’
by defining how much a given waveform step can deviate from the previous step.
Instead of jumping, the voltages ramped linearly over several steps. Although
these individual potentials were not smooth in time, they were continuous,
which was sufficient to ensure that the overall potential experienced by the
ion evolved smoothly.
The axial frequency was chosen to be 3.6 MHz and was held constant during much
of the transport starting at $\mathcal{E}$ and moving toward $\mathcal{C}$
(Fig. 7). The frequency was (adiabatically) linearly ramped to 4.2 MHz as the
ion approached the apex of the pseudopotential barrier, making the ion less
susceptible to rf-noise heating of the secular motion (see Sec. V.1). The
value 4.2 MHz was the maximum axial frequency attainable at the apex due to
the strong anti-confinement of the pseudopotential at that location and the
$\pm 10$ V limit of the DACs providing the control potentials. The axial
frequency then continued to increase as the ion descended the barrier,
reaching a final value of 5.7 MHz at $\mathcal{C}$. At this location, all
control potentials were 0 V and the pseudopotential provided all the trapping,
resulting in near-degenerate 5.7 MHz confinement along the $\hat{x}$ and
$\hat{z}$ directions, while the $\hat{y}$ secular frequency was 11.3 MHz. When
transporting multiple ions in the same potential well, it would be preferable
to break the frequency degeneracy at $\mathcal{C}$ to ensure well-defined axes
for the ions. In practice, the motional excitation rates when moving pairs of
ions were still relatively low despite the near degeneracy at $\mathcal{C}$
(see Table 2).
Figure 7: Predicted secular frequencies as a function of position
corresponding to the waveform in Fig. 6. The axial frequency along $\hat{z}$
was constrained to be 3.6 MHz during the majority of the transport, while the
radial frequencies were unconstrained. As the ion ascended the pseudopotential
barrier, the axial frequency linearly ramped up to 4.2 MHz. Beyond the apex of
the barrier, a second linear ramp was applied to bring the frequency up to 5.7
MHz. As the ion approached the center of the junction, the $x$ and $z$
frequencies became nearly degenerate.
## IV Transport Experiments
The transport experiments were performed with 9Be+ ions inside a vacuum system
with a pressure of $p<5\times 10^{-11}$ Torr $=7\times 10^{-9}$ Pa. A
$1.3\times 10^{-3}$ T magnetic field was applied to split the Zeeman states,
and the ions were optically pumped to the
${}^{2}S_{1/2}\left|F=2,m_{F}=-2\right>$ state (henceforth
$\left|2,-2\right>$). Manipulation of the 9Be+ ion motional and internal
states used the techniques of Refs. Monroe et al. (1995); Wineland et al.
(1998). Two-photon stimulated-Raman transitions enabled coherent transitions
between the qubit states $\left|2,-2\right>$ and $\left|1,-1\right>$ at
frequency $\omega_{0}\approx 2\pi\times 1.28$ GHz. In addition, by tuning the
difference frequency of the Raman beams to $\omega_{0}\pm\omega_{z}$ , it was
possible to drive a blue(red)-sideband transition:
$\left|2,-2\right>\left|n\right>\leftrightarrow\left|1,-1\right>\left|n\pm
1\right>$. Here $\left|n\right>$ is a Fock state of a selected motional mode.
Ground-state cooling was performed by use of a series of red-sideband pulses,
followed by repeated optical pumping to $\left|2,-2\right>$. State detection
was performed using state-dependent resonance fluorescence, where
predominantly the $\left|2,-2\right>$ state fluoresces.
Each transport began by cooling an ion (or ion pair) in $\mathcal{E}$ to the
motional ground state. The ion was then transported into or through the
junction and returned to $\mathcal{E}$. Three transports patterns were used:
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ moved to $\mathcal{C}$ and back,
while $\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$
and $\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$
moved to $\mathcal{F}$ and $\mathcal{V}$, respectively, before returning to
$\mathcal{E}$. The $\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transport moved
the ion $1.76$ mm, while
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$ and
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$ moved
the ion $3.52$ mm and $2.84$ mm, respectively. Once the ion returned to
$\mathcal{E}$, the motional excitation was determined by measuring the
asymmetry in red- and blue-sideband Raman transitions Monroe et al. (1995);
Turchette et al. (2000).
To determine the single-ion transport success rate for
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$
transports, two sets of 10,000 consecutive transport experiments were
performed Blakestad et al. (2009), but with the imaging system focused on
$\mathcal{E}$ in the first set and on $\mathcal{F}$ in the second. The first
set verified that the ion successfully returned to $\mathcal{E}$ every time.
The second set verified that the ion always reached $\mathcal{F}$ at the
intended time. Together, these sets of experiments imply the success rate for
going to $\mathcal{F}$ and returning to $\mathcal{E}$ exceeded 0.9999. The
procedure was repeated for
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$, with
the same result. The $\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transport can
not be verified in the same manner because the bridges obscure the ion at
$\mathcal{C}$, but since the ion must transport through this location to reach
$\mathcal{F}$ and $\mathcal{V}$, the reliability should be no worse.
Ion lifetime, and thus transport success probability, was ultimately limited
by ion loss resulting from background-gas collisions Wineland et al. (1998).
With this in mind, the ion loss rate during transport was not larger than that
for a stationary ion ($\sim 0.5/$hr). Having observed millions of successive
round trips for all three types of transport, combining all losses implies a
transport success probability of greater than 0.999995111For the 0.999995
success probability figure, we only verified that each transport successfully
returned the ion to $\mathcal{E}$, not whether the ion successfully moved the
ion through the junction. Given the low rates of motional excitation, and the
fact that we did verify that the ions move through the junction using 10,000
experiments, it is reasonable to assume the ion did travel through the
junction if the ion successfully returned to $\mathcal{E}$.. Since transport
comprised a small fraction of the total experiment duration, many of these
losses likely occurred when the ion was not being transported. In one
instance, more than 1,500,000 consecutive
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transports were performed with a
single ion.
Loss rates for transported ion pairs were again comparable to stationary pairs
($\sim 2$ per hour). Absolute pair loss rates were higher than those for
single ions, presumably due to multi-ion effects Wineland et al. (1998);
Walther (1993).
## V Excitation of the Secular Motion
Excitation of the ion’s motion during transport was attributed to two main
mechanisms: one due to rf noise near $\Omega_{\mathrm{rf}}$ and the other due
to excitation from the digital-to-analog converters (DACs).
### V.1 rf-noise heating
Consider a trapping rf electric field with an additional sideband term,
$\mathbf{E}_{\textrm{rf}}(\mathbf{r},t)=\mathbf{E}_{0}(\mathbf{r})\left[\cos{\Omega_{\mathrm{rf}}t}+\xi_{\textrm{N}}\cos{(\Omega_{\mathrm{rf}}\pm\omega_{z})t}\right],$
(15)
where $\xi_{\textrm{N}}\ll 1$, and $\Omega_{\mathrm{rf}}\pm\omega_{z}$ is at
one of the two axial motional sidebands of the ion. In Blakestad et al.
(2009), it was shown that the two terms will beat at $\omega_{z}$ to produce a
force that can excite the ion’s motion. If the second term is not coherent,
but instead is broad-spectrum noise, this will lead to excitation of the axial
motion at a rate of
$\begin{split}\dot{\bar{n}}_{z}=&~{}\frac{q^{4}}{16m^{3}\Omega_{\mathrm{rf}}^{4}\hbar\omega_{z}}\left[\frac{\partial}{\partial
z}E^{2}_{0}(z)\right]^{2}\times\\\
&\quad\left(\frac{S_{V_{\textrm{N}}}(\Omega_{\mathrm{rf}}+\omega_{z})}{V_{\mathrm{rf}}^{2}}+\frac{S_{V_{\textrm{N}}}(\Omega_{\mathrm{rf}}-\omega_{z})}{V_{\mathrm{rf}}^{2}}\right),\end{split}$
(16)
where $S_{V_{\textrm{N}}}(\Omega_{\mathrm{rf}}\pm\omega_{z})$ is the voltage-
noise spectral density at either the upper or lower rf sideband, and
$V_{\mathrm{rf}}$ is the amplitude of the trapping rf potential being applied
to the rf electrodes. $E_{0}(z)$ is the axial rf electric field amplitude at
the location of the ion. This heating mechanism is proportional to the _slope_
of the pseudopotential and is significant only in places with a large slope,
such as the pseudopotential barriers near the junction (but not in, for
example, $\mathcal{E}$).
This heating mechanism was verified in Ref. Blakestad et al. (2009) by
measuring the heating rate at various locations along the pseudopotential
barrier between $\mathcal{E}$ and $\mathcal{C}$, while spectrally-dense white
noise (centered on the lower sideband, $\Omega_{\mathrm{rf}}-\omega_{z}$) was
injected onto the trap rf drive. Figure 8 plots the ratio of measured heating
rate to estimated injected $S_{V_{\textrm{N}}}$ and theoretical values of this
ratio according to Eq. (16) based on simulations of trap potentials, for the
ion held at several positions between $\mathcal{E}$ and $\mathcal{C}$. A plot
with the theoretical values multiplied by a scaling factor ($=1.4$) is also
included. The deviation of the scaling factor from 1 is not unreasonable due
to the difficulty of accurately measuring a variety of experimental
parameters.
Figure 8: The ratio of heating rate $\dot{\bar{n}}_{z}$ to voltage noise
spectral density $S_{V_{\textrm{N}}}(\Omega_{\mathrm{rf}}-\omega_{z})$ for
various locations along the trap axis ($\mathcal{C}$ is located at 0 $\mu$m).
This figure is reproduced from Ref. Blakestad et al. (2009). The theoretical
prediction used a pseudopotential modeled from electrode geometry and is shown
both with and without a scaling parameter ($=1.4$). The simulated
pseudopotential is overlaid in the background, in units of eV. Since heating
was gradient dependent, we saw very little heating at the peak of the
pseudopotential barrier, even though this was the point of maximum (axial) rf
electric field and therefore maximum axial rf micromotion. Nearly identical
pseudopotential barriers were present on the other three legs of the junction.
The motional excitation for full junction transports to $\mathcal{C}$ was
observed to decrease as the ion speed was increased, which minimized the
exposure to the rf noise while on a pseudopotential slope. This continued up
to a maximum speed limit, due to the slow DACs, above which the other
excitation mechanism (below) began to dominate. At the optimum speed, the ion
spent only approximately $50~{}\mu$s on each barrier (above 10% of the barrier
height).
Another approach to mitigate rf noise is to suppress the sideband noise with
better filtering of the applied rf trapping potential. In Ref. Blakestad et
al. (2009), the large rf potential ($V_{\mathrm{rf}}\approx 200$ Vpeak at
$\Omega_{\mathrm{rf}}\approx 2\pi\times 83$ MHz) was provided by a series of
tank resonators, which suppressed noise at the motional sidebands ($\pm 3.6$
MHz). The primary resonator was a quarter-wave step-up resonator Jefferts et
al. (1995) with a loaded $Q=42$ and corresponding bandwidth (FWHM) of 2 MHz.
This resonator extended into the vacuum, with the trap attached to the voltage
anti-node. A second half-wave resonator with $Q=145$ was attached, in series,
to the input of the primary resonator, with a 3 dB attenuator in between to
decouple the two resonators. The resonant frequencies of the two resonators
were tuned to be equal. This network resulted in an estimated ambient
$S_{V_{\textrm{N}}}(\Omega_{\mathrm{rf}}\pm\omega_{z})$ of -177 dBc at the
ion. In the work reported here, the second resonator was replaced with a pair
of half-wave tank resonating cavities. This filter pair provided more than 38
dB suppression at frequencies $\Omega_{\mathrm{rf}}\pm 2\pi\times 3.6$ MHz
(when not coupled to the primary resonator), an additional suppression of
approximately 10 dB over the half-wave filter used in Ref. Blakestad et al.
(2009). $S_{V_{\textrm{N}}}$ at the ion was not re-measured with this new
filter pair, but observed reductions in excitation during transport were
consistent with a 10 dB drop in rf noise.
### V.2 DAC update noise
Another primary source of motional excitation was attributed to the 16-bit,
$\pm 10$ V DACs that supplied the waveform potentials to the electrodes. The
DAC voltages were updated at a constant rate $R_{\mathrm{DAC}}$ ($\leq 500$
kHz), resulting in Fourier components that could excite the ion’s motion if
$2\pi\times R_{\mathrm{DAC}}=\omega_{z}/J$, for any integer $J$.
This effect was observed by first preparing the ion in the motional ground
state at $\mathcal{E}$ and then transporting toward $\mathcal{C}$. Instead of
proceeding all the way to $\mathcal{C}$, the transport was stopped (at
$z=-300~{}\mu$m) before the axial frequency began to ramp up. Thus, the local
potential-well frequency remained approximately constant at
$\omega_{z}=2\pi\times 3.6$ MHz. The ion was then returned to $\mathcal{E}$. A
red-sideband Raman $\pi$-pulse for $n=0$ to $n=1$ excitation was applied to
determine if the ion remained in the ground state Monroe et al. (1995);
Turchette et al. (2000). If the ion was excited out of the ground state during
transport, the Raman pulse had a certain probability to transfer the ion into
the $\left|1,-1\right>$ state, which did not fluoresce during detection. If
the ion remained in the ground state, this side-band pulse had no effect and
the ion remained in the bright $\left|2,-2\right>$ state. Thus, fluorescence
detection after the side-band pulse could distinguish an excited ion from a
non-excited ion.
Figure 9: Plot showing the the number of fluorescence photons detected in a
duration of $200~{}\mu$s following round-trip transport and a subsequent red
sideband pulse, for various DAC update rates $R_{\mathrm{DAC}}$. Before
transport, the ion was prepared in the motional ground state and then
transported through a specific waveform, where $\omega_{z}=2\pi\times 3.6$ MHz
was maintained during the entire transport. If the ion remained in the ground
state after transport, the ion fluorescence was at its maximum value
(approximately 8 photon counts detected), but when the ion became motionally
excited, the fluorescence dropped. As can be seen, the motion was excited at
specific update frequencies that correspond to
$R_{\mathrm{DAC}}=\omega_{z}/(2\pi J)$ for $J=8$ to $14$ (marked by the
vertical red lines).
This experiment was performed for various values of $R_{\mathrm{DAC}}$, and
the results are shown in Fig. 9. It was difficult to extract the ion’s exact
motional state after the transport, but the correlations between the ion’s
motional excitation and $\omega_{z}$ corresponding to a harmonic of
$R_{\mathrm{DAC}}$ were evident. The energy gain exhibited a resonance at
several values for $R_{\mathrm{DAC}}=\omega_{z}/(2\pi J)$ with $J=8$ to $14$.
When the number of update steps was increased, while the update rate was held
constant (which resulted in an increased transport duration), the bandwidth of
these resonances decreased, as expected from a coherent excitation.
Use of an update rate that was incommensurate with the motional frequency
($R_{\mathrm{DAC}}\neq\omega_{z}/(2\pi J)$) minimized this energy gain.
However, increasing the transport speed (using the same update rate) required
a reduction in the number of waveform steps, which caused the resonances to
broaden. Minimizing the rf-noise heating required fast transport, so at the
speed that gave the lowest rf-noise excitation rates, the DAC heating
resonances were so broad that they overlapped, and there was no achievable
$R_{\mathrm{DAC}}$ that would not result in energy gain. The DAC heating
effect was further compounded by the fact that the axial frequency was not
constant during a full junction transport, making it impossible to achieve
$R_{\mathrm{DAC}}\neq\omega_{z}/(2\pi J)$ for any constant $R_{\mathrm{DAC}}$.
The update frequency $R_{\mathrm{DAC}}=480$ kHz appeared to be most favorable
and was used for the results here and in Ref. Blakestad et al. (2009).
Faster DACs capable of $R_{\mathrm{DAC}}>\omega_{z}/2\pi$ should significantly
suppress this motional excitation. Alternatively, aggressive filtering of the
DAC output can combat this problem. The results in Ref. Blakestad et al.
(2009) used the RC filter network shown in Fig. 10(a), which provided
suppression by two orders-of-magnitude over the range of $\omega_{z}/2\pi$
values used during transport (3.6 to 5.7 MHz), but was not sufficient to
completely suppress the DAC heating. Increasing the RC time constant would
increase the filtering but would also slow down the rate at which the ion can
be transported. Instead, these simple RC filters were replaced with the
approximately third-order Butterworth filter Tietze and Schenk (2008) shown in
Fig. 10(b). (The output impedance of the DAC was $<0.1~{}\Omega$ and
contributed minimally to the filter response.)
Figure 10: The control voltages were provided by 40 independent DACs (only one
shown here). The DAC output was filtered prior to being applied to the trap
electrodes, through a two stage filter seen in (a). The control potentials
were referenced to the grounded vacuum system, which served as the rf ground
as well. (b) After DAC-update noise was observed to excite the secular motion
of the ions, the external filters were replaced by the approximate
$3^{\mathrm{rd}}$-order ‘Butterworth’ filter shown here.
A Butterworth filter has a frequency response given by
$G(\omega)=\frac{1}{\left|B_{n}(i\omega/\omega_{0})\right|}=\frac{1}{\sqrt{1+(\omega/\omega_{0})^{2n}}},$
(17)
where $B_{n}(s)$ is the $n^{\mathrm{th}}$-order Butterworth polynomial and
$\omega_{0}$ is the corner frequency. If $n=1$, the frequency response reduces
to a RC frequency response. In the experiments here, such higher-order filters
provide stronger noise suppression at $\omega_{z}$ while still allowing fast
transport. A comparison of the theoretical response functions for the RC
filters used in Ref. Blakestad et al. (2009) and the Butterworth filters used
here is shown in Fig. 11. The internal vacuum RC components already on the
filter board were taken into account when planning the Butterworth filter, but
the external filter components were designed to dominate the filter’s response
in the frequency range of concern. Thus, the filter was approximately third-
order, despite the presence of four components (including the filter board
capacitor) with frequency-dependent impedances. The new filters increased the
noise filtering by 22 dB at 3.6 MHz and 26 dB at 5.7 MHz. Furthermore, the
electric-field noise at the ion due to Johnson noise in the resistive elements
of the new filters was less than that for the previous filters for all
frequencies of interest. For the transport durations used, these filters did
not appreciably distort the waveform.
Figure 11: Theoretical transfer function $G(\omega)$ versus frequency for the
original RC filter (dashed) and the improved approximately-‘Butterworth’
filter (solid), where the new filter had a faster roll-off at high frequency.
Since the pass-band extended farther for the new filter, the transport speed
could be increased while providing more filtering at the secular frequency
(3.6 MHz to 5.7 MHz). Both traces include the RC components inside the vacuum.
### V.3 Anomalous noise heating
The ‘anomalous heating’, which is thought to arise from noisy electric
potentials on the surface of the trap Turchette et al. (2000), was measured to
be 40 quanta/s for $\omega_{z}/2\pi=3.6$ MHz at $\mathcal{E}$ and was not a
significant source of excitation during the transport experiments. For
example, we estimate that it should have contributed only 0.007 quanta for
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transport. To compare the
measurements of various ion traps, it has been common to express the heating
in terms of the electric field noise with the expression Turchette et al.
(2000):
$\dot{\bar{n}}_{z}=\frac{q^{2}}{4m\hbar\omega_{z}}S_{E}(\omega_{z}),$ (18)
where $S_{E}(\omega_{z})$ is the spectral density of electric field
fluctuations at the secular frequency. From the results here we found
$S_{E}(2\pi\times 3.6~{}$MHz$)=2.2\times
10^{-13}~{}(\mathrm{V}/\mathrm{m})^{2}\mathrm{Hz}^{-1}$, where the distance of
the ion to the nearest electrode surface was $160~{}\mu$m. This result, when
compared to other traps as in Refs. Amini et al. (2010); Deslauriers et al.
(2004); Epstein et al. (2007); Daniilidis et al. (2011), was significantly
below that of most other room-temperature ion traps. The cause of this
relatively low heating rate is not known, but surface preparation could be a
contributing factor.
### V.4 Other heating mechanisms
In the experiments here, the transport was slow and the trapping potential
changed slowly compared to the motional frequencies, so we did not expect non-
adiabatic excitation of the motion. This was supported by observations that
the excitation did not decrease as the transport was slowed. For very slow
transport, the heating actually increased because the ion spent more time
crossing the rf barriers, resulting in increased rf-noise heating.
Furthermore, no reduction in heating was observed when a gradual (sinusoidal)
velocity profile was used instead of a constant velocity over the entire
transport.
The waveforms were produced assuming a specific value of $V_{\mathrm{rf}}$ and
corresponding pseudopotential. In theory, if the actual $V_{\mathrm{rf}}$ does
not match the assumed $V_{\mathrm{rf}}$, the axial trapping potential will not
be as intended at the barriers. In practice, there was an optimal value for
the rf power which resulted in the lowest excitation rates and likely
corresponded to the assumed $V_{\mathrm{rf}}$. The rf power was prone to slow
drifts over many minutes (likely due to temperature drifts in the resonators)
which resulted in modest increases in motional excitation; it was necessary to
occasionally adjust the rf power (every 10 to 30 minutes) and hold it to
within $<1~{}\%$ to achieve the lowest motional-excitation rates. In practice,
this was performed by ensuring that the radial secular frequencies at
$\mathcal{E}$ remained constant.
### V.5 Motional excitation rates
The motional excitation for single-ion transports was measured by use of
sideband asymmetry measurements Monroe et al. (1995) after a single pass
through the junction, and the results are summarized in Table
LABEL:tab:heatingrates. These results were significantly better than those in
Ref. Blakestad et al. (2009), which are listed for comparison. In Ref.
Blakestad et al. (2009), rf noise was estimated to contribute 0.1 to 0.5
quanta of excitation per pass over a pseudopotential barrier, which explained
between 3 and $30\%$ of the excitation seen. The remainder of the excitation
was attributed primarily to DAC update noise. The new trap rf filters and
control electrode Butterworth filters produced the observed reduction in
excitation rates.
| This work | Ref. Blakestad et al. (2009)
---|---|---
| Duration | Energy gain | Duration | Energy gain
Transport | ($\mu$s) | (quanta/trip) | ($\mu$s) | (quanta/trip)
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ | 350 | $0.053\pm 0.003$ | 310 | $3.2\pm 1.8$
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$ | 910 | $0.18\pm 0.02$ | 630 | $7.9\pm 1.5$
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$ | 950 | $0.18\pm 0.02$ | 870 | $14.5\pm 2.0$
Table 1: The axial-motion excitation $\Delta\bar{n}$ for a single 9Be+ ion for
three different transports through the X-junction. The results of this work,
as well as that of Ref. Blakestad et al. (2009), are given for
comparison333The transport durations given in Ref. Blakestad et al. (2009)
were reported in error. The correct values are $140~{}\mu$s for transporting
from $\mathcal{E}$ to $\mathcal{C}$, $300~{}\mu$s to go from $\mathcal{E}$ to
$\mathcal{V}$, and $420~{}\mu$s to go from $\mathcal{E}$ to $\mathcal{F}$.
This error did not affect any other results in Ref. Blakestad et al. (2009)..
The transport duration includes $20~{}\mu$s for the ion to remain stationary
at the intermediate destination ($30~{}\mu$s for the data from Ref. Blakestad
et al. (2009)), before returning to $\mathcal{E}$. The energy gain per trip is
stated in units of quanta in a 3.6 MHz trapping well where $\Delta\bar{n}=0.1$
quantum corresponds to 1.6 neV.
The transport durations, which were optimized for minimal excitation, are also
given in Table LABEL:tab:heatingrates. The tabulated durations correspond to
the full transport duration including returning to $\mathcal{E}$ (rather than
the half-transport reported in Ref. Blakestad et al. (2009). The durations
also include a $20~{}\mu$s wait at the half-way point ($\mathcal{C}$,
$\mathcal{F}$, or $\mathcal{V}$, depending on the transport) for the new
results and a $30~{}\mu$s wait for those from Ref. Blakestad et al. (2009).
Moving pairs of ions in the same trapping well would be useful for both
sympathetic cooling and efficient ion manipulation Kielpinski et al. (2002).
This type of transport was demonstrated by use of pairs of 9Be+ ions and the
measured motional excitation is reported in Table 2. Excitation in both the
center-of-mass (COM) and stretch modes was measured. Additional heating
mechanisms for multiple ions Wineland et al. (1998); Walther (1993) may
explain the higher energy gain observed for the pair. For
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$
transport, the two-ion crystal must rotate from the $\hat{z}$ axis to the
$\hat{x}$ axis and back. For the waveforms used, the potential was nearly the
same in the $\hat{x}$ and $\hat{z}$ directions at $\mathcal{C}$. Therefore the
axes were not well defined throughout the transport, which can lead to an
uncontrolled rotation of axes. It is possible that the discrepancy in the
excitation between
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$ and
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$ for two
ions may have resulted from this uncontrolled rotation at $\mathcal{C}$.
Transport | Energy gain (quanta/trip)
---|---
| This work | This work | Ref. Blakestad et al. (2009)
| COM | Stretch | COM
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ | $0.39\pm 0.03$ | $0.13\pm 0.02$ | $5.4\pm 1.2$
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{F}$-$\mathcal{C}$-$\mathcal{E}$ | $0.67\pm 0.05$ | $0.53\pm 0.05$ | $16.6\pm 1.8$
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{V}$-$\mathcal{C}$-$\mathcal{E}$ | $0.72\pm 0.06$ | $0.14\pm 0.02$ | $53.0\pm 1.2$
Table 2: The axial-motion excitation $\Delta\bar{n}$ for a pair of 9Be+ ions
transported in the same trapping well. Values for both axial modes of motion
(COM and stretch) are reported. The energy gain per trip is stated in units of
quanta where the COM frequency is 3.6 MHz and the stretch frequency is 6.2
MHz. Results from Ref. Blakestad et al. (2009) are also given, though only the
COM mode excitation was investigated.
We expect (and observed) less excitation of the stretch mode relative to the
COM mode, for two reasons. First, the stretch mode frequency was higher than
that of the COM mode ($\omega_{\mathrm{STR}}=\sqrt{3}\omega_{\mathrm{COM}}$).
Thus, the filters on the rf and control potentials were more effective at
suppressing noise that could excite the stretch mode. Second, a stretch mode
can be excited only by a differential force on the two ions, while the COM
mode is excited by a force common to both ions. Given the proximity of the
ions to each other (a few micrometers) compared to the distance of the ions to
the trap electrodes, the relative amplitude of differential forces acting on
the ions are expected to be less than common forces.
## VI Mitigation of Magnetic Field Fluctuations
So far, we have discussed the suppression of undesired excitation of motional
degrees of freedom. We now discuss how magnetic-field fluctuations affecting
internal-state (qubit) coherence are suppressed in the X-junction trap array.
Decoherence of superpositions of the $\left|2,-2\right>$ and
$\left|1,-1\right>$ qubit basis states occurs both during transport and while
the qubit is stationary. Previous experiments demonstrated that junction
transport contributed negligibly to decoherence Blakestad et al. (2009).
Magnetic field fluctuations form the dominant contribution to qubit dephasing,
yielding typical values (in this trap and others) of less than $100~{}\mu$s
Langer et al. (2005). Use of a magnetic-field-insensitive qubit configuration
enables extension of the coherence time to approximately 10 s and can be used
with some gate operations such as the Mølmer-Sørenson gate Mølmer and Sørensen
(1999), but excludes implementation of $\sigma_{z}\sigma_{z}$ gates Leibfried
et al. (2003); Lee et al. (2005); Langer et al. (2005).
To suppress the effects of magnetic field fluctuations, we enclosed the trap
and field coils inside a high-magnetic-susceptibility mu-metal shield and
implemented an active magnetic field stabilization system. The shield (Fig.
12) was designed for compatibility with the existing trap vacuum envelope and
optical systems, and for ease of installation without the need to lift the
trap apparatus from the supporting table. A cylindrical body and approximately
hemispherical dome were selected based on general guidelines for magnetic
shielding and manufacturing constraints. The main body and baseplate of the
shield were constructed of $3.2$ mm thick, single-layer mu-metal in order to
provide maximum shielding of low-frequency magnetic field fluctuations and to
suppress magnetic saturation of the mu-metal. This latter constraint arose
because part of the field coils defining the quantization axis of the qubits
were located approximately 1 cm from the walls, where a calculated field of
$4\times 10^{-3}$ T was expected for typical operating conditions.
Figure 12: Mu-metal magnetic shield. The main dome enclosed both the trap and
the magnetic-field coils. Cylindrical tubulation extended along a glass vacuum
envelope, which corresponds to the $\hat{z}$ direction at the trap. Reentrant
flanges minimized field leakage around the imaging and optical access points.
Openings in the shield for optical components or laser beams were outfitted
with a reentrant flange fastened to the main body of the shield. The flanges
extended both outwards and inwards in the shield in order to maximize flux-
line redirection. A target length-to-diameter ratio of 5:1 guided design but
was typically not achieved due to geometric constraints arising from the
exterior dimensions of the vacuum envelope and the desire to position optical
elements as close as possible to the shield for maximum beamline stability.
These flanges were designed to be modular, allowing for redesign and
replacement if increased shielding became necessary. Magnetic continuity was
achieved for all mating flanges by use of internally threaded fasteners,
producing a snug contact fit.
The structure was designed to provide a minimum 22 dB shielding of low-
frequency fields. This was confirmed by use of a pickup coil and detecting 60
Hz fluctuations. The minimum suppression measured for the shield alone was
$>20$ dB parallel to the $\hat{z}$ axis of the trap, and the direction of the
largest access opening in the shield. Shielding in excess of 60 dB at 60 Hz
was measured in transverse directions.
The $1.3\times 10^{-3}$ T quantization field is oriented $45^{\circ}$ with
respect to the vacuum envelope axis. Initial measurements identified current
instability due to power-supply ripple as dominating measured decoherence once
the shield was installed. We implemented a custom current-regulation system
based on a proportional-integral-differential feedback circuit, a series
current-sense resistor, and a low-current field-effect transistor. To minimize
the effect of thermal drifts in the electronic circuit on magnetic field
stability, we selected special low-TC (thermal coefficient) components and
temperature-stabilized the enclosure. The most critical components were the
gain and sense resistors; these were selected to be low-TC metal foil
resistors with less than $2$ ppm/K and less than $3$ ppm/K stability,
respectively. A four-terminal current-sense resistor was selected with high-
power-handling construction ($0.1~{}\Omega$ for the main coil and
$0.25~{}\Omega$ for the transverse shim coils). Similar care was taken to
select low-TC difference amplifiers for the input stage and a low-TC voltage
reference. All sense and feedback components were thermally sunk to an Al
enclosure that was thermally stabilized by use of Peltier coolers and a
commercial temperature controller with milliKelvin stability. Stabilization
reduced current ripple from $\sim 1$ mA to $\sim 30~{}\mu$A on the main field-
coil current of 1.2 A. Net magnetic field fluctuations due to current ripple
at the location of the ions were $\sim 26$ nT.
Measurements of the dephasing time including both the magnetic shield and the
stabilization circuitry demonstrated extension of the qubit coherence to
$1.41\pm 0.09$ ms, more than 15$\times$ longer than that without shielding and
current stabilization, and sufficient for multiple transports before the qubit
dephased. A spin-echo pulse doubles the coherence time to $2.99\pm 0.04$ ms,
indicating that slow shot-to-shot field fluctuations are small, and that
decoherence is dominated by fluctuations on a millisecond time scale.
## VII Mode Energy Exchange
The secular modes of the ions were constrained to change throughout the
transports, both in frequency and orientation. For most parts of the
transport, the splittings between the mode frequencies were sufficiently large
and the transport speed was sufficiently slow that modes changed adiabatically
and energy did not transfer between modes. However, at $\mathcal{C}$, the two
principle axes that lie in the $(x,z)$ plane were designed to have nearly
degenerate secular frequencies
($\omega^{\prime}_{x}\approx\omega^{\prime}_{z}\approx 2\pi\times 5.7$ MHz),
which could lead to mode-mixing. The third mode along $\hat{y}$ had a
significantly higher frequency $\omega^{\prime}_{y}=2\pi\times 11.3$ MHz, and
would remain decoupled from the $x$ and $z$ modes. Since the radial modes were
only Doppler laser-cooled before transport, $x/z$ mode mixing would increase
the excitation of the axial mode during transport. We employed two approaches
that would minimize such axial excitation. First, the duration during which
the ion was at $\mathcal{C}$ could be adjusted such to minimize the energy
transfer between modes (by using a duration that corresponded to a full cycle
of the mixing process). Alternately, a potential could be applied to various
electrodes, which we will call the shim potential, to sufficiently break the
degeneracy (in practice,
$\left|\omega^{\prime}_{x}-\omega^{\prime}_{z}\right|>2\pi\times 400$ kHz
could be achieved) and suppress the mixing. Both methods were effective and
yielded similar transport excitation, though the second approach was used for
the results in Tables LABEL:tab:heatingrates and 2.
However, in separate experiments, we explored a method for controlling energy
transfer between the motional modes of a single ion by using field shims near
the junction to tune $\omega^{\prime}_{x}$ and $\omega^{\prime}_{z}$ to near-
degeneracy. Ideally, a demonstration of the method would work as follows.
Prior to transport from $\mathcal{E}$, the ion is cooled to the axial ground
state $\left|n_{z}=0\right>$ along $\hat{z}$ and prepared in Doppler-cooled
thermal states in the transverse modes. If the relative orientation of the
modes remains stationary as the ion approaches $\mathcal{C}$, the modes should
not exchange energy, even if they become degenerate. However, if the $x$ and
$z$ mode directions diabatically (fast compared to $1/\Delta\omega$) rotate
$45^{\circ}$ to new directions given by $x^{\prime}=(x+z)/\sqrt{2}$ and
$z^{\prime}=(x-z)/\sqrt{2}$, we would expect the initial $x$ oscillation to
project onto the new mode basis with half of the energy going into each of the
new modes. If
$\Delta\omega^{\prime}\equiv\omega^{\prime}_{x}-\omega^{\prime}_{z}\neq 0$,
the two oscillations would then begin acquiring a relative phase
$\phi=\Delta\omega^{\prime}t$, where $t$ is the period spent at $\mathcal{C}$.
By then quickly transporting away from $\mathcal{C}$ towards $\mathcal{E}$
such that the mode axes rotate diabatically by $-45^{\circ}$ back to their
original orientation, the oscillations would project back onto the original
oscillator basis. If the wait period is such that $\phi=\pi\times M$ (where
$M$ is an integer), the motion originally in the $x$ mode would project back
into the same mode. If, however, $\phi=\frac{\pi}{2}\times(2M-1)$, then the
$x$-motion would project into the $z$ mode; that is, the energy would exchange
between $x$ and $z$ modes.
We demonstrated the basic features of this exchange as follows. To tune the
$\omega^{\prime}_{x}$ and $\omega^{\prime}_{z}$ close to degeneracy, an
external shim potential with an adjustable amplitude was applied. The shim
potential consisted of various contributions from 17 control electrodes near
$\mathcal{C}$, each multiplied by the overall scaling factor $A$. These
contributions were selected so that the net shim potential would primarily
alter the frequency splitting without significantly affecting other trapping
parameters (such as the position of the trapping minimum and the $y$ mode
frequency).
Figure 13: (a) Average motional excitation $(\bar{n}_{z})$ in the axial mode
after an $\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transport, versus the
duration at $\mathcal{C}$ (wait period). The black trace indicates exchange of
energy between the $z$ mode, prepared near the ground state (at
$\mathcal{E}$), and the $x$ mode, prepared in a thermal state via Doppler
cooling. The smaller blue trace represents the identical preparation, except
the transport was performed twice. During the first transport, the wait period
was set to maximize the energy transfer from the radial mode to the axial
mode, followed by returning the ion to $\mathcal{E}$. After re-cooling the
axial mode to the ground state, the round-trip transfer was repeated. The
contrast was decreased (blue trace), indicating that less transverse mode
energy was available for transfer to the $z$ mode and therefore indicating
cooling of the $x$ mode. (b) and (c) The exchange contrast and exchange
frequency (respectively), plotted versus the shim-potential scaling factor
$A$.
The black trace in Fig. 13(a) shows $\bar{n}_{z}$ for the axial mode after
$\mathcal{E}$-$\mathcal{C}$-$\mathcal{E}$ transport, as the wait period at
$\mathcal{C}$ was varied. We derive $\bar{n}_{z}$ from sideband measurements,
as described above, and assume a thermal distribution. Although this
assumption may not be strictly valid, it should give a reasonable
approximation for $\bar{n}_{z}<1$. Oscillations between an excited and near-
ground state energy are visible. The projection process began prior to the ion
reaching $\mathcal{C}$ and transport was too slow for the projection to be
perfectly diabatic. Thus, the phase of the exchange oscillation in Fig. 13(a)
is not well determined and was observed to depend on both the exchange
frequency and the details of the approach to $\mathcal{C}$ (including speed
and trajectory). In practice, it was difficult to maintain a constant phase
for more than a few minutes, as drifts in the potential, likely caused by
transient charge buildup and dissipation on the electrodes and also
pseudopotential amplitude changes, caused the exchange frequency to drift over
that time scale.
Figure 13(b) displays the oscillation contrast
$\Delta\bar{n}_{z}=\textrm{max}(\bar{n}_{z})-\textrm{min}(\bar{n}_{z})$ for
various scaling factors, $A$, of the shim potential, while Fig. 13(c) gives
the frequency of those oscillations versus the shim scaling factor. In
separate experiments, the two mode frequencies at $\mathcal{C}$ were measured
as a function of $A$ by driving excitations with an oscillatory potential
applied to the control electrodes. The difference between the two mode
frequencies,
$\Delta\omega^{\prime}=\left|\omega^{\prime}_{x}-\omega^{\prime}_{z}\right|$,
was observed to match the oscillation frequency of the exchange process.
Figure 13(c) suggests $\Delta\omega^{\prime}$ is high on the extreme ends of
the $A$ range, while Fig. 13(b) shows a reduction in contrast in these regions
of high $\Delta\omega^{\prime}$, likely due to the reduction in diabaticity
when $\Delta\omega^{\prime}$ was large. This conclusion was supported by the
observation that the contrast decreased as the ion transport speed was
reduced. However, there was a maximum speed, above which contrast no longer
increased, because other sources of excitation began to obscure the
oscillatory signal.
From Fig. 13(b), we see that the fringe contrast was also minimized for shim
scaling factors near $A=-0.15$, where $\Delta\omega^{\prime}$ was small. This
reduction in contrast can be explained as coinciding with the condition where
the initial mode orientation is identical to the rotated mode orientation and
thus the modes do not mix when projected, which is a condition not necessarily
related to $\Delta\omega^{\prime}$. (We note that the $A$ value for minimum
exchange frequency does not match that for minimum contrast.) Unfortunately,
it was not possible to verify this, as we could not measure the mode
orientation at $\mathcal{C}$, due to lack of laser-beam access.
In the case where the energy from the $x$ mode was transferred into the $z$
mode, the ion could be returned to $\mathcal{E}$ for a second round of ground-
state cooling of the $z$ mode. In the experiment, the exchange process in
$\mathcal{C}$ was repeated and the results are shown as the blue trace in Fig.
13(a), where a noticeable decrease in the ion’s axial excitation was observed
compared to the first experiment without the second stage of cooling. The
small relative phase shift for the two traces in Fig. 13(a) was due to the
slow drift of $\Delta\omega^{\prime}$ over the several minutes required to
take the two traces.
Ideally, all of the energy would be transferred from the $x$ mode into the $z$
mode, and the subsequent cooling of the $z$ mode would leave both modes in the
ground state, leading to no oscillation during the second trip into
$\mathcal{C}$. In practice, complete transfer was inhibited for two primary
reasons. First, the ions were not being transported fast enough to make a
clean diabatic projection of the motion onto the switched axes. Second, for
complete energy transfer, the projection should be onto axes rotated by $\pm
45^{\circ}$. Any other angles would have resulted in incomplete transfer of
energy. Attempts were made to adjust additional shims in hopes of realizing
configurations closer to $\pm 45^{\circ}$. However, as
$\Delta\omega^{\prime}\rightarrow 0$, it, again, becomes difficult to predict
the mode orientation with our idealized computer models and we could not
experimentally determine the mode axes in $\mathcal{C}$.
Nevertheless, we observed a clear and easily reproducible reduction in maximum
oscillation amplitude (Fig. 13(a)) from $\mathrm{max}(\bar{n}_{z})=0.68\pm
0.08$ to $\mathrm{max}(\bar{n}_{z})=0.40\pm 0.05$ after the second round of
cooling, indicating the radial mode energy was being reduced. The use of
additional rounds of exchange followed by cooling reduced
$\mathrm{max}(\bar{n}_{z})$ further, but after three or four exchange rounds,
other sources of excitation offset the energy reduction.
When optimized, this technique might be used to cool all modes of a single ion
to the ground state, while having the ability only to ground-state cool a
single mode, as for the laser beam configuration used here. A junction is not
required; all that is needed is a trap that can diabatically change the
relevant mode orientations by $\pm 45^{\circ}$, which could be possible in
many trap configurations.
## VIII Conclusion
In conclusion, we have demonstrated that transport through a two-dimensional
trap array incorporating a junction can be highly reliable and excite an ion’s
motion by less than one quantum. This is a significant improvement over prior
work with junction arrays Hensinger et al. (2006); Blakestad et al. (2009) and
suggests the viability of trap arrays incorporating junctions for use in
large-scale ion-based quantum information processing. In addition, we have
implemented a mu-metal shield and current stabilization to reduce qubit
decoherence. We have also examined a technique for transferring energy between
motional modes.
DAC-update noise can be mitigated with the use of more appropriate filters
such as the Butterworth filters used here and/or faster DAC update rates.
Noise on the trapping rf potential can result in motional excitation at
pseudopotential barriers as described in Sec. V.1. The junction design
criteria in Ref. Amini et al. (2010) included minimizing these barriers.
However, the results here show that the slope of the pseudopotential barrier
is more important than the barrier height for suppressing motional excitation,
suggesting that suppression of barrier height may not be a necessary
constraint in future designs. Also, as observed here, with proper rf
filtering, significant barrier slopes can be tolerated without causing
significant heating.
The technique for determining the waveforms described in this report can be
extended to incorporate multiple trapping wells by expanding Eq. (9) to
include multiple minima. Transport procedures such as the ion exchange in Ref.
Splatt et al. (2009) are also amenable to these solving techniques. Separating
and combining of trapping wells requires consideration of the potential’s
quartic term Home and Steane (2006). Therefore, $\mathcal{P}$ in Eq. (8) could
be expanded to include fourth-order derivatives.
With the use of multiple junctions, the techniques described here could help
provide a path toward transfer of information in a large-scale ion-based
quantum processor and enable an increased number of qubits in quantum
algorithm experiments. To do this, waveforms must be expanded to incorporate
many trapping wells. Also, it is likely that a sympathetic cooling ion species
will need to be co-trapped with the qubit ions to allow removal of the
motional excitation from electronic noise, multiple junction transports, and
separating and re-combining wells Home et al. (2009); Hanneke et al. (2009);
Jost (2010). If sympathetic-cooling ions are present, it may be advantageous
to transport both ion species through a junction in a single local trapping
well. Since the pseudopotential (and micromotion) are mass dependent, the
qubit and cooling ions will experience different potentials, which could lead
to additional motional excitation. If such excitation is excessive, it should
still be possible to separate the ions into individual wells by species and
pass the different species through junctions separately, followed by
recombination.
We thank J.M. Amini, K.R. Brown, and J. Britton for contributions to the
apparatus and R. Bowler and J.P. Gaebler for comments on the manuscript. This
work was supported by IARPA, ONR, NSA, and the NIST Quantum Information
program. This is a publication of NIST and not subject to U.S. copyright.
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|
arxiv-papers
| 2011-06-24T16:03:05 |
2024-09-04T02:49:20.044569
|
{
"license": "Public Domain",
"authors": "R.B. Blakestad, C. Ospelkaus, A.P. VanDevender, J.H. Wesenberg, M.J.\n Biercuk, D. Leibfried, and D.J. Wineland",
"submitter": "Brad Blakestad",
"url": "https://arxiv.org/abs/1106.5005"
}
|
1106.5008
|
# On non-relativistic $Q\bar{Q}$ potential via Wilson loop in Galilean space-
time
Haryanto M. Siahaan haryanto.siahaan@gmail.com,
haryanto.siahaan@home.unpar.ac.id Theoretical and Computational Physics Group,
Department of Physics, Faculty of Information Technology and Sciences,
Parahyangan Catholic University, Bandung 40141 - INDONESIA
###### Abstract
We calculate the static Wilson loop from string/gauge correspondence to obtain
the $Q\bar{Q}$ potential in non-relativistic quantum field theory, i.e. CFT
with Galilean symmetry. We analyze the convexity conditions bachas for
$Q\bar{Q}$ potential in this theory, and obtain restrictions for the
acceptable dynamical exponent $z$.
Wilson loop, Galilean symmetry, $Q\bar{Q}$ pair, potential, holography
###### pacs:
11.25.Tq
## I Introduction
It has been shown by Maldacena that large N superconformal gauge theories have
a dual description in terms of string theory in AdS space Maldacena:1997re .
This proposal was realized by Maldacena to compute the energy between quark
$(Q)$ and anti-quark $(\bar{Q})$ pairs Maldawilson . His method was to
calculate expectation values of an operator similar to the Wilson loop in the
large N limit of field theories. Maldacena’s idea was improved later by Rey,
Theisen, and Yee Rey:1998ik . It turns Wilson loop into a physical gauge
invariant property that can be read from the string picture. The $Q\bar{Q}$
energy in the large N superconformal ${\cal N}=4$ Yang-Mills theory can be
obtained from the Wilson loop of the corresponding string in AdS space. It is
proposed that quark and anti-quark pairs live on the boundary, connected by a
U-shaped string in the bulk. In the discussion on this spacetime, the energy
has a non-confining Coulomb-like behavior, as expected for a conformal field
theory. Later this approach was applied to many other spaces and models, as
summarized in ref. sonwilson .
Recently, gravity duals for a certain Galilean-invariant conformal field
theory has attracted some attention in theoretical high energy physics
community Hartnoll:2009sz ; Son:2005rv ; Nishida:2007pj ; Son:2008ye ;
Balasubramanian:2008dm . A special case when we take the dynamical exponent
$z=2$ of this theory (whose isometry is the Schrodinger group $Sch(d-1)$) is
considered to be the basis in constructing duality between gravity and unitary
Fermi gas. However, our interest in this paper is the theory with an arbitrary
dynamical exponent $z$, i.e. Galilean invariant CFT. In this general scheme,
one can discuss the non-relativistic version of the AdS/CFT dictionary, i.e.
the operator-state correspondence between the particle on the boundary and the
string in the bulk. Scaling transformation in this non-relativistic conformal
symmetry can be written as Son:2008ye ; Balasubramanian:2008dm ;
Akhavan:2008ep
$\displaystyle x^{i}\to\lambda x^{i},t\to\lambda^{z}t.$ (1)
The asymptotic metric in this case can be written as
$\displaystyle
ds^{2}=\frac{{R^{2}}}{{r^{2}}}\left({-\frac{{dt^{2}}}{{r^{2\left({z-1}\right)}}}+dtd\xi+\left({dx^{i}}\right)^{2}+dr^{2}}\right)+ds_{X_{5}}^{2}$
(2)
where $R$ is the characteristic radius of space-time, $\xi$ is a compact
light-like coordinate, $x^{i}$ for $i=1$,…,$d$ together with $t$ are the
space-time coordinates on the boundary where (2) is mapped at $r=0$, and
finally $ds_{X_{5}}^{2}$ is the metric of a suitable internal manifold
geometry which allows (2) to be a solution of the supergravity equations of
motion. The extra dimension $\xi$ is usually associated with quantum numbers
interpreted as the particle number. However, the relation between translation
in $\xi$ and its interpretation as particle number operator is still an
unclear topic Kluson:2010prd ; Balabsnumber2010 . Thus we just set this time-
like extra dimension $\xi$ to be constant.
The holographic Wilson loop in non-relativistic CFT had been studied by Kluson
in ref. Kluson:2010prd . He assumed general time dependence of $\xi$ and also
the moving $Q\bar{Q}$ pair cases in the context of non-relativistic quantum
field theory. His study was devoted to the space-time with Galilean symmetry
111From now on this will be abbreviated as Galilean space-time.. Nevertheless,
he still does not include analysis of convexity conditions (12) and (13) yet.
One needs to verify these conditions in $Q\bar{Q}$ potential discussions to
make sure that the corresponding potential function $V\left(L\right)$ is a
monotone non-decreasing and convex function of the separation $L$. The goal of
this paper is to verify these conditions for $Q\bar{Q}$ potential, which is
obtained by calculating the Wilson loop in the string picture in Galilean
space-time. Furthermore, we would like to see the restrictions which may
appear for acceptable dynamical exponent $z$.
This paper is organized as follows. In section 2, we will perform calculations
to acquire the $Q\bar{Q}$ potential energy in Galilean space-time. In section
3, we will derive some conditions for acceptable $z$ due to convexity
inequality. Finally in section 4, there is a summary of our findings.
## II $Q\bar{Q}$ potential in non-relativistic CFT with Galilean symmetry
We will start with the Nambu-Goto action
$\displaystyle S=-\frac{1}{{2\pi\alpha^{\prime}}}\int{d\tau d\sigma\sqrt{-\det
G_{MN}\partial_{\alpha}x^{M}\partial_{\beta}x^{N}}}$ (3)
for metric (2) where $x^{M}=\left({t,r,\xi,x^{i}}\right)$, $G_{MN}$ is space-
time metric in (2), and impose suitable ansatzs in describing static strings,
i.e. $t=x^{0}=\tau$, $r=r\left(\sigma\right)$, $x=x\left(\sigma\right)$, and
$\xi={\rm{constant}}$. Kluson in ref. Kluson:2010prd has considered a more
general case for an extra time-like dimension $\xi$ as a $\tau$-dependent
variable, but we can simply set $\xi$ to be constant (for example as
disucussed in ref. Akhavan:2008ep ) since the $Q\bar{Q}$ potential would
depend on their separation distance 222A distance between $Q$ and $\bar{Q}$ in
our $3+1$ dimensional world, i.e. on the boundary of the Galilean bulk, see
Fig.1. only. The corresponding action can be written as
$\displaystyle
S=-\frac{T}{{2\pi\alpha^{\prime}}}\int{d\sigma\sqrt{f^{2}\left(r\right)\left({\left({r^{\prime}}\right)^{2}+\left({x^{\prime}}\right)^{2}}\right)}}$
(4)
for $f\left(r\right)=R^{2}r^{-(z+1)}$ and we have used
$\left({}\right)^{\prime}\equiv\partial_{\sigma}\left({}\right)$. Variable $T$
in (4) is the loop period and can be written this way due to the time
translation invariance of action (3) for metric (2). We have followed a
standard prescription that has been used in some literature, for example refs.
sonwilson ; nunez ; filho ; caceres ; kinar ; arias , in obtaining the action
(4) as well as the corresponding $Q\bar{Q}$ potential as a function of
$Q\bar{Q}$ pair’s distance. Though the metric (2.1) is not diagonal, but
action (4) leads us to a problem of Wilson loop computation which can be
started by finding a geodesic in the effective 2-dimensional geometry arias
$\displaystyle\left({ds_{eff}}\right)^{2}=f^{2}\left(r\right)\left({dx^{2}+dr^{2}}\right).$
(5)
The equation of motion (geodesic line) from (4) is
$\displaystyle\frac{{dx}}{{dr}}=\pm\frac{{f\left({r_{0}}\right)}}{{\sqrt{f^{2}\left(r\right)-f^{2}\left({r_{0}}\right)}}}.$
(6)
$r_{0}$ is the maximum position of the U-shaped string with respect to the
$r$-coordinate (bulk radius, see Fig. 1). From (6) one can obtain the
separation distance of quark and anti-quark on the boundary, by integrating
the geodesic with respect to $r$. Since the boundary is at $r=0$, then the
separation as the function of $r_{0}$ can be obtained by the following
integration
$\displaystyle
L\left({r_{0}}\right)=2\int\limits_{0}^{r_{0}}{\frac{{f\left({r_{0}}\right)}}{{\sqrt{f^{2}\left(r\right)-f^{2}\left({r_{0}}\right)}}}dr}.$
(7)
Related to the expression for the $Q\bar{Q}$ separation above, one may provide
such an illustration as depicted in Fig. 1.
Figure 1: $Q\bar{Q}$ pair on the boundary as each ends of string.
Inserting $f\left(r\right)=R^{2}r^{-(z+1)}$ to (7) and using the beta function
in our computation give the following exact result
$\displaystyle
L\left({r_{0},z}\right)=2\int\limits_{0}^{r_{0}}{\frac{{r^{z+1}}}{{\sqrt{r_{0}^{2z+2}-r^{2z+2}}}}}=\frac{{2r_{0}\sqrt{\pi}\Gamma\left({\frac{{z+2}}{{2z+2}}}\right)}}{{\Gamma\left({\frac{1}{{2z+2}}}\right)}}.$
(8)
Then we follow a general prescription in refs. sonwilson ; filho ; kinar ;
arias to compute the energy between quark and anti-quark. We have a general
form of total $Q\bar{Q}$ energy as
$\displaystyle
E\left({r_{0}}\right)=\frac{1}{{\pi\alpha^{\prime}}}\int\limits_{0}^{r_{0}}{\frac{{f^{2}\left(r\right)}}{{\sqrt{f^{2}\left(r\right)-f^{2}\left({r_{0}}\right)}}}dr}-2m_{Q}$
(9)
where $m_{Q}$ is considered as the energy of non interacting quark nunez ;
filho ; kinar ; arias . Thus the $Q\bar{Q}$ potential can be written as
$V_{Q\bar{Q}}\left({r_{0}}\right)=E\left({r_{0}}\right)-2m_{Q}$
$\displaystyle=\frac{1}{{\pi\alpha^{\prime}}}\int\limits_{0}^{r_{0}}{\frac{{f^{2}\left(r\right)}}{{\sqrt{f^{2}\left(r\right)-f^{2}\left({r_{0}}\right)}}}dr}$
(10)
which can also be computed by the use of beta function. The potential is
$\displaystyle
V_{Q\bar{Q}}\left({r_{0},z}\right)=2R^{2}r_{0}^{z+1}\int\limits_{0}^{r_{0}}{\frac{{dr}}{{r^{z+1}\sqrt{r_{0}^{2z+2}-r^{2z+2}}}}}=\frac{{2R^{2}\sqrt{\pi}}}{{r_{0}^{z}\left({2z+2}\right)}}\frac{{\Gamma\left({\frac{{-z}}{{2z+2}}}\right)}}{{\Gamma\left({\frac{1}{{2z+2}}}\right)}}.$
(11)
In the next section we will see the compatibility of the potential (11) with
convexity conditions.
## III Convexity conditions and string embeddings
There are some conditions that should be satisfied by any potential which
describes interaction between quark and anti-quark whose name ’convexity’
conditions bachas ; arias
$\displaystyle\frac{{dV}}{{dL}}>0$ (12)
and
$\displaystyle\frac{{d^{2}V}}{{dL^{2}}}\leq 0.$ (13)
Condition (12) means quark and anti-quark are attractive everywhere, and (13)
tells us that the potential is a monotone non-increasing function of their
separation. These conditions can be verified as follows
$\displaystyle\frac{{dV_{Q\bar{Q}}\left({r_{0},z}\right)}}{{dL\left({r_{0},z}\right)}}=\frac{{dV_{Q\bar{Q}}\left({r_{0},z}\right)}}{{dr_{0}}}\frac{{dr_{0}}}{{dL\left({r_{0},z}\right)}}=\frac{{-zR^{2}}}{{r_{0}^{z+1}\left({2z+2}\right)}}\frac{{\Gamma\left({\frac{{-z}}{{2z+2}}}\right)}}{{\Gamma\left({\frac{{z+2}}{{2z+2}}}\right)}}>0$
(14)
and
$\frac{{d^{2}V_{Q\bar{Q}}\left({r_{0},z}\right)}}{{dL\left({r_{0},z}\right)^{2}}}=\frac{{d\left({\frac{{dV_{Q\bar{Q}}\left({r_{0},z}\right)}}{{dL\left({r_{0},z}\right)}}}\right)}}{{dr_{0}}}\frac{{dr_{0}}}{{dL\left({r_{0},z}\right)}}$
$\displaystyle=\frac{{zR^{2}}}{{4\sqrt{\pi}r_{0}^{z+2}}}\frac{{\Gamma\left({\frac{1}{{2z+2}}}\right)\Gamma\left({\frac{{-z}}{{2z+2}}}\right)}}{{\left({\Gamma\left({\frac{{z+2}}{{2z+2}}}\right)}\right)^{2}}}\leq
0.$ (15)
The two last equations are inequalities for physically accepted $z$ based on
convexity conditions for the $Q\bar{Q}$ pair.
In ref. yoshida-hartnoll , the authors present simple embeddings of duals for
nonrelativistic critical points, where the dynamical critical exponent can
take many values $z\neq 2$ 333I thank Koushik Balasubramanian to give me know
this work.. They find that $z=1$ and $z\geq 3/2$ as the possible dynamical
critical exponents that allow string embeddings in gauge/gravity dual picture.
From their paper yoshida-hartnoll , we could learn that our $f\left(r\right)$
would depend on the coordinates of the internal manifold $X_{5}$ 444I thank
reviewer for pointing this out to me.. Hartnoll and Yoshida write the non-
compact part of the metric which can accommodate a large number of values of
$z$ by the following ansatz 555We follow the form of metric by Balasubramanian
and McGreevy Balasubramanian:2008dm . $f\left({X_{5}}\right)$ in ref. yoshida-
hartnoll is $h^{2}\left({X_{5}}\right)$ in this paper.
$\displaystyle
ds^{2}=\frac{{R^{2}}}{{r^{2}}}\left({-\frac{{dt^{2}}}{{h^{2}\left({X_{5}}\right)r^{2\left({z-1}\right)}}}+dtd\xi+\left({dx^{i}}\right)^{2}+dr^{2}}\right)$
(16)
which modifies our previous $f\left(r\right)$ from $R^{2}r^{-(z+1)}$ to
$R^{2}r^{-(z+1)}h\left({X_{5}}\right)^{-1}$. Nevertheless, the function
$h\left({X_{5}}\right)$ would not appear in (8) and (11). Thus our findings on
the restrictions for $z$ can be applied to the work of Hartnoll and Yoshida in
ref. yoshida-hartnoll . One can verify that conditions (14) and (15) are
fulfilled for $z=1$, and also for $z\geq 3/2$. The negativity of
$\Gamma\left({{\textstyle{{-z}\over{2z+2}}}}\right)$ for $z\geq 1$ guarantees
both (14) and (15) are satisfied.
## IV Summary
We have calculated the potential between $Q$ and $\bar{Q}$ in the non-
relativistic quantum field theory by using the Wilson loop analysis in the
gauge/gravity correspondence in the Galilean bulk. Our findings are
inequalities (14) and (15) for physically acceptable dynamical exponent $z$
from convexity conditions. Yoshida and Hartnoll yoshida-hartnoll have found
families of $z$ for string embeddings in Galilean space-time, i.e. $z=1$ and
$z\geq 3/2$, which agree with inequalities (14) and (15) above.
## Acknowledgments
I would thank LPPM-UNPAR for supporting my research. I also thank to my
colleagues in physics department of Parahyangan Catholic University for all
their supports, and to Frank Landsman of PPB-UNPAR for correcting my
manuscript.
## References
* (1) J. M. Maldacena, _“The large N limit of superconformal field theories and supergravity,”_ Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].
* (2) J. M. Maldacena, _“Wilson loops in large N field theories,”_ Phys. Rev. Lett. 80, 4859 (1998) [arXiv:hep-th/9803002].
* (3) S. J. Rey and J. T. Yee, _“Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity,”_ Eur. Phys. J. C 22 (2001) 379 [arXiv:hep-th/9803001].
* (4) J. Sonnenschein, _“What does the string/gauge correspondence teach us about Wilson loops?”_ [arXiv:hep-th/0003032]
* (5) S. A. Hartnoll, _“Lectures on holographic methods for condensed matter physics,”_ Class. quant. Grav.26 :224002 (2009) [arXiv:0903.3246 [hep-th]].
* (6) D. T. Son and M. Wingate, _“General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas,”_ Annals Phys. 321, 197 (2006) [arXiv:cond-mat/0509786].
* (7) Y. Nishida and D. T. Son, _“Nonrelativistic conformal field theories,”_ Phys. Rev. D 76, 086004 (2007) [arXiv:0706.3746 [hep-th]].
* (8) D. T. Son, _“Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry,”_ Phys. Rev. D 78, 046003 (2008) [arXiv:0804.3972 [hep-th]].
* (9) K. Balasubramanian and J. McGreevy, _“Gravity duals for non-relativistic CFTs,”_ Phys. Rev. Lett. 101, 061601 (2008) [arXiv:0804.4053 [hep-th]].
* (10) A. Akhavan, M. Alishahiha, A. Davody and A. Vahedi, _“Non-relativistic CFT and Semi-classical Strings,”_ JHEP 0903 (2009) 053 [arXiv:0811.3067 [hep-th]].
* (11) J. Klusoň, _“Open String in Non-Relativistic Background,”_ Phys. Rev. D81, 106006 (2010) [arXiv:0912.4587 [hep-th]].
* (12) K. Balasubramanian and J. McGreevy, _“The particle number in Galilean holography,”_ JHEP 1101 (2011) 137 [arXiv:1007.2184 [hep-th]].
* (13) C. Bachas, _“Convexity Of The Quarkonium Potential,”_ Phys. Rev. D 33 (1986) 2723.
* (14) C. Nunez, M. Piai, A. Rago, _“Wilson Loops in string duals of Walking and Flavored Systems,”_ Phys. Rev. D 81, 086001 (2010) [arXiv:0909.0748 [hep-th]].
* (15) H. Boschi-Filho and N. R. F. Braga, _“Wilson Loops for a quark anti-quark pair in D3-brane space,”_ JHEP 03 (2005) 051 [arXiv:hep-th/0411135].
* (16) E. Cáceres, M. Natsuume and T. Okamura, _“Screening length in plasma winds,”_ JHEP 0610 (2006) 011 [arXiv:hep-th/0607233].
* (17) Y. Kinar, E. Schreiber, J. Sonnenschein, _“ $Q\bar{Q}$ Potential from Strings in Curved Spacetime - Classical Results,”_ Nucl. Phys. B 566 (2000) 103-125 [arXiv:hep-th/9811192].
* (18) R. E. Arias and G. A. Silva, _“Wilson loops stability in the gauge/string correspondence,”_ JHEP 1001 (2010)023 [arXiv:0911.0662 [hep-th]].
* (19) S. A. Hartnoll and K. Yoshida, _“Families of IIB duals for nonrelativistic CFTs,”_ JHEP 0812 (2008)071 [arXiv:0810.0298 [hep-th]]
|
arxiv-papers
| 2011-06-24T16:14:54 |
2024-09-04T02:49:20.057574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Haryanto M. Siahaan",
"submitter": "Haryanto Siahaan",
"url": "https://arxiv.org/abs/1106.5008"
}
|
1106.5178
|
# The Open Annotation Collaboration (OAC) Model
Bernhard Haslhofer1, Rainer Simon2, Robert Sanderson3 and Herbert van de
Sompel3 1Cornell Information Science, Ithaca, USA
_bernhard.haslhofer@cornell.edu_ 2Austrian Institute of Technology, Vienna,
Austria
_rainer.simon@ait.ac.at_ 3Los Alamos National Laboratory, Los Alamos, USA
_{ rsanderson,herbertv}@lanl.gov_
###### Abstract
Annotations allow users to associate additional information with existing
resources. Using proprietary and closed systems on the Web, users are already
able to annotate multimedia resources such as images, audio and video. So far,
however, this information is almost always kept locked up and inaccessible to
the Web of Data. We believe that an important step to take is the integration
of multimedia annotations and the Linked Data principles. This should allow
clients to easily publish and consume, thus exchange annotations about
resources via common Web standards. We first present the current status of the
Open Annotation Collaboration, an international initiative that is currently
working on annotation interoperability specifications based on best practices
from the Linked Data effort. Then we present two use cases and early
prototypes that make use of the proposed annotation model and present lessons
learned and discuss yet open technical issues.
## I Introduction
Large scale media portals such as Youtube and Flickr allow users to attach
information to multimedia objects by means of annotations. Web portals hosting
multi-lingual collections of millions of digitized items such as Europeana are
currently investigating how to integrate the knowledge of end users with
existing digital curation processes. Annotations are also becoming an
increasingly important component in the cyber-infrastructure of many scholarly
disciplines.
A Web-based annotation model should fulfill several requirements. In the age
of video blogging and real-time sharing of geo-located images, the notion of
solely textual annotations has become obsolete. Therefore, multimedia Web
resources should be annotatable and also be able to be annotated onto other
resources. Users often discuss multiple segments of a resource, or multiple
resources, in a single annotation and thus the model should support multiple
targets. An annotation framework should also follow the Linked Open Data
guidelines to promote annotation sharing between systems. In order to avoid
inaccurate or incorrect annotations, it must take the ephemeral nature of Web
resources into account.
Annotations on the Web have many facets: a simple example could be a textual
note or a tag (cf., [1]) annotating an image or video. Things become more
complex when a particular paragraph in an HTML document annotates a segment
(cf., [2]) in an online video or when someone draws polygon shapes on tiled
high-resolution image sets. If we further extend the annotation concept, we
could easily regard a large portion of Twitter tweets as annotations on Web
resources. Therefore, in a generic and Web-centric conception, we regard an
annotation as association created between one _body_ resource and other
_target_ resources, where the body must be somehow _about_ the target.
Annotea [3] already defines a specification for publishing annotations but has
several shortcomings: (i) it was designed for the annotation of Web pages and
provides only limited means to address segments in multimedia objects, (ii) if
clients want to access annotations they need to be aware of the Annotea-
specific protocol, and (iii) Annotea annotations do not take into account that
Web resources are very likely to have different states over time.
Throughout the years several Annotea extensions have been developed to deal
with these and other shortcomings: Koivunnen [4] introduced additional types
of annotations, such as Bookmark and Topic. Schroeter and Hunter [5] proposed
to express segments in media-objects by using _context_ resources in
combination with formalized or standardized descriptions to represent the
context, such as SVG or complex datatypes taken from the MPEG-7 standard.
Based on that work, Haslhofer et al. [6] introduce the notion of _annotation
profiles_ as containers for content- and annotation-type specific Annotea
extensions and suggested that annotations should be dereferencable resources
on the Web, which follow the Linked Data principles. However, these extensions
were developed separate from each other and inherit the above-mentioned
Annotea shortcomings.
In this paper, we describe how the Open Annotations Collaboration (OAC), an
effort aimed at establishing annotation interoperability, tackles these
issues. We describe two annotation use cases — image and historic map
annotation – for which we have implemented the OAC model and report on lessons
learned and open issues. We also briefly summarize related work in this area
and give outlook on our future work.
## II The Open Annotation Collaboration
The Open Annotation Collaboration (OAC) is an international group with the aim
of providing a Web-centric, interoperable annotation environment that
facilitates cross-boundary annotations, allowing multiple servers, clients and
overlay services to create, discover and make use of the valuable information
contained in annotations. To this end, a Linked Data based data model has been
adopted.
### II-A Open Annotation Data Model
The OAC data model tries to pull together various extensions of Annotea into a
cohesive whole. The Web architecture and Linked Data guidelines are
foundational principles, resulting in a specification that can be applied to
annotate any set of Web resources. At the time of this writing, the
specification, which is available at
http://www.openannotation.org/spec/alpha3/, is still under development.
Following its predecessors, the OAC model, shown in Figure 1, has three
primary classes of resources. In all cases below, the oac namespace prefix
expands to http://www.openannotation.org/ns/.
* •
The oac:Body of the annotation (node B-1). This resource is the comment,
metadata or other information that is created about another resource. The Body
can be any Web resource, of any media format, available at any URI. The model
allows for exactly one Body per Annotation.
* •
The oac:Target of the annotation (node T-1). This is the resource that the
Body is about. Like the Body, it can be any URI identified resource. The model
allows for one or more Targets per Annotation.
* •
The oac:Annotation (node A-1). This resource stands for a document identified
by an HTTP URI that describes at least the Body and Target resources involved
in the annotation as well as any additional properties and relationships
(e.g., dcterms:creator). Dereferencing an annotation’s HTTP URI returns a
serialization in a permissible RDF format.
If the Body of an annotation is identified by a dereferencable HTTP URI, as it
is the case in Twitter, various blogging platforms, or Google Docs, it can
easily be referenced from an annotation. If a client cannot create URIs for an
annotation Body, for instance because it is an offline client, they can assign
a unique non-resolvable URI (called a URN) as the identifier for the Body
node. This approach can still be reconciled with the Linked Data principles as
servers that publish such annotations can assign HTTP URIs they control to the
Bodies, and express equivalence between the HTTP URI and the URN.
The OAC model also allows to include textual information directly in the
annotation document by adding the representation of a resource as plain text
to the Body via the cnt:chars property and defining the character encoding
using cnt:characterEncoding [7].
Figure 1: OAC Baseline Data Model.
### II-B OAC and Linked Data
Several Linked Data principles have influenced the approach taken by Open
Annotation Collaboration. The promotion of a publish/discover approach for
handling Annotations as opposed to the protocol-oriented approach taken by
Annotea stands out. But also the emphasis on the use (wherever possible) of
HTTP URIs for resources involved in Annotations reflects a Linked Data
philosophy. Earlier versions of the data model considered an Annotation to be
an OAI-ORE Aggregation [8] and hence a conceptual non-information resource.
Community feedback led to a revision of this approach as it was deemed
artificial, and the complexities regarding handling the difference between
information resource and non-information resource at the protocol level were
considered a hindrance to potential adoption.
### II-C Addressing Media Segments
Many annotations concern parts, or segments, of resources rather than the
entirety of the resource. While simple segments of resources can be identified
and referenced directly using the emerging W3C Media Fragment specification
[9] or media-type-specific fragment identifiers as defined in RFCs, there are
many use cases for segments that currently cannot be identified using this
proposal. One example is the annotation of historic maps, where users need to
draw polygon shapes around the geographic areas they want to address with
their notes. Sanderson et al. [10] describe another use case where annotators
express the relationships between images, texts and other resources in
medieval manuscripts by means of line segments. For these and other use cases,
which require the expression of complex media segment information, the current
W3C Media Fragments specification is insufficient.
For this reason, OAC introduces the so-called oac:ConstraintTarget node
(CT-1), which constrains the annotation target to a specific segment that is
further described by an oac:Constraint (C-1) node. The description of the
constraint depends on the application scenario and on the (media) type of the
annotated target resource. For example, an SVG path could be used to describe
a region within an image.
### II-D Robust Annotations in Time
It must be stressed that different agents may create the Annotation, Body and
Target at different times. For example, Alice might create an annotation
saying that Bob’s YouTube video annotates Carol’s Flickr photo. Also, being
regular Web resources, the Body and Target are likely to have different
representations over time. Some annotations may apply irrespective of
representation, while others may pertain to specific representations. In order
to provide the ability to accurately interpret annotations past their
publication, the OAC Data Model introduces three ways to express temporal
context. The manner in which these three types of Annotation use the oac:when
property, which has a datetime as its value, distinguishes them.
A _Timeless Annotation_ applies irrespective of the evolving representations
of Body and Target; it can be considered as if the Annotation references the
semantics of the resources. For example, an annotation with a Body that says
“This is the front page of CNN” remains accurate as representations of the
Target http://cnn.com/ change over time. Timeless Annotations don’t make use
of the oac:when property.
A _Uniform Time Annotation_ has a single point in time at which all the
resources involved in the Annotation should be considered. This type of
Annotation has the oac:when property attached to the Annotation. For example,
if Alice recurrently publishes a cartoon at http://example.org/cartoon that
comments on a story on the live CNN home page, an Annotation that has the
cartoon as Body and the CNN home page as Target would need to be handled as a
Uniform Time Annotation in order to provide the ability to match up correct
representations of Body and Target.
A _Varied Time Annotation_ has a Body and Target that need to be considered at
different moments in time. This type of Annotation uses the oac:when property
attached to an oac:TimeConstraint node (a specialization of oac:Constraint)
for both Body and Target. If, in the aforementioned cartoon example, Alice
would have the habit to publish her cartoon at http://example.org/cartoon when
the mocked article is no longer on the home page, but still use http://cnn.com
as the Target of her Annotation, the Varied Time Annotation approach would
have to be used.
This temporal information can be used in the Memento framework to recreate the
Annotation as it was intended by reconstructing it with the time-appropriate
Body and Target(s). Previous versions of Web resources exist in archives such
as the Internet Archive, or within content management systems such as
MediaWiki’s article history, however they are divorced from their original
URI. Memento proposes a simple extension of HTTP in order to connect the
original and archived resources. It leverages existing HTTP capabilities in
order to support accessing resource versions through the use of the URI of a
resource and a datetime as the indicator of the required version. In the
framework, a server that host versions of a given resource exposes a TimeGate,
which acts as a gateway to the past for a given Web resource. In order to
facilitate access to a version of that resource, the TimeGate supports HTTP
content negotiation in the datetime dimension. Several mechanisms support
discovery of TimeGates, including HTTP Links that point from a resource to its
TimeGate(s) [11].
## III Use Cases and Annotation Prototypes
We describe two annotation use cases for which we have implemented early
prototypes that publish annotations as Linked Data on the Web following the
OAC approach.
### III-A The OAC/Djatoka Demonstrator
The OAC/Djatoka Demonstrator implements the current OAC data model for image
resources. It uses the Djatoka image server [12] as its primary platform,
which provides panning and zooming functionality for images using a JPEG 2000
image tiling system. The demonstrator enables both creation and viewing of
annotations, with both inline and external resources.
SVG elements can be dynamically created and manipulated using javascript such
that they describe a region of interest. This information is then encoded as
an oac:SvgConstraint with a UUID URN identifier, which constrains the full
image. The SVG elements may be resized or repositioned by the user, and scale
or translate respectively with zooming or panning operations. Instead of part
of the image, the Target may also be one of the other Annotations, enabling a
threaded discussion.
The Body of the annotation is either an external Web resource, or a string
encoded using cnt:ContentAsText [7]. The use of external resources allows the
embedded rendering of image, video and audio within the display, and the re-
use of third party hosting services.
The resulting graph is serialized to RDFa and published to online services
such as Blogger. The annotations are then harvested by a graph database that
subscribes to the feeds of known annotators. The database system [13] makes
the annotations searchable via the target URI, creation date and content’s
text.
This prototype affirmed the OAC strategies for strictly adhering to the Web
architecture, allowing any resource to be the Body of an Annotation rather
than just text, and the feasibility of an environment in which resources are
discovered and harvested rather than transmitted according to a strict
protocol, such as in Annotea.
A video of the OAC/Djatoka annotation prototype is available at
http://www.openannotation.org/demos/.
### III-B Historic Map Annotation with YUMA
YUMA111https://github.com/yuma-annotation/ is an open source annotation
framework for the annotation of online multimedia resources. It consists of a
server backend and multiple Web clients, each dedicated to a specific media
type. At the moment, the YUMA client-suite encompasses clients for the
annotation of images, audio, video, and, as a special case of images, maps.
Demonstrations of the different YUMA clients are available at:
http://dme.ait.ac.at/annotation/.
The YUMA Map Annotation Tool, which is shown in Figure 2, enables scholars to
annotate high-resolution scans of historic maps. It provides similar panning
and zooming functionality as the Djatoka annotation service, as well as
drawing tools for annotating specific areas on the map. In addition to
conventional free-text annotation, YUMA also supports _Semantic Tagging_. When
creating or editing annotations, users can make them semantically more
expressive by adding references to relevant Linked Data resources on the Web:
e.g. links to geographical resources on Geonames or resources from DBpedia. To
support users in this task, the tool employs a semi-automatic approach. Based
on the annotated geographical area, as well as on an analysis of the
annotation text, potentially related Linked Data resources are proposed
automatically in the form of a tag cloud. The user is prompted to verify the
proposed resources, or can simply ignore them. For those resources that have
been user-verified, the system adds the link to the annotation. Additionally,
the system dereferences the resource and stores relevant properties as part of
the annotation metadata: e.g. alternative language labels, spelling variants,
or geographical coordinates. This information can later be exploited to
facilitate advanced search functionalities such as multilingual, synonym, or
geographical search [14].
Figure 2: YUMA Map Annotation Tool Screenshot.
The YUMA server backend exposes annotations as Linked Data on the Web
following the OAC model. As illustrated with an example annotation in Figure
3, each Annotation resource has its own dereferencable HTTP URI. The textual
annotation Body is embedded directly into the Annotation document and has a
unique non-resolvable URI (URN) as identifier. The annotation Body is about a
Web resource - in this case a high-resolution zoomable image, published as a
Zoomify222http://zoomify.com/ tileset. The tileset is identified by its XML
metadata descriptor file, which acts as the annotation Target. The annotated
region within the zoomable image is defined by means of an oac:SvgConstraint
resource, which allows us to add an SVG snippet expressing the boundaries of
that region, to that resource.
Figure 3: Sample YUMA OAC Annotation.
Because of the limited expressiveness of oac:Body in the Alpha3 OAC
specification, user-contributed links are currently loosely attached to the
Body. This results in a loss of semantics because some of the proposed links
annotate certain text segments or named entities in the annotation Body. We
plan to change this representation as soon as OAC provides means for
representing structured information in annotation Bodies.
## IV Related Work
Annotations have a long research history, and unsurprisingly the research
perspectives and interpretations of what an _annotation_ is supposed to be
vary widely. Agosti et al. [15] provide a comprehensive study on the contours
and complexity of annotations. A representative discussion on how annotations
can be used in various scholarly disciplines in given by Bradley [16]. He
describes how annotations can support interpretation development by collecting
notes, classifying resources, and identifying novel relationships between
resources.
Besides Annotea other annotation models have been proposed: [17] built the
MPEG-7 compliant COMM Ontology. OAC, in contrast, is a resource-centric
annotation model, which is more light-weight because it doesn’t have a
background in the automated feature extraction and representation. The M3O
Ontology [18] allows the integration of annotations with SMIL and SVG
documents. OAC, in contrast, treats annotations as first-class resources on
the Web, which would not be part but _about_ a presentation.
Early related work on the issue of describing segments in multimedia resources
can be traced back to research on linking in hypermedia documents [19]. For
describing segments using a non-URI based mechanism one can use MPEG-7 Shape
Descriptors (cf. [20]) or terms defined in a dedicated multimedia ontology.
SVG [21] and MPEG-21 [22] introduced XPointer-based URI fragment definitions
for linking to segments in multimedia resources.
## V Conclusions and Future Work
We apply a generic and Web-centric conception to the various facets
annotations can have and regard an annotation as association created between
one _body_ resource and other _target_ resources, where the body must be
somehow _about_ the target. This conception lead to the specification of the
OAC model, which originates from activities in the Open Annotation
Collaboration and aims at building an interoperable environment for publishing
annotations on the Web. We also presented and provided pointers to two
prototypes that currently implement the OAC specification.
The OAC specification is currently in Alpha3 stage and our future work will
focus on the following issues: support for structured bodies that go beyond
resource referencing and string-literal representation, extension mechanisms
for addressing complex media segments in various media types, and processing
of constraints.
We will further pursue the integration of the OAC segment identification
approach with the W3C Media Fragment Identification mechanism. Since it is
hardly possible to address all possible segment shapes in a fragment
identification specification, we propose an additional fragment key/value pair
for the spatial dimension, which enables fragment identification by reference
in W3C Media Fragment URIs. The key could be _ptr_ , _ref_ or something
similar and the value a URI. The URI points to a resource, which provides
further information about the properties of the spatial region/segment. We
suggested this to the Media Fragment Working Group and hope that this issue
will be addressed in future W3C Media Fragments URI recommendations.
As a final result, we expect a light-weight annotation model that is
straightforward to use in basic annotation use cases but provides extension
points for more complex annotation scenarios.
## Acknowledgment
The work has partly been supported by the European Commission as part of the
eContentplus program (EuropeanaConnect) and by a Marie Curie International
Outgoing Fellowship within the 7th Europeana Community Framework Programme”.
The development of OAC is funded by the Andrew W. Mellon foundation.
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* [18] C. Saathoff and A. Scherp, “Unlocking the semantics of multimedia presentations in the web with the multimedia metadata ontology,” in _Proceedings of the 19th international conference on World wide web_ , ser. WWW ’10. New York, NY, USA: ACM, 2010, pp. 831–840.
* [19] L. Hardman, D. C. A. Bulterman, and G. van Rossum, “The amsterdam hypermedia model: adding time and context to the dexter model,” _Commun. ACM_ , vol. 37, no. 2, pp. 50–62, 1994.
* [20] F. Nack and A. T. Lindsay, “Everything You Wanted to Know About MPEG-7: Part 2,” _IEEE MultiMedia_ , pp. 64–73, October - December 1999\.
* [21] W3C SVG Working Group, _Scalable Vector Graphics (SVG) — XML Graphics for the Web_ , W3C, 2003, available at: http://www.w3.org/Graphics/SVG/.
* [22] ISO/IEC, _Multimedia framework (MPEG-21) – Part 17: Fragment Identification of MPEG Resources_ , International Organization for Standardization, Geneva, Switzerland, Dec. 2006.
|
arxiv-papers
| 2011-06-25T22:55:17 |
2024-09-04T02:49:20.068440
|
{
"license": "Public Domain",
"authors": "Bernhard Haslhofer, Rainer Simon, Robert Sanderson, Herbert van de\n Sompel",
"submitter": "Bernhard Haslhofer",
"url": "https://arxiv.org/abs/1106.5178"
}
|
1106.5186
|
# Learning Shape and Texture Characteristics of CT Tree-in-Bud Opacities for
CAD Systems
Ulaş Bağcı, Jianhua Yao, Jesus Caban, Anthony F. Suffredini, Tara N. Palmore,
Daniel J. Mollura Radiology and Imaging Sciences
National Institutes of Health ulasbagc@gmail.com
###### Abstract.
Although radiologists can employ CAD systems to characterize malignancies,
pulmonary fibrosis and other chronic diseases; the design of imaging
techniques to quantify infectious diseases continue to lag behind. There
exists a need to create more CAD systems capable of detecting and quantifying
characteristic patterns often seen in respiratory tract infections such as
influenza, bacterial pneumonia, or tuborculosis. One of such patterns is Tree-
in-bud (TIB) which presents thickened bronchial structures surrounding by
clusters of micro-nodules. Automatic detection of TIB patterns is a
challenging task because of their weak boundary, noisy appearance, and small
lesion size. In this paper, we present two novel methods for automatically
detecting TIB patterns: (1) a fast localization of candidate patterns using
information from local scale of the images, and (2) a Möbius invariant feature
extraction method based on learned local shape and texture properties. A
comparative evaluation of the proposed methods is presented with a dataset of
39 laboratory confirmed viral bronchiolitis human parainfluenza (HPIV) CTs and
21 normal lung CTs. Experimental results demonstrate that the proposed CAD
system can achieve high detection rate with an overall accuracy of 90.96%.
###### Contents
1. 1 Introduction
2. 2 Methodology
3. 3 Feature Extraction
4. 4 Experimental Results
5. 5 Conclusion
## 1\. Introduction
As shown by the recent pandemic of novel swine-origin H1N1 influenza,
respiratory tract infections are a leading cause of disability and death. A
common image pattern often associated with respiratory tract infections is TIB
opacification, represented by thickened bronchial structures locally
surrounded by clusters of 2-3 millimeter micro-nodules. Such patterns
generally represent disease of the small airways such as infectious-
inflammatory bronchiolitis as well as bronchiolar luminal impaction with
mucus, pus, cells or fluid causing normally invisible peripheral airways to
become visible [1]. Fig. 1 shows TIB patterns in a chest CT.
The precise quantification of the lung volume occupied by TIB patterns is a
challenging task limited by significant inter-observer variance with
inconsistent visual scoring methods. These limitations raise the possibility
that radiologists’ assessment of respiratory tract infections could be
enhanced through the use of computer assisted detection (CAD) systems.
However, there are many technical obstacles to detecting TIB patterns because
micro-nodules and abnormal peripheral airway structures have strong shape and
appearance similarities to TIB patterns and normal anatomic structures in the
lungs.
Figure 1. (Left) CT image with a significant amount TIB patterns. (Right)
Labelled TIB patterns (blue) in zoomed window on the right lung.
In this work, we propose a new CAD system to evaluate and quantify respiratory
tract infections by automatically detecting TIB patterns. The main
contributions of the paper are two-fold: (1) A candidate selection method that
locates possible abnormal patterns in the images. This process comes from a
learning perspective such that the size, shape, and textural characteristics
of TIB patterns are learned a priori. The candidate selection process removes
large homogeneous regions from consideration which results in a fast
localization of candidate TIB patterns. The local regions enclosing candidate
TIB patterns are then used to extract shape and texture features for automatic
detection; (2) another novel aspect in this work is to extract Möbius
invariant local shape features. Extracted local shape features are combined
with statistical texture features to classify lung tissues. To the best of our
knowledge, this is the first study that uses automatic detection of TIB
patterns for a CAD system in infectious lung diseases. Since there is no
published work on automatic detection of TIB patterns in the literature, we
compare our proposed CAD system on the basis of different feature sets
previously shown to be successful in detecting lung diseases in general.
## 2\. Methodology
The proposed CAD methodology is illustrated in Fig. 2. First, lungs are
segmented from CT volumes. Second, we use locally adaptive scale based
filtering method to detect candidate TIB patterns. Third, segmented lung is
divided into local patches in which we extract invariant shape features and
statistical texture features followed by support vector machine (SVM)
classification. We extract features from local patches of the segmented lung
only if there are candidate TIB patterns in the patches. The details of the
proposed methods are presented below.
Figure 2. The flowchart of the proposed CAD system for automatic TIB
detection.
I. Segmentation: Segmentation is often the first step in CAD systems. There
are many clinically accepted segmentation methods in clinics [2, 3]. In this
study, fuzzy connectedness (FC) image segmentation algorithm is used to
achieve successful delineation [2]. In FC framework, left and right lungs are
“recognized” by automatically assigned seeds, which initiate FC segmentation.
II. Learning characteristics of TIB patterns: From Fig. 1, we can readily
observe that TIB patterns have intensity characteristics with high variation
towards nearby pixels, and such regions do not usually exceed a few
millimetre(mm) in length. In other words, TIB patterns do not constitute
sufficiently large homogeneous regions. Non-smooth changes in local gradient
values support this observation. As guided by these observations, we conclude
that (a) TIB patterns are localized only in the vicinity of small homogeneous
regions, and (b) their boundaries have high curvatures due to the nature of
its complex shape.
III. Candidate Pattern Selection: Our candidate detection method comes from a
learning perspective such that we assign every internal voxel of the lung a
membership value reflecting the size (i.e., scale) of the homogeneous region
that the voxel belongs to. To do this, we use a locally adaptive scale based
filtering method called ball-scale (or b-scale for short) [2]. b-scale is the
simplest form of a locally adaptive scale where the scene is partitioned into
several scale levels within which every voxel is assigned the size of the
local structure it belongs. For instance, voxels within the large homogeneous
objects have highest scale values, and the voxels nearby the boundary of
objects have small scale values. Because of this fact and the fact in II.(a),
we draw the conclusion that TIB patterns constitute only small b-scale values,
hence, it is highly reasonable to consider voxels with small b-scale values as
candidate TIB patterns. Moreover, it is indeed highly practical to discard
voxels with high b-scale values from candidate selection procedure. Fig. 2
(candidate selection) and Fig. 3(b) show selected b-scale regions as candidate
TIB patterns. A detailed description of the b-scale algorithm is presented in
[2].
## 3\. Feature Extraction
For a successful CAD system for infectious lung diseases, there is a need to
have representative features characterizing shape and texture of TIB patterns
efficiently. Since TIB is a complex shape pattern consisting of curvilinear
structures with nodular structures nearby (i.e., a budding tree), we propose
to use local shape features (derived from geometry of the local structures)
combined with grey-level statistics (derived from a given local patch).
It has been long known that curvatures play an important role in the
representation and recognition of intrinsic shapes. However, similarity of
curvature values may not necessarily be equivalent to intrinsic shape
similarities, which causes a degradation in recognition and matching
performance. To overcome this difficulty, we propose to use Willmore energy
functional [4] and several different affine invariant shape features
parametrically related to the Willmore energy functional.
### 3.0.1. Willmore Energy:
The Willmore energy of surfaces plays an important role in digital geometry,
elastic membranes, and image processing. It is closely related to Canham-
Helfrich model, where surface energy is defined as
$\mathcal{S}=\int_{\Sigma}\alpha+\beta(H)^{2}-\gamma KdA.$ (1)
This model is curvature driven, invariant under the the group of Möbius
transformations (in particular under rigid motions and scaling of the surface)
and shown to be very useful in energy minimization problems. Invariance of the
energy under rigid motions leads to conservation of linear and angular
momenta, and invariance under scaling plays a role in setting the size of
complex parts of the intrinsic shapes (i.e., corners, wrinkles, folds). In
other words, the position, grey-level characteristics, size and orientation of
the pattern of interest have minimal effect on the extracted features as long
as the suitable patch is reserved for the analysis. In order to have simpler
and more intuitive representation of the given model, we simply set $\alpha=0$
and $\beta=\gamma=1$, and the equation turns into the Willmore energy
functional,
$\mathcal{S}_{w}=\int_{\Sigma}(H^{2}-K)dA=\int_{\Sigma}|H|^{2}dA-\int_{\partial\Sigma}|K|ds,$
(2)
where $H$ is the mean curvature vector on $\Sigma$, $K$ the Gaussian curvature
on $\partial\Sigma$, and $dA$, $ds$ the induced area and length metrics on
$\Sigma$, $\partial\Sigma$ (representing area and boundary, respectively).
Since homogeneity region that a typical TIB pattern appears is small in size,
total curvature (or energy) of that region is high and can be used as a
discriminative feature.
In addition to Willmore energy features, we have included seven different
local shape features in the proposed CAD system. Let $\kappa_{1}$ and
$\kappa_{2}$ indicate eigenvalues of the local Hessian matrix for any given
local patch, the following shape features are extracted: 1) mean curvature
($H$), 2) Gaussian curvature ($K$), 3) shape index ($SI$), 4) elongation
($\kappa_{1}/\kappa_{2}$), 5) shear ($(\kappa_{1}-\kappa_{2})^{2}/4$), 6)
compactness ($1/(\kappa_{1}\kappa_{2})$), and 7) distortion
($\kappa_{1}-\kappa_{2}$). Briefly, the $SI$ is a statistical measure used to
define local shape of the localized structure within the image [5].
Elongation indicates the flatness of the shape. Compactness feature measures
the similarity between shape of interest and a perfect ellipse. Fig. 3(c) and
(d) show mean and Gaussian curvature maps from which all the other local shape
features are extracted. Fig. 3(e) and (f) show Willmore energy map extracted
from Fig. 3(a).
Based on the observation in training, TIB patterns most likely occur in the
regions inside the lung with certain ranges (i.e, blue and yellow regions).
This observation facilitates one practically useful fact in the algorithm
that, in the feature extraction process, we only extract features if and only
if at least “one” b-scale pattern exists in the local region as well as
Willmore energy values of pixels lie in the interval observed from training.
Moreover, considering the Willmore energy has a role as hard control on
feature selection and computation, it is natural to investigate their ability
to segment images. We present a segmentation framework in which every voxel is
described by the proposed shape features. A multi-phase level set [6] is then
applied to the resulting vectorial image and the results are shown in Fig.
3(g). First and second columns of the Fig. 3(g) show segmented structures and
the output homogeneity maps showing segmented regions in different grey-level,
respectively. Although segmentation of small airway structures and
pathological patterns is not the particular aim of this study, the proposed
shape features show promising results due to their discriminative power.
g
Figure 3. a. CT lung, b. selected b-scale patterns, c. mean Curvature map
($H$), d. Gaussian Curvature ($K$), e. Willmore energy map, f. zoomed (e). g.
Multi-phase level set segmentation based on the proposed shape features is
shown in three different slices from the same patient’s chest CT scan.
Texture features: Spatial statistics based on Grey-Level Co-occurrence Matrix
(GLCM) [7] are shown to be useful in discriminating patterns pertaining to
lung diseases. As texture can give a lot of insights into the classification
and characterization problem of poorly defined lesions, regions, and objects,
we combine our proposed shape based invariants with GLCM based features. We
extract 18 GLCM features from each local patch including autocorrelation,
entropy, variance, homogeneity, and extended features of those. Apart from the
proposed method, we also compare our proposed method with well known texture
features: steerable wavelets (computed over 1 scale and 6 orientations with
derivative of Gaussian kernel), GLCM, combination of shape and steerable
wavelets, and considering different local patch size.
## 4\. Experimental Results
39 laboratory confirmed CTs of HPIV infection and 21 normal lung CTs were
collected for the experiments. The in-plane resolution is affected from
patients’ size and varying from 0.62mm to 0.82mm with slice thickness of 5mm.
An expert radiologist carefully examined the complete scan and labeled the
regions as normal and abnormal (with TIB patterns). As many regions as
possible showing abnormal lung tissue were labeled (see Table 1 for details of
the number of regions used in the experiments). After the proposed CAD system
is tested via two-fold cross validations with labeled dataset, we present
receiver operator characteristic (ROC) curves of the system performances.
Figure 4. Comparison of CAD performances via ROC curves of different feature sets. Table 1. Accuracy ($A_{z}$) of the CAD system with given feature sets. Features | Dimension | Patch Size | # of patches | # of patches | Area under
---|---|---|---|---|---
| | | (TIB) | (Normal) | ROC curve: $A_{z}$
Shape & GLCM | 8+18=26 | 17x17 | 14144 | 12032 | 0.8991
Shape & GLCM | 8+18=26 | 13x13 | 24184 | 20572 | 0.9038
Shape & GLCM | 8+18=26 | 9x9 | 50456 | 42924 | 0.9096
Shape | 8 | 17x17 | 14144 | 12032 | 0.7941
Shape | 8 | 13x13 | 24184 | 20572 | 0.7742
Shape | 8 | 9x9 | 50456 | 42924 | 0.7450
Steer. Wavelets& Shape | 6x17x17+8=1742 | 17x17 | 14144 | 12032 | 0.7846
Steer. Wavelets& Shape | 6x13x13+8=1022 | 13x13 | 24184 | 20572 | 0.7692
Steer. Wavelets& Shape | 6x9x9+8=494 | 9x9 | 50456 | 42924 | 0.7908
Steer. Wavelets | 6x17x17=1734 | 17x17 | 14144 | 12032 | 0.7571
Steer. Wavelets | 6x13x13=1014 | 13x13 | 24184 | 20572 | 0.7298
Steer. Wavelets | 6x9x9=486 | 9x9 | 50456 | 42924 | 0.7410
GLCM | 18 | 17x17 | 14144 | 12032 | 0.7163
GLCM | 18 | 13x13 | 24184 | 20572 | 0.7068
GLCM | 18 | 9x9 | 50456 | 42924 | 0.6810
Table 2. p-values are shown in confusion matrix. p-Confusion | Shape | Steer.& | Steer. | GLCM
---|---|---|---|---
Matrix | | Shape | |
Shape& | | | |
GLCM | 0.0171 | 0.0053 | 0.0056 | 0.0191
Shape | – | 0.0086 | 0.0094 | 0.0185
Steer.& | – | – | 0.0096 | 0.0175
Shape | | | |
Steer. | – | – | – | 0.0195
Table 1 summarizes the performance of the proposed CAD system as compared to
different feature sets. The performances are reported as the areas under the
ROC curves $(A_{z})$. Note that shape features alone are superior to other
methods even though the dimension of the shape feature is only 8. The best
performance is obtained when we combined shape and GLCM features. This is
expected because spatial statistics are incorporated into the shape features
such that texture and shape features are often complementary to each other. In
what follows, we select the best window size for each feature set and plot
their ROC curves all in Fig. 4. To have a valid comparison, we repeat
candidate selection step for all the methods, hence, the CAD performances of
compared feature sets might perhaps have lower accuracies if the candidate
selection part is not applied. Superiority of the proposed features is clear
in all cases. To show whether the proposed method is significantly different
than the other methods, we compared the performances through paired t-tests,
and the p-values of the tests are summarized in Table 2. Note that
statistically significant changes are emphasized by $p<.01$ and $p<.05$.
## 5\. Conclusion
In this paper, we have proposed a novel CAD system for automatic TIB pattern
detection from lung CTs. The proposed system integrates 1) fast localization
of candidate TIB patterns through b-scale filtering and scale selection, and
2) combined shape and textural features to identify TIB patterns. Our proposed
shape features illustrate the usefulness of the invariant features, Willmore
energy features in particular, to analyze TIB patterns in Chest CT. In this
paper, we have not addressed the issue of quantitative evaluation of severity
of diseases by expert observers. This is a challenging task for complex shape
patterns such as TIB opacities, and subject to further investigation.
## References
* [1] Eisenhuber, E.: _The tree-in-bud sign._ Radiology 222(3), 771–772 (2002)
* [2] Saha, P.K., Udupa, J.K., Odhner, D.: _Scale-based fuzzy connected image segmentation: Theory, algorithms, and validation._ Computer Vision Image Understanding 77, 145–174 (2000)
* [3] Hu, S., Hoffman, E.A., Reinhardt, J.M.: _Automatic lung segmentation for accurate quantification of volumetric X-ray CT images._ Transactions on Medical Imaging 20(6), 490–498 (2001)
* [4] Bobenko, E.I., Shroder, P.: _Discrete Willmore Flow._ In Eurographics Symposium on Geometry Processing, 101–110 (2005)
* [5] Lillholm, M., and Griffin, L.D.: _Statistics and category systems for the shape index descriptor of local 2nd order natural image structure._ Image and Vision Computing 27(6), 771–781 (2009)
* [6] Ayed, I.B., Mitiche, A., Belhadj, Z.: _Polarimetric image segmentation via maximum-likelihood approximation and efficient multiphase level-sets_. Transactions on Pattern Analysis and Machine Intelligence 28(9), 1493–1500 (2006)
* [7] Haralick,R.M., Shanmugam, K., Dinstein, I.: _Textural Features for Image Classification._ IEEE Transactions on Systems, Man, and Cybernetics 3(6), 610–-621 (1973)
|
arxiv-papers
| 2011-06-26T03:35:08 |
2024-09-04T02:49:20.075299
|
{
"license": "Public Domain",
"authors": "Ulas Bagci, Jianhua Yao, Jesus Caban, Anthony F. Suffredini, Tara N.\n Palmore, Daniel J. Mollura",
"submitter": "Ula\\c{s} Ba\\u{g}ci",
"url": "https://arxiv.org/abs/1106.5186"
}
|
1106.5230
|
# Opportunistic Power Control for Multi-Carrier Interference Channels
††thanks: Manuscript received April 18, 2011. This work was supported in part
by Tarbiat Modares University, and in part by Iran Telecommunications Research
Center , Tehran, Iran under PhD Research Grant 89-09-95.
Mohammad R. Javan and Ahmad R. Sharafat The authors are with the Department of
Electrical and Computer Engineering, Tarbiat Modares University, P. O. Box
14155-4838, Tehran, Iran. Corresponding author is A. R. Sharafat (email:
sharafat@modares.ac.ir).
###### Abstract
We propose a new method for opportunistic power control in multi-carrier
interference channels for delay-tolerant data services. In doing so, we
utilize a game theoretic framework with novel constraints, where each user
tries to maximize its utility in a distributed and opportunistic manner, while
satisfying the game’s constraints by adapting its transmit power to its
channel. In this scheme, users transmit with more power on good sub-channels
and do the opposite on bad sub-channels. In this way, in addition to the
allocated power on each sub-channel, the total power of all users also depends
on channel conditions. Since each user’s power level depends on power levels
of other users, the game belongs to the _generalized_ Nash equilibrium (GNE)
problems, which in general, is hard to analyze. We show that the proposed game
has a GNE, and derive the sufficient conditions for its uniqueness. Besides,
we propose a new pricing scheme for maximizing each user’s throughput in an
opportunistic manner under its total power constraint; and provide the
sufficient conditions for the algorithm’s convergence and its GNE’s
uniqueness. Simulations confirm that our proposed scheme yields a higher
throughput for each user and/or has a significantly improved efficiency as
compared to other existing opportunistic methods.
###### Index Terms:
Game theory, multi-carrier interference channel, generalized Nash equilibrium,
opportunistic power control, pricing.
## I Introduction
Frequency spectrum of a network can be divided into many orthogonal sub-
channels that can be shared by all users. However, shared usage of spectrum by
a user produces interference to other users. As the number of users increases,
users’ received interference levels increase as well, resulting in fewer users
that can achieve their required quality of service (QoS). In such instances,
efficient use of spectrum becomes more important, meaning that users should
transmit at their lowest possible power levels while satisfying their required
data rates.
Radio resource allocation in multi-carrier wireless networks aims to allocate
available resources to users in such a way that under some system and service
constraints, a performance measure for each user, e.g., the total throughput
or the total transmit power, is optimized. In distributed resource allocation,
each user chooses its strategy (required resources) independent of other
users. Game theory [1] is commonly used in the literature to analyze
distributed resource allocation algorithms by formulating the problem as a
non-cooperative game, where each user competes with other users with a view to
optimizing its own utility. The game settles at its Nash equilibrium (NE),
where no user can improve its utility by unilaterally changing its strategy.
In the context of distributed resource allocation, the problem of power
minimization under throughput constraint is considered in [2, 3]. In this
problem, each user tries to minimize its total consumed power over all sub-
channels while maintaining its total throughput above a predefined threshold.
In such a game, since the strategy space of each user depends on the chosen
strategies of other users, the game cannot be analyzed by way of the
conventional Nash equilibrium, and the so called _generalized Nash
equilibrium_ (GNE) should be used [4]. In [2], the game is simulated, and in
[3], the game is mathematically analyzed and the sufficient conditions for the
existence and uniqueness of GNE, as well as the sufficient conditions for the
convergence of the distributed algorithm are presented.
The counterpart of this problem in single carrier systems is known as the
target SINR-tracking power-control algorithm (TPC) [5]. In TPC, each user
tries to choose its transmit power in a distributed manner with a view to
satisfying its predefined target SINR. An important issue in TPC is its
feasibility, meaning that the algorithm converges only if a power vector
exists such that the target SINRs of all users are satisfied. In TPC, users
adapt their transmit power levels to their channel conditions, i.e., each user
increases its transmit power when its channel is bad, and does the reverse
when its channel is good. Likewise, minimizing the transmit power under the
data rate constraint may not be feasible, as there may not be a power vector
that satisfies the data rate constraints for all users, meaning that no GNE
may exist. Also, since users consume more power when their channels are bad,
they increase their interference to others, which in turn forces others to
increase their transmit power levels, and thus aggravating the situation
further for all users.
To avoid such a case for delay-tolerant services, it is worthwhile for users
to reduce their transmit power (which would result in lower data rates) when
their channels are bad, and do the reverse when their channels are good. This
is the opportunistic power control (OPC) that was introduced and analyzed in
[6] for single carrier systems. The alternative OPC for multi-carrier systems
is that users try to maximize their total throughput over all sub-channels in
a distributed manner when the total transmit power for each user is
constrained. This game has been extensively analyzed in [7, 8, 9, 10, 11], and
different conditions for the uniqueness of NE and for the convergence of the
distributed algorithms are obtained. The drawbacks of this scheme include
causing interference without any gain by a transmitting user when its channel
is bad, and convergence to an inefficient NE because of self-serving and
independent users.
In [7], the data rate maximization game is analyzed and the sufficient
conditions for NE’s uniqueness are obtained. In addition, by interpreting the
waterfilling solution as a projector and using its contraction property [12],
in [8, 9], the sufficient conditions for the convergence of the iterative
waterfilling solution are derived. Using linear complementarity [10]
reformulation of the Karush-Kuhn-Tucker (KKT) condition [13] of the rate
maximization game, in [14], the linear complementarity conditions are
converted to the affine variational inequality [15], and the sufficient
conditions for NE’s uniqueness and convergence of the iterative waterfilling
algorithm are obtained. In [11], the sufficient conditions for NE’s uniqueness
and convergence of the iterative waterfilling algorithm are established by
interpreting the waterfilling solution as a piecewise affine function [15].
In a non-cooperative game, since each user selfishly tries to optimize its own
utility, the equilibrium of the game may not be a desirable one. In such
cases, pricing is an effective mechanism to control users’ behaviors and
achieve a more efficient NE. The proposed pricing in [16, 17] for the rate
maximization game is a linear function of the transmit power. In [16], only
numerical results are presented that provide some insight on NE’s uniqueness
and convergence of the distributed algorithm, whereas in [17], a mathematical
analysis based on the notion of variational inequalities and non-linear
complementarity is presented to obtain the sufficient conditions for NE’s
existence and for convergence of the distributed algorithm.
In this paper, we propose an OPC for multi-carrier interference channels using
a game theoretic framework. In our scheme, each user opportunistically tries
to maximize its total transmit power over all sub-channels in a distributed
manner with a view to maximizing its data rate while satisfying the given
constraint on the total transmit power that depends on interference levels in
sub-channels. In doing so, higher transmit power levels are allocated to sub-
channels with low interference, and lower power levels to high interference
sub-channels. This is in contrast to the rate maximization game with total
power constraint irrespective of interference levels on sub-channels [2, 14].
For the proposed game, we analyze the existence of GNE. Similar to the power
minimization game with data rate constraint, the strategy space of a user in
our proposed scheme depends on the strategies chosen by other users. However,
we will show that at least one GNE is guaranteed to exist for the proposed
scheme. In addition, we obtain the sufficient conditions for GNE’s uniqueness,
and for the convergence of the distributed algorithm. We also introduce
pricing to the rate maximization game for controlling selfish users and
providing incentives for behaving in an opportunistic manner. In doing so,
pricing is a function of the user’s transmit power and the interference
experienced by that user. Furthermore, we obtain the sufficient conditions for
NE’s uniqueness, and for the the distributed algorithm’s convergence by
utilizing variational inequalities as in [14, 3, 8, 17]. By way of simulation,
we evaluate the performance of our proposed OPC, as well as that of the
pricing-based data rate maximization problem, and show that the total
throughput of users is increased as compared to when pricing is not applied.
This paper is organized as follows. A brief review of the TPC and the OPC
algorithms is provided in Section II. The opportunistic power control problem
is studied in Section III, and our pricing is introduced in Section IV.
Simulation results are presented in Section V, and conclusions are in Section
VI.
## II TPC and OPC in Single Carrier Systems
Consider a single carrier wireless network with $M$ active users that are
spread in its coverage area. Let the channel gain between the transmitter of
user $j$ and the receiver of user $i$ be $G_{i,j}$, and noise power at the
receiver of user $i$ be $\eta_{i}$. When user $i$ transmits at power level
$p_{i}$, its SINR, denoted by $\gamma_{i}$ is
$\gamma_{i}=\frac{{G_{i,i}}{p_{i}}}{\sum\limits_{j\neq
i}{{G_{i,j}}{p_{j}}}+\eta_{i}}.$ (1)
We denote the effective interference by
$I_{i}=\frac{\sum\limits_{j\neq i}{{G_{i,j}}{p_{j}}}+\eta_{i}}{G_{i,i}}.$ (2)
The value of $I_{i}$ depends on power levels of users (i.e.,
$I_{i}(\textbf{p})$), but for simplicity in notation, we use $I_{i}$.
Consider a case in which each user chooses its power level in a distributed
and iterative manner with a view to maintaining its SINR above a predefined
threshold, i.e., $\gamma_{i}\geq\hat{\gamma}_{i}$. In this case, a distributed
algorithm for achieving a given fixed target SINRs (TPC) as in [5] is
$p_{i}(n+1)=\frac{\hat{\gamma}_{i}}{\gamma_{i}(n)}p_{i}(n).$ (3)
When the system is feasible, i.e., if there exists a power vector
$\textbf{p}=[p_{1},\cdots,p_{M}]$ such that the SINR constraints are
satisfied, this algorithm converges to the solution of the following problem
$\displaystyle\text{min}\sum p_{i}$ $\displaystyle\text{subject
to:}~{}\gamma_{i}\geq\hat{\gamma}~{}\qquad\forall i,$
where all constraints are satisfied with the equality.
To maintain a predefined SINR, a user with a bad channel transmits at high
power and causes interference to other users with no apparent benefit to
itself. To increase the system throughput, for delay-tolerant services, it is
better that users with bad channels reduce their transmit power even to 0, and
users with good channels do the reverse, and both groups adapt their data
rates to their respective transmit power levels. This is the OPC algorithm,
defined by
$p_{i}(n+1)=\frac{\zeta_{i}}{I_{i}(n)},$ (5)
where $\zeta_{i}$ is a predefined constant and $I_{i}(n)$ is the effective
interference experienced by user $i$ in iteration $n$ as defined in (2). As
such, each user transmits at a rate given by
$R_{i}=\log(1+\gamma_{i}),$ (6)
where $\gamma_{i}$ is defined in (1). From (5), it is clear that when the
channel is bad, i.e., when $I_{i}$ is high, the user decreases its transmit
power, and when the channel is good, i.e., when $I_{i}$ is low, it does the
opposite. For a given transmit power level, the OPC’s throughput is higher
than that of TPC.
## III Opportunistic Power Control
### III-A Problem Formulation
We now formulate the opportunistic power control problem for multi-carrier
interference channels via a game theoretic framework. We begin by considering
a special power control problem that is the TPC’s counterpart in multi-carrier
systems. In this problem, each user tries to minimize its total transmit power
over all sub-channels in a distributed manner, while maintaining its total
data rate above a given threshold $\hat{R}_{i}$. This problem is stated by
$\displaystyle\min_{\textbf{p}_{i}\geq 0}~{}\sum\limits_{l}p_{i}^{l}$
$\displaystyle\text{subject
to:}~{}\sum\log(1+\frac{{G_{i,i}^{l}}{p_{i}^{l}}}{\sum\limits_{j\neq
i}{{G_{i,j}^{l}}{p_{j}^{l}}}+\eta_{i}^{l}})\geq\hat{R}_{i},$
where $p_{i}^{l}$ is the transmit power of user $i$ over sub-channel $l$, and
$G_{i,j}^{l}$ is the channel gain from the transmitter of user $i$ to the
receiver of user $j$ on sub-channel $l$.
In this game, each user tries to choose an optimal power vector
$\textbf{p}_{i}=[p_{i}^{1},\cdots,p_{i}^{L}]^{\text{T}}$, where $L$ is the
number of sub-channels that are utilized to satisfy the data rate constraint,
such that the data rate constraint in (III-A) is satisfied with the equality.
Similar to TPC, in this game, a user with a bad channel increases its transmit
power to satisfy its rate constraint, which can cause more interference to
other users, resulting in higher transmit power levels. To make this algorithm
opportunistic, we use a new constraint and reformulate the objective function.
The basic idea is similar to OPC, i.e., users increase their transmit power in
good sub-channels, and do the opposite in bad sub-channels, but in a more
profound manner. This will lead to a higher total throughput and a lower
transmit power. We first define the utility of each user $i$ as its total
power over all sub-channels. In opportunistic power control for multi-carrier
interference channels, each user chooses a strategy from its strategy space
that maximizes its utility. In other words, each user consumes more power to
achieve a higher data rate. As each user attempts to maximize its utility (its
total power consumed over all sub-channels), we impose a constraint on each
user’s transmit power, and provide it with incentives to behave
opportunistically. One choice for the constraint would be
$\sum\limits_{l}(p_{i}^{l}I_{i}^{l})\leq\hat{\varsigma}_{i},$ (8)
where $\hat{\varsigma}_{i}$ is a predefined upper bound for
$\sum\limits_{l}(p_{i}^{l}I_{i}^{l})$, and
$I_{i}^{l}=\frac{\sum\limits_{j\neq
i}{{G_{i,j}^{l}}{p_{j}^{l}}}+\eta_{i}^{l}}{G_{i,i}^{l}}=\sum\limits_{j\neq
i}\hat{G}_{i,j}^{l}p_{j}^{l}+\hat{\eta}_{i}^{l}.$ (9)
Note that, the value of $I_{i}^{l}$ depends on power levels of users (i.e.,
$I_{i}^{l}(\textbf{p})$, but for simplicity in notation, we use $I_{i}^{l}$.
Considering (8), one can observe that users would allocate more power on good
sub-channels. In addition, the total power consumed by users depends on sub-
channel conditions. With this constraint and the objective functions as the
total power over all sub-channels, each user needs to solve a linear program
in which its entire transmit power is assigned only to the best sub-channels,
which causes instability in the iterative algorithm. Moreover, when effective
interference levels are the same on some sub-channels, the problem would have
numerous solutions. This means that the convergence analysis of this problem
is very difficult, and convergence would be guaranteed in a very restrictive
set of channel conditions. Due to the above problems, instead of (8), we
define a new constraint for each user as
$\sum\limits_{l}(p_{i}^{l}I_{i}^{l})^{2}\leq\varsigma_{i},$ (10)
where $\varsigma_{i}$ is a predefined upper bound for
$\sum\limits_{l}(p_{i}^{l}I_{i}^{l})^{2}$, and $I_{i}^{l}$ is defined in (9).
Similar to (8), applying the constraint (10) would cause each user to transmit
at higher power levels on good sub-channels, and the total transmit power will
depend on the channel conditions of sub-channels. The strategy space of each
user is
$\mathcal{P}_{i}(\textbf{p}_{-i})=\\{\textbf{p}_{i}:p_{i}^{l}\geq
0,\sum\limits_{l}(p_{i}^{l}I_{i}^{l})^{2}\leq\varsigma_{i}\\},$ (11)
where $\textbf{p}_{-i}$ is the strategies of all users other than user $i$.
The problem is formulated by the game
$\mathcal{G}^{\text{o}}=\langle\mathcal{M},\mathcal{P}_{i}(\textbf{p}_{-i}),\\{u_{i}\\}\rangle$,
where $\mathcal{M}$ is the set of users in the system,
$\mathcal{P}_{i}(\textbf{p}_{-i})$ is the strategy space of user $i$ defined
in (11), and $u_{i}$ is the utility of user $i$. Each user aims to solve
$\displaystyle\max_{\textbf{p}_{i}\geq 0}~{}\sum\limits_{l}p_{i}^{l}$
$\displaystyle\text{subject
to:}~{}\textbf{p}_{i}\in\mathcal{P}_{i}(\textbf{p}_{-i}),$
where $\mathcal{P}_{i}(\textbf{p}_{-i})$ is defined in (11).
### III-B Game Analysis
In the opportunistic power control game, each user solves (III-A) in a
distributed manner. Since the strategy space of user $i$, i.e.,
$\mathcal{P}_{i}(\textbf{p}_{-i})$, depends on the strategies of other users,
this game belongs to the generalized Nash equilibrium (GNE) problems [4]. In
such games, in addition to the utility function, users’ interactions affect
their strategy choices. However, this dependence makes the problem hard to
analyze. In what follows, we present an analysis of (III-A). When users
iteratively solve (III-A), the game settles at a GNE as defined below.
Definition 1: The strategy vector
$\textbf{p}=[\textbf{p}_{1}^{\text{*T}},\cdots,\textbf{p}_{M}^{\text{*T}}]^{\text{T}}$
is a GNE for the game $\mathcal{G}^{\text{o}}$ if
$\sum\limits_{l}p_{i}^{l}\leq\sum\limits_{l}p_{i}^{l*},~{}~{}\qquad\forall\textbf{p}_{i}\in\mathcal{P}_{i}(\textbf{p}_{-i}^{*}),~{}\forall
i.$ (13)
When the strategies of other users are fixed, user $i$ solves the optimization
problem (III-A), which is a convex optimization. To do so, we consider its
Lagrangian given by
$L_{i}=\sum\limits_{l}p_{i}^{l}-\lambda_{i}(\sum\limits_{l}(p_{i}^{l}I_{i}^{l})^{2}-\varsigma_{i}).$
(14)
For a fixed but arbitrary non-negative $\textbf{p}_{-i}$, we take the
derivative of Lagrangian $L_{i}$ with respect to $p_{i}^{l}$, and write
$\frac{\partial L_{i}}{\partial
p_{i}^{l}}=1-2\lambda_{i}p_{i}^{l}(I_{i}^{l})^{2}=0.$ (15)
Note that at the optimal point, the constraint (10) is satisfied with the
equality. Hence, the solution to (III-A) is
$p_{i}^{l}=\frac{1}{2\lambda_{i}(I_{i}^{l})^{2}},$ (16)
where $\lambda_{i}$ is obtained such that (10) is satisfied with the equality.
Comparing (16) with (5), one can see their similarity. When the effective
interference experienced by user $i$ on sub-channel $l$ is high, i.e., when
the interference from other users is high and/or the direct channel gain from
the transmitter of user $i$ to its corresponding receiver is low, user $i$
consumes less power on that sub-channel, and when the sub-channel is good,
user $i$ increases its transmit power. The corresponding data rate on that
sub-channel is $R_{i}^{l}=\log{(1+\gamma_{i}^{l})}$, where $\gamma_{i}^{l}$ is
the SINR of user $i$ on sub-channel $l$. If all sub-channels for user $i$ are
bad, its total transmit power is reduced, which helps users with good sub-
channels to transmit at higher rates. This means that in our proposed scheme,
a user opportunistically benefits from the reduced transmit power levels of
other users who experience bad sub-channels.
In the game $\mathcal{G}^{\text{o}}$, each user updates its transmit power
over all sub-channels via (16) in a distributed and iterative manner. If the
power updates converge, they will converge to a GNE of the game. Since the
strategy of a user depends on other users’ strategies, we cannot use the
existing analysis in the literature that are developed for conventional games.
Instead, we use the following theorem to prove that the proposed game always
has at least one GNE.
Theorem 1 [4]: Let
$\mathcal{G}^{\text{GNEP}}=\langle\mathcal{M},\mathcal{S}_{i}(\textbf{s}_{-i}),\\{u_{i}\\}\rangle$
be given, where $\mathcal{M}$ is the set of players,
$\mathcal{S}_{i}(\textbf{s}_{-i})$ is the strategy set of player $i$ that
depends on the strategies of other users, i.e., on $\textbf{s}_{-i}$, and
$u_{i}$ is the utility function of user $i$. Suppose that
* a)
There exist $M$ nonempty, convex and compact sets
$\mathcal{K}_{i}\subset\mathbb{R}^{L}$ such that for every
$\textbf{s}\in\mathbb{R}^{ML}$ with $\textbf{s}_{i}\in\mathcal{K}_{i}$ for
every $i$, the set $\mathcal{S}_{i}(\textbf{s}_{-i})$ is nonempty, closed, and
convex, $\mathcal{S}_{i}(\textbf{s}_{-i})\subseteq\mathcal{K}_{i}$, and
$\mathcal{S}_{i}$, as a point-to-set map, is both upper and lower semi-
continuous;
* b)
For every player $i$, the function $u_{i}(\cdot,\textbf{s}_{-i})$ is quasi-
concave on $\mathcal{S}_{i}(\textbf{s}_{-i})$, which is required in our case,
as each user tries to maximize its own utility.
Then a GNE exists for $\mathcal{G}^{\text{GNEP}}$.
From Theorem 1, we have the following theorem for the existence of GNE in
$\mathcal{G}^{\text{o}}$.
Theorem 2: The game $\mathcal{G}^{\text{o}}$ always admits at least one GNE.
Proof: Consider the set $\mathcal{P}_{i}(\textbf{p}_{-i})$ as defined in (11).
Since each $\hat{\eta}_{i}^{l}$ in (9) is positive, we define the set
$\mathcal{K}_{i}=\\{\textbf{p}_{i}:0\leq p_{i}^{l}\leq\kappa_{i}^{l}\\}$,
where $\kappa_{i}^{l}=\frac{\sqrt{\varsigma_{i}}}{\hat{\eta}_{i}^{l}}$. One
can easily see that the assumptions of Theorem 1 are satisfied, i.e., the game
$\mathcal{G}^{\text{o}}$ always has at least one GNE. $\blacksquare$
Although Theorem 2 states that a GNE always exists for
$\mathcal{G}^{\text{o}}$, it nevertheless may not be unique. However, when GNE
is not unique, the distributed algorithm may not converge to a GNE, and may
toggle between two GNEs. Below, in Theorem 3 we provide the sufficient
conditions for GNE’s uniqueness.
Theorem 3: The GNE in the game $\mathcal{G}^{\text{o}}$ is unique if matrix A
defined as
$[\textbf{A}]_{i,j}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{\varsigma_{i}}}\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})&\mbox{\text{if}
$i=j$},\\\
-3\sqrt{\varsigma_{i}}\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}&\mbox{\text{if}
$i\neq j$},\end{array}\right.$ (17)
is a P-matrix111A matrix $\textbf{A}\in\mathbb{R}^{n\times n}$ is called a
P-matrix if all of its principal minors are positive [10]., where
$\underline{q}_{i}^{l}=(\underline{p}_{i}^{l}\underline{I}_{i}^{l})^{2}-\sigma$,
$\sigma$ is a small positive constant,
$\underline{I}_{i}^{l}=\hat{\eta}_{i}^{l}$, and $\underline{p}_{i}^{l}$ is the
minimum power level of user $i$ on sub-channel $l$.
Proof: See Appendix A. $\blacksquare$
We also provide another sufficient condition for GNE’s uniqueness in Theorem 4
below.
Theorem 4: The GNE in the game $\mathcal{G}^{\text{o}}$ is unique if
$\rho(\textbf{B})<1,$ (18)
where $\rho(\textbf{B})$ is the spectral radius222The spectral radius of
matrix B is its maximum absolute eigenvalue. of B, and
$[\textbf{B}]_{i,j}=\left\\{\begin{array}[]{ll}~{}0&\mbox{\text{if} $i=j$},\\\
3~{}{\varsigma_{i}}\frac{\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}}{\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})}&\mbox{\text{if}
$i\neq j$},\end{array}\right.$ (19)
Proof: See Appendix B. $\blacksquare$
When users in the network solve (III-A), each user chooses its transmit power
iteratively in a distributed manner, and simultaneously with other users
according to (16), i.e., $p_{i}^{l}(n+1)=P(\textbf{p}(n))$, where
$P(\cdot)=\frac{1}{2\lambda_{i}(I_{i}^{l})^{2}}$ and $\textbf{p}(n)$ is the
transmit power levels of users at iteration $n$. In the following theorem, we
provide conditions for convergence of this distributed algorithm.
Theorem 5: The distributed iterative power update function converges to the
unique GNE of the game under the same condition as in Theorem 3.
Proof: See Appendix C. $\blacksquare$
As stated earlier, global convergence of the distributed algorithm is
guaranteed if its GNE is unique. Therefore, it is not surprising that the
conditions for convergence of the algorithm are the same as those of
uniqueness of its GNE.
## IV Pricing for Opportunistic Power Control
In game theoretic distributed schemes, each user selfishly chooses a strategy
for optimizing its utility. However, this may cause unacceptable consequences
for other users, but can be controlled via pricing in the game. Pricing is set
in such a way to attain certain desirable characteristics, such as introducing
opportunistic behavior in our case for rate maximization under total power
constraint that depends on interference levels in sub-channels. We denote the
game with no pricing by
$\mathcal{G}^{\text{r}}=\langle\mathcal{M},\mathcal{P}_{i},\\{u_{i}\\}\rangle$,
where $\mathcal{M}$ is the set of users, $\mathcal{P}_{i}$ is the strategy
space of user $i$ defined by
$\mathcal{P}_{i}=\\{\textbf{p}_{i}:\textbf{p}_{i}\geq\textbf{0},\sum\limits_{l}p_{i}^{l}<P_{i}\\}$,
and $u_{i}$ is the total throughput of user $i$ over all sub-channels. Since
the strategy space of each user is independent of other users, this is the
conventional game where each user aims to solve
$\displaystyle\hskip
34.14322pt\max_{\textbf{p}_{i}\geq\textbf{0}}~{}\sum\limits_{l}\log(1+\frac{G_{i,i}^{l}p_{i}^{l}}{\sum\limits_{j\neq
i}{{G_{i,j}^{l}}{p_{j}^{l}}}+\eta_{i}^{l}})$ $\displaystyle\text{Subject
to:}~{}~{}\sum\limits_{l}p_{i}^{l}\leq P_{i}.$
The solution to (IV) is the well known waterfilling, given by
$p_{i}^{l}=\left[\mu_{i}-\frac{\sum\limits_{j\neq
i}{{G_{i,j}^{l}}{p_{j}^{l}}}+\eta_{i}^{l}}{G_{i,i}^{l}}\right]^{+}\forall l,$
(21)
where $[\cdot]=\max(\cdot,0)$, and $\mu_{i}$ is chosen such that the
constraint in (IV) is satisfied with the equality.
The game $\mathcal{G}^{\text{r}}$ has been extensively studied in the
literature, and conditions for the uniqueness of its NE and for the
convergence of the distributed algorithm are provided in [14, 7, 8, 9].
However, as stated earlier, due to the distributed nature of optimization and
selfish behavior of users, the output of the game may not be a desirable one.
Therefore, if users’ behavior is controlled, it may be possible to achieve
certain improvements in utilizing resources such as higher throughputs and
lower power levels. To this end, we propose a pricing mechanism that takes
into account the transmit power levels of users as well as the interference
they experience. The proposed pricing is defined by
$C(\textbf{p})=\lambda_{i}\sum\limits_{l}p_{i}^{l}I_{i}^{l},$ (22)
where $I_{i}^{l}$ is the effective interference experienced by user $i$ on
sub-channel $l$ as defined in (9), and $\lambda_{i}$ is the pricing for user
$i$. When pricing (22) is applied, each user is priced more when its transmit
power and/or its effective received interference are increased. Thus, users
would allocate more power on sub-channels whose interference levels are low.
When pricing (22) is applied to the data rate maximization problem, each user
in the game
$\mathcal{G}^{\text{p}}=\langle\mathcal{M},\mathcal{P}_{i},\\{u_{i}\\}\rangle$
aims to solve
$\displaystyle\hskip
34.14322pt\max_{\textbf{p}_{i}\geq\textbf{0}}~{}\sum\limits_{l}\log(1+\frac{p_{i}^{l}}{\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}{p_{j}^{l}}}+\hat{\eta}_{i}^{l}})-\lambda_{i}\sum\limits_{l}p_{i}^{l}I_{i}^{l}$
$\displaystyle\text{Subject to:}~{}~{}\sum\limits_{l}p_{i}^{l}\leq P_{i}.$
The solution to (IV), and the existence of NE for the game
$\mathcal{G}^{\text{p}}$ are provided in the following theorem.
Theorem 6: The game $\mathcal{G}^{\text{p}}$ always admits at least one NE.
Moreover, each user chooses its transmit power over each sub-channel according
to
$p_{i}^{l}=\left[\frac{1}{\mu_{i}+\lambda_{i}I_{i}^{l}}-I_{i}^{l}\right]^{+},$
(24)
where $I_{i}^{l}$ is defined in (9), and $\mu_{i}$ is so chosen to satisfy the
constraint (IV).
Proof: The strategy space of users are non-empty, compact and convex subset of
$L$-dimensional Euclidian space. The utility function of each user is a
continuous function of the power vector p, and is quasi-concave function of
users’ power levels $\textbf{p}_{i}$. Therefore, the game
$\mathcal{G}^{\text{p}}$ always has at least one NE. The solution to the
optimization problem (IV) which is (24) can be readily obtained using the KKT
conditions for (IV). $\blacksquare$
In the game $\mathcal{G}^{\text{p}}$, each user updates its transmit power in
a distributed and iterative manner by using (24), where the water level
$\frac{1}{\mu_{i}+\lambda_{i}I_{i}^{l}}$ depends on the interference that
users receive in their sub-channels. Since $\mu_{i}$ has the same value for
all sub-channels of user $i$, it is clear that the water levels in those sub-
channels whose interference is higher than those of others is lower, meaning
that each user assigns a smaller power level or no power to those sub-
channels. The following theorem provides the conditions for NE’s uniqueness.
Theorem 7: The NE of $\mathcal{G}^{p}$ is unique if the matrix D defined below
is a P-matrix.
$[\textbf{D}]_{i,j}=\left\\{\begin{array}[]{ll}~{}1&\mbox{\text{if} $i=j$},\\\
-\max_{l}\left(\frac{{\hat{G}_{i,j}^{l}}\left(1+\lambda_{i}{\overline{\psi}_{i}^{l}}^{2}\right){\overline{\psi}_{j}^{l}}}{{\underline{\psi}_{i}^{l}}}\right)&\mbox{\text{if}
$i\neq j$},\end{array}\right.$ (25)
where $\underline{\psi}_{i}^{l}=\hat{\eta}_{i}^{l}$ and
$\overline{\psi}_{i}^{l}=\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{P_{j}}}+\hat{\eta}_{i}^{l}$
and $P_{j}$ is defined in (IV).
Proof: See Appendix D. $\blacksquare$
Users in $\mathcal{G}^{\text{p}}$ update their power levels in a distributed
and iterative manner according to (24). In the following theorem, we provide
the sufficient condition for convergence of the distributed algorithm.
Theorem 8: Suppose that the matrix D in (25) is a P-matrix. When users update
their power levels simultaneously according to (24), the distributed algorithm
converges to the unique NE of the game.
Proof: See Appendix E. $\blacksquare$
## V Simulation Results
We now present simulation results for the proposed opportunistic power control
game as well as the pricing mechanism. The system under study is the uplink of
a multi-carrier network consisting of one base station and 5 users.
For the opportunistic power control, the system has 20 sub-channels. The
interfering channel gain from the transmitter of user $j$ to the receiver of
user $i$ on sub-channel $l$ is chosen randomly from $(0,\frac{0.1}{i})$, where
$i$ denotes the user number, and the normalized noise power is set to 0.01
Watts. Accordingly, the channel conditions become better from user 1 to user
5. For an instance of the network realization, we run the proposed algorithm
and show the results in Fig. 1. Note that user 1 whose sub-channels are bad
consumes less power and achieves a low data rate, and user 5 with the best
sub-channels transmits with high power and achieves a high data rate.
Next, we fix the channel conditions of all users except user 5 for which in
four steps, we gradually deteriorate its channel conditions. The results are
shown in Fig. 2. As expected, user 5 decreases its transmit power levels, and
as the conditions for other users improve, they increase their transmit power
levels.
As stated earlier, the behavior of the proposed algorithm is in the opposite
direction of tracking a target data rate as in (III-A). We repeat our
simulations and compare the results to those of (III-A) in Fig. 3. As can be
seen, in (III-A), as the channel condition of user 5 deteriorates, its
transmit power is increased to achieve its target data rate. This is in
contrast to our proposed algorithm, where this user decreases its transmit
power to reduce its interference to other users. Note that, in our proposed
opportunistic scheme, at step 4, there is a 37 percent reduction in the
transmit power as compared to that of the power minimization game (III-A), but
the total data rate is reduced by 9 percent, which shows the efficiency of our
proposed scheme. In addition, using (III-A) may cause the system to become
infeasible, whereas this would not happen in our proposed game.
In the sequel, we present simulation results for the proposed pricing. The
network setup is similar to the previous simulations except that now we have
10 sub-channels. First we show the effect of pricing on the users’ transmit
power levels in Fig. 4. Note that pricing affects those users with bad
channels more than other users, and forces such users to reduce their transmit
power levels more than those users with good channels, as we desired.
In the next two simulations, we compare the effects of our proposed pricing
with those of fixed pricing used in [17]. We first run the algorithm with our
proposed pricing, and after convergence, use the multiplication of pricing and
the effective interference over each sub-channel as the fixed pricing in [17].
We then use the fixed pricing for subsequent steps in which the channel for
user 5 gradually deteriorates. The power levels are shown in Fig. 5, and data
rates in Fig. 6. Note that, at step 6 for our proposed pricing, the total data
rates of users is about 3 percent less than that of the fixed pricing, but the
total transmit power levels is about 15 percent less than that of the fixed
pricing.
Next simulations reverse the previous one, meaning that the channels for user
5 are gradually improved at successive steps. The power levels are shown in
Fig. 7, and data rates in Fig. 8. Note that at step 6, the total data rates of
users as well as the total transmit power levels are about 10 percent higher
as compared to those of the fixed pricing.
## VI Conclusions
We proposed an opportunistic power control for multi-carrier systems, in which
each sub-channel is shared among all users. In such a power control framework,
each user transmits at lower power levels on bad sub-channels, and does the
opposite on good sub-channels. We showed that in the proposed game there
always exists a generalized Nash equilibrium, and provided the sufficient
conditions for GNE’s uniqueness and for convergence of the distributed
algorithm. Furthermore, we proposed a pricing mechanism for the data rate
maximization problem when the total transmit power of each user is constrained
depending on the interference levels on sub-channels. In such cases, we also
provided the sufficient conditions for GNE’s uniqueness, and for convergence
of the distributed algorithm. By way of simulations, we demonstrated the
improved performances of our proposed schemes as compared to those of existing
algorithms.
## Appendix A Proof of Theorem 3
We utilize variational inequalities to prove the uniqueness of NE for the game
(III-A). To do so, we use the following definition for variational
inequalities.
Definition 2 [15]: The variational inequality denoted by
$VI(\mathcal{K},\textbf{F})$, where $\mathcal{K}$ is a subset of
$\mathbb{R}^{n}$ and $\textbf{F}:\mathcal{K}\rightarrow\mathbb{R}^{n}$, is to
find a vector $\textbf{x}\in\mathcal{K}$ such that
$(\textbf{y}-\textbf{x})^{\text{T}}\textbf{F}(\textbf{x})\geq 0$ for all
$\textbf{y}\in\mathcal{K}$.
For the opportunistic power control game (III-A), the strategy space of each
user depends on the strategies chosen by other users. Therefore, we cannot
directly apply the variational inequality formulation to the game, and some
reformulations are needed. Let $\mu_{i}^{l}$ be the multiplier corresponding
to the nonnegativity constraint and $\lambda_{i}$ the multiplier of the power
constraint (10). The KKT conditions of the optimization problem (III-A) are
$\displaystyle\hskip
85.35826pt-1+2\lambda_{i}p_{i}^{l}(I_{i}^{l})^{2}-\mu_{i}^{l}=0,~{}~{}\forall
l,i,$ (27) $\displaystyle\mu_{i}^{l}\geq 0~{}\bot~{}p_{i}^{l}\geq
0,~{}~{}\forall l,i,$ $\displaystyle\lambda_{i}\geq
0~{}\bot~{}\sum\limits_{l}({p_{i}^{l}I_{i}^{l}})^{2}\leq\varsigma_{i},~{}~{}\forall
i,$ (28)
where $\textbf{a}\bot\textbf{b}$ means that vectors a and b are perpendicular.
Note that $\lambda_{i}>0$, otherwise the condition (27) will lead to
$\mu_{i}^{l}<0$, which contradicts (27). This means that the constraint
$\sum\limits_{l}{p_{i}^{l}I_{i}^{l}}^{2}\leq\varsigma_{i}$ is satisfied with
equality. By eliminating the multipliers $\mu_{i}^{l}$, the KKT conditions can
be reformulated as a nonlinear complementarity problem
$\displaystyle\hskip 85.35826ptp_{i}^{l}\geq
0~{}\bot~{}-1+2\lambda_{i}p_{i}^{l}(I_{i}^{l})^{2}\geq 0,~{}~{}\forall l,i,$
(30) $\displaystyle\lambda_{i}\geq
0~{}\bot~{}\sum\limits_{l}({p_{i}^{l}I_{i}^{l}})^{2}-\varsigma_{i}=0,~{}~{}\forall
i.$
We define the variable $q_{i}^{l}$ by
$q_{i}^{l}=({p_{i}^{l}I_{i}^{l}})^{2},$ (31)
and use it to reformulate (10) as
$\sum\limits_{l}q_{i}^{l}=\varsigma_{i}.$ (32)
Note that $q_{i}^{l}=0$ if and only if $p_{i}^{l}=0$. On the other hand, for
each value of $\mathbf{q}^{l}=[q_{i}^{l}]_{i=1}^{M}$, the corresponding values
of $p_{i}^{l}$ can be obtained using the OPC algorithm. Hence, we write
$p_{i}^{l}=\theta_{i}^{l}(\mathbf{q}^{l})$. Since $\theta_{i}^{l}(\cdot)$ is a
continuous function of $\mathbf{q}^{l}$ [6], and considering $p_{i}^{l}$ and
$I_{i}^{l}$ as functions of $\mathbf{q}^{l}$, we reformulate the conditions
(30) and (30) by
$\displaystyle\hskip 85.35826ptq_{i}^{l}\geq
0~{}\bot~{}-1+2\lambda_{i}p_{i}^{l}(I_{i}^{l})^{2}\geq 0,~{}~{}\forall l,i,$
(34) $\displaystyle\lambda_{i}\geq
0~{}\bot~{}\sum\limits_{l}q_{i}^{l}=\varsigma_{i},~{}~{}\forall i,.$
Note that the variable $q_{i}^{l}$ is nonnegative, i.e., $q_{i}^{l}>0$. One
can see from (16) that $p_{i}^{l}$ is always positive, i.e., $p_{i}^{l}>0$,
and since $I_{i}^{l}>0$, we have $q_{i}^{l}>0$. The maximum power level for
each user on each sub-channel is
$\overline{p}_{i}^{l}=\frac{\sqrt{\varsigma_{i}}}{\hat{\eta}_{i}^{l}}$.
Now suppose that all users except user $i$ transmit at their maximum power
levels only on sub-channel $l$. The minimum power level of user $i$ on sub-
channel $l$, denoted by $\underline{p}_{i}^{l}$, is obtained from (16).
Therefore, $q_{i}^{l}>\underline{q}_{i}^{l}$, where
$\underline{q}_{i}^{l}=(\underline{p}_{i}^{l}\underline{I}_{i}^{l})^{2}-\sigma$
and $\sigma$ is a small positive constant, and
$\underline{I}_{i}^{l}=\hat{\eta}_{i}^{l}$. From the above, we change the
conditions (34) and (34) as
$\displaystyle\hskip 85.35826ptq_{i}^{l}-\underline{q}_{i}^{l}\geq
0~{}\bot~{}-1+2\lambda_{i}p_{i}^{l}(I_{i}^{l})^{2}\geq 0,~{}~{}\forall l,i,$
(36) $\displaystyle\lambda_{i}\geq
0~{}\bot~{}\sum\limits_{l}q_{i}^{l}=\varsigma_{i},~{}~{}\forall i.$
The conditions in (36) and (36) are not equivalent to (34) and (34). However,
all solutions to (34) and (34) are also solutions to (36) and (36). The change
in (34) may yield additional solutions to (36) and (36). Thus, solutions to
(36) and (36) consist of all solutions to (34) and (34) plus possible other
solutions. Note that $\lambda_{i}>0$. We further reformulate (36) and (36)
into a more suitable form as
$\displaystyle\hskip 85.35826ptq_{i}^{l}-\underline{q}_{i}^{l}\geq
0~{}\bot~{}\xi_{i}-\log(p_{i}^{l})+\log(q_{i}^{l})\geq 0,~{}~{}\forall l,i,$
(38)
$\displaystyle\xi_{i}\in\mathbb{R},~{}~{}\sum\limits_{l}q_{i}^{l}=\varsigma_{i},~{}~{}\forall
i.$
We use log transform because we wish to eliminate the multiplication of the
power $p_{i}^{l}$ and the effective interference $I_{i}^{l}$. It is obvious
that (38) and (38) are the KKT conditions of the variational inequality
$VI(\mathcal{X},\textbf{F})$ where
$\mathcal{X}=\prod\limits_{i}\mathcal{X}_{i}$ and
$\mathcal{X}_{i}=\\{\mathbf{q}_{i}\in\mathbb{R}^{L}:q_{i}^{l}\geq\underline{q}_{i}^{l},\sum\limits_{l}q_{i}^{l}=\varsigma_{i}\\}$,
and $F_{i}^{l}=\log(q_{i}^{l})-\log(p_{i}^{l})$. With this modification, we
now provide conditions for GNE’s uniqueness. Note that all GNEs of the game
are solutions to $VI(\mathcal{X},\textbf{F})$. However, the variational
inequality may have additional solutions. Hence, the conditions for uniqueness
of the solution to $VI(\mathcal{X},\textbf{F})$ guarantee GNE’s uniqueness as
well. This condition, however, may be excessive, since solutions to
$VI(\mathcal{X},\textbf{F})$ may not be the GNE of the game.
Let $\widehat{\textbf{q}}=\mathbf{q}(1)$ and
$\widetilde{\textbf{q}}=\mathbf{q}(2)$ be two solutions to
$VI(\mathcal{X},\textbf{F})$. This means that for each user $i$, we have
$\sum\limits_{l}(q_{i}^{l}(2)-q_{i}^{l}(1))\left(-\log({p}_{i}^{l}(1))+\log(q_{i}^{l}(1))\right)\geq
0,$ (39)
$\sum\limits_{l}(q_{i}^{l}(1)-q_{i}^{l}(2))\left(-\log({p}_{i}^{l}(2))+\log(q_{i}^{l}(2))\right)\geq
0.$ (40)
In addition, from the definition of $q_{i}^{l}$, we have
$\log(q_{i}^{l}(2))-\log(q_{i}^{l}(1))=2\left(\log({p}_{i}^{l}(2))-\log({p}_{i}^{l}(1))+\log({I}_{i}^{l}(2))-\log({I}_{i}^{l}(1))\right).$
(41)
We add (39) and (40), and write
$\sum\limits_{l}(q_{i}^{l}(1)-q_{i}^{l}(2))\left(\log({p}_{i}^{l}(2))-\log({p}_{i}^{l}(1))+2\log({I}_{i}^{l}(2))-2\log({I}_{i}^{l}(1))\right)\geq
0.$ (42)
From the mean value theorem, we know that there exists a $q_{i}^{l}(1)\leq
q_{i}^{l}\leq q_{i}^{l}(2)$ such that
$q_{i}^{l}(1)-q_{i}^{l}(2)=q_{i}^{l}\left(\log(q_{i}^{l}(1))-\log(q_{i}^{l}(2))\right).$
(43)
Therefore, using (41) and (43), (42) can be written as
$\displaystyle\hskip
28.45274pt\sum\limits_{l}q_{i}^{l}\left(\log({p}_{i}^{l}(1))-\log({p}_{i}^{l}(2))+\log({I}_{i}^{l}(1))-\log({I}_{i}^{l}(2)\right)$
(44)
$\displaystyle\left(\log({p}_{i}^{l}(2))-\log({p}_{i}^{l}(1))+2\log({I}_{i}^{l}(2))-2\log({I}_{i}^{l}(1))\right)\geq
0.$
Rearranging (44) and using Schwarz’s inequality, we get
$\sqrt{\sum\limits_{l}q_{i}^{l}\left(\log({p}_{i}^{l}(1))-\log({p}_{i}^{l}(2))\right)^{2}}\leq
3\sqrt{\sum\limits_{l}q_{i}^{l}\left(\log({I}_{i}^{l}(1))-\log({I}_{i}^{l}(2))\right)^{2}},$
(45)
or equivalently
$\|\sqrt{\mathbf{q}}(\log({\textbf{p}}(1))-\log({\textbf{p}}(2)))\|\leq
3\|\sqrt{\mathbf{q}}(\log({\textbf{I}}(1))-\log({\textbf{I}}(2)))\|,$ (46)
where $\|\cdot\|$ denotes the Euclidian norm. Again, we use the mean value
theorem for $\log({p}_{i}^{l})$ and $\log({I}_{i}^{l})$, and write
$\sqrt{\sum\limits_{l}q_{i}^{l}(\frac{{p}_{i}^{l}(1)-{p}_{i}^{l}(2)}{p_{i}^{l}})^{2}}\leq
3\sqrt{\sum\limits_{l}q_{i}^{l}(\frac{{I}_{i}^{l}(1)-{I}_{i}^{l}(2)}{I_{i}^{l}})^{2}},$
(47)
and
$\sqrt{\sum\limits_{l}q_{i}^{l}(\frac{{p}_{i}^{l}(1)-{p}_{i}^{l}(2)}{p_{i}^{l}})^{2}}\leq
3\sqrt{\sum\limits_{l}q_{i}^{l}(\frac{1}{I_{i}^{l}}\sum\limits_{j\neq
i}\hat{G}_{i,j}^{l}({p}_{j}^{l}(1)-{p}_{j}^{l}(2)))^{2}}.$ (48)
Note that $p_{i}^{l}\leq\frac{\sqrt{\varsigma_{i}}}{\hat{\eta}_{i}^{l}}$,
$I_{i}^{l}\geq\hat{\eta}_{i}^{l}$, and $\underline{q}_{i}^{l}\leq
q_{i}^{l}\leq\varsigma_{i}$. From (49), we obtain
$\frac{1}{\sqrt{\varsigma_{i}}}\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})\sqrt{\sum\limits_{l}({p}_{i}^{l}(1)-{p}_{i}^{l}(2))^{2}}\leq
3\sum\limits_{j\neq
i}\sqrt{\varsigma_{i}}\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}\sqrt{\sum\limits_{l}({p}_{j}^{l}(1)-{p}_{j}^{l}(2))^{2}}.$
(49)
Defining
$[\textbf{a}]_{i}=\sqrt{\sum\limits_{l}(p_{i}^{l}(1)-p_{i}^{l}(2))^{2}}$ and
considering the matrix A defined in (17), we obtain
$\textbf{A}\textbf{a}\leq\textbf{0}$. Therefore, if the matrix A is a
P-matrix, we have $\textbf{a}=\textbf{0}$, and hence the proof.
## Appendix B Proof of Theorem 4
From (49), one obtains
$\sqrt{\sum\limits_{l}({p}_{i}^{l}(1)-{p}_{i}^{l}(2))^{2}}\leq
3\sum\limits_{j\neq
i}\frac{\sqrt{\varsigma_{i}}\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}}{\frac{1}{\sqrt{\varsigma_{i}}}\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})}\sqrt{(\sum\limits_{l}({p}_{j}^{l}(1)-{p}_{j}^{l}(2)))^{2}},$
(50)
If the matrix
$[\widehat{\textbf{{B}}}]_{i,j}=\left\\{\begin{array}[]{ll}~{}1&\mbox{\text{if}
$i=j$},\\\
-3\frac{\sqrt{\varsigma_{i}}\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}}{\frac{1}{\sqrt{\varsigma_{i}}}\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})}&\mbox{\text{if}
$i\neq j$},\end{array}\right.$ (51)
is a P-matrix, GNE is unique. From the P-property of the matrix, this is
equivalent to the spectral condition in (18).
## Appendix C Proof of Theorem 5
Note that at each iteration, say $n+1$, the parameters $q_{i}^{l}$ and power
vectors are related by
$q_{i}^{l}(n+1)=(p_{i}^{l}(n+1)I_{i}^{l}(\textbf{p}(n))^{2}.$ (52)
Since they are the solutions of the game (III-A), we have
$\displaystyle\hskip 85.35826ptq_{i}^{l}(n+1)-\underline{q}_{i}^{l}\geq
0,~{}~{}\xi_{i}^{n+1}-\log(p_{i}^{l}(n+1))+\log(q_{i}^{l}(n+1))\geq 0,$ (54)
$\displaystyle\xi_{i}^{n+1}\in\mathbb{R},~{}~{}\sum\limits_{l}q_{i}^{l}(n+1)=\varsigma_{i}.$
Therefore, with ${p}_{i}^{l}(1)=p_{i}^{l}(n+1)$ and
${p}_{i}^{l}(2)=p_{i}^{l*}$, where $p_{i}^{l*}$ is the GNE of the game, one
can follow the same line as in proof of Theorem 3 to obtain the following
inequality
$\frac{1}{\sqrt{\varsigma_{i}}}\min_{l}(\sqrt{\underline{q}_{i}^{l}}\hat{\eta}_{i}^{l})\sqrt{\sum\limits_{l}(p_{i}^{l}(n+1)-p_{i}^{l*})^{2}}\leq
3\sum\limits_{j\neq
i}\sqrt{\varsigma_{i}}\max_{l}\frac{\hat{G}_{i,j}^{l}}{\hat{\eta}_{i}^{l}}\sqrt{(\sum\limits_{l}(p_{j}^{l}(n+1)-p_{j}^{l*}))^{2}}.$
(55)
Using (55) and the P-property of matrix A defined in (17), one can easily
derive the condition.
## Appendix D Proof of Theorem 7
Let $\mu_{i}$ be the multiplier corresponding to the power constraint (IV).
The KKT conditions for the optimization problem (IV) can be reformulated as
the following complementarity problem
$\displaystyle\hskip 34.14322ptp_{i}^{l}\geq
0~{}\bot~{}-(\frac{1}{\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{p_{j}^{l}}}+\hat{\eta}_{i}^{l}})+\lambda_{i}I_{i}^{l}+\mu_{i}\geq
0,~{}~{}\forall l,i,$ (57) $\displaystyle\mu_{i}\geq
0~{}\bot~{}\sum\limits_{l}p_{i}^{l}\leq P_{i},~{}~{}\forall i.$
These are the KKT conditions for $VI(\mathcal{X},\textbf{F})$, where
$\mathcal{X}=\prod\limits_{i}\mathcal{X}_{i}$ and
$\mathcal{X}_{i}=\\{\textbf{p}_{i}\in\mathbb{R}^{L}:p_{i}^{l}\geq
0,\sum\limits_{l}p_{i}^{l}\leq P_{i}\\}$, and
$F_{i}^{l}(\textbf{p})=-(\frac{1}{\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{p_{j}^{l}}}+\hat{\eta}_{i}^{l}})+\lambda_{i}I_{i}^{l}.$
(58)
Since each set $\mathcal{X}_{i}$ is closed and convex, and $F_{i}^{l}(\cdot)$
is continuous, $VI(\mathcal{X},\textbf{F})$ has a solution. Since the set
$\mathcal{X}$ is a Cartesian product of some independent closed and convex
sets, it is known [15] that $VI(\mathcal{X},\textbf{F})$ has a unique solution
if $\textbf{F}(\cdot)$ is uniformly P-function, which means that there exists
a constant $c$ such that for every $\textbf{p}\in\mathcal{X}$ and
$\textbf{p}^{\prime}\in\mathcal{X}$, we have
$\max_{i}(\textbf{p}_{i}-\textbf{p}_{i}^{\prime})(\textbf{F}_{i}(\textbf{p})-\textbf{F}_{i}(\textbf{p}))\geq
c\|\textbf{p}-\textbf{p}^{\prime}\|^{2}.$ (59)
Therefore, it suffices to prove that the function $F(\cdot)$ is uniformly
P-function. For $\textbf{F}(\cdot)$, we have
$F_{i}^{l}(\textbf{p})-F_{i}^{l}(\textbf{p}^{\prime}))=\frac{\sum\limits_{j}{{\hat{G}_{i,j}^{l}}({p_{j}^{l}}-{p_{j}^{l}}^{\prime})}}{(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}}+\hat{\eta}_{i}^{l})(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}^{\prime}}+\hat{\eta}_{i}^{l})}+\lambda_{i}\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}({p_{j}^{l}}-{p_{j}^{l}}^{\prime})}.$ (60)
We define the following variables
$\psi_{i}^{l}=\sqrt{(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}}+\hat{\eta}_{i}^{l})(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}^{\prime}}+\hat{\eta}_{i}^{l})},$
(61)
and write
$\displaystyle\hskip
2.84544pt(\textbf{p}_{i}-\textbf{p}_{i}^{\prime})(\textbf{F}_{i}(\textbf{p})-\textbf{F}_{i}(\textbf{p}))=\sum\limits_{l}(p_{i}^{l}-{p_{i}^{l}}^{\prime})\frac{\sum\limits_{j}{{\hat{G}_{i,j}^{l}}({p_{j}^{l}}-{p_{j}^{l}}^{\prime})}}{(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}}+\hat{\eta}_{i}^{l})(\sum\limits_{j}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}}^{\prime}}+\hat{\eta}_{i}^{l})}$
(65) $\displaystyle+\lambda_{i}\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}({p_{j}^{l}}-{p_{j}^{l}}^{\prime})}=\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})^{2}}{{\psi_{i}^{l}}^{2}}+\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}\left(1+\lambda_{i}{\psi_{i}^{l}}^{2}\right)({p_{j}^{l}}-{p_{j}^{l}}^{\prime})}}{{\psi_{i}^{l}}^{2}}$
$\displaystyle\geq\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})^{2}}{{\psi_{i}^{l}}^{2}}-\sum\limits_{j\neq
i}\left|\sum\limits_{l}(p_{i}^{l}-{p_{i}^{l}}^{\prime})(p_{j}^{l}-{p_{j}^{l}}^{\prime})\frac{{\hat{G}_{i,j}^{l}}\left(1+\lambda_{i}{\psi_{i}^{l}}^{2}\right)}{{\psi_{i}^{l}}^{2}}\right|$
$\displaystyle\geq\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})^{2}}{{\psi_{i}^{l}}^{2}}-\sum\limits_{j\neq
i}\left(\max_{l}\left(\frac{{\hat{G}_{i,j}^{l}}\left(1+\lambda_{i}{\psi_{i}^{l}}^{2}\right){\psi_{j}^{l}}}{{\psi_{i}^{l}}}\right)\sqrt{\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})^{2}}{{\psi_{i}^{l}}^{2}}}\sqrt{\sum\limits_{l}\frac{(p_{j}^{l}-{p_{j}^{l}}^{\prime})^{2}}{{\psi_{j}^{l}}^{2}}}\right),$
where we applied Schwartz’ inequality to (65). By some manipulations, one can
obtain the following inequality
$(\textbf{p}_{i}-\textbf{p}_{i}^{\prime})(\textbf{F}_{i}(\textbf{p})-\textbf{F}_{i}(\textbf{p}))\geq[\textbf{d}]_{i}[\textbf{D}\textbf{d}]_{i},$
(66)
where d is a vector whose $i^{\text{th}}$ element is
$[\textbf{d}]_{i}=\sqrt{\sum\limits_{l}\frac{(p_{i}^{l}-{p_{i}^{l}}^{\prime})^{2}}{{\psi_{i}^{l}}^{2}}}$
and D is defined in (25). From (66) and the P-property assumption on matrix D,
one can show that $\textbf{F}(\cdot)$ is uniformly P-function [10, 17].
## Appendix E Proof of Theorem 8
To prove the convergence of the algorithm, we follow the same line as in the
proof of Theorem 5. Given the power profile $\textbf{p}(n)$ at iteration $n$,
users update their power levels according to (24). This means that their power
levels $p_{i}^{l}(n+1)$ must satisfy the following optimality condition
$\sum\limits_{l}(p_{i}^{l}-p_{i}^{l}(n+1))(-\frac{1}{\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}(n)}}}+\hat{\eta}_{i}^{l}}+\lambda
I_{i}^{l}(p(n))).$ (67)
The NE must also satisfy a similar condition, i.e.,
$\sum\limits_{l}(p_{i}^{l}-{p_{i}^{l}}^{*})(-\frac{1}{\sum\limits_{j\neq
i}{{\hat{G}_{i,j}^{l}}{{p_{j}^{l}}^{*}}}+\hat{\eta}_{i}^{l}}+\lambda
I_{i}^{l}(p^{*})).$ (68)
Adding these two inequalities and following the same steps as in Theorem 5,
this theorem is proved.
## References
* [1] D. Fudenberg and J. Tirole, _Game Theory_. Cambridge, MA: MIT Press, 1991.
* [2] Z. Han, Z. Ji, and K. J. R. Liu, “Non-cooperative resource competition game by virtual referee in multi-cell OFDMA networks,” _IEEE Journal on Selected Areas in Communications_ , vol. 25, no. 6, pp. 1 –10, August 2007.
* [3] J.-S. Pang, G. Scutari, F. Facchinei, and C. Wang, “Distributed power allocation with rate constraints in Gaussian parallel interference channels,” _IEEE Transactions on Information Theory_ , vol. 54, no. 8, pp. 2868 –2878, August 2008.
* [4] F. Facchinei and C. Kanzow, “Generalized Nash equilibrium problems,” _4OR_ , vol. 5, pp. 173– 210, 1993.
* [5] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” _IEEE Transactions on Vehicular Technology_ , vol. 42, no. 4, pp. 641–646, November 1993.
* [6] K. Leung and C. W. Sung, “An opportunistic power control algorithm for cellular network,” _IEEE/ACM Transactions on Networking_ , vol. 14, no. 3, pp. 470 –478, June 2006.
* [7] G. Scutari, D. P. Palomar, and S. Barbarossa, “Optimal linear precoding strategies for wideband noncooperative systems based on game theory - Part I: Nash equilibria,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 3, pp. 1230– 1249, March 2008.
* [8] ——, “Optimal linear precoding strategies for wideband noncooperative systems based on game theory Part II: Algorithms,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 3, pp. 1250 –1267, March 2008\.
* [9] ——, “Asynchronous iterative water-filling for Gaussian frequency-selective interference channels,” _IEEE Transactions on Information Theory_ , vol. 54, no. 7, pp. 2868 –2878, July 2008.
* [10] R. W. Cottle, J.-S. Pang, and R. E. Stone, _The Linear Complementarity Problem_. Cambridge Academic Press, 1992\.
* [11] K. W. Shum, K.-K. Leung, and C. W. Sung, “Convergence of iterative waterfilling algorithm for Gaussian interference channels,” _IEEE Journal on Selected Areas in Communications_ , vol. 25, no. 6, pp. 1091– 1100, August 2007.
* [12] D. P. Bertsekas and J. N. Tsitsiklis, _Parallel and Distributed Computation: Numerical Methods_ , 2nd ed. Athena Scientific, 1989.
* [13] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
* [14] Z.-Q. Luo and J.-S. Pang, “Analysis of iterative waterfilling algorithm for multiuser power control in digital subscriber lines,” _EURASIP Journal on Applied Signal Processing_ , vol. 2006, pp. 1 –10, May 2006.
* [15] F. Facchinei and J.-S. Pang, _Finite-Dimensional Variational Inequalities and Complementarity Problem_. Springer-Verlag, New York, 2003.
* [16] F. Wang, M. Krunz, and S. Cui, “Price-based spectrum management in cognitive radio networks,” _IEEE Journal on Selected Topics in Signal Processing_ , vol. 2, no. 1, p. 74 87, February 2008.
* [17] J.-S. Pang, G. Scutari, D. P. Palomar, and F. Facchinei, “Design of cognitive radio systems under temperature-interference constraints: A variational inequality approach,” _IEEE Transactions on Signal Processing_ , vol. 58, no. 6, pp. 3251–3271, June 2010.
Figure 1: Data rates and power levels of users in our proposed opportunistic
power control algorithm. Channel conditions for user $i+1$ is better than that
of user $i$. Figure 2: Data rates and power levels of users in our proposed
opportunistic power control algorithm when channel conditions for user 5
gradually worsens. Figure 3: Comparison of power levels in our algorithm and
those in (III-A) when channel conditions for user 5 gradually worsens. Figure
4: The effect of pricing on the converged power levels. Figure 5: Comparison
of converged values of power levels using fixed pricing and those of our
proposed pricing when channel conditions for user 5 gradually worsens. Figure
6: Comparison of converged data rates using fixed pricing and those of our
proposed pricing when channel conditions for user 5 gradually worsens. Figure
7: Comparison of converged power levels using fixed pricing and those of our
proposed pricing when channel conditions for user 5 gradually improves. Figure
8: Comparison of converged data rates using fixed pricing and those of our
proposed pricing when channel conditions for user 5 gradually improves.
|
arxiv-papers
| 2011-06-26T15:44:45 |
2024-09-04T02:49:20.082235
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohammad R. Javan and Ahmad R. Sharafat",
"submitter": "Mohammad Reza Javan",
"url": "https://arxiv.org/abs/1106.5230"
}
|
1106.5334
|
IPPP/11/35
DCPT/11/70
The small $x$ gluon and $b\bar{b}$ production at the LHC
E.G. de Oliveiraa, A.D. Martina and M.G. Ryskina,b
a Institute for Particle Physics Phenomenology, University of Durham, Durham,
DH1 3LE
b Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300,
Russia
###### Abstract
We study open $b\bar{b}$ production at large rapidity at the LHC in an attempt
to pin down the gluon distribution at very low $x$. For the LHC energy of 7
TeV, at next-to-leading order (NLO), there is a large factorization scale
uncertainty. We show that the uncertainty can be greatly reduced if events are
selected in which the transverse momenta of the two $B$-mesons balance each
other to some accuracy, that is
$|\boldsymbol{p}_{1T}+\boldsymbol{p}_{2T}|<k_{0}$. This will fix the scale
$\mu_{F}\simeq k_{0}$, and will allow the LHCb experiment, in particular, to
study the $x$-behaviour of gluon distribution down to $x\sim 10^{-5}$, at
rather low scales, $\mu\sim 2$ GeV. We evaluate the expected cross sections
using, for illustrative purposes, various recent sets of Parton Distribution
Functions.
## 1 Introduction
The data from HERA and the Tevatron do not constrain the behaviour of the low
$x$ gluon density, $g(x,Q^{2})$. Indeed, if $Q^{2}\sim 4~{}\rm GeV^{2}$, then
already for $x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}10^{-3}$
there is a significant difference between the gluon distributions found in the
different global PDF analyses. On the other hand, this is just the region
sampled by the underlying events at the LHC.
It appears attractive to use the inclusive $b\bar{b}$ production at the LHC to
study the behaviour of the gluon distribution in the very low-$x$ region.
Indeed, due the rather large mass of the $b$ quark, the process may be
described in the framework of perturbative QCD. The dominant contribution
arises from the $gg\to b\bar{b}$ hard subprocess. Its cross section has the
following structure
$d\sigma/d^{3}p~{}=~{}\int dx_{1}dx_{2}~{}g(x_{1},\mu_{F})~{}|{\cal
M}(p;\mu_{F},\mu_{R})|^{2}~{}g(x_{2},\mu_{F})\ ,$ (1)
where the gluon densities, $g(x_{i},\mu_{F})$, are taken at some factorization
scale $\mu_{F}$, and the matrix element squared, $|{\cal M}|^{2}$, describes
the cross section of the elementary $gg\to b\bar{b}$ subprocess. The process
samples gluons which carry momenta fraction $x_{i}$ of the initial protons,
where
$x_{1,2}=\frac{m_{\rm hard}}{\sqrt{s}}\exp(\pm y).$ (2)
At the LHC energy of $\sqrt{s}=7$ TeV, and rather large rapidity111Rapidities
in this range are optimal for the LHCb experiment [1]., $y\sim 5$, of the
whole system produced in hard subprocess, one can probe the gluon densities
with $x\sim 10^{-5}$. For this estimate we have taken the mass created in
‘hard subprocess’ $m_{\rm hard}=10$ GeV. Recall that at present there are no
data in this small $x$ domain and different global parton analysis predict
quite different gluons, especially close to the input scale for parton
evolution, see, for example, [2, 3]. It therefore appears that the LHC, and
the LHCb experiment in particular, offers a golden opportunity to make a
precise determination of the gluon in this important low $x$ domain. However,
first we must face the problem of the choice of factorization and
renormalization scales.
A factorization scale $\mu_{F}$ is needed to separate the contributions hidden
in the incoming Parton Distribution Functions (PDFs) from those that included
in the hard matrix element $|{\cal M}|^{2}$. Here, the gluon $g(x,\mu_{F})$ is
the PDF that we are concerned with. Contributions with low gluon virtuality
$q^{2}<\mu_{F}^{2}$ are included in the PDF, while those with
$q^{2}>\mu_{F}^{2}$ are assigned to the matrix element. The second scale, the
renormalization scale $\mu_{R}$, in (1) is necessary to fix the small value of
QCD coupling, $\alpha_{s}(\mu_{R})$, and to justify the perturbative QCD
approach. In principle, if all contributions (NLO, NNLO, etc.) are included,
then calculated cross section would not depend on the values chosen for both
of the scales $\mu_{R}$ and $\mu_{F}$.
## 2 Problems associated with the choice of scales
Figure 1: The NLO predictions for the cross section of $b\bar{b}$ production
obtained using the FONLL program [4] from MSTW08 [2] (continuous curves) and
CT10 [3] (dashed curves) parton sets, at the LHC energies of 7 TeV and 14 TeV,
as a function of pseudo-rapidity $\eta_{B}$ with scale $\mu_{F}=m_{\perp}$ and
$m_{b}$=4.75 GeV; compared with LHCb data at 7 TeV [1]. The predictions using
MSTW08 partons are also shown for four choices of factorization scale:
$\mu_{F}=2m_{\perp},~{}m_{\perp},~{}m_{\perp}/2,~{}m_{\perp}/4$. The
renormalization scale is set to $\mu_{R}=m_{\perp}.$
However, one faces difficulties in the description of the new LHC data [1] for
$b\bar{b}$ production. We list these below.
* •
The NLO QCD prediction strongly depends on the choice of factorization scale,
see Fig. 1. For example, the result obtained with the choice
$\mu_{F}=2m_{\perp}$ is more than twice larger than that for the case of
$\mu_{F}=m_{\perp}/2$, where here
$m_{\perp}\equiv\sqrt{p^{2}_{T}+m^{2}_{b}}$).
* •
Moreover, at the NLO, we have a sizeable contribution from the $2\to 3$
($gg\to b\bar{b}g$) subprocess, where one additional gluon is emitted in the
hard collision. This leads to a considerable smearing of the $x$ domain where
we sample the incoming gluons. The smearing is especially strong if we adopt a
low factorization scale, because then there is a large phase space allowed for
gluon emission from the matrix element. Note that the probability of emission
is enhanced by two large logarithms222The $\ln(1/x)$ enhancement is the main
origin of the scale uncertainty observed in the collinear NLO approach at very
small $x$. If we were to decrease the factorization scale $\mu_{F}$, then we
have to move gluons with $p_{gT}\sim\mu_{F}$ from the PDF to the matrix
element. The problem is that, at very low $x$, there may be several gluons
emitted in the PDF, while only one gluon emission is allowed in the NLO matrix
element. This spoils the compensation between the variations of $|{\cal
M}|^{2}$ and the PDF, which should provide (and, indeed, in the larger $x$
region, does provide) the stability of the results under scale variations.:
$\ln(m^{2}_{\perp}/\mu_{F}^{2})$ and $\ln(1/x)$. In particular, Fig. 2 shows
that if we choose a scale $\mu_{F}=m_{\perp}/4$ then the major contribution
comes from $x\sim 10^{-2}$, and not from $x\sim 10^{-5}$ as we had hoped. From
this viewpoint it would be better to take a large $\mu_{F}$.
* •
On the other hand, to differentiate between the low $x$ gluons it would be
better to work with a relatively low $\mu_{F}$, where the difference between
the different global PDF analyses is larger. At high scales $\mu_{F}$, a large
fraction of low $x$ gluons comes from a region of much larger333Recall that a
parton loses $x$ during DGLAP evolution. $x$ in input distribution, where the
input distribution is already well constrained by existing data. Therefore, at
larger $\mu_{F}$, predictions for LHCb $b\bar{b}$ production, based on
different PDF sets, become close to each other.
* •
Moreover, recall that since for a large scale $\mu_{F}\sim 2m_{\perp}$ up to
the half of the cross section originates from rather heavy virtual gluons444We
have seen from Fig. 1 that
$\sigma(\mu_{F}=2m_{\perp})>2\sigma(\mu_{F}=m_{\perp}/2)$., the original
perturbative calculation, which assumes that the gluon virtuality $q^{2}$ is
small in comparison with the quark mass (or $m_{\perp}$), becomes
inconsistent.
* •
Finally, at NLO, we also have an unavoidable uncertainty in the prediction of
$b\bar{b}$ production arising from the choice of the renormalization scale,
$\mu_{R}$, which we will discuss in Section 6.
However, despite these difficulties, we show that it is possible to use the
LHC $b\bar{b}$ data to make a measurement of the shape of the gluon PDF in the
interval $10^{-5}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}10^{-2}$.
Figure 2: The distribution of the values of $x_{2}$ of the gluons sampled in
NLO $b\bar{b}$ production with $\eta_{b}=5.5$ at the LHC energy of 7 TeV,
after the cross section has been integrated over $x_{1}$ and $p_{T}$. The
distributions are shown for the MSTW08 [2] and CT10 [3] parton sets for three
values of the factorization scale, namely $\mu_{F}=m_{\perp}$ (upper pair of
curves), $m_{\perp}/2$ and $m_{\perp}/4$ (lower pair of curves at small $x$).
Note that there is a difference between $y$ and $\eta$, namely
$\eta-y\simeq~{}$ln$(m_{\perp}/p_{T})$. This difference is accounted for in
the FONLL calculation [4] used to produce this plot. The renormalisation scale
is taken to be $\mu_{R}=m_{\perp}$.
## 3 Fixing $\mu_{F}$ by a cut on the vector sum of $p_{T}$’s
To overcome the problems associated with the choice of renormalization scale,
we may ask for the measurement of the cross section of $b\bar{b}$ events in
which the two quarks balance each other in the transverse momentum plane to
some accuracy; that is ${\boldsymbol{p}}_{1T}\simeq-{\boldsymbol{p}}_{2T}$. In
other words, to seek events which satisfy a cut on the vector sum of the
transverse momenta of the heavy quarks,
$|\boldsymbol{p}_{1T}+\boldsymbol{p}_{2T}|<k_{0}\ .$ (3)
Of course, in this way we will lose some of the cross section, but this should
not be a problem with the available high LHC luminosity. Another point is that
one cannot measure the quark momentum directly. However the momentum of the
$B$-meson can be measured555Even for the $B\to D\mu\nu$ decay, exploited in
[1], one can restore the full 4-momentum of the $B$-meson based on three
constraints: we know the $B$-meson mass and its direction (two angles: $\phi$
and $\theta$), since the position of the $B$-meson decay vertex is observed in
the detector., and due to the strong leading effect666The $B$-meson carries
more than 80% [5, 6] of the original $b$ quark momentum. In this paper, we
present results for $b\bar{b}$ production. in $B$-meson production, the event
selection, proposed in (3), can be performed with sufficient accuracy for
those events with reasonably small transverse momenta of the $B$-mesons, say,
$|\boldsymbol{p}_{TB}|<5$ GeV. Moreover in order to better constrain the $x$
values of the gluons in our selected events, we may put an additional cut on
the pseudo-rapidities of the $B$-mesons, say,
$|\eta_{1}-\eta_{2}|<1\ ~{}~{}~{}~{}\mbox{or}\
~{}~{}~{}~{}\eta-0.5<\eta_{1},\eta_{2}<\eta+0.5.$ (4)
Here, for illustration, we use the latter cut and present results as a
function of $\eta$.
With the above kinematics it is natural to choose $\mu_{F}=k_{0}$. At first
sight, in this way we appear to have excluded any gluon emission due to the
NLO matrix element; a gluon with a transverse momentum, $p_{gT}$, less than
$\mu_{F}$ should be included in the PDF, while one with $p_{gT}>\mu_{F}$
spoils the cut (3). However, this is not true at NLO. DGLAP evolution is
written in terms of parton virtualities $q^{2}=q^{2}_{T}/(1-z)$, where $z$ is
the fraction of parent parton momentum carried by the next (in this case,
final) parton. So, a relatively soft gluon with $p_{gT}<\mu_{F}$ may
correspond to $q^{2}>\mu^{2}_{F}$, and thus be assigned to the matrix element.
However, this will happen mainly for large $z$ close to 1, that is, in a
situation where the emission of an additional (and now soft) gluon does not
change the mass $m_{\rm hard}$ of the ‘hard block’ too much; and thus does not
smear out the low $x$ of the gluon sampled by the process.
## 4 Procedure to calculate the effect of the proposed cuts
To demonstrate how effective the proposed cuts will be in determining the
gluon PDF at small $x$, we calculate the expected NLO cross section using two
different recent sets of parton distributions, namely MSTW08 [2] and CT10 [3],
for an LHC energy of $\sqrt{s}=7$ TeV. To implement the cuts we adapt the
subroutines for the matrix elements squared of LO and NLO $b\bar{b}$
production that are given in the public MCFM program [7]. These subroutines
use the expressions given in [8] for the NLO loop corrections to the $2\to 2$
subprocess, and the expressions given in [9] for the $2\to 3$ subprocess. Note
that the infrared divergences arising from the emission of very soft gluons
are regularized by the so-called ‘plus’ prescription, which means that the
singularities in the integrands, as the momentum fraction $z\to 0$, are tamed
by
$\int_{0}^{1}dzf(z)\left[\frac{1}{z}\right]_{+}=\int_{0}^{1}dz\frac{f(z)-f(0)}{z}.$
(5)
The ‘plus’ prescription is well justified at very low $z$, and corresponds to
the Bloch-Nordsieck procedure for soft gluon radiation and for the inclusive
cross section, where the $f(0)$ terms, added in the virtual loop and in the
real NLO contributions, cancel each other. However, in the $2\to 3$ matrix
element not only soft gluons are emitted. If $z$ is not small, the
cancellation may be spoiled by our cuts. Therefore, to calculate the real
$2\to 3$ contribution we impose the restriction that the virtuality of any
external (gluon or $b$-quark) line, after gluon emission, must be larger than
the factorization scale $\mu_{F}^{2}$. All contributions with smaller
virtualities are included either in the incoming parton distributions or in
the quark fragmentation functions. The remaining $2\to 3$ contribution, with
exactly the same kinematics, due to the additional $f(0)$ term, goes to the
NLO loop correction to cancel the corresponding (unphysical, if $z$ is not too
small) term in the loop correction to the $2\to 2$ subprocess777The
contribution corresponding to $f(0)$ was calculated taking the matrix element
for very soft gluon emission, keeping the momenta of incoming partons and the
outgoing $b$-quark fixed. After this, the $1/z$ singular factor was replaced
by the corresponding $1/z$ factor for the $2\to 3$ event which satisfies all
of our proposed cuts. coming from the ‘plus prescription’.
As default parameters we take the renormalization scale to be
$\mu_{R}=m_{b}=4.75$ GeV, and the factorization scale to be $\mu_{F}=2$ GeV.
Also we take $k_{0}=2$ GeV in (3) for the cut on the vector sum of the
transverse momentum of the outgoing $b$ quarks.
## 5 Factorization scale dependence
To illustrate the dependence of the predictions for $b\bar{b}$ production on
the choice of the factorization scale, $\mu_{F}$, we evaluate the cross
section for the production of $b$ and $\bar{b}$ quarks with both of their
pseudo-rapidities in the interval $5<\eta_{1,2}<6$, first using $\mu_{F}=2$
GeV, and then for $\mu_{F}=4$ GeV. We repeat the exercise for the interval
$2<\eta_{1,2}<3$. We use the CT10 NLO set of partons [3], which are available
for $Q>Q_{0}=1.3$ GeV. For both choices of rapidity intervals, the cross
section calculated with the higher scale, $\mu_{F}=4$ GeV, is about 3 - 4
times larger than that calculated with $\mu_{F}=2$ GeV. Such a strong
factorization scale dependence is due to the behaviour of the incoming parton
densities. In the small $x$ domain, relevant for the LHC, the summation of the
double logarithmic terms,
$\Sigma_{n}c_{n}(\alpha_{s}\ln(1/x)\ln(\mu^{2}_{F}/Q^{2}_{0}))^{n},$ (6)
in the DGLAP evolution, leads to an
$\exp\left(\sqrt{(4N_{c}\alpha_{s}/\pi)\ln(1/x)\ln(\mu^{2}_{F}/Q^{2}_{0})}\right)$
(7)
growth of the gluon density with increasing $\mu_{F}$. The exponential growth
comes from the sum over the possibilities of emitting different numbers of
gluons. The growth cannot be compensated by the ‘hard’ matrix element, which
at NLO level, allows for the emission of only one gluon. This double-
logarithmic effect is the main source of the strong factorization scale
dependence of the predictions for the single $b$-quark inclusive cross
section.
On the other hand, if we choose a ‘large’ value of the scale, $\mu_{F}>k_{0}$,
then we invalidate our proposed ‘$p_{T}$’ cut (3). Recall that an integrated
parton density at a scale $\mu_{F}$ includes the effects of all partons with
transverse momenta $k_{t}<\mu_{F}$; and the transverse momentum of an incoming
parton with a ‘large’ $k_{t}$ will spoil the $p_{T}$ balance in (3). To
control the transverse momenta of the incoming partons we may use unintegrated
parton distributions888We also included the contributions from the incoming
quarks, which are, however, negligibly small in the low $x$ region of
interest., $f_{g}(x,k_{t},\mu_{F})$, and then integrate them over all
$k_{t}<k_{0}$. These distributions can be obtained to NLO accuracy from the
conventional integrated PDFs following the prescription of Ref. [10]. However,
recall that integrated PDFs are not available at low scales, $k$, less than
$Q_{0}$. Therefore we replace that part of the integral over the unintegrated
PDFs with $k^{2}=k_{t}^{2}/(1-z)<Q_{0}^{2}$ by the ‘integrated’ value
$xg(x,Q_{0})T(Q_{0},\mu_{F})$, where the Sudakov factor, $T$, accounts for the
probability not to emit an extra parton, and thus not to enlarge $k_{t}$,
during the evolution from $Q_{0}$ to $\mu_{F}$. The $T$ factor is given by
$T(Q_{0},\mu_{F})~{}=~{}{\rm
exp}\left(-\int^{\mu_{F}}_{Q_{0}}\frac{d\kappa^{2}}{\kappa^{2}}\frac{\alpha_{s}(\kappa^{2})}{2\pi}\int^{1}_{0}dz~{}zP_{gg}^{\rm
LO+NLO}(z)\right)$ (8)
where the precise form of the splitting function $P_{gg}^{\rm LO+NLO}(z)$ is
given in [10].
With such a procedure, for fixed $k_{0}$, the main double-logarithmic effects
are correctly included via the unintegrated PDF, $f_{g}$, while the single-log
dependence of $f_{g}$ on $\mu_{F}$ is mainly compensated by the NLO loop
corrections in the ‘hard subprocess’ cross section. The net effect of this
procedure, based on ‘unintegrated PDFs’, is a great reduction in the
dependence on the choice of the factorization scale. For example, changing the
scale $\mu_{F}$ from 2 to 4 GeV now leads to less than 20(35)% decrease in the
prediction of the $b\bar{b}$ cross section in the intervals $2<\eta<3$ (and
$5<\eta<6$), rather than the factor of 4 (3) increase, see Fig. 4 which we
will introduce below. Note that within a 30% accuracy, the result obtained
using the unintegrated PDFs coincides with that calculated in the conventional
NLO collinear framework with $\mu_{F}=k_{0}$.
## 6 Renormalization scale dependence
The dependence of the cross section on the choice of renormalization scale,
$\mu_{R}$, arises from the running QCD coupling, $\alpha_{s}(\mu_{R})$. In
general, the variation of the QCD coupling should be compensated by the
logarithmic terms, $\ln(\mu_{R}/\mu_{F})$, in the NLO (and higher order)
virtual loop corrections [11]. Unfortunately, in the region of interest, say,
$\mu_{F}=2$ GeV and $\mu_{R}\simeq(1-2)m_{b}$, the corresponding NLO
contribution, that is the factor $\alpha_{s}^{3}\ln(\mu_{R}/\mu_{F})$, is
practically constant, and does not compensate the variation of the LO term,
$|{\cal M}^{\rm LO}|^{2}\propto\alpha_{s}^{2}(\mu_{R})$. Therefore the result
calculated with $\mu_{R}=2m_{b}$ turns out to be about 30% smaller than that
with $\mu_{R}=m_{b}$. Taken together with the uncertainty in the value of the
mass, $m_{b}$, of the $b$-quark, this leads to an unavoidable uncertainty
($\sim 50\%$), at NLO level, in the normalization of the $b\bar{b}$ cross
section999For our computations, we choose the conventional value,
$\mu_{R}=m_{b}=4.75$ GeV [6, 12] which was used to describe $b\bar{b}$
production at the Tevatron..
Recall that at these scales the NLO corrections are rather large, larger than
the size of the LO contribution. Thus it is not evident that the much more
complicated NNLO calculation will improve the accuracy significantly.
## 7 Results for $b\bar{b}$ production after cuts
Nevertheless, in spite of the uncertainties in the normalization that we
discussed above, the expected ratio of the cross sections measured in
different rapidity intervals is quite stable and is driven entirely by the $x$
behaviour of the gluon. We illustrate below how this enables LHC $b\bar{b}$
data to determine the shape of the gluon PDF in the interval $10^{-5}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}10^{-3}$.
Recall that after imposing the cuts of (3) and (4), the variation of the mass,
$m_{\rm hard}$, created in the hard subprocess, is strongly limited. The
contribution of the $2\to 3$ subprocess never exceeds 40% of the whole cross
section; typically it only amounts to about 1/3. Moreover, this $2\to 3$
contribution arises from relatively soft gluon emission, which does not change
$m_{\rm hard}$ very much. An important consequence is that these $b\bar{b}$
events in different intervals of pseudo-rapidity, $\eta$, sample the gluon in
rather narrow intervals of $x$, which allows a precise study of the shape of
gluon $x$ distribution.
Figure 3: The $b\bar{b}$ cross section as a function of the momentum fraction
of the proton carried by the slowest gluon, after cuts (3) and (4) have been
imposed, for four different pseoudo-rapidity intervals taking
$\eta=2.5,~{}3.5,~{}4.5,~{}5.5$ in (4). In each case we show the predictions
obtained using the NLO MSTW08 [2] and CT10 [3] parton sets.
We illustrate this in Fig. 3. It shows the distributions of the momentum
fraction $x_{2}$ (where $x_{2}<x_{1}$) carried by the gluon in $b\bar{b}$
events if cuts (3) and (4) are imposed. The predictions are shown for four
different intervals of the pseudo-rapidity, $\eta=2.5,~{}3.5,~{}4.5,~{}5.5$ in
(4), corresponding to the $b$ and $\bar{b}$ quarks both having rapidities in
the intervals (2,3), (3,4), (4,5) and (5,6) respectively. In each case we show
the results obtained using MSTW08 and CT10 parton sets. We take $\mu_{F}=2$
GeV and $k_{0}=2$ GeV. We see, for instance, that the $b\bar{b}$ events,
selected by the rapidity cut (3,4), sample the gluon in quite a narrow range
about $x=10^{-4}$.
Fig. 3 is obtained using integrated conventional PDFs. Fig. 4 shows the effect
of using the procedure based on unintegrated PDFs (as introduced in Section
5). The figure, obtained via CT10 partons, illustrates two effects. First, it
shows the reduction in sensitivity to a change in factorization scale (from
$\mu_{F}=2$ to 4 GeV) if unintegrated partons are used. Second, it shows that
for our default choice, $\mu_{F}=2$ GeV, the predictions obtained using
integrated and unintegrated PDFs are reasonably similar.
Figure 4: The $b\bar{b}$ cross sections as a function of the momentum fraction
carried by the slowest gluon, after cuts (3) and (4) have been imposed,
predicted using integrated and unintegrated PDFs for two choices of
factorization scale, $\mu_{F}=2$ and 4 GeV. The prefix “u” indicates
unintegrated PDFs are used. The left and right plots correspond to the
rapidity intervals (5,6) and (2,3) specified by taking $\eta=5.5$ and
$\eta=2.5$ in (4). The $\mu_{F}=4$ GeV prediction obtained from integrated
partons is not shown for the latter interval since it about 4 times higher
than that for $\mu_{F}=2$ GeV. CT10 NLO partons [3] are used.
We have emphasized that, despite normalization uncertainties, the expected
ratio of the cross sections measured in, say, the pseudo-rapidity intervals
(2,3) and (5,6) is quite stable and is driven entirely by the $x$ behaviour of
the gluon. If the gluon had a pure power behavior, $xg(x,\mu_{F})\propto
x^{-\lambda}$, then we would observe a flat $\eta$ dependence. This would
follow since
$x_{1,2}\simeq\frac{m_{\rm hard}}{\sqrt{s}}\exp(\pm\eta)~{}~{}~{}~{}~{}{\rm
giving}~{}~{}~{}~{}~{}x_{1}g(x_{1})~{}x_{2}g(x_{2})={\rm constant}.$ (9)
A non-trivial, that is non-flat, $\eta$ behaviour of the cross section will
reflect the curvature (or a deviation from the power law) of the $x$
dependence of the gluon, and may be used to distinguish between the different
sets of ‘global’ PDFs.
For example, in the case of CT10 integrated/conventional partons the expected
ratio for $\mu_{F}=2$ GeV is 2.86 ($\pm$ 15% if we use the unintegrated PDF
with $\mu_{F}=2$ or 4 GeV), while for the case of integrated MSTW08 partons
the analogous ratio is 3.95.
Therefore a study of the $\eta$ dependence of the $b\bar{b}$ events, selected
by the cuts, is a valuable way to study the small $x$ behaviour of the gluon.
In Fig. 5(a) we present the $\eta$ dependence of the cross section expected
for six different NLO sets of PDFs, to demonstrate the sensitivity of such a
method to the small $x$ behaviour of gluons. For this plot, $x_{2}$ varies
from $x_{2}\sim 3\times 10^{-4}$ (corresponding to $\eta=2.5$) to $x_{2}\sim
2\times 10^{-5}$ (corresponding to $\eta=5.5$). This corresponds to the
pseudo-rapidity range relevant to the LHCb experiment. In this small $x$
region the PDFs are unconstrained by existing data. In particular, we note
that the gluon distribution is well determined at $Q^{2}=5~{}\rm GeV^{2}$ to
within 20$\%$ (typically less than 10$\%$) only if $x$ is in the range
$10^{-3}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}0.3$, see
Fig. 16 of [2]. Since the gluons are unconstrained at very low $x$, we should
anticipate the large spread of the predictions, shown in Fig. 5(a), for the
$\eta$ dependence of the cross section after the cuts are imposed. Indeed, if
the errors on the PDFs were to be included, then the various predictions would
overlap.
Recall that there is an overall normalization scale uncertainty of about
50$\%$ (see Section 6). Even allowing for this, the observed $b\bar{b}$ cross
section, $d\sigma/d\eta$, after cuts, will be provide valuable information
about the gluon in this completely unexplored low $x$ domain. However, the
normalization uncertainty does not affect the shape of the prediction of the
cross section versus $\eta$, so the $b\bar{b}$ data are capable of yielding
even more precise information. Simply for illustration, we compare in Fig.
5(b) the shapes of the predictions of all the parton sets after they have been
normalized to 1 at $\eta=2.5$. Of course, the present huge PDF errors mean
such a comparison cannot distiguish between them. Rather the $b\bar{b}$ data
will determine the gluon distribution to 50$\%$ and its shape to much better
accuracy. Clearly the shape has increasing discriminatory power as $\eta$
increases.
Figure 5: (a) The $\eta$ dependence of the $b\bar{b}$ cross section, after the
cuts (3) and (4) have been imposed, obtained using six different NLO sets of
partons MSTW08, CT10, NNPDF21, ABKM09-4, HERAPDF01 and GJR08VF [2, 3, 13]; (b)
the predictions normalized to one at $\eta=2.5$.
We emphasize again that we have used the integrated PDFs obtained from the
various ‘global’ analyses at face value, simply to illustrate that $b\bar{b}$
data at the high values of $\eta$, accessible at LHCb, offer a powerful probe
of the gluon distribution at small $x$, provided that the cuts given in (3)
and (4) are applied. The LHC data will be able to probe a low $x$, low $Q^{2}$
domain well beyond the range of the data fitted in the ‘global’ PDF analyses.
Of course, in this domain, we have no reason to trust any of the predictions
obtained from ‘extrapolated’ PDFs. We merely show the curves obtained from six
different PDF sets to illustrate the potential ability to constrain the gluon
in the $10^{-5}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}10^{-3}$
domain by measuring $b\bar{b}$ events at high $\eta$, with cuts (3) and (4)
imposed.
## 8 Conclusion
At LHC energies, the $b\bar{b}$ cross section predicted within the
conventional NLO collinear approach has huge uncertainties (up to a factor of
4) arising from the variation of the factorization scale, $\mu_{F}$. This
uncertainty can be strongly reduced by selecting events where the transverse
momenta of the two $B$-mesons balance each other to some accuracy. If the sum
of the momenta $|\boldsymbol{p}_{1T}+\boldsymbol{p}_{2T}|<k_{0}$, then the
scale $\mu_{F}\simeq k_{0}$. This offers the possibility to use $b\bar{b}$
production at the LHC, particularly in the LHCb experiment, to study the
$x$-dependence of the gluon distribution down to $x\sim 10^{-5}$ at rather low
scales, $\mu^{2}\sim 4$ GeV2, where the present HERA and Tevatron data do not
constrain the behaviour of the parton densities, and where different parton
analyses propose quite different gluons.
We have considered the renormalization and factorization scale dependences of
the $b\bar{b}$ cross section after the cuts (3) and (4), on the transverse
momenta and rapidities of the $B$-mesons, have been imposed. In this way, we
have demonstrated how such events may determine the behaviour of the gluon
distribution down to $x\sim 10^{-5}$. In Fig. 5 we showed the differences
obtained using six different set of PDFs101010With our cuts, a negative gluon
distribution leads to a negative $b\bar{b}$ cross section and so such a PDF
set is rejected in the corresponding kinematic domain. This happens, for
example, for MSTW08 gluons at very low $x$ and low scales, where the PDFs have
been extrapolated well below the region of the data fitted in the global
analysis. Of course, in this domain a negative gluon should not be taken
literally; it is a way to account for (negative) absorptive effects which
become essential at very low $x$ and low scales. For the results obtained from
integrated partons in Figs. 3 and 5, the MSTW08 gluon distribution is positive
throughout the relevant domain, but the predictions based on unintegrated
partons do sample a small region where the gluon is negative. simply for
illustration, bearing in mind that extrapolations of existing PDFs into this
domain are unreliable.
## Acknowledgements
We thank Graeme Watt for useful discussions. MGR thanks the IPPP at Durham
University for hospitality. EGdeO is supported by CNPq (Brazil) under contract
201854/2009-0, and MGR is supported by the grant RFBR 11-02-00120-a, and by
the Federal Program of the Russian State RSGSS-65751.2010.2.
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S. Alekhin, J. Blumlein, S. Klein and S. Moch, arXiv:0908.3128; Phys. Rev.
D81, 014032 (2010), arXiv:0908.2766 [hep-ph].
|
arxiv-papers
| 2011-06-27T09:14:16 |
2024-09-04T02:49:20.093052
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E.G. de Oliveira, A.D. Martin and M.G. Ryskin",
"submitter": "Alan D. Martin",
"url": "https://arxiv.org/abs/1106.5334"
}
|
1106.5423
|
# A law of large numbers for weighted plurality
Joe Neeman111Department of Statistics, U.C. Berkeley. joeneeman@gmail.com
###### Abstract
Consider an election between $k$ candidates in which each voter votes randomly
(but not necessarily independently) and suppose that there is a single
candidate that every voter prefers (in the sense that each voter is more
likely to vote for this special candidate than any other candidate). Suppose
we have a voting rule that takes all of the votes and produces a single
outcome and suppose that each individual voter has little effect on the
outcome of the voting rule. If the voting rule is a weighted plurality, then
we show that with high probability, the preferred candidate will win the
election. Conversely, we show that this statement fails for all other
reasonable voting rules.
This result is an extension of one by Häggström, Kalai and Mossel, who proved
the above in the case $k=2$.
## 1 Introduction
For elections between two candidates, it is well known that voting rules in
which every voter has a small effect are good rules in the sense that they
“aggregate information well:” if every voter has a small bias towards the same
candidate then that candidate will win with overwhelming probability. When
voters vote independently, this fact was noted by Margulis [4] and Russo [5],
whose results were later strengthened by Kahn, Kalai and Linial [3] and by
Talagrand [6].
When the voters are not independent, the situation is more complicated. It is
no longer true, then, that every reasonable voting rule aggregates well. In
fact, [2] show that if we want the aggregation to hold for every distribution
of the voters, then weighted majority functions are the only option. We extend
their result to the non-binary case.
The author would like to thank Elchanan Mossel for suggesting this problem and
providing fruitful discussions.
## 2 Definitions and results
In the introduction, we made a few allusions to “reasonable” voting rules. Let
us now say precisely what that means: we will require that our voting rules do
not have a built-in preference for any alternative. This is a common
assumption, and its definition is standard (see, eg. [1]). In what follows,
the notation $[k]$ stands for the set $\\{0,\dots,k-1\\}$.
###### Definition 2.1.
A function $f:[k]^{n}\to[k]$ is _neutral_ if $f(\sigma(x))=\sigma(f(x))$ for
all $x\in[k]^{n}$ and all permutations $\sigma$ on $[k]$, where
$\sigma(x)_{i}=\sigma(x_{i})$.
Note that in the case $k=2$, a function is neutral if, and only if, it is
anti-symmetric according to the definition in [2].
###### Example 2.2
When $k=2$ and $n$ is odd, then the simple majority function (for which
$f(x)=1$ if $\\#\\{i:x_{i}=1\\}>\\#\\{i:x_{i}=0\\}$) is neutral. On the other
hand, if $n$ is even then in order to fully specify the simple majority
function, we need to say what happens in the case of a tie; the choice of tie-
breaking rule will determine whether the resulting function is neutral. For
example, if we define $f(x)=x_{1}$ for every tied configuration $x$, then $f$
is neutral. On the other hand, if $f(x)=1$ for every tied configuration $x$,
then $f$ is not neutral.
The example can be extended to $k\geq 3$. In this case, consider the tie-
breaking rule $f(x)=x_{i}$ where $i$ is the smallest possible number for which
$x_{i}$ is equal to one of the tied alternatives. This tie-breaking rule is
neutral, and it is more natural than setting $f(x)=x_{1}$ because it
guarantees that the output of $f$ is one of the tied alternatives.
### 2.1 Weighted plurality functions
Let us say precisely what we mean by a weighted plurality function. The
definition that we take here generalizes the definition from [2] of a weighted
majority function.
###### Definition 2.3.
A function $f:[k]^{n}\to[k]$ is a _weighted plurality function_ if there exist
weights $w_{1},\dots,w_{n}\in\mathbb{R}_{\geq 0}$ such that $\sum_{i}w_{i}=1$
and for all $a,b\in[k]$, $f(x)=a$ implies that
$\sum_{i:x_{i}=a}w_{i}\geq\sum_{i:x_{i}=b}w_{i}.$
Note that the above definition does not prescribe a particular behavior if a
tie occurs between two alternatives. If the weights are chosen so that ties
never occur, then the weighted plurality function is clearly neutral.
Moreover, for any set of weights we can construct a neutral weighted plurality
function with those weights by following the tie-breaking rule outlined in
Example 2.2.
### 2.2 The influence of a voter
The final notion that we need before stating our result is a way to quantify
the power of a single voter. When $k=2$, the notion of _effect_ is well-
established and can be found, for example, in [2]. However, there does not
seem to be a well-established way of quantifying the effect of voters for non-
binary social choice functions. Here, we propose a definition that closely
resembles the one used in [2] for binary functions.
###### Definition 2.4.
Let $f$ be a function $[k]^{n}\to[k]$ and fix a probability distribution $P$
on $[k]^{n}$. The _effect of voter $i$_ is
$e_{i}(f,P)=\sum_{j=1}^{k}P(f(X)=j|X_{i}=j)-P(f(X)=j|X_{i}\neq j),$
where $X$ is a random variable distributed according to $P$.
Note that for the case $k=2$, the preceding definition reduces to
$e_{i}(f,P)=2(P(f(X)=1|X_{i}=1)-P(f(X)=1|X_{i}=0)),$
which is just twice the definition in [2] of a voter’s effect. Also, the
effect is closely related to the correlation between the voters and the
outcome:
$\displaystyle P(f(X)=j|X_{i}=j)-P(f(X)=j|X_{i}\neq j)$
$\displaystyle=\frac{\operatorname{Cov}(\mathbbm{1}_{\\{f=j\\}},\mathbbm{1}_{\\{X_{i}=j\\}})}{P(X_{i}=j)P(X_{i}\neq
j)}$ $\displaystyle\geq
4\operatorname{Cov}(\mathbbm{1}_{\\{f=j\\}},\mathbbm{1}_{\\{X_{i}=j\\}})$
and so
$e_{i}(f,P)\geq
4\sum_{j}\operatorname{Cov}(\mathbbm{1}_{\\{f=j\\}},\mathbbm{1}_{\\{X_{i}=j\\}}).$
###### Example 2.5
The simplest example of $e_{i}(f,P)$ is when $P$ is a product measure (ie. the
$X_{i}$ are independent) and the function $f$ does not depend on its $i$th
coordinate; in that case, $P(f(X)=j|X_{i}=j)=P(f(X)=j|X_{i}\neq j)$ for all
$j$ and so $e_{i}(f,P)=0$. On the other hand, if $P$ is a distribution such
that $X_{1}=X_{2}=\cdots=X_{n}$ with probability 1, and if $f$ is a plurality
function, then $P(f(X)=j|X_{i}=j)=1$ for all $j$, while $P(f(X)=j|X_{i}\neq
j)=0$; hence, $e_{i}(f,P)=1$ for all $i$.
For a less trivial example, suppose that the $X_{i}$ are independent and
uniformly distributed on $[k]$. Let $f$ be an unweighted plurality function.
Then the Central Limit Theorem implies that $e_{i}(f,P)=O(\frac{1}{\sqrt{n}})$
as $n\to\infty$.
On the other hand, suppose that $f$ is still an unweighted plurality function
and the $X_{i}$ are independent, but now $P(X_{i}=1)>P(X_{i}=j)+\delta$ for
some $\delta>0$ and all $j\neq 1$. Then Hoeffding’s inequality implies that
$P(f(X)=1|X_{i})\geq 1-2\exp(-\delta^{2}n/4)$ for sufficiently large $n$,
regardless of the value of $X_{i}$. In particular, this implies that
$e_{i}(f,P)=O(\exp(-\delta^{2}n/4))$. Compared to the case where the $X_{i}$
are uniformly distributed, this demonstrates that $e_{i}(f,P)$ can depend
strongly on $P$, even when $P$ is restricted to being a product measure.
### 2.3 The main result
Our main theorem is the following:
###### Theorem 2.6.
1. (a)
For every $\delta>0$ and $\epsilon>0$, there is a $\tau>0$ such that for every
weighted plurality function $f$ with weights $w_{i}$ and every probability
distribution $P$ on $[k]^{n}$, if $e_{i}(f,P)\leq\tau$ and there is a set
$A\subset[n]$ such that
$\sum_{i}w_{i}P(X_{i}=a)\geq\sum_{i}w_{i}P(X_{i}=b)+\delta$ for all $i\in[n]$,
all $a\in A$ and all $b\not\in A$, then $P(f(X)\in A)\geq 1-\epsilon$.
2. (b)
If $f$ is not a weighted plurality function then there exists a probability
distribution $P$ on $[k]^{n}$ such that $P(X_{i}=2)>P(X_{i}=1)$ for all
$i\in[n]$ but $P(f(X)=1)=1$ (and hence $e_{i}(f,P)=0$ for all $i$).
We remark that the Theorem is constructive in the sense that we can give an
algorithm (based on solving a linear program) which either constructs some
weights $w_{i}$ witnessing the fact that $f$ is a weighted plurality, or a
probability distribution $P$ satisfying part (b).
Parts (a) and (b) of Theorem 2.6 are converse to one another in the following
sense: under the hypothesis of small effects, part (a) says that if there is a
gap between the popularity of the most popular alternatives $A$ and the less
popular alternatives $A^{c}$ then a weighted plurality function will choose an
alternative in $A$. Part (b) shows that this property fails for every function
that is not a weighted plurality. Note that part (a) has an important special
case, which is closer to the statement of [2]: if $P(X_{i}=a)\geq
P(X_{i}=b)+\delta$ for all $i\in[n]$ and all $b\neq a$, then $f(X)=a$ with
high probability if the effects are small enough.
The remainder of the paper is devoted to the proof of Theorem 2.6.
###### Proof of Theorem 2.6 (a).
This part of the proof follows very closely the argument in [2]. Suppose that
$f$ is a weighted plurality function with weights $w_{i}$. The first step is
to show that $f$ is “correlated” in some sense with each voter: define
$p_{ij}=P(X_{i}=j)$ and let $W_{j}$ be the (random) weight assigned to
alternative $j$: $W_{j}=\sum_{i:X_{i}=j}w_{i}$. Then
$\displaystyle\mathbb{E}\sum_{i=1}^{n}w_{i}\sum_{j=1}^{k}\mathbbm{1}_{\\{f(X)=j\\}}(\mathbbm{1}_{\\{X_{i}=j\\}}-p_{ij})$
$\displaystyle=\mathbb{E}\left(\sum_{i,j}w_{i}\mathbbm{1}_{\\{f(X)=j\\}}\mathbbm{1}_{\\{X_{i}=j\\}}-\sum_{i,j}\mathbbm{1}_{\\{f(X)=j\\}}w_{i}p_{ij}\right)$
$\displaystyle=\mathbb{E}\sum_{i,j}w_{i}\mathbbm{1}_{\\{f(X)=j\\}}\mathbbm{1}_{\\{X_{i}=j\\}}-\sum_{j}P(f=j)\mathbb{E}W_{j}.$
(1)
Now, let $\alpha_{j}=P(f=j)$ and set
$\tilde{\alpha}_{j}=\alpha_{j}/(\sum_{i\in A}\alpha_{i})$ for $j\in A$ and
$\tilde{\alpha}_{j}=0$ otherwise. The first term of (2.3) is just
$\mathbb{E}\sum_{i,j}w_{i}\mathbbm{1}_{\\{f(X)=j\\}}\mathbbm{1}_{\\{X_{i}=j\\}}=\mathbb{E}\sum_{j}\mathbbm{1}_{\\{f(X)=j\\}}W_{j}\\\
\geq\mathbb{E}\sum_{j}\mathbbm{1}_{\\{f(X)=j\\}}\sum_{i}\tilde{\alpha}_{i}W_{i}=\sum_{i}\tilde{\alpha}_{i}\mathbb{E}W_{j}$
(2)
since the winning alternative always has at least as much weight as any convex
combination of alternatives. Since $\min_{j\in
A}\mathbb{E}W_{j}\geq\max_{j\not\in A}\mathbb{E}W_{j}+\delta$, we can plug (2)
into (2.3) to obtain
$\displaystyle\eqref{eq:corr}$
$\displaystyle\geq\sum_{j}\tilde{\alpha}_{j}\mathbb{E}W_{j}-\sum_{j}\alpha_{j}\mathbb{E}W_{j}$
$\displaystyle\geq\sum_{j\in A}(\tilde{\alpha}_{j}-\alpha_{j})\delta$
$\displaystyle=\delta P(f\not\in A).$
Recalling that $e_{i}(f,P)\geq
4\sum_{j}\operatorname{Cov}(\mathbbm{1}_{\\{f=j\\}},\mathbbm{1}_{\\{X_{i}=j\\}})$,
we have
$\displaystyle\delta P(f\not\in A)$
$\displaystyle\leq\mathbb{E}\sum_{i=1}^{n}w_{i}\sum_{j=1}^{k}\mathbbm{1}_{\\{f(X)=j\\}}(\mathbbm{1}_{\\{X_{i}=j\\}}-p_{ij})$
$\displaystyle\leq\frac{1}{4}\sum_{i}w_{i}e_{i}(f,P)$
$\displaystyle\leq\frac{\tau}{4}$
and so one direction of the theorem is proved once we take $\tau$ small enough
that $\epsilon\geq\tau/(4\delta)$. ∎
The proof of the second part of the theorem follows the idea of [2], in that
we use linear programming duality to find a witness for $f$ being a weighted
plurality function. However, the details of the proof are quite different,
since [2] uses a well-known linear program (the fractional vertex cover of a
hypergraph) which does not extend beyond $k=2$.
The proof idea is this: we will write down a linear program and its dual. If
the primal program has a large enough value it will turn out that $f$ is a
weighted plurality function. Otherwise, the dual has a small value and the
dual variables witness the claim of Theorem 2.6 (b). In particular, note that
this proof provides the algorithm that we mentioned after the statement of
Theorem 2.6.
First we make a trivial observation that will simplify our linear program
considerably: if a function is neutral, it is easier to check whether it is a
weighted plurality because it is not necessary to try all possible
combinations of $a,b\in[k]$:
###### Proposition 2.7.
Suppose $f:[k]^{n}\to[k]$ is neutral. Then $f$ is a weighted plurality if and
only if there exist weights $w_{1},\dots,w_{n}\in\mathbb{R}$ such that
$f(x)=1$ implies that
$\sum_{i:x_{i}=1}w_{i}\geq\sum_{i:x_{i}=2}w_{i}.$
We can write a linear program for checking whether a given neutral function
$f$ is a weighted plurality. The variables for this program are $t$; $w_{i}$
for each $i\in[n]$ and $g_{x}$ for each $x\in[k]^{n}$ for which $f(x)=1$. In
standard form, the primal program is the following:
maximize $\displaystyle t_{+}-t_{-}$ subject to $\displaystyle g_{x}\geq
0\text{ for all }x\in[k]^{n}\text{ such that }f(x)=1$ $\displaystyle w_{i}\geq
0\text{ for all }i\in[n]$ $\displaystyle t_{+}\geq 0\text{ and }t_{-}\geq 0$
$\displaystyle\sum_{i}w_{i}=1$
$\displaystyle\sum_{i:x_{i}=1}w_{i}-\sum_{i:x_{i}=2}w_{i}-g_{x}-(t_{+}-t_{-})=0\text{
for all }x\in[k]^{n}\text{ with }f(x)=1.$
###### Proposition 2.8.
Let $t^{*}$ be the value of the above linear program. If $t^{*}\geq 0$ then
$f$ is a weighted plurality function.
###### Proof.
Let $w_{i}$, $g_{x}$, $t_{+}$ and $t_{-}$ be feasible points such that
$t_{+}-t_{-}\geq 0$. Then, for all $x$ with $f(x)=1$,
$\sum_{i:x_{i}=1}w_{i}-\sum_{i:x_{i}=2}w_{i}=g_{x}+(t_{+}-t_{-})\geq 0$
and so $f$ satisfies the conditions of Proposition 2.7. ∎
Now consider the dual program; since the primal is in standard form, the dual
is easy to write down. Let the dual variables be $a$ and $q_{x}$ for all $x$
such that $f(x)=1$. Then the dual program is:
minimize $\displaystyle a_{+}-a_{-}$ subject to
$\displaystyle\sum_{x:f(x)=1}q_{x}\leq-1$
$\displaystyle\sum_{x:f(x)=1}(\mathbbm{1}_{\\{x_{i}=1\\}}-\mathbbm{1}_{\\{x_{i}=2\\}})q_{x}+(a_{+}-a_{-})\geq
0\text{ for all }i\in[n]$ $\displaystyle q_{x}\leq 0\text{ for all }x\text{
such that }f(x)=1$ $\displaystyle a_{+}\leq 0\text{ and }a_{-}\leq 0.$
###### Proposition 2.9.
Let $a^{*}$ be the value of the above dual program. If $a^{*}<0$ then there
exists a probability distribution on $[k]^{n}$ such that
$P(X_{i}=2)>P(X_{i}=1)$ for all $i$ but $f(X)=1$ almost surely.
###### Proof.
Choose a feasible point with $a_{+}-a_{-}<0$ and define
$p_{x}=-q_{x}/(\sum_{x}q_{x})$. Then $p_{x}\geq 0$ and $\sum_{x}p_{x}=1$, so
we can define a probability distribution by $P(X=x)=p_{x}$ when $f(x)=1$ and
$P(X=x)=0$ otherwise. Under this distribution, $f(X)=1$ with probability 1. On
the other hand, with $a_{+}-a_{-}<0$ the constraints of the dual program imply
that
$\sum_{x:f(x)=1}\mathbbm{1}_{\\{x_{i}=1\\}}q_{x}>\sum_{x:f(x)=1}\mathbbm{1}_{\\{x_{i}=2\\}}q_{x}$
for all $i$. Thus,
$P(X_{i}=1)=\sum_{x:f(x)=1}\mathbbm{1}_{\\{x_{i}=1\\}}p_{x}<\sum_{x:f(x)=1}\mathbbm{1}_{\\{x_{i}=2\\}}p_{x}=P(X_{i}=2)$
for all $i$. ∎
To conclude the proof of Theorem 2.6, note that both the primal and dual
programs are feasible and bounded and so $a^{*}=t^{*}$.
## References
* [1] S.J. Brams and P.C. Fishburn. Voting procedures. Handbook of social choice and welfare, 1:173–236, 2002.
* [2] O. Häggström, G. Kalai, and E. Mossel. A law of large numbers for weighted majority. Advances in Applied Mathematics, 37(1):112–123, 2006.
* [3] J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science, pages 68–80. IEEE Computer Society, 1988.
* [4] G. Margulis. Probabilistic characteristic of graphs with large connectivity. Problems Info. Transmission, 10:174–179, 1977.
* [5] L. Russo. An approximate zero-one law. Probability Theory and Related Fields, 61(1):129–139, 1982.
* [6] M. Talagrand. On Russo’s approximate zero-one law. The Annals of Probability, 22(3):1576–1587, 1994.
|
arxiv-papers
| 2011-06-27T15:53:48 |
2024-09-04T02:49:20.102082
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joe Neeman",
"submitter": "Joe Neeman",
"url": "https://arxiv.org/abs/1106.5423"
}
|
1106.5472
|
# Quartet fixed point theorems for nonlinear contractions in partially ordered
metric spaces
Erdal Karapinar erdal karapınar,
Department of Mathematics, Atilim University 06836, İncek, Ankara, Turkey
erdalkarapinar@yahoo.com ekarapinar@atilim.edu.tr
###### Abstract.
The notion of coupled fixed point is introduced in by Bhaskar and
Lakshmikantham in [2]. Very recently, the concept of tripled fixed point is
introduced by Berinde and Borcut [1]. In this manuscript, by using the mixed
$g$ monotone mapping, some new quartet fixed point theorems are obtained. We
also give some examples to support our results.
###### Key words and phrases:
Fixed point theorems, Nonlinear contraction, Partially ordered, Quartet Fixed
Point, mixed g monotone
###### 2000 Mathematics Subject Classification:
47H10,54H25,46J10, 46J15
## 1\. Introduction and Preliminaries
In 2006, Bhaskar and Lakshmikantham [2] introduced the notion of coupled fixed
point and proved some fixed point theorem under certain condition. Later,
Lakshmikantham and Ćirić in [8] extended these results by defining of
$g$-monotone property. After that many results appeared on coupled fixed point
theory (see e.g. [3, 4, 6, 5, 11, 10]).
Very recently, Berinde and Borcut [1] introduced the concept of tripled fixed
point and proved some related theorems. In this manuscript, the quartet fixed
point is considered and by using the mixed $g$-monotone mapping, existence and
uniqueness of quartet fixed point are obtained.
First we recall the basic definitions and results from which quartet fixed
point is inspired. Let $(X,d)$ be a metric space and $X^{2}:=X\times X$. Then
the mapping $\rho:X^{2}\times X^{2}\rightarrow[0,\infty)$ such that
$\rho((x_{1},y_{1}),(x_{2},y_{2})):=d(x_{1},x_{2})+d(y_{1},y_{2})$ forms a
metric on $X^{2}$. A sequence $(\\{x_{n}\\},\\{y_{n}\\})\in X^{2}$ is said to
be a double sequence of $X$.
###### Definition 1.
(See [2]) Let $(X,\leq)$ be partially ordered set and $F:X\times X\rightarrow
X$. $F$ is said to have mixed monotone property if $F(x,y)$ is monotone
nondecreasing in $x$ and is monotone non-increasing in $y$, that is, for any
$x,y\in X$,
$\displaystyle x_{1}\leq x_{2}\Rightarrow F(x_{1},y)\leq F(x_{2},y),\ \
\mbox{for}\ x_{1},x_{2}\in X,\ \ \mbox{and}\ $ $\displaystyle y_{1}\leq
y_{2}\Rightarrow F(x,y_{2})\leq F(x,y_{1}),\ \mbox{for}\ y_{1},y_{2}\in X.$
###### Definition 2.
(see [2]) An element $(x,y)\in X\times X$ is said to be a coupled fixed point
of the mapping $F:X\times X\rightarrow X$ if
$F(x,y)=x\ \mbox{and}\ \ F(y,x)=y.$
Throughout this paper, let $(X,\leq)$ be partially ordered set and $d$ be a
metric on $X$ such that $(X,d)$ is a complete metric space. Further, the
product spaces $X\times X$ satisfy the following:
$(u,v)\leq(x,y)\Leftrightarrow u\leq x,\ y\leq v;\ \ \mbox{for all}\ \
(x,y),(u,v)\in X\times X.$ (1.1)
The following two results of Bhaskar and Lakshmikantham in [2] were extended
to class of cone metric spaces in [5]:
###### Theorem 3.
Let $F:X\times X\rightarrow X$ be a continuous mapping having the mixed
monotone property on $X$. Assume that there exists a $k\in[0,1)$ with
$d(F(x,y),F(u,v))\leq\frac{k}{2}\left[d(x,u)+d(y,v)\right],\ \mbox{for all}\
u\leq x,\ y\leq v.$ (1.2)
If there exist $x_{0},y_{0}\in X$ such that $x_{0}\leq F(x_{0},y_{0})$ and
$F(y_{0},x_{0})\leq y_{0}$, then, there exist $x,y\in X$ such that $x=F(x,y)$
and $y=F(y,x)$.
###### Theorem 4.
Let $F:X\times X\rightarrow X$ be a mapping having the mixed monotone property
on $X$. Suppose that $X$ has the following properties:
1. $(i)$
if a non-decreasing sequence $\\{x_{n}\\}\rightarrow x$, then $x_{n}\leq x,\
\forall n;$
2. $(i)$
if a non-increasing sequence $\\{y_{n}\\}\rightarrow y$, then $y\leq y_{n},\
\forall n.$
Assume that there exists a $k\in[0,1)$ with
$d(F(x,y),F(u,v))\leq\frac{k}{2}\left[d(x,u)+d(y,v)\right],\ \mbox{for all}\
u\leq x,\ y\leq v.$ (1.3)
If there exist $x_{0},y_{0}\in X$ such that $x_{0}\leq F(x_{0},y_{0})$ and
$F(y_{0},x_{0})\leq y_{0}$, then, there exist $x,y\in X$ such that $x=F(x,y)$
and $y=F(y,x)$.
Inspired by Definition 1, the following concept of a $g$-mixed monotone
mapping introduced by V. Lakshmikantham and L.Ćirić [8].
###### Definition 5.
Let $(X,\leq)$ be partially ordered set and $F:X\times X\rightarrow X$ and
$g:X\rightarrow X$. $F$ is said to have mixed $g$-monotone property if
$F(x,y)$ is monotone $g$-non-decreasing in $x$ and is monotone $g$-non-
increasing in $y$, that is, for any $x,y\in X$,
$g(x_{1})\leq g(x_{2})\Rightarrow F(x_{1},y)\leq F(x_{2},y),\ \ \mbox{for}\
x_{1},x_{2}\in X,\ \ \mbox{and}\ $ (1.4) $g(y_{1})\leq g(y_{2})\Rightarrow
F(x,y_{2})\leq F(x,y_{1}),\ \mbox{for}\ y_{1},y_{2}\in X.$ (1.5)
It is clear that Definition 13 reduces to Definition 9 when $g$ is the
identity.
###### Definition 6.
An element $(x,y)\in X\times X$ is called a couple point of a mapping
$F:X\times X\rightarrow X$ and $g:X\rightarrow X$ if
$F(x,y)=g(x),\ \ \ \ F(y,x)=g(y).$
###### Definition 7.
Let $F:X\times X\rightarrow X$ and $g:X\rightarrow X$ where $X\neq\emptyset$.
The mappings $F$ and $g$ are said to commute if
$g(F(x,y))=F(g(x),g(y)),\ \ \ \mbox{for all}\ x,y\in X.$
###### Theorem 8.
Let $(X,\leq)$ be partially ordered set and $(X,d)$ be a complete metric space
and also $F:X\times X\rightarrow X$ and $g:X\rightarrow X$ where
$X\neq\emptyset$. Suppose that $F$ has the mixed $g$-monotone property and
that there exists a $k\in[0,1)$ with
$d(F(x,y),F(u,v))\leq\frac{k}{2}\left[\frac{d(g(x),g(u))+d(g(y),g(v))}{2}\right]$
(1.6)
for all $x,y,u,v\in X$ for which $g(x)\leq g(u)$ and $g(v)\leq g(y)$. Suppose
$F(X\times X)\subset g(X)$, $g$ is sequentially continuous and commutes with
$F$ and also suppose either $F$ is continuous or $X$ has the following
property:
$\mbox{if a non-decreasing sequence}\ \\{x_{n}\\}\rightarrow x,\ \mbox{then}\
x_{n}\leq x,\ \mbox{for all}\ n,$ (1.7) $\mbox{if a non-increasing sequence}\
\\{y_{n}\\}\rightarrow y,\ \mbox{then}\ y\leq y_{n},\ \mbox{for all}\ n.$
(1.8)
If there exist $x_{0},y_{0}\in X$ such that $g(x_{0})\leq F(x_{0},y_{0})$ and
$g(y_{0})\leq F(y_{0},x_{0})$, then there exist $x,y\in X$ such that
$g(x)=F(x,y)$ and $g(y)=F(y,x)$, that is, $F$ and $g$ have a couple
coincidence.
Berinde and Borcut [1] introduced the following partial order on the product
space $X^{3}=X\times X\times X$:
$(u,v,w)\leq(x,y,z)\mbox{ if and only if }x\geq u,\ y\leq v,\ z\geq w,$ (1.9)
where $(u,v,w),(x,y,z)\in X^{3}$. Regarding this partial order, we state the
definition of the following mapping.
###### Definition 9.
(See [1]) Let $(X,\leq)$ be partially ordered set and $F:X^{3}\rightarrow X$.
We say that $F$ has the mixed monotone property if $F(x,y,z)$ is monotone non-
decreasing in $x$ and $z$, and it is monotone non-increasing in $y$, that is,
for any $x,y,z\in X$
$\begin{array}[]{r}x_{1},x_{2}\in X,\ x_{1}\leq x_{2}\Rightarrow\
F(x_{1},y,z)\leq F(x_{2},y,z),\\\ y_{1},y_{2}\in X,\ y_{1}\leq
y_{2}\Rightarrow\ F(x,y_{1},z)\geq F(x,y_{2},z),\\\ z_{1},z_{2}\in X,\
z_{1}\leq z_{2}\Rightarrow\ F(x,y,z_{1})\leq F(x,y,z_{2}).\\\ \end{array}$
(1.10)
###### Theorem 10.
(See [1]) Let $(X,\leq)$ be partially ordered set and $(X,d)$ be a complete
metric space. Let $F:X\times X\times X\rightarrow X$ be a mapping having the
mixed monotone property on $X$. Assume that there exist constants
$a,b,c\in[0,1)$ such that $a+b+c<1$ for which
$d(F(x,y,z),F(u,v,w))\leq ad(x,u)+bd(y,v)+cd(z,w)$ (1.11)
for all $x\geq u,\ y\leq v,\ z\geq w$. Assume that $X$ has the following
properties:
* $(i)$
if non-decreasing sequence $x_{n}\rightarrow x$, then $x_{n}\leq x$ for all
$n,$
* $(ii)$
if non-increasing sequence $y_{n}\rightarrow y$, then $y_{n}\geq y$ for all
$n$,
If there exist $x_{0},y_{0},z_{0}\in X$ such that
$x_{0}\leq F(x_{0},y_{0},z_{0}),\ \ y_{0}\geq F(y_{0},x_{0},y_{0}),\ \ \
z_{0}\leq F(x_{0},y_{0},z_{0})$
then there exist $x,y,z\in X$ such that
$F(x,y,z)=x\mbox{ and }F(y,x,y)=y\mbox{ and }F(z,y,x)=z$
The aim of this paper is introduce the concept of quartet fixed point and
prove the related fixed point theorems.
## 2\. Quartet Fixed Point Theorems
Let $(X,\leq)$ be partially ordered set and $(X,d)$ be a complete metric
space. We state the definition of the following mapping. Throughout the
manuscript we denote $X\times X\times X\times X$ by $X^{4}$.
###### Definition 11.
(See [7]) Let $(X,\leq)$ be partially ordered set and $F:X^{4}\rightarrow X$.
We say that $F$ has the mixed monotone property if $F(x,y,z,w)$ is monotone
non-decreasing in $x$ and $z$, and it is monotone non-increasing in $y$ and
$w$, that is, for any $x,y,z,w\in X$
$\begin{array}[]{r}x_{1},x_{2}\in X,\ x_{1}\leq x_{2}\Rightarrow\
F(x_{1},y,z,w)\leq F(x_{2},y,z,w),\\\ y_{1},y_{2}\in X,\ y_{1}\leq
y_{2}\Rightarrow\ F(x,y_{1},z,w)\geq F(x,y_{2},z,w),\\\ z_{1},z_{2}\in X,\
z_{1}\leq z_{2}\Rightarrow\ F(x,y,z_{1},w)\leq F(x,y,z_{2},w),\\\
w_{1},w_{2}\in X,\ w_{1}\leq w_{2}\Rightarrow\ F(x,y,z,w_{1})\geq
F(x,y,z,w_{2}).\\\ \end{array}$ (2.1)
###### Definition 12.
(See [7]) An element $(x,y,z,w)\in X^{4}$ is called a quartet fixed point of
$F:X\times X\times X\times X\rightarrow X$ if
$\begin{array}[]{rl}F(x,y,z,w)=x,&F(x,w,z,y)=y,\\\
F(z,y,x,w)=z,&F(z,w,x,y)=w.\end{array}$ (2.2)
###### Definition 13.
Let $(X,\leq)$ be partially ordered set and $F:X^{4}\rightarrow X$. We say
that $F$ has the mixed $g$-monotone property if $F(x,y,z,w)$ is monotone
$g$-non-decreasing in $x$ and $z$, and it is monotone $g$-non-increasing in
$y$ and $w$, that is, for any $x,y,z,w\in X$
$\begin{array}[]{r}x_{1},x_{2}\in X,\ g(x_{1})\leq g(x_{2})\Rightarrow\
F(x_{1},y,z,w)\leq F(x_{2},y,z,w),\\\ y_{1},y_{2}\in X,\ g(y_{1})\leq
g(y_{2})\Rightarrow\ F(x,y_{1},z,w)\geq F(x,y_{2},z,w),\\\ z_{1},z_{2}\in X,\
g(z_{1})\leq g(z_{2})\Rightarrow\ F(x,y,z_{1},w)\leq F(x,y,z_{2},w),\\\
w_{1},w_{2}\in X,\ g(w_{1})\leq g(w_{2})\Rightarrow\ F(x,y,z,w_{1})\geq
F(x,y,z,w_{2}).\\\ \end{array}$ (2.3)
###### Definition 14.
An element $(x,y,z,w)\in X^{4}$ is called a quartet coincidence point of
$F:X^{4}\rightarrow X$ and $g:X\rightarrow X$ if
$\begin{array}[]{rl}F(x,y,z,w)=g(x),&F(y,z,w,x)=g(y),\\\
F(z,w,x,y)=g(z),&F(w,x,y,z)=g(w).\end{array}$ (2.4)
Notice that if $g$ is identity mapping, then Definition 2.3 and Definition2.4
reduce to Definition 2.1 and Definition2.2, respectively.
###### Definition 15.
Let $F:X^{4}\rightarrow X$ and $g:X\rightarrow X$. $F$ and $g$ are called
commutative if
$g(F(x,y,z,w))=F(g(x),g(y),g(z),g(w)),\ \mbox{ for all }x,y,z,w\in X.\\\ $
(2.5)
For a metric space $(X,d)$, the function $\rho:X^{4}\times
X^{4}\rightarrow[0,\infty)$, given by,
$\rho((x,y,z,w),(u,v,r,t)):=d(x,u)+d(y,v)+d(z,r)+d(w,t)$
forms a metric space on $X^{4}$, that is, $(X^{4},\rho)$ is a metric induced
by $(X,d)$.
Let $\Phi$ denote the all functions $\phi:[0,\infty)\rightarrow[0,\infty)$
which is continuous and satisfy that
* $(i)$
$\phi(t)<t$
* $(i)$
$\lim_{r\rightarrow t+}\phi(r)<t$ for each $r>0$.
The aim of this paper is to prove the following theorem.
###### Theorem 16.
Let $(X,\leq)$ be partially ordered set and $(X,d)$ be a complete metric
space. Suppose $F:X^{4}\rightarrow X$ and there exists $\phi\in\Phi$ such that
$F$ has the mixed $g$-monotone property and
$d(F(x,y,z,w),F(u,v,r,t))\leq\phi\left(\frac{d(x,u)+d(y,v)+d(z,r)+d(w,t)}{4}\right)$
(2.6)
for all $x,u,y,v,z,r,w,t$ for which $g(x)\leq g(u)$, $g(y)\geq g(v)$,
$g(z)\leq g(r)$ and $g(w)\geq g(t)$. Suppose there exist
$x_{0},y_{0},z_{0},w_{0}\in X$ such that
$\begin{array}[]{c}g(x_{0})\leq F(x_{0},y_{0},z_{0},w_{0}),\ \ g(y_{0})\geq
F(x_{0},w_{0},z_{0},y_{0}),\\\ \ \ g(z_{0})\leq F(z_{0},y_{0},x_{0},w_{0}),\ \
g(w_{0})\geq F(z_{0},w_{0},x_{0},y_{0}).\\\ \end{array}$ (2.7)
Assume also that $F(X^{4})\subset g(X)$ and $g$ commutes with $F$. Suppose
either
* $(a)$
$F$ is continuous, or
* $(b)$
$X$ has the following property:
* $(i)$
if non-decreasing sequence $x_{n}\rightarrow x$, then $x_{n}\leq x$ for all
$n,$
* $(ii)$
if non-increasing sequence $y_{n}\rightarrow y$, then $y_{n}\geq y$ for all
$n$,
then there exist $x,y,z,w\in X$ such that
$\begin{array}[]{c}F(x,y,z,w)=g(x),\ \ \ \ F(x,w,z,y)=g(y),\\\
F(z,y,x,w)=g(z),\ \ \ \ F(z,w,x,y)=g(w).\end{array}$
that is, $F$ and $g$ have a common coincidence point.
###### Proof.
Let $x_{0},y_{0},z_{0},w_{0}\in X$ be such that (2.7). We construct the
sequences $\\{x_{n}\\}$, $\\{y_{n}\\}$, $\\{z_{n}\\}$ and $\\{w_{n}\\}$ as
follows
$\begin{array}[]{c}g(x_{n})=F(x_{n-1},y_{n-1},z_{n-1},w_{n-1}),\\\
g(y_{n})=F(x_{n-1},w_{n-1},z_{n-1},y_{n-1}),\\\
g(z_{n})=F(z_{n-1},y_{n-1},x_{n-1},w_{n-1}),\\\
g(w_{n})=F(z_{n-1},w_{n-1},x_{n-1},y_{n-1}).\\\ \end{array}$ (2.8)
for $n=1,2,3,....$.
We claim that
$\begin{array}[]{rl}g(x_{n-1})\leq g(x_{n}),&g(y_{n-1})\geq g(y_{n}),\\\
g(z_{n-1})\leq g(z_{n}),&g(w_{n-1})\geq g(w_{n}),\mbox{ for all }n\geq 1.\\\
\end{array}$ (2.9)
Indeed, we shall use mathematical induction to prove (2.9). Due to (2.7), we
have
$\begin{array}[]{c}g(x_{0})\leq F(x_{0},y_{0},z_{0},w_{0})=g(x_{1}),\ \
g(y_{0})\geq F(x_{,}w_{0},z_{0},y_{0})=g(y_{1}),\\\ \ \ g(z_{0})\leq
F(z_{0},y_{0},x_{0},w_{0})=g(z_{1}),\ \ g(w_{0})\geq
F(z_{0},w_{0},x_{0},y_{0})=g(w_{1}).\\\ \end{array}$
Thus, the inequalities in (2.9) hold for $n=1$. Suppose now that the
inequalities in (2.9) hold for some $n\geq 1$. By mixed $g$-monotone property
of $F$, together with (2.8) and (2.3) we have
$\begin{array}[]{c}g(x_{n})=F(x_{n-1},y_{n-1},z_{n-1},w_{n-1})\leq
F(x_{n},y_{n},z_{n},w_{n})=g(x_{n+1}),\\\
g(y_{n})=F(x_{n-1},w_{n-1},z_{n-1},y_{n-1})\geq
F(x_{n},w_{n},z_{n},y_{n})=g(y_{n+1}),\\\
g(z_{n})=F(z_{n-1},y_{n-1},x_{n-1},w_{n-1})\leq
F(z_{n},y_{n},x_{n},w_{n})=g(z_{n+1}),\\\
g(w_{n})=F(z_{n-1},w_{n-1},x_{n-1},y_{n-1})\geq
F(z_{n-1},w_{n-1},x_{n-1},y_{n-1})=g(w_{n+1}),\\\ \end{array}$ (2.10)
Thus, (2.9) holds for all $n\geq 1$. Hence, we have
$\begin{array}[]{c}\cdots g(x_{n})\geq g(x_{n-1})\geq\cdots\geq g(x_{1})\geq
g(x_{0}),\\\ \cdots g(y_{n})\leq g(y_{n-1})\leq\cdots\leq g(y_{1})\leq
g(y_{0}),\\\ \cdots g(z_{n})\geq g(z_{n-1})\geq\cdots\geq g(z_{1})\geq
g(z_{0}),\\\ \cdots g(w_{n})\leq g(w_{n-1})\leq\cdots\leq g(w_{1})\leq
g(w_{0}),\\\ \end{array}$ (2.11)
Set
$\begin{array}[]{c}\delta_{n}=d(g(x_{n}),g(x_{n+1}))+d(g(y_{n}),g(y_{n+1}))+d(g(z_{n}),g(z_{n+1}))\\\
+d(g(w_{n}),g(w_{n+1}))\end{array}$
We shall show that
$\delta_{n+1}\leq 4\phi(\frac{\delta_{n}}{4}).$ (2.12)
Due to (2.6), (2.8) and (2.11), we have
$\begin{array}[]{rl}d(g(x_{n+1}),g(x_{n+2}))&=d(F(x_{n},y_{n},z_{n},w_{n}),F(x_{n+1},y_{n+1},z_{n+1},w_{n+1}))\\\
&\phi\left(\frac{d(g(x_{n}),g(x_{n+1}))+d(g(y_{n}),g(y_{n+1}))+d(g(z_{n}),g(z_{n+1}))+d(g(w_{n}),g(w_{n+1}))}{4}\right)\\\
&\leq\phi(\frac{\delta_{n}}{4})\end{array}$ (2.13)
$\begin{array}[]{rl}d(g(y_{n+1}),g(y_{n+2}))&=d(F(y_{n},z_{n},w_{n},x_{n}),F(y_{n+1},z_{n+1},w_{n+1},x_{n+1}))\\\
&\leq\phi\left(\frac{d(g(y_{n}),g(y_{n+1}))+d(g(z_{n}),g(z_{n+1}))+d(g(w_{n}),g(w_{n+1}))+d(g(x_{n}),g(x_{n+1}))}{4}\right)\\\
&\leq\phi(\frac{\delta_{n}}{4})\end{array}$ (2.14)
$\begin{array}[]{rl}d(g(z_{n+1}),g(z_{n+2}))&=d(F(z_{n},w_{n},x_{n},y_{n}),F(z_{n+1},w_{n+1},x_{n+1},y_{n+1}))\\\
&\leq\phi\left(\frac{d(g(z_{n}),g(z_{n+1}))+d(g(w_{n}),g(w_{n+1}))+d(g(x_{n}),g(x_{n+1}))+d(g(y_{n}),g(y_{n+1}))}{4}\right)\\\
&\leq\phi(\frac{\delta_{n}}{4})\end{array}$ (2.15)
$\begin{array}[]{rl}d(g(w_{n+1}),g(w_{n+2}))&=d(F(w_{n},x_{n},y_{n},z_{n}),F(w_{n+1},x_{n+1},y_{n+1},z_{n+1}))\\\
&\phi\left(\frac{d(g(w_{n}),g(w_{n+1}))+d(g(x_{n}),g(x_{n+1}))+d(g(y_{n}),g(y_{n+1}))+d(g(z_{n}),g(z_{n+1}))}{4}\right)\\\
&\leq\phi(\frac{\delta_{n}}{4})\end{array}$ (2.16)
Due to (2.13)-(2.16), we conclude that
$d(x_{n+1},x_{n+2})+d(y_{n+1},y_{n+2})+d(z_{n+1},z_{n+2})+d(w_{n+1},w_{n+2})\leq
4\phi(\frac{\delta_{n}}{4})$ (2.17)
Hence we have (2.12).
Since $\phi(t)<t$ for all $t>0$, then $\delta_{n+1}\leq\delta_{n}$ for all
$n$. Hence $\\{\delta_{n}\\}$ is a non-increasing sequence. Since it is
bounded below, there is some $\delta\geq 0$ such that
$\lim_{n\rightarrow\infty}\delta_{n}=\delta+.$ (2.18)
We shall show that $\delta=0$. Suppose, to the contrary, that $\delta>0$.
Taking the limit as $\delta_{n}\rightarrow\delta+$ of both sides of (2.12) and
having in mind that we suppose $\lim_{t\rightarrow r}\phi(r)<t$ for all $t>0$,
we have
$\delta=\lim_{n\rightarrow\infty}\delta_{n+1}\leq\lim_{n\rightarrow\infty}4\phi(\frac{\delta_{n}}{4})=\lim_{\delta_{n}\rightarrow\delta+}4\phi(\frac{\delta_{n}}{4})<4\frac{\delta}{4}<\delta$
(2.19)
which is a contradiction. Thus, $\delta=0$, that is,
$\lim_{n\rightarrow\infty}[d(x_{n},x_{n-1})+d(y_{n},y_{n-1})+d(z_{n},z_{n-1})+d(w_{n},w_{n-1})]=0.$
(2.20)
Now, we shall prove that $\\{g(x_{n})\\}$,$\\{g(y_{n})\\}$,$\\{g(z_{n})\\}$
and $\\{g(w_{n})\\}$ are Cauchy sequences. Suppose, to the contrary, that at
least one of $\\{g(x_{n})\\}$,$\\{g(y_{n})\\}$,$\\{g(z_{n})\\}$ and
$\\{g(w_{n})\\}$ is not Cauchy. So, there exists an $\varepsilon>0$ for which
we can find subsequences $\\{g(x_{n(k)})\\}$, $\\{g(x_{n(k)})\\}$ of
$\\{g(x_{n})\\}$ and $\\{g(y_{n(k)})\\}$, $\\{g(y_{n(k)})\\}$ of
$\\{g(y_{n})\\}$ and $\\{g(z_{n(k)})\\}$, $\\{g(z_{n(k)})\\}$ of
$\\{g(z_{n})\\}$ and $\\{g(w_{n(k)})\\}$, $\\{g(w_{n(k)})\\}$ of
$\\{g(w_{n})\\}$ with $n(k)>m(k)\geq k$ such that
$\begin{array}[]{r}d(g(x_{n(k)}),g(x_{m(k)}))+d(g(y_{n(k)}),g(y_{m(k)}))\\\
+d(g(z_{n(k)}),g(z_{m(k)}))+d(g(w_{n(k)}),g(w_{m(k)}))\geq\varepsilon.\end{array}$
(2.21)
Additionally, corresponding to $m(k)$, we may choose $n(k)$ such that it is
the smallest integer satisfying (2.21) and $n(k)>m(k)\geq k$. Thus,
$\begin{array}[]{r}d(g(x_{n(k)-1}),g(x_{m(k)}))+d(g(y_{n(k)-1}),g(y_{m(k)}))\\\
+d(g(z_{n(k)-1}),g(z_{m(k)}))+d(g(w_{n(k)-1}),g(w_{m(k)}))<\varepsilon.\end{array}$
(2.22)
By using triangle inequality and having (2.21),(2.22) in mind
$\begin{array}[]{rl}\varepsilon&\leq
t_{k}=:d(g(x_{n(k)}),g(x_{m(k)}))+d(g(y_{n(k)}),g(y_{m(k)}))\\\
&+d(g(z_{n(k)}),g(z_{m(k)}))+d(g(w_{n(k)}),g(w_{m(k)}))\\\ &\leq
d(g(x_{n(k)}),g(x_{n(k)-1}))+d(g(x_{n(k)-1}),g(x_{m(k)}))\\\
&+d(g(y_{n(k))},g(y_{n(k)-1}))+d(g(y_{n(k)-1}),g(y_{m(k)}))\\\ &\ \
+d(g(z_{n(k)}),g(z_{n(k)-1}))+d(g(z_{n(k)-1}),g(z_{m(k)}))\\\
&+d(g(w_{n(k)}),g(w_{n(k)-1}))+d(g(w_{n(k)-1}),g(w_{m(k)}))\\\
&<d(g(x_{n(k)}),g(x_{n(k)-1}))+d(g(y_{n(k)}),g(y_{n(k)-1}))+\\\
&d(g(z_{n(k)}),g(z_{n(k)-1}))+d(g(w_{n(k)}),g(w_{n(k)-1}))+\varepsilon.\end{array}$
(2.23)
Letting $k\rightarrow\infty$ in (2.23) and using (2.20)
$\lim_{k\rightarrow\infty}t_{k}=\lim_{k\rightarrow\infty}\left[\begin{array}[]{c}d(g(x_{n(k)}),g(x_{m(k)}))+d(g(y_{n(k)}),g(y_{m(k)}))\\\
+d(g(z_{n(k)}),g(z_{m(k)}))+d(g(w_{n(k)}),g(w_{m(k)}))\end{array}\right]=\varepsilon+$
(2.24)
Again by triangle inequality,
$\begin{array}[]{rl}t_{k}&=d(g(x_{n(k)}),g(x_{m(k)}))+d(g(y_{n(k)}),g(y_{m(k)}))\\\
&+d(g(z_{n(k)}),g(z_{m(k)}))+d(g(w_{n(k)}),g(w_{m(k)}))\\\ &\leq
d(g(x_{n(k)}),g(x_{n(k)+1}))+d(g(x_{n(k)+1}),g(x_{m(k)+1}))+d(g(x_{m(k)+1}),g(x_{m(k)}))\\\
&\
+d(g(y_{n(k)}),g(y_{n(k)+1}))+d(g(y_{n(k)+1}),g(y_{m(k)+1}))+d(g(y_{m(k)+1}),g(y_{m(k)}))\\\
&\
+d(g(z_{n(k)}),g(z_{n(k)+1}))+d(g(z_{n(k)+1}),g(z_{m(k)+1}))+d(g(z_{m(k)+1}),g(z_{m(k)}))\\\
&\
+d(g(w_{n(k)}),g(w_{n(k)+1}))+d(g(w_{n(k)+1}),g(w_{m(k)+1}))+d(g(w_{m(k)+1}),g(w_{m(k)}))\\\
&\leq\delta_{n(k)+1}+\delta_{m(k)+1}+d(g(x_{n(k)+1}),g(x_{m(k)+1}))+d(g(y_{n(k)+1}),g(y_{m(k)+1}))\\\
&\ \
+d(g(z_{n(k)+1}),g(z_{m(k)+1}))+d(g(w_{n(k)+1}),g(w_{m(k)+1}))\end{array}$
(2.25)
Since $n(k)>m(k)$, then
$\begin{array}[]{c}g(x_{n(k)})\geq g(x_{m(k)})\mbox{ and }g(y_{n(k)})\leq
g(y_{m(k)}),\\\ g(z_{n(k)})\geq g(z_{m(k)})\mbox{ and }g(w_{n(k)})\leq
g(w_{m(k)}).\\\ \end{array}$ (2.26)
Hence from (2.26), (2.8) and (2.6), we have,
$\begin{array}[]{rl}d(g(x_{n(k)+1}),g(x_{m(k)+1}))&=d(F({x_{n(k)}},{y_{n(k)}},{z_{n(k)}},{w_{n(k)}}),F({x_{m(k)}},{y_{m(k)}},{z_{m(k)}},{w_{m(k)}}))\\\
&\leq\phi\left(\begin{array}[]{l}\frac{1}{4}[d(g({x_{n(k)}}),g({x_{m(k)}}))+d(g({y_{n(k)}}),g({y_{m(k)}}))\\\
+d(g({z_{n(k)}}),g({z_{m(k)}}))+d(g({w_{n(k)}}),g({w_{m(k)}}))]\\\
\end{array}\right)\end{array}$ (2.27)
$\begin{array}[]{rl}d(g(y_{n(k)+1}),g(y_{m(k)+1}))&=d(F({y_{n(k)}},{z_{n(k)}},{w_{n(k)}},{x_{n(k)}}),F({y_{m(k)}},{z_{m(k)}},{w_{m(k)}},{x_{m(k)}}))\\\
&\leq\phi\left(\begin{array}[]{l}\frac{1}{4}[d(g({y_{n(k)}}),g({y_{m(k)}}))+d(g({z_{n(k)}}),g({z_{m(k)}}))\\\
+d(g({w_{n(k)}}),g({w_{m(k)}}))+d(g({x_{n(k)}},{x_{m(k)}}))]\\\
\end{array}\right)\\\ \end{array}$ (2.28)
$\begin{array}[]{rl}d(g(z_{n(k)+1}),g(z_{m(k)+1}))&=d(F({z_{n(k)}},{w_{n(k)}},{x_{n(k)}},{y_{n(k)}}),F({z_{m(k)}},{w_{m(k)}},{x_{m(k)}},{y_{m(k)}}))\\\
&\leq\phi\left(\frac{1}{4}[\begin{array}[]{l}d(g(z_{n(k)}),g({z_{m(k)}}))+d(g({w_{n(k)}}),g({w_{m(k)}}))\\\
+d(g({x_{n(k)}}),g({x_{m(k)}}))+dg(({y_{n(k)}}),g({y_{m(k)}}))]\\\
\end{array}\right)\\\ \end{array}$ (2.29)
$\begin{array}[]{rl}d(g(w_{n(k)+1}),g(w_{m(k)+1})&=d(F({w_{n(k)}},{x_{n(k)}},{y_{n(k)}},{z_{n(k)}}),F({w_{m(k)}},{x_{m(k)}},{y_{m(k)}},{z_{m(k)}})\\\
&\leq\phi\left(\begin{array}[]{l}\frac{1}{4}[d(g({w_{n(k)}}),g({w_{m(k)}}))+d(g({x_{n(k)}}),g({x_{m(k)}}))\\\
+d(g({y_{n(k)}}),g({y_{m(k)}}))+d(g({z_{n(k)}}),g({z_{m(k)}}))]\\\
\end{array}\right)\\\ \end{array}$ (2.30)
Combining (2.25) with (2.27)-(2.30), we obtain that
$\begin{array}[]{rl}t_{k}&\leq\delta_{n(k)+1}+\delta_{m(k)+1}+d(g(x_{n(k)+1}),g(x_{m(k)+1})+d(g(y_{n(k)+1}),g(y_{m(k)+1}))\\\
&\ \ +d(g(z_{n(k)+1}),g(z_{m(k)+1}))+d(g(w_{n(k)+1}),g(w_{m(k)+1})))\\\
&\leq\delta_{n(k)+1}+\delta_{m(k)+1}+t_{k}+4\phi\left(\frac{t_{k}}{4}\right)\\\
&<\delta_{n(k)+1}+\delta_{m(k)+1}+t_{k}+4\frac{t_{k}}{4}\end{array}$ (2.31)
Letting $k\rightarrow\infty$, we get a contradiction. This shows that
$\\{g(x_{n})\\}$,$\\{g(y_{n})\\}$ ,$\\{g(z_{n})\\}$ and $\\{g(w_{n})\\}$ are
Cauchy sequences. Since $X$ is complete metric space, there exists $x,y,z,w\in
X$ such that
$\begin{array}[]{c}\lim_{n\rightarrow\infty}g(x_{n})=x\mbox{ and
}\lim_{n\rightarrow\infty}g(y_{n})=y,\\\
\lim_{n\rightarrow\infty}g(z_{n})=z\mbox{ and
}\lim_{n\rightarrow\infty}g(w_{n})=w.\\\ \end{array}$ (2.32)
Since $g$ is continuous, (2.32) implies that
$\begin{array}[]{c}\lim_{n\rightarrow\infty}g(g(x_{n}))=g(x)\mbox{ and
}\lim_{n\rightarrow\infty}g(g(y_{n}))=g(y),\\\
\lim_{n\rightarrow\infty}g(g(z_{n}))=g(z)\mbox{ and
}\lim_{n\rightarrow\infty}g(g(w_{n}))=g(w).\\\ \end{array}$ (2.33)
From (2.10) and by regarding commutativity of $F$ and $g$,
$\begin{array}[]{c}g(g(x_{n+1}))=g(F(x_{n},y_{n},z_{n},w_{n}))=F(g(x_{n}),g(y_{n}),g(z_{n}),g(w_{n})),\\\
g(g(y_{n+1}))=g(F(x_{n},w_{n},z_{n},y_{n}))=F(g(x_{n}),g(w_{n}),g(z_{n}),g(y_{n})),\\\
g(g(z_{n+1}))=g(F(z_{n},y_{n},x_{n},w_{n}))=F(g(z_{n}),g(y_{n}),g(x_{n}),g(w_{n})),\\\
g(g(w_{n+1}))=g(F(z_{n},w_{n},x_{n},y_{n}))=F(g(z_{n}),g(w_{n}),g(x_{n}),g(y_{n})),\\\
\end{array}$ (2.34)
We shall show that
$\begin{array}[]{c}F(x,y,z,w)=g(x),\ \ \ \ F(x,w,z,y)=g(y),\\\
F(z,y,x,w)=g(z),\ \ \ \ F(z,w,x,y)=g(w).\end{array}$
Suppose now $(a)$ holds. Then by (2.8),(2.34) and (2.32), we have
$\begin{array}[]{rl}g(x)&=\displaystyle\lim_{n\rightarrow\infty}g(g(x_{n+1}))=\lim_{n\rightarrow\infty}g(F(x_{n},y_{n},z_{n},w_{n}))\\\
&=\displaystyle\lim_{n\rightarrow\infty}F(g(x_{n}),g(y_{n}),g(z_{n}),g(w_{n}))\\\
&=F(\displaystyle\lim_{n\rightarrow\infty}g(x_{n}),\displaystyle\lim_{n\rightarrow\infty}g(y_{n}),\displaystyle\lim_{n\rightarrow\infty}g(z_{n}),\displaystyle\lim_{n\rightarrow\infty}g(w_{n}))\\\
&=F(x,y,z,w)\\\ \end{array}$ (2.35)
Analogously, we also observe that
$\begin{array}[]{rl}g(y)&=\displaystyle\lim_{n\rightarrow\infty}g(g(y_{n+1}))=\lim_{n\rightarrow\infty}g(F(x_{n},w_{n},z_{n},y_{n})\\\
&=\displaystyle\lim_{n\rightarrow\infty}F(g(x_{n}),g(w_{n}),g(z_{n}),g(y_{n}))\\\
&=F(\displaystyle\lim_{n\rightarrow\infty}g(x_{n}),\displaystyle\lim_{n\rightarrow\infty}g(w_{n}),\displaystyle\lim_{n\rightarrow\infty}g(z_{n}),\displaystyle\lim_{n\rightarrow\infty}g(y_{n}))\\\
&=F(x,w,z,y)\\\ \end{array}$ (2.36)
$\begin{array}[]{rl}g(z)&=\displaystyle\lim_{n\rightarrow\infty}g(g(z_{n+1}))=\lim_{n\rightarrow\infty}g(F(z_{n},y_{n},x_{n},w_{n}))\\\
&=\displaystyle\lim_{n\rightarrow\infty}F(g(z_{n}),g(y_{n}),g(x_{n}),g(w_{n}))\\\
&=F(\displaystyle\lim_{n\rightarrow\infty}g(z_{n}),\displaystyle\lim_{n\rightarrow\infty}g(y_{n}),\displaystyle\lim_{n\rightarrow\infty}g(x_{n}),\displaystyle\lim_{n\rightarrow\infty}g(w_{n}))\\\
&=F(z,y,x,w)\\\ \end{array}$ (2.37)
$\begin{array}[]{rl}g(w)&=\displaystyle\lim_{n\rightarrow\infty}g(g(w_{n+1}))=\lim_{n\rightarrow\infty}g(F(z_{n},w_{n},x_{n},y_{n}))\\\
&=\displaystyle\lim_{n\rightarrow\infty}F(g(z_{n}),g(w_{n}),g(x_{n}),g(y_{n}))\\\
&=F(\displaystyle\lim_{n\rightarrow\infty}g(z_{n}),\displaystyle\lim_{n\rightarrow\infty}g(w_{n}),\displaystyle\lim_{n\rightarrow\infty}g(x_{n}),\displaystyle\lim_{n\rightarrow\infty}g(y_{n}))\\\
&=F(z,w,x,y)\\\ \end{array}$ (2.38)
Thus, we have
$\begin{array}[]{c}F(x,y,z,w)=g(x),\ \ \ \ F(y,z,w,x)=g(y),\\\
F(z,,w,x,y)=g(z),\ \ \ \ F(w,x,y,z)=g(w).\end{array}$
Suppose now the assumption $(b)$ holds. Since $\\{g(x_{n})\\},\
\\{g(z_{n})\\}$ is non-decreasing and $g(x_{n})\rightarrow x,\
g(z_{n})\rightarrow z$ and also $\\{g(y_{n})\\},\ \\{g(w_{n})\\}$ is non-
increasing and $g(y_{n})\rightarrow y,\ g(w_{n})\rightarrow$, then by
assumption $(b)$ we have
$g(x_{n})\geq x,\ \ g(y_{n})\leq y,\ \ g(z_{n})\geq z,\ \ g(w_{n})\leq w$
(2.39)
for all $n$. Thus, by triangle inequality and (2.34)
$\begin{array}[]{l}d(g(x),F(x,y,z,w))\leq
d(g(x),g(g(x_{n+1})))+d(g(g(x_{n+1})),F(x,y,z,w))\\\ \leq
d(g(x),g(g(x_{n+1})))+\phi\left(\frac{1}{4}\left[\begin{array}[]{c}d(g(g(x_{n}),g(x)))+d(g(g(y_{n}),g(y)))\\\
+d(g(g(z_{n}),g(z)))+d(g(g(w_{n}),g(w)))\end{array}\right]\right)\\\
\end{array}$ (2.40)
Letting $n\rightarrow\infty$ implies that $d(g(x),F(x,y,z,w))\leq 0$. Hence,
$g(x)=F(x,y,z,w)$. Analogously we can get that
$F(y,z,w,x)=g(y),F(z,w,x,y)=g(z)\mbox{ and }F(w,x,y,z)=g(w).$
Thus, we proved that $F$ and $g$ have a quartet coincidence point. ∎
## References
* [1] V. Berinde and M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, _Nonlinear Analysis_ , 74(15), 4889–4897 (2011).
* [2] Bhaskar, T.G., Lakshmikantham, V.: Fixed Point Theory in partially ordered metric spaces and applications _Nonlinear Analysis_ , 65, 1379–1393 (2006).
* [3] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, _Nonlinear Analysis_ , 74, 983- 992(2011).
* [4] B. Samet, Coupled fixed point theorems for a generalized Meir Keeler contraction in partially ordered metric spaces, _Nonlinear Analysis_ , 74(12), 4508 4517(2010).
* [5] E. Karapınar, Couple Fixed Point on Cone Metric Spaces, _Gazi University Journal of Science_ , 24(1),51-58(2011).
* [6] E. Karapınar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, _Comput. Math. Appl._ , 59 (12), 3656- 3668(2010).
* [7] E. Karapınar, N.V.Luong, Quartet Fixed Point Theorems for nonlinear contractions, submitted.
* [8] Lakshmikantham, V., Ćirić, L.: : Couple Fixed Point Theorems for nonlinear contractions in partially ordered metric spaces _Nonlinear Analysis_ , 70, 4341-4349 (2009).
* [9] Nieto, J. J., Rodriguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. _Order_ 2005(22), 3, 223–239 (2006).
* [10] Binayak S. Choudhury, N. Metiya and A. Kundu, Coupled coincidence point theorems in ordered metric spaces, _Ann. Univ. Ferrara_ , 57, 1 -16(2011).
* [11] B.S. Choudhury, A. Kundu : A coupled coincidence point result in partially ordered metric spaces for compatible mappings. _Nonlinear Anal._ TMA 73, 2524 2531 (2010)
|
arxiv-papers
| 2011-06-27T18:38:24 |
2024-09-04T02:49:20.108594
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Erdal Karapinar",
"submitter": "Erdal Karapinar",
"url": "https://arxiv.org/abs/1106.5472"
}
|
1106.5498
|
# The host galaxies of radio-loud AGN: colour structure
E. J. A. Mannering1, D. M. Worrall1, M. Birkinshaw1
1H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol
BS8 1TL, UK E-mail: l.mannering@bristol.ac.uk
###### Abstract
We construct a sample of 3,516 radio-loud host galaxies of active galactic
nuclei (AGN) from the optical Sloan Digital Sky Survey (SDSS) and Faint Images
of the Radio Sky at Twenty cm (FIRST). These have 1.4 GHz luminosities in the
range $10^{23}-10^{25}$ WHz-1, span redshifts $0.02<z<0.18$, are brighter than
$r^{*}_{petro}<17.77$ mag and are constrained to ‘early-type’ morphology in
colour space ($u^{*}-r^{*}>2.22$ mag). Optical emission line ratios (at
$>3\sigma$) are used to remove type 1 AGN and star-forming galaxies from the
radio sample using BPT diagnostics. For comparison, we select a sample of
35,160 radio-quiet galaxies with the same $r^{*}$-band magnitude-redshift
distribution as the radio sample. We also create comparison radio and control
samples derived by adding the NRAO VLA Sky Survey (NVSS) to quantify the
effect of completeness on our results.
We investigate the effective radii of the surface brightness profiles in the
SDSS $r$ and $u$ bands in order to quantify any excess of blue colour in the
inner region of radio galaxies. We define a ratio $R=r_{e}(r)/r_{e}(u)$ and
use maximum likelihood analysis to compare the average value of $R$ and its
intrinsic dispersion between both samples. $R$ is larger for the radio-loud
AGN sample as compared to its control counterpart, and we conclude that the
two samples are not drawn from the same population at $>99\%$ significance.
Given that star formation proceeds over a longer time than radio activity, the
difference suggests that a subset of galaxies has the predisposition to become
radio loud. We discuss host galaxy features that cause the presence of a
radio-loud AGN to increase the scale size of a galaxy in red relative to blue
light, including excess central blue emission, point-like blue emission from
the AGN itself, and/or diffuse red emission. We favour an explanation that
arises from the stellar rather than the AGN light.
## 1 Introduction
Considerable uncertainties remain as to what controls the apparent link
between the activity of active galactic nuclei (AGN) during the time that a
black hole is fed and star formation. The accretion disk surrounding a super-
massive black hole (SMBH) emits highly energetic radiation and particles, and
can form powerful winds and/or collimated, relativistic jets (e.g., van
Breugel et al., 2004). The radiation and outflows might then affect the
interstellar medium, triggering star formation which might be detectable as an
excess of blue light from the central regions of galaxies. Alternatively, the
activity of the AGN may be sparked by specific events in the galaxy’s past.
For example, there is morphological evidence that activity in radio galaxies
might be triggered by mergers and galaxy interactions (e.g., González Delgado
et al., 2006), which in turn could contribute to central blue light through
enhanced star formation. In either case, we might expect an observable
association between AGN activity, for which in this work we use radio loudness
as a tracer, with bluer stellar continuum in the central regions of the
galaxy.
Previous work has hinted at such a relationship. For example, Mahabal et al.
(1999) examined the central light of 30 radio galaxies as compared to 30
normal galaxies from the Molongo Reference Catalogue and found excess blue
light in the inner regions of the radio galaxies as compared with the control
sample. This central excess is unrelated to the conventional colour gradient
of the broader distribution of stars. Gonzalez-Perez et al. (2010) studied
colour variations within galaxies from the Sloan Digital Sky Survey (SDSS;
York et al., 2000) DR7, finding marginally steeper colour gradients in massive
galaxies with nuclear activity and concluded that this is due to a higher
fraction of young stars in their central regions.
In the present paper we expand the work of Mahabal et al. (1999), by combining
optical information from SDSS with the FIRST and NVSS radio surveys (Becker et
al., 1995; Condon et al., 1998) to construct a local ($0.02<z<0.18$) sample of
powerful radio-loud AGN host galaxies (R-AGN), and a control sample of
‘normal’ early-type ellipticals. We define $R$ as the ratio of SDSS $r^{*}$ to
$u^{*}$ band de Vaucouleurs effective radii and compare its value between the
samples.
The paper is structured as follows. In Section 2 we briefly describe the
surveys used to construct our samples. Separation into the radio-loud and
control samples is discussed in Section 3. Section 4 outlines the maximum
likelihood analysis used to compare the average values of $R$ for the two
samples. In Section 5 we present a discussion of our findings in the context
of radio-AGN fuelling mechanisms and the properties of the host galaxies.
Throughout, we use a flat $\Lambda$CDM cosmology with $\Omega_{m_{0}}=0.3$ and
$\Omega_{\Lambda 0}=0.7$. We adopt $H_{0}=70$ km s-1 Mpc-1.
## 2 Data Sources
### 2.1 SDSS
SDSS is an imaging and spectroscopic survey covering $\sim$10,000 deg2,
primarily in the northern hemisphere. Its 7th data release (SDSS-DR7;
Abazajian et al., 2009) contains over 350 million entries, of which about one
million are confirmed galaxies with spectroscopic follow up.
In the photometric sample, flux densities are measured simultaneously in five
broadband filters ($u,g,r,i,z$), with effective wavelengths $3551$, $4686$,
$6165$, $7481$ and $8931$Å (Fukugita et al., 1996). All magnitudes are given
on the ABv system (Oke & Gunn, 1983), and are determined using Petrosian
apertures (see Blanton et al., 2001; Graham et al., 2005, for a complete
definition of the Petrosian system). In common with Ivezić et al. (2002), we
refer to SDSS measured magnitudes as $u^{*},g^{*},r^{*},i^{*}$ and $z^{*}$ due
to the uncertainty in the absolute calibration of the SDSS photometric system,
which has been assessed at $\lesssim$0.03 mag (Stoughton & Lupton, 2002). For
a complete guide to the photometric system, see Fukugita et al. (1996), Gunn
et al. (2001) and Stoughton & Lupton (2002).
#### 2.1.1 Main galaxy sample (MGS)
We use the main galaxy survey spectroscopic sample (MGS, Strauss et al.,
2002). The MGS algorithm selected extended sources brighter than
$r^{*}_{petro}=17.77$ mag with Petrosian half-light surface brightness
$\mu_{50}\leq 24.5$ mag arcsec-2, providing $\sim$ 90 galaxies deg-2 (Blanton
et al., 2001). Candidate galaxies were then observed spectroscopically.
$99.9\%$ of the resulting redshifts have velocity errors $<$ 30 km s-1. The
survey is unbiased, except for galaxies with close companions. Around $6\%$ of
galaxies satisfying the photometric target criteria were not included for
spectroscopic follow up due to a companion galaxy within the 55″ minimum fibre
separation, although some of these galaxies have been subsequently observed.
The MGS contains $\sim$680,000 spectroscopically confirmed galaxies. The full
target selection is described in Strauss et al. (2002).
We corrected for Galactic extinction using the SDSS ‘reddening’ parameter,
which was derived from maps of the infrared emission from dust across the sky
in accordance with Schlegel, Finkbeiner & Davis (1998). We applied
$k$-corrections using the SDSS-derived ‘kcorr’ parameter as detailed in
Blanton & Roweis (2007). Typical corrections $k_{u},k_{g},k_{r}$ were
$(0-0.4)$, $(0-0.3)$, and $(0-0.1)$ mag.
### 2.2 FIRST
The FIRST survey (Becker et al., 1995) utilized the VLA (at 1.4 GHz in B
array) to map the radio sky over $\sim$9000 deg2 in the Northern hemisphere
and in a $2\overset{{}^{\circ}}{.}5$ wide strip along the celestial equator.
It has $\sim 8400$ deg2 overlap with SDSS, contains $\sim 97$ sources deg-2 at
the 1 mJy survey threshold and reaches an rms sensitivity of $\sim$0.15 mJy
beam-1. In this configuration, the VLA has a synthesized beam of 5.4″ FWHM,
providing accurate flux densities for small-scale radio structures, but
underestimating the flux densities of sources extended to several arcminutes.
At the 1 mJy survey threshold, individual sources have 90% confidence
astrometric errors $\lesssim 1\arcsec$. We adopted a mean spectral index of
$\alpha=0.7$ (where $S_{\nu}\propto\nu^{-\alpha}$) and obtained rest-frame 1.4
GHz power densities by applying k-corrections assuming the usual form
$L_{\nu,rest}=(1+z)^{(\alpha-1)}S_{\nu}\,4\pi D_{L}^{2}$ (1)
where $D_{L}$ is the luminosity distance.
### 2.3 NVSS
The NRAO VLA Sky Survey (NVSS, Condon et al., 1998) mapped the radio sky (at
1.4 GHz in D array) north of $-40\overset{{}^{\circ}}{}$ declination. The
survey is complete down to a point source flux density of $\sim$ 2.5 mJy. The
synthesized beam of 45″ is much larger than FIRST, providing more accurate
flux measurements for highly extended sources. Astrometric accuracy ranges
from 1″ for bright sources to 7″ for the faintest detections. The entire
survey contains over 1.8 million unique sources brighter than 2.5 mJy.
$k$-corrections are applied as in §2.2.
## 3 Sample selection
### 3.1 Cross-correlation: FIRST-SDSS
We initially derived a FIRST-SDSS sample of radio galaxies and a comparison
control sample with no nearby detectable radio counterpart. Figure
LABEL:fig:idlsep shows the distribution of angular separations between SDSS
objects and their nearest FIRST counterpart. We find that at $\sim$ 5″ the
distribution becomes dominated by random matches. Hence we adopt a match
criterion of 2″ angular separation to avoid such false matches. Our cross-
matched catalogue contains $25,931$ galaxies and we denote this as our
preliminary ‘radio’ sample. All unmatched sources are initial candidates for
our control sample of radio-quiet galaxies.
We define our efficiency as the fraction of matches in our radio sample which
are physically real (including contamination from line of sight false matches,
which cannot be accounted for here). In order to evaluate the number of random
false matches in our sample, we offset the RA and DEC of the FIRST sources by
$1^{\circ}$ and re-matched to our set of $680,056$ SDSS galaxies, selecting
objects within a 2″ radius. We found on average $58$ random matches,
corresponding to 0.3% contamination of the FIRST-SDSS match sample ($>$99%
efficiency, Figure LABEL:fig:idlsep, red dotted line).
We also need to know the completeness of our matching criteria - the fraction
of correct matches we recover. A factor affecting this is the possible
exclusion of large lobe-dominated radio sources (Kimball & Ivezić, 2008). Both
lobes will be included in FIRST but may be excluded in the cross-matched
subset for lobe-core distances $>$ 2″, if the core is weak. We estimated
$\sim$ 8$\%$ of real matches between the SDSS subset and FIRST that are
excluded from our radio sample, by assuming the radio lobes are outside a 2″
radius from the optical core, but within a 5″ radius (see Figure
LABEL:fig:idlsep). Our matching algorithm efficiency and completeness are
similar to Kimball & Ivezić (2008), who match FIRST to SDSS-DR6 positions
within a 2″ radius for sources with $z\leq$ 2 (95% efficiency and 98%
completeness111Kimball & Ivezić (2008) estimate their completeness and
efficiency by fitting a gaussian to the nearest-neighbour distribution
(representing physical matches) plus a rising linear function (representing
random matchings).).
However, we could not account for sources with FIRST lobes $>5$″ from the
optical core or the population of FIRST double-lobed sources with no
detectable radio core. Ivezić et al. (2002) cross-correlated all FIRST sources
with SDSS, then identified potential double-lobed radio sources with
undetected cores222Via comparing the mid-points of FIRST pairs to SDSS sources
within a separation $<\,90$″ and accepted all matches with offsets $<\,3$″.,
estimating these contribute less than $10\%$ of all radio sources (Best et
al., 2005a, estimate $\sim 5\%$ of their radio-loud AGN sample had no FIRST
detection).
Becker et al. (1995) estimated the FIRST catalogue to be $95\%$ complete at 2
mJy and $80\%$ complete down to the survey limit of 1 mJy. Figure 2 shows
integrated radio power against redshift for the preliminary radio sample
(25,931 sources). The solid black lines trace the 1 mJy and 2 mJy peak flux
density thresholds, the dotted and dashed lines show cuts we applied in
redshift and radio power (see §3.3 and §3.4), and the red points are the
resultant selection of radio sources. It is noted that for sources that are
not well-described by an elliptical Gaussian model, the integrated flux
density as derived by FIRST may be an inaccurate measure of the true value.
Such sources with radio powers corresponding to flux densities below the 1 mJy
threshold can be seen in Figure 2.
We note that although the two cuts improve the sample completeness (all red
points are above 1 mJy), many of our sources are below the 2 mJy line, and so
our sample completeness cannot be $>80\%$. Combining this with our estimation
of completeness in our selection criteria ($\sim 90\%$), we estimate the
actual completeness of our preliminary radio sample to be $\sim 72\%$ for
sources brighter than 1 mJy. The incompleteness is dominated by statistical
effects rather than the $10\%$ effect of sources missing due to weak cores.
Figure 2: Integrated FIRST radio power vs redshift for the preliminary ‘radio’
sample (see text). The solid black lines trace peak radio power at the 1 mJy
detection threshold and at 2 mJy, the dashed lines denote the redshift limits
imposed in §3.3 and the dotted line denotes the radio power cut imposed in
§3.4. The red points are the remaining radio sample after these cuts are
applied.
To produce a preliminary control sample, we select sources from the SDSS
subset which fall within the same spatial region as FIRST ($\sim 8000$ deg2).
We selected all objects from this subset which are not matched to a FIRST
source within 2″ of their optical core, providing $\sim$625,664 radio-quiet
galaxies. The efficiency of this sample, based on Figure LABEL:fig:idlsep is
$>99.9\%$ (i.e., $<1\%$ contamination by radio sources detectable at the FIRST
flux density limit) and the sample completeness is $>92\%$.
### 3.2 Cross-correlation: FIRST-NVSS-SDSS
For FIRST sources substantially larger than $5$″ some flux density is resolved
out, leading to an increase in the survey threshold for extended objects and
flux density underestimates for larger objects (Lu et al., 2007). Our FIRST-
SDSS radio sample is large due to the number of FIRST candidate sources near
the 1 mJy survey limit. Its limitations are potential flux density
underestimates, and incompleteness to extended radio sources, notably
extended, double-lobed sources with no detectable radio core within several
arcseconds of its optical counterpart.
To attempt to quantify these limitations, we created a complementary sample of
radio-loud AGN host galaxies using both NVSS and FIRST. Best et al. (2005a)
note that the 45″ resolution of NVSS is large enough that $\sim 99\%$ of all
radio sources are contained within a single component, allowing for a higher
sample completeness. However, NVSS is less deep than FIRST and consequently
the matched sample contains fewer galaxies. The NVSS-FIRST-SDSS matched sample
therefore does not benefit from such good statistics.
We used the Unified Radio Catalogue constructed by Kimball & Ivezić (2008)
(see sample ‘C’ in their table 8), selecting sources which are detected by
both FIRST and NVSS, matched to within 25″. This selection yields a radio flux
density limited ($S_{\mathrm{1.4GHz}}>2.5$ mJy) catalogue (hereafter FIRST-
NVSS) containing 141,881 sources. We cross-matched these to the $\sim 680,056$
MGS sources adopting the same 2″ match criterion used in §3.1. Our cross-
matched catalogue contains 5,719 galaxies and we denote this as the
preliminary ‘comparison radio’ (hereafter CR) sample. Integrated flux
densities as derived by Kimball & Ivezić (2008) are adopted in our analysis.
We estimated our matching criterion of 2″ to be $>99\%$ complete, and $>99\%$
efficient (8 random matches were found when the FIRST-NVSS RA and DEC were
offset by $1^{\circ}$). The FIRST-NVSS catalogue used to create the CR sample
is 99$\%$ complete and matched with $96\%$ efficiency (see table 2 of Kimball
& Ivezić, 2008). We therefore estimated the comparison radio sample to be
$>99\%$ complete and $>95\%$ efficient (see Table 1). 5 of the 5,719 galaxies
in the CR sample reside in the control sample derived in §3.1 and these were
removed. 4,935 of the CR sample (86$\%$) are also in the FIRST-SDSS sample.
Matched | $S_{\mathrm{lim}}$(mJy) | Objects | Completeness | Efficiency
---|---|---|---|---
surveys | | | |
FIRST-SDSS | 1.0 | 25,931 | $72\%$ | $>99\%$
FIRST-NVSS-SDSS | 2.5 | 5,719 | $99\%$ | $>95\%$
Table 1: Radio flux density limits, $S_{\mathrm{lim}}$, of both the radio
sample (FIRST-SDSS) and the candidate CR sample (FIRST-NVSS-SDSS), the number
of objects in each and the estimated completeness and efficiency.
### 3.3 Redshift range
In the following sections (§3.3-3.8) we discuss the secondary selection
criteria applied to the FIRST-SDSS sample. The CR sample follows the same
path, and a summary of the final CR sample properties are presented alongside
those of the final radio sample in §3.9.
We imposed a redshift cut of $0.02<z<$ 0.18 to ensure that the galaxy spatial
structure could be well examined. For $z<0.02$, redshift is not a reliable
distance indicator. Figure LABEL:fig:lumredradio shows less than 2% of matched
sources have redshift $<0.02$. The MGS magnitude limit of
$r^{*}_{petro}<$17.77 corresponds to L${}_{r^{*}}>2.6\times 10^{23}$ W Hz-1 at
$z=0.18$, and our sample should be complete to this optical luminosity
density.
We restricted the SDSS-derived redshift confidence (zconfidence) to be greater
than $95\%$, which cuts out 12% of objects from the preliminary radio sample
and 10% of objects from the preliminary control sample. After redshift
selection, $499,418$ radio-quiet galaxies remained in the control sample,
$\sim$70% of the parent MGS, whilst the radio sample contained $17,024$
galaxies. The average error in redshift for both samples is $<0.002$ and the
redshift distributions of these two samples were indistinguishable at the 1%
level on the basis of a two-tailed K-S test.
### 3.4 Radio Power
Best et al. (2005a) suggested that typical radio-loud AGN have powers above
$10^{23}$ W Hz-1 at 1.4 GHz and Condon (1989) shows that
P${}_{\mathrm{1.4\,GHz}}=10^{23}$ W Hz-1 separates the spiral starburst
population from the AGN - E/S0 population in local galaxy fields. We therefore
restrict the radio sample to galaxies harbouring radio sources with
P${}_{\mathrm{1.4\,GHz}}>10^{23}$ W Hz-1, which selects 5,119 galaxies (30%)
from the radio sample. Figure LABEL:fig:lumredradio shows that this threshold
in radio power is well matched to our redshift selection. We name this set of
5,119 galaxies the ‘radio-loud’ sample. The mean redshift of the radio sample
increased from $\sim 0.095$ to $\sim 0.136$ when the radio power cut was
applied.
AGN can be split on the basis of large-scale radio structure into FRI type,
where radio brightness decreases outwards from the centre and FRII type, with
edge-brightened lobes and hot spots (Fanaroff & Riley, 1974). FRIIs are more
luminous P${}_{178\mathrm{\,MHz}}\geq 1.3\times 10^{26}$ W Hz-1 and are rare
in a low-redshift sample such as ours (Figure LABEL:fig:lumredradio), since
this 178 MHz power corresponds to P${}_{\mathrm{1.4\,GHz}}\approx 3.1\times
10^{25}$ WHz-1 for a typical spectral index of $\alpha=0.7$.
### 3.5 Removing type 1 AGN
In unified AGN models (e.g, Antonucci, 1993) the appearance of the central
black hole and associated continuum of an AGN differ only in the viewing angle
at which it is observed. Sources viewed face on (type 1) show broad emission
lines that are absent in those observed edge on and where the broad emission
line region is obscured by a dusty torus (type 2). Type 1 AGN are excluded
from this study, as the optical continuum can be dominated by relativistically
boosted non-thermal emission, which may overwhelm measurements of the host
galaxy’s properties.
The SDSS spectral classification pipeline automatically flags and excludes
quasars from the MGS, but we chose to verify its reliability and the relative
numbers of type 1 AGN remaining in our sample. We followed the method outlined
by Masci et al. (2010) to identify broad line emission. H$\alpha$ or H$\beta$
emission lines exceeding 1000 km s-1 (FWHM) with a $S/N>3$ and
H$\alpha$/H$\beta$ EW $>5$Å were classified as broad line. Galaxies with both
broad H$\alpha$ and H$\beta$ emission were classified as type 1 AGN. 3
galaxies of the 5,119 radio-loud sample have broad H$\alpha$ and H$\beta$
emission lines and were removed from the sample.
184 galaxies (4$\%$) of the remaining 5,116 radio-loud sample have broad
H$\alpha$ emission, but do not have broad $H\beta$ emission.
Osterbrock (1989) classified these objects as type ‘1.9’ AGN, which have
substantial but not complete obscuration of the central continuum source.
Kauffmann et al. (2003) determine that the contribution to the observed
continuum is not significant in these sources, so we retain these within our
sample. The control sample contained 10 type 1 AGN and 1835 type ‘1.9’ AGN. We
removed the type 1 AGN to leave 499,408 galaxies within the control sample.
### 3.6 Removing star-forming galaxies
Type 2 AGN have narrow permitted and forbidden lines and their stellar
continuum is often similar to normal starforming galaxies. Yun et al. (2001)
show a tight correlation between far infra-red luminosity (indicative of star
formation) and P${}_{\mathrm{1.4\,GHz}}$. This will cause a level of
contamination by star-forming galaxies if P1.4GHz is used as the sole tracer
of galaxies hosting a radio-loud AGN. We should therefore remove the small
subset of radio-loud galaxies in which the radio power arises from star
formation (SF) and not from an AGN.
AGN separation from SF galaxies in the local Universe can be achieved via
optical emission line ratio diagnostics (Baldwin, Phillips & Terlevich, 1981,
herein BPT). Emission-line ratios probe the ionizing source: for AGN, non-
thermal continuum from the accretion disc around a black hole and in star-
forming galaxies (SFGs), photoionization via hot massive stars. However, Best
et al. (2005b) find no correlation between a galaxy being radio-loud and
whether it is optically classified as an AGN. In agreement with this, we found
no correlation333Spearman’s rank correlation result is $\pm$0.06 or less
between the radio-flux and optical emission-line flux ratios for our radio
galaxy sample. between radio flux at 1.4 GHz and the optical line ratios
[NII]/H$\alpha$ or [OIII]/H$\beta$ in our sample. Hence, a substantial
fraction of radio-loud AGN would not be selected using BPT diagnostics, and
were we to apply them to the radio sample it would be biased towards radio
galaxies with particularly strong optical emission lines.
We instead remove galaxies which are strongly identified as non-AGN, i.e.
star-forming. Despite this method leaving a small fraction of star-forming
galaxies which are faint in optical line emission, all radio-loud AGN, whether
optically bright or otherwise, will remain in the sample. This decrease in
efficiency of the sample is preferential to a drastic decrease in
completeness. A similar problem was identified by Sadler et al. (2002), who
defined a sample of radio-loud AGN from the 2dFGRS catalogue. They found
approximately half of the sample have absorption spectra similar to those of
inactive giant ellipticals, and therefore would be mostly missed by optical
AGN emission-line selection.
In classifying galaxies as AGN or star-forming, we utilized the demarcation
criterion of Kauffmann et al. (2003)
$\mathrm{log}(\mathrm{[OIII]}/\mathrm{H}\beta)>0.61/\\{\mathrm{log}([\mathrm{NII}]/\mathrm{H}\alpha)-0.05\\}+1.3$
(2)
plotted as the dashed line on Figure 4.
Figure 4: BPT diagram for galaxies within the radio-loud sample with emission
line ratios. 1,069 galaxies from the 5,116 radio-loud sample have emission
line ratios and are plotted as grey crosses where the size of the symbol does
not indicate the sizes of the error bars, which vary widely. The dashed curve
(eq. 2) indicates the demarcation given by Kauffmann et al. (2003) between
optical AGN (above the line) and SF galaxies (below the line). 766 galaxies
plotted have emission line ratios with $>3\sigma$ significance. The density of
these galaxies is shown by contours, 85% of the 766 galaxies lie above the
line as optically classified AGN. The remaining 15% (118 galaxies) lie below
the curve (red circles) and we classified these as star-forming. We also show
the maximum uncertainty for a point on the Kauffmann separator with S/N of 3
(dotted lines). 69 objects detected at S/N $>3$ (black crosses) lie between
the Kauffmann separator and the upper maximum uncertainty line.
Figure 4 shows the standard line ratio diagnostics for galaxies from the
radio-loud sample with all four emission lines catalogued in the SDSS (grey
crosses, 1,069 objects of the 5,116). 766 of these 1,069 galaxies have both
optical emission-line ratios at S/N $\geq 3$ and their density on the BPT
diagram is shown as contours. 118 (11%) lie below the demarcation line and are
marked with red circles (112 out of these 118 have all four emission-lines
with S/N $\geq 5$). We removed these 118 optically selected SF galaxies from
the radio-loud sample, to leave a sample of 4,998 predominantly radio-loud AGN
hosts.
We then estimated the residual contamination expected in the radio AGN sample
from star-forming galaxies by plotting the maximum uncertainty for a point on
the Kauffmann separator with S/N of 3 (Fig 4, dotted lines). 69 objects
detected at S/N $>3$ (black crosses) lie between the Kauffmann separator and
the upper maximum uncertainty line. Therefore the contamination expected in
the radio AGN sample from star-forming galaxies is $\sim 11\%$.
Our radio sample contains predominantly FRI galaxies, which are usually hosted
by giant elliptical galaxies and on average have weak or no optical nuclear
emission lines (Lin et al., 2010, and references therein). Within our sample
of radio-loud AGN, 79% of objects do not have optical emission line fluxes. We
estimate $\approx 11\%$ contamination of non-AGN (e.g. SFG) if galaxies
without SDSS emission lines are similar to galaxies with bright line emission.
5 galaxies without emission line fluxes $>$ 3$\sigma$ lie below the
demarcation line in Figure 4 and are potentially star forming, but without
reliable line information we retain these within our sample.
As discussed by Best et al. (2005b), a potential shortfall of spectral
classification of emission-line radio-loud AGN is that emission-line AGN
activity is often accompanied by star formation (e.g., Kauffmann et al.,
2003). This star formation will give rise to radio emission, even if the AGN
itself is radio quiet. For these sources, the optical spectrum could still be
dominated by a (radio-quiet) AGN leading to classification as an emission-line
radio-loud AGN.
Morić et al. (2010) (hereafter M10) derive a population of sources matched
from all NVSS galaxies in the Unified Radio Catalog (Kimball & Ivezić, 2008),
the SDSS-MGS and IRAS data. The matched sources are divided into star-forming,
composite and AGN using standard BPT diagnostics. Star formation rates are
derived via broad-band spectral fitting to the NUV-NIR SDSS photometry, and
the _average_ fractional star formation/AGN contribution to the radio power is
estimated (see their table 2). They find that in 203 composite galaxies,
81.3$\%$ of the total radio power is due to star formation. The variation of
this fraction with radio power is not specified.
Following M10, we defined 206 composite galaxies in our own radio sample
(27$\%$ of the emission-line galaxies confirmed at S/N $\geq 3$), using the
diagnostics of Kewley et al. (2001) and Kauffmann et al. (2003). If the
average fractional contribution is independent of total radio luminosity, then
we estimated an upper limit of $\sim 27\%$ of our radio sample may have radio
power boosted by star formation but possess a radio-quiet AGN. However,
$\lesssim 14\%$ of the composite sample defined by Morić have log[P1.4(W
Hz-1)] $>23$. Mauch & Sadler (2007) find SF galaxies tend to have median
log[P1.4(W Hz-1)] = 22.13, whereas AGN have a median log[P1.4(W Hz-1)] =
23.04. Therefore, our luminosity cut will have significantly reduced the
numbers of galaxies where star-formation is the principal contributer to the
total radio power, and we expect far less than $27\%$ contamination from
radio-quiet, optically-loud AGN.
We also cannot account for the population of ‘composite’ radio-loud host
galaxies with ongoing star formation that have been lost from our sample
through this selection technique. Mauch & Sadler (2007) estimate that $\sim
10\%$ of local radio sources may have a starforming spectrum but have radio
flux densities dominated by a radio-loud AGN.
### 3.7 Colour bimodality
As low-redshift radio-loud AGN are hosted predominantly by elliptical
galaxies, we attempted to constrain the control sample to contain only early-
type galaxies. The colour distributions of galaxies have been shown to be
highly bimodal (Yan et al., 2006; Yan et al., 2010; Strateva et al., 2001).
Figure 5 shows a colour-colour plot in $g$*-$r$* against $u$*-$g$* as explored
by Strateva et al. (2001) who define an optimal colour separator444On a
spectroscopically classified galaxy sample, early-types are recovered at
$\gtrsim$98% completeness and $\gtrsim$83% reliability. between early and late
types of $u^{*}-r^{*}\geq 2.22$ mag. Early-types (E, S0, Sa) populate the
upper right region, and late types (Sb, Sc, Irr) occupy the lower left.
Figure 5: Colour-colour plot for the entire control sample (grey) and 34606
visually classified ellipticals from Galaxy Zoo-control sample match (log10
contours). The dashed line is $u^{*}-r^{*}\geq 2.22$ mag from Strateva et al.
(2001) and defines a morphological colour separator. 80% of the Galaxy Zoo-MGS
sample lie above the line. The $u^{*}$, $g^{*}$ and $r^{*}$ magnitudes are
derived from SDSS columns: petrocounts $-$ reddening $-$ kcorr.
In order to test the completeness of this demarcation, we used the 62,190
galaxies visually classified as ellipticals in Galaxy Zoo data (Lintott et
al., 2010). We cross-matched these to the control sample, selecting objects
within 1″ radius. 34,606 matches were found, and 27,569 of these lie above
$u$*-$r$*$>$2.22 mag. Therefore, this demarcation gives $\sim 80\%$
reliability (Fig 5, contours), suggesting $\sim 20\%$ of the early-type galaxy
population may be excluded through applying this criterion. 147,275 visually
classified ‘spiral’ galaxies from the Galaxy Zoo data are also present in our
control sample. 22% of these lie above $u$*-$r$*$>$2.22, i.e. we expect $\sim
22\%$ contamination in the early-type sample. Of the 499,408 galaxies in the
control sample, $\sim 40\%$ were classified as early-types, using the Strateva
et al. colour selection, to leave 199,391 early-type galaxies in the control
sample, morphologically selected in colour space at $\sim 80\%$ completeness
and $\sim 78\%$ efficiency based on Galaxy Zoo visual analysis.
We explore the 4,998 radio-loud AGN selected galaxies in colour space. Figure
6 shows the radio-loud AGN are predominately early types, with $>70\%$ lying
above $u$*-$r$*$>$2.22, i.e. the radio-loud AGN sources are predominantly
hosted by ellipticals as defined in colour space. The distribution in colour
space is similar to that of the control sample, despite the AGN colours
possibly influencing the light. In agreement with this result, Griffith &
Stern (2010) found radio-selected AGN have a high incidence of being hosted by
early-type galaxies.
The 118 galaxies we flagged as star forming and removed from the radio-loud
sample in §3.6 are shown as crosses in Figure 6, and 98% of these sit below
the line. We select the radio-loud AGN above the colour cut, to give 3,516
‘early-type’ radio-loud AGN hosts.
Figure 6: Colour-colour plot of radio-loud AGN (contoured, binsize =
$0.1\times 0.1\,$mag, 4 equally spaced levels, max density contour = 400/bin),
and SF galaxies as defined in §3.6 (crosses) The dashed line is
$u^{*}-r^{*}\geq 2.22$ from Strateva et al. (2001) and defines a morphological
colour separator.
### 3.8 Redshift distributions
(a) Redshift distributions
(b) Absolute $r^{*}$ magnitude distributions
Figure 7: The fractional distributions of (a) redshift and (b) $r^{*}$ band
luminosity. The solid black line shows the radio-loud AGN distributions. The
red dashed line shows the control sample distribution prior to nearest
neighbour matching in redshift-magnitude space. The blue dotted line shows the
resultant sample randomly selected to match the redshift-magnitude
distributions of the radio sample.
At this stage we re-examined the redshift distributions of the samples, using
the KS test. The probability of the control sample and the radio-loud AGN
being drawn from the same parent sample in redshift was $<0.1\%$: the
distributions in redshift differ (Figure 7(a)). Evolutionary differences in
host galaxy properties should be small over this entire redshift range,
however we chose to re-sample the control sample redshift distribution to
match that of the radio-loud AGN sample.
We also examined the intrinsic $r^{*}$ band luminosity of the two samples.
Figure 7(b) shows the radio sample is globally brighter in $r^{*}$ than the
control sample. It has been well established that radio-loud AGN are hosted
preferentially in the brightest and most massive elliptical galaxies (Best et
al., 2005b; Mauch & Sadler, 2007). Since we wanted to test whether radio-loud
AGN harbour an excess of blue emission in their centres, we chose to re-sample
the control sample to have the same $r^{*}$ band optical properties so as to
avoid any possible correlation of colour gradient with galaxy luminosity.
We generated a matched control sample by selecting the 10 galaxies from the
whole sample that lie closest to each radio-loud AGN in magnitude-redshift
space. The final control sample contains 35,160 galaxies, whose magnitude and
redshift distributions are shown as dotted lines in Figure 7. This final
selection should allow the distribution of observables in the radio and
control samples to be compared directly.
### 3.9 Comparison radio sample (CR)
The criteria applied to the ‘radio’ sample in §3.3-3.8 were also applied to
the 5,719 objects in the CR sample (see §3.2).
We imposed a redshift cut of $0.02<z<0.18$ and restricted the redshift
confidence parameter given by SDSS to $>95\%$, leaving 3,727 sources. We
examined the CR sample radio powers based on NVSS and FIRST. Figure 8 shows
the NVSS and FIRST luminosities for the 3,727 CR sources (grey crosses). Those
with P${}_{\mathrm{1.4\,GHz}}^{\mathrm{NVSS}}>10^{23}$ W Hz-1 are highlighted
by blue squares (50$\%$). We restricted the CR sample to galaxies harbouring
radio sources with P${}_{\mathrm{1.4\,GHz}}^{\mathrm{NVSS}}>10^{23}$ W Hz-1,
which selects 1,847 (50$\%$) of the sample.
Of those 1,847 sources, 251 have FIRST derived powers $<10^{23}$ W Hz-1, and
were therefore excluded in the radio sample selection in §3.4.
Figure 8: FIRST vs NVSS derived luminosities for the 3,737 sources in the
candidate CR sample (grey crosses). Those with
P${}_{\mathrm{1.4\,GHz}}^{\mathrm{NVSS}}>10^{23}$ W Hz-1 are highlighted by
blue squares (1,847). Of those, 251 sources with
P${}_{\mathrm{1.4\,GHz}}^{\mathrm{FIRST}}<10^{23}$ W Hz-1 are shown as red
crosses. These are the sources excluded in the radio sample’s luminosity cut
(§3.4).
We removed 2 type 1 AGN and 48 star-forming galaxies confirmed at S/N $>3$
from the CR sample, leaving 1,797 radio-loud narrow-line AGN hosts. 4$\%$ of
the sample are classified as type ‘1.9’ AGN, with substaintial but not full
obscuration of the central source. We applied the Strateva et al. colour cut
($u$*-$r$*$>$2.22 mag) detailed in §3.7 to select 1,295 early-type galaxies
from the CR sample.
The control sample was cross-matched with the FIRST-NVSS catalogue to within
2″and the 5 galaxies found were removed to ensure the comparison control
sample is comprised solely of ‘radio-quiet’ AGN elliptical hosts. 199,386
galaxies remain in the CR-control sample.
We selected the 10 galaxies from the control sample that lie closest to each
CR radio-loud AGN in magnitude-redshift space. The final CR-control sample
contains 12,950 sources, matched to the CR sample in $r^{*}$-band magnitude
and redshift distributions.
### 3.10 Summary of Final Samples
Radio-loud ‘early type’ AGN (R-AGN) - 3,516 sources
After matching FIRST to the MGS ($r^{*}_{petro}<17.77$ mag, $0.02<z<0.18$)
within 2″, we classified as radio-loud AGN hosts with
P${}_{\mathrm{1.4\,GHz}}>10^{23}$ W Hz-1. Type 1 AGN exhibiting both broad
H$\alpha$ and H$\beta$ emission lines were removed from the sample. Optical
emission line ratios (if confirmed at 3$\sigma$) were used to remove star-
forming galaxies using the demarcation of Kauffmann et al. (2003). We then
created a subset with $u^{*}$-$r^{*}>2.22$ mag i.e. ‘early-types’ to remain
consistent with the colour cut applied to the control sample.
Control sample - 35,160 sources
We removed all galaxies from SDSS-MGS ($r^{*}_{petro}<17.77$ mag,
$0.02<z<0.18$) with FIRST and NVSS counterparts. Type 1 AGN exhibiting both
broad H$\alpha$ and H$\beta$ emission lines were removed from the sample.
Galaxies were constrained in colour to bias towards early-type morphology
where a comparison with results of the Galaxy Zoo program suggests we have
$\gtrsim 80\%$ completeness and $78\%$ efficiency. The resultant sample
comprises radio-quiet, ‘normal’, predominantly elliptical galaxies. We then
selected a subsample to match the redshift and optical $r^{*}$-band magnitude
distributions of the R-AGN sample, thus removing any redshift bias and
potential correlation of colour gradient with galaxy luminosity.
Comparison radio (CR) - 1,295 sources
We cross-matched the FIRST-NVSS catalogue derived by Kimball & Ivezić (2008)
with the MGS ($r^{*}_{petro}<17.77$ mag, $0.02<z<0.18$) to within 2″, and
classified radio-loud AGN hosts as those sources exhibiting
P${}_{\mathrm{1.4\,GHz}}^{\mathrm{NVSS}}>10^{23}$ W Hz-1. Type 1 AGN
exhibiting both broad H$\alpha$ and H$\beta$ emission lines were removed from
the sample. Optical emission line ratios (if confirmed at 3$\sigma$) were used
to remove star-forming galaxies using the demarcation of Kauffmann et al.
(2003). We then created a subset with $u^{*}$-$r^{*}>$2.22 mag i.e. ‘early-
types’ to remain consistent with the colour cut applied to the control sample.
CR-control sample - 12,950 sources
All selection criteria as above for the control sample, we also removed 5
galaxies within the control sample that were also present in the CR sample. We
then selected a subsample to match the redshift and optical $r^{*}$-band
magnitude distributions of the CR sample.
We therefore have two radio-loud AGN samples, both with a comparison control
sample. The CR sample is derived using both radio catalogues, and is a
brighter radio sample (the NVSS flux density completeness limit is $2.5$ mJy)
for comparison with our deeper ($>1$ mJy), larger but more incomplete R-AGN
sample derived solely from the FIRST catalogue.
## 4 analysis
The surface brightness profile of an early-type galaxy can be described by the
Sérsic equation
$I(r)=I(r_{e})e^{-b[(r/r_{e})^{1/n}-1]}$ (3)
where $r_{e}$ is the effective radius (scale length) and $I(r_{e})$ is the
corresponding effective surface brightness. $b\approx 2n-1/3$ is chosen so
$r_{e}$ contains half the light in the galaxy. n $\approx$ 4 for bright
ellipticals decreasing to n $\approx$ 2 as luminosity decreases (Caon et al.,
1993). SDSS chooses to fix $n=4$, fitting the ‘de Vaucouleurs profile’,
truncating the profile beyond 7 $r_{e}$. The model fitted by SDSS has an
arbitrary axis ratio and position angle and is convolved with a double-
Gaussian representation of the PSF. The fitting then yields, among other
properties; the effective radius, $r_{e}$ and error, $\Delta r_{e}$. In
general, the error for the $u$ model is greater than other bands. It is noted
that SDSS’s fitting algorithm generates some weak discretization of model
parameters, especially in $r_{e}(r)$ and $r_{e}(u)$. Objects with scale
lengths lying in discrete bands in $r_{e}$ tend to have poorer goodness of
fit. This is not included in the error estimates, $\Delta r_{e}$, which were
derived from count statistics on the image. In our work with the $r_{e}$
values, we consider $\Delta r_{e}$ rather than DeV_l (the likelihood of the
model fit) due to our reluctance to remove data with poor de Vaucouleurs $u$
band fitting. A poor goodness of fit to the deVaucouleurs profile may be
indicative of a cuspy $u$ band and removing such objects could bias results.
SDSS denotes the fracDeV parameter as the fraction of the total flux
contributed by the de Vaucouleurs component in a linear combination of a de
Vaucouleurs and an exponential model to find the best fit. It is noted that
the likelihood values in the $r^{*}$-band are intrinsically poor (all samples
have mean values $\sim\,0.01$), but the likelihood ratios still pick out
reliable best-fit parameters. The fracDeV parameter ($f$ hereafter) is
correlated with the Sérsic index; n = 1 corresponds to $f$ = 0, n = 4
corresponds to $f$ = 1 (Kuehn & Ryden, 2005, and references therein).
Following Vincent & Ryden (2005), galaxies with $0.5\leq f\leq 0.9$
($2.0\lesssim$ n $\lesssim 3.3$) are labeled “de/ex” galaxies and galaxies
with $f$ $\geq 0.9$ (n $\gtrsim 3.3$) are labeled “de” galaxies. We chose not
to remove objects with $f$ $<\,0.5$, but instead created subsets of each
sample with $f$ $>\,0.5$ for comparison.
We investigate the colour structure of AGN host galaxies principally through
the distribution of $R$, which we define as $R=r_{e}(r)/r_{e}(u)$, which is
the ratio of $r$ to $u$ de Vaucouleurs effective radii. This should avoid any
intrinsic scale differences in $r$ and $u$ between individual galaxies.
### 4.1 Diversity of intrinsic distributions.
The distributions of $R$ for the R-AGN and control sample are shown in Figure
9. Both samples have a few ($<0.001\%$) strong outliers ($R>100$) which
significantly disturb the mean. The medians of the R-AGN and control
distributions are 0.837 and 0.793 respectively. Using a two-tailed Kolmogorov-
Smirnov test we find the probability of the control and the R-AGN sample being
drawn from the same distribution to be small ($<0.1\%$).
Similarly, the CR sample’s $R$ distribution also shows a higher median
compared to the CR-control sample (0.851 and 0.797 respectively). The KS test
also finds the probability of the CR and CR-control sample being drawn from
the same distribution in $R$ is vanishingly small ($<0.1\%$).
Figure 9: Normalised distributions of $R$ for the radio-loud AGN (R-AGN)
sample (solid) and the control sample (red dashed).
### 4.2 Maximum likelihood analysis
The fractional errors in individual values of $r_{e}(u)$ are relatively large,
which makes proving an intrinsic difference between the radio-loud AGN samples
and their respective control sample distributions challenging. We used a
maximum likelihood analysis to compare the average values of $R$ for the AGN
and control samples, taking into account measurement errors according to
Maccacaro et al. (1988) and Worrall (1989), in order to quantify the
difference between the distributions of $R$.
Under the assumption that both the AGN and control samples have normally-
distributed intrinsic values of $R$, then the population mean, $\overline{R}$,
and intrinsic dispersion, $\sigma$, of either can be determined from
individual values $R_{i}\pm\sigma_{i}$ by minimizing the function
$\centering{{\large
S}={\displaystyle\sum\limits_{i}}\left[\dfrac{(R_{i}-\overline{R})^{2}}{(\sigma^{2}+\sigma_{i}^{2})}+\mathrm{ln}(\sigma^{2}+\sigma_{i}^{2})\right]}.\@add@centering$
(4)
(a) R-AGN and control (b) CR and CR-control
Figure 10: 90% (solid line) and 99% (dashed/dotted line) confidence contours
of the mean ratio of scale lengths ($R=r_{e}(r)$/$r_{e}(u)$) and intrinsic
dispersion for radio-loud AGN galaxies and ‘normal’ early types. The control
and radio-loud ‘early-type’ AGN are distinct populations, as are the CR and
CR-control samples.
Figure 10 shows the results of minimization of Eq 4 across $\overline{R}$ and
$\sigma$ parameter space, for the two samples. The 90% and 99% joint-
confidence contours for $\overline{R}$ and $\sigma$ are given by
S${}_{min}+\Delta$S for $\Delta$S$=4.61$ and 9.21 respectively. Table 2 shows
the best fit $\overline{R},\sigma$ planes for both distributions. A clear
separation between radio-loud ‘early type’ AGN and ‘normal’ control galaxies
can be seen for both the R-AGN and CR samples, and we conclude the two are not
drawn from the same population at $\gg 99\%$ confidence.
We applied the cut $f>0.5$ in the $r^{*}$-band, and re-examine the results for
both radio-AGN samples. This cut will ensure the reliability of the de
Vaucouleur’s scale lengths used to derive $R$, and this subset is used as a
comparison to the full samples to verify whether galaxies with shapes that
differ significantly from the de Vaucouleurs profile perturb our main result.
Table 2 also shows the fraction of each sample with ‘good’ de Vaucouleurs
$r^{*}$-band fits and their $\overline{R}$ values (see Figure 11). Although
the values do not significantly differ from the full sample, the confidence
contours now overlap in the smaller, but more complete, CR sample. However,
the R-AGN and control samples are still seen to come from distinct
populations.
Sample | $\overline{R}$ | $\sigma$ | $f>0.5$ | $\overline{R}_{f}$ | $\sigma_{f}$
---|---|---|---|---|---
R-AGN | 0.76$\pm$0.02 | 0.23$\pm$0.01 | 53$\%$ | 0.77$\pm$0.02 | 0.25$\pm$0.02
Control | 0.719$\pm$0.005 | 0.206$\pm$0.003 | 49$\%$ | 0.722$\pm$0.006 | 0.208$\pm$0.005
CR | 0.78$\pm$0.02 | 0.20$\pm$0.02 | 44$\%$ | 0.77$\pm$0.03 | 0.19$\pm$0.03
CR-Control | 0.731$\pm$0.007 | 0.204$\pm$0.006 | 49$\%$ | 0.730$\pm$0.010 | 0.206$\pm$0.007
Table 2: Best fit $\overline{R}$ and $\sigma$ corresponding to $90\%$ error,
for the R-AGN and CR sample, and their relative control samples. The fraction
of each sample with $r^{*}$-band $f>0.5$ is shown, and the resultant mean and
standard deviation, $\overline{R}_{f}$ and $\sigma_{f}$.
(a) R-AGN and control (b) CR and CR-control
Figure 11: 90% (solid line) and 99% (dashed/dotted line) confidence contours
of the mean ratio of scale lengths ($R=r_{e}(r)$/$r_{e}(u)$) and intrinsic
dispersion for the subset of galaxies with $f>0.5$. The 1,851 radio-loud AGN
galaxies and 17,338 ‘normal’ early types are shown in panel (a), the 575 CR
galaxies and 6,364 CR-control galaxies are shown in panel (b). The control and
radio-loud ‘early-type’ AGN remain distinct populations. The smaller CR and
CR-control sample lead to larger confidence contours that overlap, but these
remain distinct at $89\%$ confidence.
## 5 Discussion
Our results confirm that radio-loud AGN are hosted by brighter, bigger
galaxies on average (see Figure 7). Our technique has shown the presence of an
AGN is associated with an increase of the scale size of a galaxy in red light
relative to blue light. This suggests that radio galaxies have a bluer central
bulge and/or more diffusely distributed red light compared to their radio-
quiet counterparts. With the analysis so far, we cannot tell whether this
difference is from AGN driven or diffuse star-like light within the bulge, or
more distributed red light.
The CR sample is also significantly different in $\overline{R}$ relative to
the CR-control sample, so that our R-AGN sample’s lower completeness in
extended sources does not significantly affect the result. The CR sample
dispersion is smaller than that of the R-AGN sample, whereas all four control
samples have values of $\sigma$ that are similar. We applied a flux cut of
$>4$ mJy to the R-AGN sample (61$\%$), and found $\overline{R}=0.80\pm 0.02$
and $\sigma=0.21\pm 0.01$. The increased scatter about $R$ is caused by
fainter radio sources, which tend to be more heterogenous, and/or have
underestimates of errors on $r_{e}$ values in more distant objects.
The subset of CR galaxies constrained to have good $r^{*}$-band de Vaucouleurs
fits ($f>0.5$) does not contain enough radio galaxies to show strongly a
significant separation from the CR-control sample (see Figure 11(b)). We
present discussion predominantly on the R-AGN and control sample, which
display a clear separation in both Figure 10(a) and 11(a). We concluded the
two are not drawn from the same population at $\gg 99\%$ confidence. This
sample is incomplete to radio sources with lobes $>5.4$″ and the population of
galaxies with a weak radio AGN core but powerful lobes. However it is large
due to the depth of the FIRST survey. The CR results for all Figures in the
discussion are similar to the results presented for the R-AGN sample. We find
no significant deviation between the two in any subsequent parameter examined,
except in $\sigma$, as discussed above.
Figure 12: Radio luminosity density vs $R$ for the CR sample. We found no
correlation ($>99\%$ certainty) between the two.
Mahabal et al. (1999, hereafter MKM99) used a complementary
ratio555$r_{e}(B)/r_{e}(R)$ to demonstrate the ubiquity of inner blue
components in a sample of only 30 radio galaxies and 30 control galaxies, and
argued that the blue light is due to star formation associated with the
presence of a radio source (e.g. Antón et al., 2008). Figure 4 of MKM99 (power
at 408 MHz vs log [$r_{e}(B)/r_{e}(R)$]) shows a trend for more powerful radio
galaxies to have steeper colour gradients as indicated by the ratio of scale
lengths. In contrast to this, when considering only the brightest
(P${}_{\mathrm{1.4\,GHz}}>10^{25}$ WHz-1) radio sources in the CR sample (for
which the powers are more accurate) we find no correlation between radio power
and $R$. We also find no correlation over the full range of radio powers and
$R$ as shown in Figure 12: the Spearman’s rank correlation result is $\pm
0.07$ or less between P${}_{\mathrm{1.4\,GHz}}$ and $R$. We infer that a high
value of $R$ (and possible blue central bulge) is not associated with the
power of the radio source.
There are several possible scenarios to be considered to explain the larger
$R$ in the R-AGN sample.
### 5.1 AGN light
Central point-like $u^{*}$ light may be indicative of a blue AGN. We
investigated the nature of the R-AGN excess $R$, using the goodness of the
deVaucouleurs fit. The SDSS deVaucouleurs fitting procedure returns the
likelihood (between $0$ and $1$) associated with the model from the $\chi^{2}$
fit. If the $u^{*}$-band goodness of fit for the R-AGN sample approaches 1 for
low $R$ and 0 for high $R$ then this is indicative of an AGN/core component
(as the AGN point-like component would perturb a deVaucouleurs fit) driving a
blue excess and high $R$. If no structure is seen in $R$ vs deVaucouleurs
$u^{*}$-band goodness of fit in either sample, then any blue excess is more
likely driven by some sort of diffuse starlight in the central bulge.
The $u^{*}$-band data for both the R-AGN and control samples are generally
well fit by a deVaucouleurs profile (the R-AGN/control median likelihood
values are 0.86 and 0.83 respectively), and there is no trend for the
$u^{*}$-band or $r^{*}$-band fits to be worse at large $R$ in the R-AGN than
the control sample. We can conclude that the high $R$ in the R-AGN sample is
not due to any point-like AGN component.
### 5.2 Radio AGN triggering star formation
It is generally agreed that AGN feedback regulates the supply of cold gas for
star formation by supplying energy to the ISM and IGM. Some radio-loud star-
forming AGN show a connection between the suppression of star formation and
the strength of the radio jets, by heating and expelling the surrounding gas
(Nesvadba et al., 2008). Schawinski et al. (2009) looked at a sample of low
redshift SDSS early-type galaxies for which late-time star formation is being
quenched. They found that molecular gas disappears less than 100 Myr after the
onset of accretion onto the central black hole. These galaxies were not
associated with radio jets, but show that low-luminosity AGN episodes are
sufficient to suppress residual star-formation in early-type galaxies.
Figure 13: Residual fractional number density distributions (R-AGN - control),
in 2 kpc bins, for the $u^{*}$-band and $r^{*}$-band scale radii. There is a
higher fraction of blue cores in the R-AGN sample as compared to the control
sample at $r_{e}\sim 10$ kpc. This plot does not demonstrate the correlation
between the red and blue scale radii (see Figure 14). The measurement errors
are larger in the $u^{*}$ band which may account for the higher variability in
the residuals.
The residual fractional number density distribution of scale lengths are shown
in Figure 13, for the control sample subtracted from the R-AGN sample. There
is a clear excess of blue cores ($0.1-10$ kpc) in the R-AGN sample as compared
to the control sample. There are also more small $r_{e}(r^{*})$ in the R-AGN
sample, ranging from $4-10$ kpc. This plot does not demonstrate the
correlation between the red and blue emission, which is shown in Figure 14,
but it does demonstrate that the presence of a radio AGN seems not to have
suppressed star-formation in the central regions of its host galaxy.
Figure 14: Normalised number densities of $r_{e}(u^{*})$ and $r_{e}(r^{*})$
for the R-AGN (solid) and control sample (red dashed). The dotted - dashed
line is the locus $r_{e}(r^{*})=0.719r_{e}(u^{*})$, galaxies above this line
become bluer inward. Contours show the fractional number densities of each
sample with levels at 0.004, 0.008 and 0.011. There is a larger fraction of
the R-AGN sample above the line, possibly denoting a tendency to become bluer
inward more often than the control sample.
Figure 14 plots $r_{e}(u^{*})$ against $r_{e}(r^{*})$ for the R-AGN sample
(black) and the control sample (red dashed). The dotted - dashed line denotes
the locus $r_{e}(r^{*})=0.719r_{e}(u^{*})$ as defined by the control sample
(see Table 2).
At $r_{e}(r^{*})>0.719r_{e}(u^{*})$, the surface brightness in $u^{*}$
increases more rapidly toward the centre of a galaxy than in $r^{*}$ as
compared to the mean of the control sample, i.e. half the blue light from the
galaxy is contained in a smaller region than half the red light, implying the
galaxy becomes bluer inward.
The distribution of number densities in Figure 14 shows radio galaxies appear
to become bluer inwards more often than the control galaxies (66% of the R-AGN
sample are above the locus $r_{e}(r)=0.719\,r_{e}(u)$, compared to 60% of the
control sample).
There are fewer blue smaller cores in the control sample as compared with the
R-AGN sample (see also Figure 13), implying there may be a blue excess near
the AGN core for some of the radio galaxies. However, the higher $R$ values of
the R-AGN sample also seems to be driven by a higher scale length in $r^{*}$,
seen in the difference between the inner two contour levels of each sample
(see also §5.4), so we cannot definitively conclude that the increased $R$ in
the R-AGN sample is due to star formation within the central few kpc. Star-
formation within the central few kpc would contradict feedback models which
predict the suppression of star formation near an AGN.
### 5.3 Star formation in the bulge
Shabala et al. (2008) found that radio sources in massive hosts are re-
triggered more frequently than their less massive counterparts, suggesting
that the onset of an AGN quiescent phase is due to fuel depletion. AGN
activity is therefore promoted by an increase in gas in the centres of the
galaxies, which may imply a link between the AGN radio phase and star
formation in the bulge ($<10$kpc) of the host galaxy.
It has been suggested that AGN activity and a major episode of star formation
in radio-loud galaxies is triggered by the accretion of gas during major
mergers and/or tidal interactions. However AGN activity is initiated later in
the merger event than the starburst (e.g. Schawinski et al., 2007; Emonts et
al., 2006; Tadhunter et al., 2005). Our spectroscopic selection rules out
major merger events (see §2.1.1), but residual star formation may still be
present on global scales.
Observationally, there is a strong link between AGN and starbursts (Shin et
al., 2011, and references therein). Kauffmann et al. (2003) found powerful
optical AGN (as classified by the strength of the [OIII] emission
$L_{\mathrm{[OIII]}}>10^{7}L_{\odot}$) predominantly reside in ‘young bulges’.
Recent star formation can provide up to $25-40\%$ of the optical/UV continuum
in radio galaxies at low and intermediate z e.g. (e.g., Holt et al., 2007;
Tadhunter et al., 2005).
We found a higher fraction of R-AGN galaxies have $r_{e}(u^{*})<10$ kpc as
compared to the control sample (see Figure 13), perhaps indicative of star
formation in the outskirts of the bulge, fuelled by gas expelled from the
central regions by the AGN.
### 5.4 Diffuse red emission
An alternative interpretation of a larger $R$ is that the R-AGN sample has
more distributed red light. Radio-loud ‘early-type’ galaxies predominantly
reside in elliptical galaxies, which are well known to be redder than late-
types (e.g., Strateva et al., 2001). Diffuse red emission would cause a larger
deVaucouleurs scale radius in the $r^{*}$ band and an increased $R$.
Figure 14 suggests the increased $R$ in the R-AGN sample may also be
attributed to more diffuse red light; between $r_{e}(u^{*})\sim 4-10$ kpc,
$r_{e}(r^{*})$ is on average higher for the radio population. This is also
seen in the lower panel of Figure 13.
Our results show a difference in the distributions of red and blue light in
R-AGN and normal ‘early-type’ galaxy populations, though a high value of $R$
is not a property of every active galaxy. We found the higher $R$ in the R-AGN
sample is contributed to by a higher fraction of radio galaxies harbouring
small blue cores but also an increase in the numbers of radio galaxies with
more diffuse red light as compared to the control sample. Given that star
formation proceeds over a longer timescale than radio activity, this
disfavours the idea that all galaxies undergo short bursts of radio activity,
but rather implies that a subset have the predisposition to become radio-loud.
## 6 Conclusion
We cross-matched low-redshift ($0.02<z<0.18$) data from the SDSS MGS
($r^{*}_{petro}<17.77$ mag) and FIRST to within 2″, deriving a radio sample of
galaxies at $>99\%$ efficiency and $>72\%$ completeness. Radio luminosities
were in the range $10^{23}-10^{25}$ WHz-1. Type 1 AGN were removed from the
sample (identified via $H\alpha$ and $H\beta$ broad line characteristics),
4$\%$ of the sample are classified as type ‘1.9’ AGN, with substantial but not
full obscuration of the central source. Contamination from SFGs (identified
via optical emission line ratios) in the radio sample is expected at $\sim
11\%$. A control sample was defined from SDSS sources with no match to a FIRST
source within 2″ of their optical core, providing a sample of $>99\%$
efficiency and $>92\%$ completeness. At 80% reliability, the demarcation
$u^{*}-r^{*}>2.22$ mag selected ‘early-type’ galaxies in both samples. Samples
were matched in $r^{*}$-band magnitude and redshift distributions and final
sample sizes were 3,516 radio-loud AGN galaxies (R-AGN) and 35,160 control
galaxies.
We also created a complementary flux-limited sample through cross-matching
with NVSS (the CR sample). The same cuts were applied to derive a radio-loud
‘early-type’ AGN sample, except we used the more all-encompassing NVSS flux
estimates to cut in radio luminosity, thereby retaining the populations of
galaxies with weak/no core radio emission but bright, extended radio lobes.
This sample had higher completeness for comparison with the R-AGN sample. A
control sample was defined from the SDSS sources with no match to a NVSS-FIRST
source within 2″ of their optical cores. We further considered only sources
where fracDeV (f) $>0.5$ to restrict all four samples to having good
$r^{*}$-band deVaucouleurs fits.
We investigated the colour structure of AGN host galaxies through $R$, the
ratio of $r^{*}$ to $u^{*}$ de Vaucouleurs effective radii and used maximum
likelihood analysis to quantify the degree of difference in the distribution
of $R$ between samples. We concluded the radio (R-AGN/CR) samples are not
drawn from the same population as their radio-faint control samples at $\gg
99\%$ confidence: the presence of an AGN increases the scale size of a galaxy
in red light relative to blue light, on average.
Our result does not appear to be driven by the presence of blue AGN in the
radio-loud samples since the goodness of the de Vaucouleurs fits does not
become worse as $R$ increases. We found no structure in $R$ vs $u^{*}$-band
goodness of fit in either radio sample. We found an excess of blue cores in
radio-loud galaxies as compared to radio-quiet, ‘early-type’ galaxies,
implying the increased $R$ may be due to star formation in the central few
kpc, in contrast with feedback models which predict the suppression of star
formation near an AGN. Spectroscopic selection of our samples rules out major
merger events, as starbursts can be triggered by the accretion of gas/tidal
interactions. We also found radio AGN hosts to have larger red scale lengths
in relation to their blue light and note this to be a contributing factor in
an increased $R$. We cannot definitively discern whether a small blue core or
larger distribution of red light is the driving factor in this result.
Given the longer timescale for star formation than radio activity, our results
imply a subset of galaxies have the predisposition to become radio-loud,
rather than all galaxies undergoing bursts of radio activity at some stage in
their lifetimes.
## Acknowledgements
EM gratefully acknowledges support from the UK Science and Technology
Facilities Council and thanks Luke Davies and James Price for helpful
comments. We thank the anonymous referee for helpful and insightful comments
that have resulted in an improved analysis. In undertaking this research, we
made extensive use of the topcat software (Taylor, 2005). Funding for the SDSS
has been provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Science Foundation, the U.S. Department of Energy,
the National Aeronautics and Space Administration, the Japanese
Monbukagakusho, the Max Planck Society, and the Higher Education Funding
Council for England. This research makes use of the NVSS and FIRST radio
surveys, carried out using the National Radio Astronomy Observatory Very Large
Array: NRAO is operated by Associated Universities Inc., under cooperative
agreement with the National Science Foundation.
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|
arxiv-papers
| 2011-06-27T20:00:02 |
2024-09-04T02:49:20.116989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Elizabeth J.A. Mannering, Diana M. Worrall, Mark Birkinshaw",
"submitter": "Elizabeth Mannering",
"url": "https://arxiv.org/abs/1106.5498"
}
|
1106.5553
|
# Effective Field Theory Analysis on $\mu$ Problem in Low-Scale Gauge
Mediation
Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
Supersymmetric models based on the scenario of gague mediation often suffer
from the well-known $\mu$ problem. In this paper, we reconsider this problem
in low-scale gauge mediation in terms of effective field theory analysis. In
this paradigm, all high energy input soft mass can be expressed via loop
expansions. If the corrections coming from messenger thresholds are small, as
we assume in this letter, then all RG evaluations can be taken as linearly
approximation for low-scale supersymmetric breaking. Due to these
observations, the parameter space can be systematically classified and studied
after constraints coming from electro-weak symmetry breaking are imposed. We
find that some old proposals in the literature are reproduced, and two new
classes are uncovered. We refer to a microscopic model, where the specific
relations among coefficients in one of the new classes are well motivated.
Also, we discuss some primary phenomenologies.
May 2011
## 1 Introduction
### 1.1 Motivations
Among scenarios that solve the stringent constraints on experiments of flavor
violations in supersymmetric (SUSY) models, gauge mediation [1, 2, 3, 4, 5] is
an appealing candidate. In particular, SUSY models with low-scale SUSY-
breaking can be directly examined at the LHC. However, gauge mediation suffers
from a well-known fine tuning problem related to electro-weak symmetry
breaking (EWSB), i.e, the $\mu$ and B$\mu$ problem [6]. On general grounds
there is a small hierarchy between these two input scales at mediation scale,
$B_{\mu}\sim~{}(16\pi^{2})\mu^{2}$. There have been a lot of efforts on
solving this problem in the literature [7]. However, except all important
subtleties are taken into account, it is not concrete to argue whether a SUSY
model suffers such a fine tuning problem.
What is the most important essentials related to $\mu$ problem are all the
input soft masses that are determined both by the SUSY-breaking and the
messenger sectors as well as the renormalization group (RG) evaluations for
these soft masses. The specific relations in the mediation scale can be
substantially modified by the RG corrections. A typical example is SUSY models
of gaugino mediation [8], in which the high energy input parameters are three
gaugino masses (together with $L^{-1}$, $\tan\beta$ and $sign{\mu}$). It
obviously does not realize EWSB and satisfy other considerations. But the
correct and large RG corrections turns it into a viable SUSY model at electro-
weak scale.
### 1.2 Strategy
In this paper, we systematically analyze the $\mu$ problem from viewpoint of
effective field theory, where the SUSY-breaking sector is referred by spurion
superfields. We introduce two separate spurion superfields, which generalize
the minimal setup of gauge mediation where only one spurion field needs to be
considered. These spurion superfields introduce relevant dynamical scales,
$\displaystyle{}X_{G}=M_{G}\left(1+\theta^{2}\Lambda_{G}\right),~{}~{}~{}~{}~{}~{}X_{D}=M_{D}\left(1+\theta^{2}\Lambda_{D}\right)$
(1.1)
$X_{G}$ and $X_{D}$ are responsible for SUSY-breaking related to generations
of scalar scalars, gaugino masses as well as $\mu$ and $B_{\mu}$ terms of
Higgs superfields, respectively. In this note, for simplicity we assume
$M_{D}/M_{G}\sim 0.1-1$ so that the corrections coming from multiplet
messenger thresholds are small.
Using $X_{G}$ and $X_{D}$, we can express all the high energy input soft mass
terms through loop expansions 111We take the effective number of messengers as
$N_{eff,G}=1$ for the squark and gaugino masses. This value can vary when the
models derive from the minimal setup (see e.g, [9]). It thus changes the
relative ratio between squark and gaugino masses. . Note that the specific
relations $\Lambda_{G}/M_{G}\lesssim 10^{-1},\Lambda_{D}/M_{D}\lesssim
10^{-1}$ have to be imposed such that we can express
$\displaystyle{}m^{2}_{\tilde{f}_{i}}$ $\displaystyle=$
$\displaystyle\sum^{3}_{i=1}\left(C_{2}(\tilde{f},i)\frac{g^{4}_{i}}{(16\pi^{2})^{2}}\right)\Lambda^{2}_{G},$
$\displaystyle M_{i}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{i}}{(16\pi^{2})}\Lambda_{G}$ (1.2)
for squark and gaugino masses and
$\displaystyle{}m^{2}_{{H_{\mu},d}}=\sum^{2}_{i=1}\left(C_{2}(H_{\mu,d},i)\frac{g^{4}_{i}}{(16\pi^{2})^{2}}\right)\Lambda^{2}_{G}+\Lambda^{2}_{D}\left[\frac{C^{(1)}_{H_{\mu,d}}}{(16\pi^{2})}+\frac{C^{(2)}_{H_{\mu,d}}}{(16\pi^{2})^{2}}+\cdots\right],$
(1.3)
for Higgs masses squared $m^{2}_{H_{\mu}}$ and $m^{2}_{H_{d}}$. The
contributions in (1.3) arise from the ordinary gauge mediation and the
superpotential for hidden sector $X_{D}$. The $\mu$ and B$\mu$ terms, on the
other hands, only receive the contributions coming from $X_{D}$-sector 222Note
that a tree-level $\mu$ term can exist in the SUSY limit $\Lambda_{D}=0$.,
$\displaystyle{}B_{\mu}$ $\displaystyle=$
$\displaystyle\Lambda^{2}_{D}\left[\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}+\frac{C^{(2)}_{B\mu}}{(16\pi^{2})^{2}}+\cdots\right]$
$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
C^{(0)}_{\mu}M_{D}+\Lambda_{D}\left[\frac{C^{(1)}_{\mu}}{(16\pi^{2})}+\frac{C^{(2)}_{\mu}}{(16\pi^{2})^{2}}+\cdots\right],$
(1.4)
Note that when a particular loop coefficient $C^{(i)}$ is determined to be
non-zero, all higher order corrections can be neglected because of the
perturbative nature. Furthermore, a coefficient $C^{(i)}$ smaller than loop
factor $1/(16\pi^{2})$ is actually an $(i+1)$-loop effect. Following this fact
we assume all $C^{(i)}$ are bounded as,
$\displaystyle{}\frac{1}{16\pi^{2}}<C^{(i)}\lesssim 1$ (1.5)
The upper bound in (1.5) is due to the perturbativity of new Yukawa couplings
in superpotential for hidden sector $X_{D}$. Otherwise, large Yukawa couplings
will suffer the problem of Landau poles.
Having systematically analyzed the high energy input soft masses, we can
obtain their values at EW scale by taking the RG corrections into account. The
RG equations [11] for these soft masses in minimal supersymmetric standard
model (MSSM) are quite involved. In this note we discuss gauge mediation with
low-scale SUSY breaking,
$\displaystyle{}10^{2}TeV\lesssim\Lambda_{G,D}\lesssim 10^{6}TeV$ (1.6)
In this region the NLSP particle are mainly prompt decays or can be long
lived, which is of highly interest at searches of SUSY signals at colliders.
Also in the region (1.6) the RG evaluations for soft masses mainly receive
their corrections coming from $\ln(M/M_{Z})$ factor, which is of order unity .
This contribution can be taken as linear approximation [12]. Following the RG
equations at one-loop, one obtains the soft masses at EW scale via replacing
(1.3) and (1.2) with
$\displaystyle{}\mu\rightarrow\mu$ $\displaystyle=$ $\displaystyle\hat{\mu},$
$\displaystyle B_{\mu}\rightarrow~{}B_{\mu}-\frac{\delta
C_{B_{\mu}}}{(16\pi^{2})^{2}}\mu\Lambda_{G}$ $\displaystyle=$
$\displaystyle\hat{B_{\mu}},$ $\displaystyle
m^{2}_{H_{\mu}}\rightarrow~{}m^{2}_{H_{\mu}}-\frac{\delta
C_{H_{\mu}}}{(16\pi^{2})^{2}}\Lambda^{2}_{G}$ $\displaystyle=$
$\displaystyle\hat{m}^{2}_{H_{\mu}},$ (1.7) $\displaystyle
m^{2}_{H_{d}}\rightarrow~{}m^{2}_{\tilde{H}_{d}}+\frac{\delta
C_{H_{d}}}{(16\pi^{2})^{2}}\Lambda^{2}_{G}$ $\displaystyle=$
$\displaystyle\hat{m}^{2}_{H_{d}}$
where $\delta~{}C_{i}$ are both positive real numbers of order one 333The
$\ln(M)$ terms are all of order one in low-scale gauge mediation, which are
absorbed into the $\delta~{}C_{i}$..
Then, as a primary analysis about the physical parameter space, we impose the
necessary condition for EWSB,
$\displaystyle{}\left(\hat{m}^{2}_{H_{\mu}}+\mid\hat{\mu}\mid^{2}\right)\left(\hat{m}^{2}_{H_{d}}+\mid\hat{\mu}\mid^{2}\right)\thickapprox\hat{B}^{2}_{\mu}$
(1.8)
In terms of (1.2) and expressions for input soft masses through loop
expansions, we can analyze the possible parameter space composed of
$(C^{(i)}_{\mu},C^{(i)}_{B_{\mu}},C^{(i)}_{H_{\mu,d}},M_{G,D},\Lambda_{G,D})$
under constraints (1.5) order by order. An important observation which can be
used to simplify the classifications is that if $C^{(i)}_{\mu}=0$ at $i$th-
loop, then we must also have $C^{(i)}_{B_{\mu}}=0$. Reversely, the statement
is not true. This conclusion can be proved by arguments of effective field
theory (see [13] for primary analysis via SUSY algebra).
Other relevant experimental constraints on parameters in Higgs sector include
negative mass squared for $H_{\mu}$ scalar,
$\displaystyle{}\hat{m}^{2}_{H_{\mu}}<0$ (1.9)
all masses of Higgs scalars are bigger than $\mathcal{O}(115)$GeV, and the
mass of the lightest chargino has to be larger than $\mathcal{O}(100)$GeV. We
will also discuss possible implications arising from these constraints on
allowed parameter space.
### 1.3 Outlines
The outline of this paper is organized as follows. In section 2, we divide the
discussions into three classes,
$\displaystyle{}Case~{}(1)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}>>\hat{m}^{2}_{H_{d}}$
$\displaystyle Case~{}(2)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}<<\hat{m}^{2}_{H_{d}}$
$\displaystyle Case~{}(3)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}>\mid\hat{m}^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}\sim\hat{m}^{2}_{H_{d}}$
(1.10) $\displaystyle Case~{}(4)$ $\displaystyle:$
$\displaystyle~{}~{}~{}~{}\hat{\mu}^{2}<<\mid
m^{2}_{H_{\mu}}\mid,~{}~{}\hat{\mu}^{2}\sim\hat{m}^{2}_{H_{d}}$
First, we find that there is parameter space allowed for the case $(1)$, which
is not extensively studied before as far as we know. There are parameter
spaces allowed for case $(2)$ to case $(4)$. Second, the typical parameter
space for case $(3)$ and case $(4)$ corresponds to the one-loop $\mu$/two-loop
B$\mu$ as well as “lopsided gauge mediation ” [14] respectively, which both
have been discussed in the literature. Finally, the second case is totally
new. Although the spectra for soft masses in Higgs sector in this case is
similar to those in [10] , it is not covered by that approach. Here only one
substantial hierarchy among soft masses in Higgs sector is needed, in
comparison with at least two proposed in [10].
In section 3, we discuss the microscopic models of hidden sector $X_{D}$ where
the typical values in parameter space for case $(2)$ is well motivated. The
new Yukawa couplings appearing in the hidden superpotential
$W(X_{D},H_{\mu,d})$ are all around unity or smaller. No severe fine tunings
are allowed in the hidden sector. This is also different from the model
buildings for the approach in [10].
In section 4, we discuss the phenomenologies for models belonging to case
$(2)$. Following the typical parameter values, we find that Higgs scalar
expect $h^{0}$ are nearly degenerate at large $\hat{m}_{H_{d}}$ and small
$\tan\beta\thickapprox 0.1$ is favored. The direct consequence for this class
of SUSY models is that the next-to-lightest supersymmetric particle (NLSP) is
mostly Bino-like and its prompt two-body decay modes are dominated by final
state $\gamma$ and $Z^{0}$ plus missing energy, with branching ratio
$cos^{2}\theta_{W}$ and $sin^{2}\theta_{W}$ respectively. The decay channel to
$h^{0}$ plus missing energy is negligible as a result of dramatically
suppression.
In section 5, we summarize the main results in this note.
## 2 Classifications
Now we discuss the allowed parameter space in the four classes in (1.3) in
turn. First,we introduce such three dimensional parameters for later use,
$\displaystyle{}x=\Lambda_{G}/M_{G}\lesssim
10^{-1},~{}~{}~{}~{}y=\Lambda_{D}/M_{D}\lesssim
10^{-1},~{}~{}~{}~{}~{}z=\Lambda_{G}/\Lambda_{D}.$ (2.1)
The first two are small positive, real numbers as mentioned in the
introduction, while the last parameter $z$ is also real and positive, in the
range of $10^{-2}$ to $10^{2}$ in general. When $z=1$, we reproduce the
physics of gauge mediation with one spurion superfield. Since the mass
parameters in Higgs sector receive contributions both from the $X_{G}$ and
$X_{D}$ sectors either directly or via RG evaluation, it will be of use to
discuss which one is the dominant contribution.
### 2.1 Case $(1)$: Large $\hat{\mu}$ Term
In this case, the most stringent constraint comes from (1.8) , from which we
have when $\mu$ is generated at tree-level
$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}$ $\displaystyle\thickapprox$
$\displaystyle\left[\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}y^{2}+\frac{C^{(2)}_{B\mu}}{(16\pi^{2})^{2}}y^{3}+\cdots\right]-\frac{C^{(0)}_{\mu}\delta
C_{B_{\mu}}}{(16\pi^{2})^{2}}\frac{\Lambda_{G}}{M_{D}}$ (2.2)
First, consider the case for $C^{(1)}_{B_{\mu}}\neq 0$. Eq.(2.2) implies that,
$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B_{\mu}}}{16\pi^{2}}y^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C^{(1)}_{B_{\mu}}>\frac{C^{(0)}_{\mu}\delta
C_{B_{\mu}}}{16\pi^{2}}\frac{M_{D}}{\Lambda_{G}}$ (2.3)
Second, if $C^{(1)}_{B_{\mu}}=0$ and $C^{(2)}_{B_{\mu}}\neq 0$, then one
obtains,
$\displaystyle{}\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B_{\mu}}}{(16\pi^{2})^{2}}y^{3}$
(2.4)
which is not allowed according to (1.5). Note that $\delta C_{B_{\mu}}$ is a
positive real coefficient of order one. As the tree-level $\mu$ term respects
the supersymmetry, it is less of interest in comparison with the mechanism
that all mass scales are originated from softly broken SUSY, unless a
dynamical generation for tree-level $\mu$ term is realized so that it is a
natural consequence.
If $\mu$ is generated at one-loop, (2.2) is replaced by,
$\displaystyle{}\left(C^{(1)}_{\mu}\right)^{2}$ $\displaystyle\thickapprox$
$\displaystyle
16\pi^{2}\left[C^{(1)}_{B_{\mu}}+\frac{C^{(2)}_{B_{\mu}}}{(16\pi^{2})}y+\cdots\right]-\frac{C^{(1)}_{\mu}\delta
C_{B_{\mu}}}{(16\pi^{2})}\frac{\Lambda_{G}}{\Lambda_{D}}$ (2.5)
A non-zero $C^{(1)}_{B_{\mu}}$ will cause the tension for its permitted value,
as stated in (1.5). What is more promising is that the $B_{\mu}$ term is
generated at two-loop. In this case, we obtain,
$\displaystyle{}C^{(1)}_{\mu}>z,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
yC^{(2)}_{B_{\mu}}$ (2.6)
Take the constraints for case $(1)$ in (1.3) into account, we find that no
matter the mainly contribution to $\hat{m}_{H_{\mu,d}}$ comes from $X_{D}$ or
$X_{G}$ sector, there is only a set of critical values,
$\displaystyle{}(z_{*})^{2}=\frac{1}{16\pi^{2}},~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}~{}C^{(1)}_{\mu*}=1$
(2.7)
if we relax the large differences in (1.3) for case $(1)$ to some moderate
values.
### 2.2 Case $(2)$: Large $\hat{m}_{H_{d}}$
We obtain after imposing the constraint in (1.3) for this case ,
$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu,d}}^{(2)}\mid\lesssim
z^{2}\\\ C^{(0)}_{\mu}>\frac{x}{(16\pi^{2})}\\\
C^{(0)}_{\mu}<<\frac{x}{(16\pi^{2})}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}~{}~{}or~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid<z^{2}\\\ C_{H_{d}}^{(2)}>>z^{2}\\\
C^{(0)}_{\mu}>\frac{x}{(16\pi^{2})}\\\
\left(C^{(0)}_{\mu}\right)^{2}<<\frac{C^{(2)}_{H_{d}}}{(16\pi^{2})}y^{2}\\\
\cdots\\\ \end{array}~{}~{}~{}~{}~{}~{}or~{}~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid>z^{2}\\\ \mid C_{H_{d}}^{(2)}\mid<<z^{2}\\\
C^{(0)}_{\mu}<<\frac{x}{(16\pi^{2})}\\\
\left(C^{(0)}_{\mu}\right)^{2}>\frac{C^{(2)}_{H_{d}}}{(16\pi^{2})}y^{2}\\\
\cdots\\\ \end{array}$ (2.22)
if $C^{(0)}_{\mu}\neq 0$. Here the constraints neglected denote those coming
from (1.8). We see that the first and third sets of choices is obviously not
consistent. The second one implies that $\ C^{(2)}_{H_{d}}y>>1$, which is not
permitted as a result of the fact that $C^{(2)}_{H_{d}}$ is of order unity and
$y$ is smaller than one.
if $C^{(0)}_{\mu}=0$ and $C^{(1)}_{\mu}\neq 0$, we get
$\displaystyle{}\begin{array}[]{c}\mid C_{H_{d}}^{(2)}\mid\lesssim z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}>x^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<x^{2}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}or~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid\lesssim z^{2}\\\ C_{H_{d}}^{(2)}>>z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}\gtrsim z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<C_{H_{d}}^{(2)}\\\ \cdots\\\
\end{array}~{}~{}~{}~{}~{}~{}or~{}~{}~{}\begin{array}[]{c}\mid
C_{H_{\mu}}^{(2)}\mid>z^{2}\\\ C_{H_{d}}^{(2)}<<z^{2}\\\
\left(C^{(1)}_{\mu}\right)^{2}>C_{H_{d}}^{(2)}\\\
\left(C^{(1)}_{\mu}\right)^{2}<<z^{2}\\\ \cdots\\\ \end{array}$ (2.37)
The first and third classes are obviously not consistent, while the second one
is allowed. We impose the constraint in (1.8), which implies that when
$C^{(1)}_{B\mu}=0$, $\sqrt{C^{(2)}_{H_{d}}}\thickapprox z/(16\pi^{2})>>z$.
Thus this choice is not allowed. On the other hand, if $C^{(1)}_{B\mu}\neq 0$,
we obtain an additional constraint,
$\displaystyle{}C^{(1)}_{\mu}\sqrt{C^{(2)}_{H_{d}}}\thickapprox
C^{(1)}_{B\mu}>>C^{(1)}_{\mu}z$ (2.38)
which is actually consistent with the second class of choices in (2.37). For
example, we take such typical values for these parameters,
$\displaystyle{}C^{(1)}_{\mu}\sim C^{(1)}_{B\mu}\sim z\sim
0.1,~{}~{}~{}C^{(2)}_{H_{\mu}}\sim
0.01~{}~{}~{}and~{}~{}~{}~{}~{}C^{(2)}_{H_{d}}\sim 1$ (2.39)
In the next section, we will construct a class of models to motivate choices
of parameters in (2.39).
### 2.3 Case $(3)$: Standard Proposal
First, when $C^{(0)}_{\mu}\neq 0$, the only possible parameter space
corresponds to the circumstance under which the contributions to soft masses
of $H_{\mu,d}$ are dominated by the $X_{G}$ hidden sector. Otherwise, we will
always get extremely small $C^{(0)}_{\mu}$, i.e,
$C^{(0)}_{\mu}<<1/(16\pi^{2})$, which indeed is high-order effects according
to our understandings. Follow this observation, we obtain the constraints,
$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu}}^{(2)}\mid\lesssim z^{2},\\\
\mid C_{H_{d}}^{(2)}\mid\lesssim z^{2},\\\
C^{(0)}_{\mu}\thickapprox\left(\frac{x}{16\pi^{2}}\right),\\\
\left(C^{(0)}_{\mu}\right)^{2}\thickapprox\frac{C^{(1)}_{B\mu}}{(16\pi^{2})}y^{2}\\\
\end{array}$ (2.44)
Second, if $\mu$ term is generated at one-loop, there is only one possibility
needed to be considered,
$\displaystyle{}\begin{array}[]{c}\mid C_{H_{\mu}}^{(2)}\mid<z^{2},\\\ \mid
C_{H_{d}}^{(2)}\mid<z^{2},\\\ C^{(1)}_{\mu}\thickapprox z^{2},\\\
\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
16\pi^{2}C^{(1)}_{B\mu}~{}~{}~{}or~{}~{}~{}\left(C^{(1)}_{\mu}\right)^{2}\thickapprox
yC^{(2)}_{B\mu},~{}~{}~{}(C^{(1)}_{B\mu}=0)\\\ \end{array}$ (2.49)
for either one-loop or two-loop $B_{\mu}$. All other choices are directly
excluded. These two choices have been well known. As shown in (2.49), for the
one-loop generation of $B_{\mu}$ , there will be no parameter space allowed
except $C^{(2)}_{B_{\mu}}$ close to its lower bounded value, which actually a
two-loop effect. In this sense, the second class in (2.49) is often referred
as the standard proposal to solve the problem.
### 2.4 Case $(4)$: Small $\hat{\mu}$ and Large $\hat{B_{\mu}}$
Similarly to the previous discussions, the fist class of choices for
$C^{(0)}_{\mu}\neq 0$ is given by,
$\displaystyle{}C_{\mu}^{(0)}<<\frac{1}{16\pi^{2}}\frac{\Lambda_{G}}{M_{D}},~{}~{}~{}~{}~{}~{}~{}C_{\mu}^{(0)}<<\frac{1}{16\pi^{2}}y$
(2.50)
which contradicts with the statement in (1.5). Second, if $C^{(0)}_{\mu}=0$
and $C^{(1)}_{\mu}\neq 0$, we obtain,
$\displaystyle{}C_{\mu}^{(1)}<<z,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}C_{\mu}^{(1)}<<C^{(2)}_{H_{d}}$
(2.51)
These choices are argued to be consistent in a class of hidden model [14].
In summary, we reproduce two old proposals in the literature as well as
discover two new approaches. In particular , the proposal discussed in [10] is
a specific choice of the case $(2)$. This statement will be more apparent in
the next section, in which the model buildings and the parameter space is
obviously different from the discussions shown in [10].
## 3 A Hidden Model
In this section we consider the hidden superpotential for SUSY-breaking sector
$X_{D}$ associated with the Higgs sector. We refer to the hidden
superpotential as 444This superpotential is the one in [14] that has been used
to analyze the models of large $B_{\mu}$ and small $\mu$. Here we use it for
different purpose and as an illustration for constructing mass spectrum in
(2.39). It is also interesting to discuss other possibilities such as models
of MSSM singlets.
$\displaystyle{}W_{hid}=\lambda_{\mu}H_{\mu}DS+\lambda_{d}H_{d}\bar{D}\bar{S}+X_{D}D\bar{D}+\frac{1}{2}X_{D}\left(a_{S}S^{2}+a_{\bar{S}}\bar{S}^{2}+a_{S\bar{S}}S\bar{S}\right)$
(3.1)
in terms of which the typical parameter space in (2.39) for the case $(2)$ can
be realized. In (3.1), $D$ and $\bar{D}$ are bi-fundamental chiral superfileds
under the representation of the EW groups, $S$ and $\bar{S}$ are MSSM
singlets, and $X_{D}$ and $X_{S}$ refer to the relevant hidden SUSY-breaking
sector.
Integrate out the massive component fields in $S,\bar{S},D$ and $\bar{D}$, we
obtain the effective Kahler potential [15],
$\displaystyle{}K_{eff}=-\frac{1}{32\pi^{2}}Tr~{}\left[\mathcal{M}\mathcal{M}^{{\dagger}}\ln\left(\frac{\mathcal{M}\mathcal{M}^{{\dagger}}}{\Lambda^{2}}\right)\right]$
(3.2)
where
$\displaystyle{}\mathcal{M}\mathcal{M}^{{\dagger}}=\left(\begin{array}[]{cc}\mid
X_{D}\mid^{2}+\mid\lambda_{d}\mid^{2}\mid~{}H_{d}\mid^{2}&\lambda_{\mu}H_{\mu}X_{D}^{*}+\lambda_{d}H_{d}^{*}X_{S}\\\
(\lambda_{\mu}H_{\mu}X_{D}^{*}+\lambda_{d}H_{d}^{*}X_{S})^{*}&\mid
X_{S}\mid^{2}+\mid\lambda_{\mu}\mid^{2}\mid~{}H_{\mu}\mid^{2}\\\
\end{array}\right)$ (3.5)
The soft masses can be obtained via evaluating the eigenvalues of matrix
(3.5), which are given by 555For the explicit expressions for functions
$P(v,w),Q(v,w)$ and $R(v,w)$ we refer the readers to [14].,
$\displaystyle{}m^{2}_{H_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}^{2}}{16\pi^{2}}\Lambda_{D}^{2}\left[c_{\theta}^{2}P(v,w)+s_{\theta}^{2}P(\lambda~{}v,w)\right],$
$\displaystyle m^{2}_{H_{d}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{d}^{2}}{16\pi^{2}}\Lambda_{D}^{2}\left[s_{\theta}^{2}P(v,w)+c_{\theta}^{2}P(\lambda~{}v,w)\right],$
(3.6) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}\lambda_{d}}{16\pi^{2}}\Lambda_{D}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle B_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\mu}\lambda_{d}}{16\pi^{2}}\Lambda^{2}_{D}s_{\theta}c_{\theta}\left[R(\lambda~{}v,w)-R(v,w)\right]$
where the dimensionless coefficients are defined as,
$\displaystyle{}\lambda=\frac{a_{\bar{S}}-a_{S}\tan^{2}\theta}{a_{S}-a_{\bar{S}}\tan^{2}\theta},~{}~{}~{}~{}\tan
2\theta=\frac{2a_{S\bar{S}}}{a_{\bar{S}}-a_{S}},~{}~{}~{}~{}~{}v=M_{S}/M_{D},~{}~{}~{}~{}w=\Lambda_{S}/\Lambda_{D},$
(3.7)
Compare (1.3) and (1.2) with (3), we obtain the explicit expressions,
$\displaystyle{}C^{(1)}_{\mu}$ $\displaystyle=$
$\displaystyle\lambda_{\mu}\lambda_{d}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle C^{(1)}_{B_{\mu}}$ $\displaystyle=$
$\displaystyle\lambda_{\mu}\lambda_{d}s_{\theta}c_{\theta}\left[Q(\lambda~{}v,w)-Q(v,w)\right],$
$\displaystyle C^{(2)}_{H_{\mu}}$ $\displaystyle=$
$\displaystyle(16\pi^{2})\lambda_{\mu}^{2}\left[c_{\theta}^{2}P(v,w)+s_{\theta}^{2}P(\lambda~{}v,w)\right],$
(3.8) $\displaystyle C^{(2)}_{H_{d}}$ $\displaystyle=$
$\displaystyle(16\pi^{2})\lambda_{d}^{2}\left[s_{\theta}^{2}P(v,w)+c_{\theta}^{2}P(\lambda~{}v,w)\right]$
Figure 1: The values of $w$s vary as $v$s in the region where $\sin\theta\sim
0.02$, with $\lambda_{\mu}=4.8$, $\lambda_{d}=0.4$ and $\lambda=0.1$. The
dashing, solid and the bottom lines represent the curves for $\mu$, $B_{\mu}$
and $H_{\mu}$ respectively. The favored region is near $v\sim 0.2-0.3$. In
this region $C^{(2)}_{H_{d}}\sim 0.4-0.6$.
Now we discuss the parameter space that is composed of
$\lambda_{\mu,d},\lambda,v$, $\theta$ and $w$ ( with an overall scale
$\Lambda_{D}$ ), by impose the concrete choices in (2.39) on (3). In
particular, the perturbativity of Yukawa couplings in (3.1) and the absence of
obvious fine tunings between them require,
$\displaystyle{}0.1\lesssim\lambda_{\mu,d}\lesssim
5,~{}~{}~{}~{}~{}~{}~{}0.1\lesssim\lambda\lesssim
10,~{}~{}~{}~{}~{}~{}~{}~{}~{}0.1\lesssim a_{S,\bar{S}}\lesssim 5,$ (3.9)
Also the value of $v$ is restricted to be around $0.1-10$ according to the
assumption we follow.
First, we take the curves for $C^{(1)}_{\mu}$ and $C^{(2)}_{B_{\mu}}$ for
consideration. In the region of small $\sin\theta$ value where substantial
simplifications happen, the large ratio of $\lambda_{\mu}/\lambda_{d}$ is
favored. If $\lambda$ is set to be around $0.8$, these two curves overlap from
$v\sim 0.1$ to $v\sim 0.3$, during which $C^{(2)}_{H_{\mu}}$ is close to its
typical value, as shown in fig. $(1)$.
When we move to the region of parameter space where $\cos\theta$ closes to 1,
similar results can be obtained, with $a_{S}\sim 0.1a_{\bar{S}}$ and
$a_{S\bar{S}}\sim 0.1a_{\bar{S}}$ as shown in fig $(2)$. In this region, it is
more easier to obtain large $m_{H_{d}}$ term. The typical value for
$C^{(1)}_{H_{d}}$ is around $1.0-3.0$.
The main goal of this section is to construct a concrete model in which the
typical parameter space is allowed. We want to mention that the parameter
space we discover is part of the total physical one.
Figure 2: The values of $w$s vary as $v$s in the region where $\cos\theta\sim
0.02$, with $\lambda_{\mu}=4$, $\lambda_{d}=0.4$ and $\lambda=0.1$. The
dashing, solid and the bottom lines represent the curves for $\mu$, $B_{\mu}$
and $H_{\mu}$ respectively. The favored region is near $v\sim 0.3-0.4$.
## 4 Phenomenology
Now we discuss the phenomenological implications for the mass spectra
represented by the second class of models in (1.3). In the typical parameter
space, we find,
$\displaystyle{}\mid\hat{m}^{2}_{H_{\mu}}\mid<\hat{\mu}^{2}\sim
M^{2}_{r}\sim\hat{m}^{2}_{\tilde{f}}\sim 10^{-1}\hat{B}_{\mu}\sim
10^{-2}\hat{m}^{2}_{H_{d}}$ (4.1)
### 4.1 Higgs Mass Spectrum
The tree-level Higgs mass spectra are given by,
$\displaystyle{}\hat{m}^{2}_{A^{0}}=\frac{2\hat{B}_{\mu}}{\sin
2\beta}=2\mid\mu\mid^{2}+\hat{m}^{2}_{H_{\mu}}+\hat{m}^{2}_{H_{d}}\thickapprox\hat{m}^{2}_{H_{d}}$
(4.2)
The last expression is obtained in terms of specific relations in (4.1).
Eq.(4.2) also implies that $\tan\beta\thickapprox 0.1$. The masses of the
remaining neutral Higgs scalars are,
$\displaystyle{}\hat{m}^{2}_{h^{0},H^{0}}=\frac{1}{2}\left(\hat{m}^{2}_{A^{0}}+m_{Z}^{2}\mp\sqrt{(\hat{m}^{2}_{A^{0}}-m_{Z}^{2})^{2}+4m_{Z}^{2}\hat{m}^{2}_{A^{0}}\sin^{2}2\beta}\right)$
(4.3)
from which we obtain up to leading order of
$O(m^{2}_{Z}/\hat{m}^{2}_{H_{d}})$,
$\displaystyle{}\hat{m}_{h^{0}}\thickapprox\frac{1}{\sqrt{2}}m_{Z}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}\hat{m}_{H^{0}}\thickapprox\hat{m}_{H_{d}}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right)$
(4.4)
The masses for charged Higgs scalars are,
$\displaystyle{}\hat{m}_{H^{\pm}}\thickapprox\hat{m}_{H_{d}}+\mathcal{O}\left(\frac{m^{2}_{Z}}{\hat{m}^{2}_{H_{d}}}\right)$
(4.5)
These mass spectra suggest that $\Lambda_{G}\gtrsim 10^{2}$TeV.
### 4.2 Bino-like NLSP
The NLSP is either the neutrilino $\tilde{\chi}^{0}_{1}$ or chargino
$\tilde{\chi}^{+}$ for the class of models in (4.2). Using the fact that
$m_{Z}<\mid\hat{\mu}\pm M_{1}\mid$ and $m_{Z}<\mid\hat{\mu}\pm M_{2}\mid$ for
$\Lambda_{G}\gtrsim 10^{2}$TeV, we have
$\displaystyle{}\hat{m}_{\tilde{\chi}^{0}_{1}}$ $\displaystyle\thickapprox$
$\displaystyle M_{1}+\mathcal{O}\left(\frac{m^{2}_{W}}{\hat{\mu}^{2}}\right),$
$\displaystyle\hat{m}^{2}_{\tilde{\chi}_{+}}$ $\displaystyle\thickapprox$
$\displaystyle
M^{2}_{2}+\frac{1}{2}m^{2}_{W}+\mathcal{O}\left(\frac{m^{2}_{W}}{\hat{\mu}^{2}}\right)$
(4.6)
So neutrilino is the NLSP. Under the same limit, the matrix $N$ [16] that is
used to dialogize the neurtilino mass matrix is given by ,
$\displaystyle{}N_{11}$ $\displaystyle\thickapprox$ $\displaystyle 1,$
$\displaystyle N_{12}$ $\displaystyle\thickapprox$ $\displaystyle 0,$
$\displaystyle N_{13}$ $\displaystyle\thickapprox$ $\displaystyle
s_{W}s_{\beta}\frac{m_{Z}(\hat{\mu}+M_{1}\cot\beta)}{\mid\hat{\mu}\mid^{2}-M_{1}^{2}}$
(4.7) $\displaystyle N_{14}$ $\displaystyle\thickapprox$ $\displaystyle-
s_{W}c_{\beta}\frac{m_{Z}(\hat{\mu}+M_{1}\tan\beta)}{\mid\hat{\mu}\mid^{2}-M_{1}^{2}}$
which leas to the conclusion that the NLSP neutrilino is mostly Bino-like. It
promptly decays into final states of $\gamma$, $Z$ and $h$ plus missing energy
when $\Lambda_{G}\lesssim 10^{3}$TeV, which can be directly produced at
Tevatron and LHC ( see [20] for recent discussions on neutrilino NLSP at
colliders). The decay widths for these final states are given by,
$\displaystyle{}\Gamma(\tilde{\chi}^{0}_{1}\rightarrow\gamma+\tilde{G})$
$\displaystyle\thickapprox$ $\displaystyle c^{2}_{W}\mathcal{A},$
$\displaystyle\Gamma(\tilde{\chi}^{0}_{1}\rightarrow Z+\tilde{G})$
$\displaystyle\thickapprox$
$\displaystyle\left[s_{W}^{2}+\frac{1}{8}\sin^{2}2\beta\left(\frac{m_{Z}}{\hat{\mu}}\right)^{2}\left(1-\frac{m^{2}_{Z}}{\hat{m}^{2}_{\tilde{\chi}^{0}_{1}}}\right)^{4}\right]\mathcal{A}\thickapprox
s^{2}_{W}\mathcal{A},$ $\displaystyle\Gamma(\tilde{\chi}^{0}_{1}\rightarrow
h+\tilde{G})$ $\displaystyle\thickapprox$ $\displaystyle 0$ (4.8)
with
$\displaystyle{}\mathcal{A}=\frac{\hat{m}^{5}_{\tilde{\chi}_{0}}}{16\pi
M^{2}_{G}\Lambda^{2}_{G}}=\left(\frac{\hat{m}_{\tilde{\chi}_{0}}}{100GeV}\right)^{5}\left(\frac{100TeV}{\sqrt{M_{G}\Lambda_{G}}}\right)^{4}\frac{1}{0.1mm}$
(4.9)
To derive the final results in (4.2) we use the definite value
$\tan\beta\thickapprox 0.1$, as imposed by (4.2). This small value
dramatically suppresses the decay width of channel
$\tilde{\chi}^{0}_{1}\rightarrow h+\tilde{G}$. The branching ratios for the
channels involved are found to be,
$\displaystyle{}Br(\tilde{\chi}^{0}_{1}$ $\displaystyle\rightarrow$
$\displaystyle\gamma+\tilde{G})\thickapprox 74\%$ $\displaystyle
Br(\tilde{\chi}^{0}_{1}$ $\displaystyle\rightarrow$ $\displaystyle
Z+\tilde{G})\thickapprox 26\%$ (4.10)
The spectra (4.1) lead to a large $m_{A}$ (relative to $m_{Z}$) and small
$\tan\beta$. No matter the value of $\tan\beta$, large $m_{A}$ is sufficient
to reduce the Higgs search of MSSM to the SM Higgs search. In this sense, in
the model we study here phenomenologies related to the rates of Higgs
prodcutions and decays are similar with those under the decoupling limit [17].
However, as reviewed in [18], it is still possible to discover or exclude
heavy $m_{H^{\pm}}$ up to 1 TeV via $\tau$ $\mu$ decays at LHC with 300 fb-1.
To descriminate between our model and the others described by the decoupling
limit, one should turn to analyze the neutralino and chargino sectors. Due to
the smaller $\mu$ term as well as smaller gaugino masses than $m_{A}$ as shown
in (4.1), lighter neutrilinos and charginos are allowed, in comparison with
what one expects in the ordinary models that solve the $\mu$ problem, with
$\mu\sim m_{H}\sim m_{A}$. Typically, if the masses of neutrilinos and
charginos are of order $\mathcal{O}(100)$ GeV, the decays with leptonic final
states such as $\tilde{C}^{\pm}_{2}\rightarrow\tilde{N}_{1}l^{\pm}\nu$ and
$\tilde{N}_{2}\rightarrow\tilde{N}_{1}l^{+}l^{-}$ are very interesting [19].
Because these decays can be prompt and searched directly at the LHC . However,
in ordinary models with $\mu\sim m_{H}\sim m_{A}$ , one expects the masses of
neutralinos and charginos are heavier than $\mathcal{O}(200)$ GeV if we take
the decoupling limit. A double increase in $m_{\tilde{\chi}_{0}}$ will results
in the decay widths
$\Gamma(\tilde{\chi}^{0}_{1}\rightarrow\gamma/Z+\tilde{G})$ dramatically
enhanced, as shown from (4.9), which is beyond the reach of LHC.
## 5 Conclusions
In this paper $\mu$ problem in gauge mediation is studied in terms of
effective field theory analysis. As the subtleties that determine the SUSY
soft mass parameters at EW scale are the high energy input boundary values and
their RG evaluations, we use the loop expansions to capture the mainly
property of the former contribution, and restrict our discussion to negligible
multiple messenger threshold corrections as well as low-scale SUSY-breaking to
simplify the later contribution. Following these facts, we classify the
problem into four classes, and explore the parameter space allowed by imposing
primary constraints coming from EWSB. We reproduce two old proposals in the
literature as well as discover two new classes.
As illustration for model buildings, we refer to a hidden theory proposed in
[14] for different purpose. We find in some regions where no fine tunings
happen these typical values of parameters in one of the two new classes can be
realized. Also, we discuss the phenomenological implications predicted by this
new class of models. The Higgs scalars expect $h^{0}$ are heavy and of order
$m_{H_{d}}$. The NLSP is Bino-like neutrilino, whose two-body decays are
prompt and mainly composed of $\gamma$ and $Z$ plus missing energy. The
channel for neutrilno decaying into $h^{0}$ is kinetically permitted, however,
dramatically suppressed.
$\bf{Acknowledgement}$
This work is supported in part by the Fundamental Research Funds for the
Central Universities with project number CDJRC10300002.
## References
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* [3] M. Dine, A. E. Nelson, Y. Nir and Y. Shirman, “New tools for low-energy dynamical supersymmetry breaking”, Phys. Rev. D53, 2658 (1996), [arXiv:hep-ph/9507378].
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* [6] G. R. Dvali, G. F. Giudice and A. Pomarol, “The $\mu$-Problem in Theories with Gauge-Mediated Supersymmetry Breaking”, Nucl. Phys. B478, 31 (1996), [arXiv:hep-ph/9603238].
* [7] A. Delgado, G. F. Giudice and P. Slavich, “Dynamical $\mu$ Term in Gauge Mediation”, Phys. Lett. B653, 424 (2007), [arXiv:0706.3873];
T. S. Roy and M. Schmaltz, “A hidden solution to the $\mu$-$B\mu$ problem in
gauge media- tion”, Phys. Rev. D77, 095008 (2008), [arXiv:0708.3593];
H. Murayama, Y. Nomura and D. Poland, “More Visible Effects of the Hidden
Sector”, Phys. Rev. D77, 015005 (2008), [arXiv:0709.0775];
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mediation”, Phys. Lett. B660, 545 (2008), [arXiv:0711.4448].
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Z. Chacko, M. A. Luty, A. E. Nelson and E. Ponton, “Gaugino mediated
supersymmetry breaking”, JHEP0001, 003 (2000), [arXiv:hep-ph/9911323].
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* [10] C. Csaki, A. Falkowski, Y. Nomura and T. Volansky, “ New Approach to the $\mu-B\mu$ Problem of Gauge-Mediated Supersymmetry Breaking”, Phys. Rev. Lett.102, 111801 (2009), [arXiv:0809.4492].
* [11] S. P. Martin and M. T. Vaughn, “Two loop renormalization group equations for soft supersymmetry breaking couplings,” Phys. Rev. D50, 2282 (1994), [hep-ph/9311340].
* [12] S. Dimopoulos, S. D. Thomas, J. D. Wells, “Sparticle spectroscopy and electroweak symmetry breaking with gauge mediated supersymmetry breaking,” Nucl.Phys.B488, 39 (1997), [hep-ph/9609434].
* [13] Z. Komargodski, N. Seiberg, “$\mu$ and General Gauge Mediation”, JHEP0903, 072 (2009), [arXiv:0812.3900].
* [14] A. D. Simone, R. Franceschini, G. F. Giudice, D. Pappadopulo, R. Rattazzi, “Lopsided Gauge Mediation,” [arXiv:1103.6033].
* [15] M. T. Grisaru, M. Rocek and R. von Unge, “ Effective Kahler Potentials,” Phys. Lett. B383, 415 (1996), [arXiv:hep-ph/9605149].
* [16] H. E. Haber and G. L. Kane, “The Search for Supersymmetry: Probing Physics Beyond the Standard Model,” Phys.Rept.117, 75 (1985).
* [17] H. E. Haber and Y. Nir, Phys. Lett. B306, (1993) 327; J. F. Gunion and H. E. Haber, [hep-ph/0207010].
* [18] M.Carena, H. H. Eaber, “Higgs boson theory and phenomenology,” Prog. Part. Nucl. Phys.50, 63-152, [arXiv: hep-ph/0208209].
* [19] S. P. Martin, “A Supersymmetry Primer,” [arXiv: hep-ph/ 9709356].
* [20] P. Meade, M. Reece, D. Shih, “Prompt Decays of General Neutralino NLSPs at the Tevatron,” JHEP1005, 105 (2010), [arXiv:0911.4130].
|
arxiv-papers
| 2011-06-28T02:27:24 |
2024-09-04T02:49:20.128825
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1106.5553"
}
|
1106.5638
|
# Naked Singularity Formation In $f({\mathcal{R}})$ Gravity
A. H. Ziaie ah.ziaie@gmail.com Department of Physics, Shahid Beheshti
University, Evin, 19839 Tehran, Iran K. Atazadeh k-atazadeh@sbu.ac.ir
Department of Physics, Azarbaijan University of Tarbiat Moallem, 53714-161
Tabriz, Iran S. M. M. Rasouli m-rasouli@sbu.ac.ir Department of Physics,
Shahid Beheshti University, Evin, 19839 Tehran, Iran
###### Abstract
We study the gravitational collapse of a star with barotropic equation of
state $p=w\rho$ in the context of $f({\mathcal{R}})$ theories of gravity.
Utilizing the metric formalism, we rewrite the field equations as those of
Brans-Dicke theory with vanishing coupling parameter. By choosing the
functionality of Ricci scalar as $f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$,
we show that for an appropriate initial value of the energy density, if
$\alpha$ and $m$ satisfy certain conditions, the resulting singularity would
be naked, violating the cosmic censorship conjecture. These conditions are the
ratio of the mass function to the area radius of the collapsing ball,
negativity of the effective pressure, and the time behavior of the Kretschmann
scalar. Also, as long as parameter $\alpha$ obeys certain conditions, the
satisfaction of the weak energy condition is guaranteed by the collapsing
configuration.
## I Introduction
Einstein’s General theory of relativity is the classical theory of one of the
four fundamental forces, gravity, which is the weakest but most dominant force
of nature governing phenomena at large scales, and is described by a
mathematically well-founded and elegant structure i.e., differential geometry
of curved spacetime. The Einstein’s field equations, a system of non-linear
partial differential equations, relate the geometric property of spacetime to
the four-momentum (energy density and linear momentum) of matter fields
leading to precise predictions that have received considerable experimental
confirmations with high accuracy such as solar system tests (see Solar and
references therein). One of the most engrossing but open debates in general
relativity is that of the final fate of gravitational collapse with
possibility of the existence of spacetime singularities, the ultra-strong
gravity regions where the densities and spacetime curvatures blow up, leading
to a spacetime which is geodesically incomplete Hawking-Ellis and the
structure of any classical theory of fields is vanquished. A star with a mass
many times than that of the Sun would undergo a continual gravitational
collapse due to its self-gravity without achieving an equilibrium state in
contrast to a neutron star or a white dwarf. Then, according to singularity
theorems established by Hawking and Penrose Hawking-Penrose a singularity is
reached as the collapse endstate. Such a singularity may be a black hole
hidden from external observers by an event horizon or visible to the outside
Universe (naked singularity). In the latter collapse procedure, the
information on super-dense regions can be transported via suitable non-
spacelike trajectories to a distant observer. Although the occurrence of a
spacetime singularity as the final outcome of a collapse scenario is proved by
the singularity theorems, they do not specify the nature of such a
singularity. The cosmic censorship conjecture first articulated by Penrose CCC
states that a black hole is always formed in complete gravitational collapse
of reasonable matter fields, or a physically reasonable spacetime contains no
naked singularities. However, up to now many exact solutions of Einstein’s
field equations describing singularities, not hidden behind an event horizon
of spacetime, are known. A remarkable study is the one by Shapiro and
Teukolsky, who showed numerically that gravitational collapse of a spheroidal
dust may end in a naked singularity Shapiro . Also many exact solutions of
Einstein’s field equations with a variety of field-sources admitting naked
singularities have been surveyed. The examples studied so far include
gravitational collapse of a pressure-less matter Dust-Patil , radiation
Radiation-Patil , perfect fluids PF-Patil , imperfect fluids IPF-Patil and
null strange quark fluids NSQF . Beside general theory of relativity, there
exist alternative theories of gravity explaining gravitational phenomena. Such
theories have been studied for a long time Alternative . From the theoretical
standpoint there has been many attempts to correct the Einstein-Hilbert action
in order to renormalize general relativity to build a quantum theory of
gravity or at least some effective action (including the low-energy limit of
string theories), or to quantize the scalar fields in curved spacetimes Callan
. From the observational point of view, the discovery of current acceleration
of the Universe using CMB Ia supernova CMB suggests that such acceleration
may be explained within the framework of general relativity by assuming that
$76\%$ of energy content of the Universe is filled with a mysterious form of
$dark~{}energy$ with equation of state $p\sim-\rho$ (where $\rho$ and $p$ are
the energy density and pressure of the cosmic fluid, respectively). Another
possibility is to include a cosmological constant $\Lambda$ of a very small
magnitude in Einstein’s field equation, but encounters such difficulties as
the well-known cosmological constant problem and the coincidence problem.
Another alteration is $f({\mathcal{R}})$ theories of gravity FR where the
Ricci scalar in the Einstein-Hilbert Lagrangian is replaced by a general
function of it, providing alternative gravitational models for dark energy
since the explanation of the cosmic acceleration comes back to the fact that
we do not understand gravity at large scales. Such theories can describe the
transition from deceleration to acceleration in the evolution of the Universe
Nojiri-Odintsov . Moreover, the coincidence problem may be solved simply in
such theories by the Universe expansion. Also some models of modified theories
of gravity are predicted by string/M-theory considerations MString . Recently,
it has been shown that the accelerated expansion of the Universe may be the
result of a modification to the Einstein-Hilbert action in the context of
higher order gravity theories Farhoudi . Our purpose here is to consider the
gravitational collapse of a star within the framework of $f({\mathcal{R}})$
theories of gravity, whose matter content obeys the barotropic equation of
state, $p=w\rho$. We investigate the conditions under which the resulting
singularity may be naked or not. In Section II we apply the metric formalism
to the action of $f({\mathcal{R}})$ gravity Metric-Formalism and rewrite it
as that of the Brans-Dicke theory with $\omega_{BD}=0$. Choosing
$f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$ Capozzillo in Section III and
fixing the corresponding potential, we find $m$ as a function of initial
energy density and $\alpha$. In Section IV we study the behavior of the
expansion parameter which is the key factor in examining the formation or
otherwise of trapped surfaces during the dynamical evolution of the collapse
scenario. In Section VI we examine the global features of the nakedness of the
resulting singularity by investigating the behavior of the Kretschmann scalar
as a function of time. In order to fully complete the model we utilize the
Vaidya metric to match the interior spacetime to that of the exterior one.
## II $f({\mathcal{R}})$ Field equations
We begin by the general action in modified theories of gravity given by
${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}f({\mathcal{R}})+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$ (1)
where $\kappa=8\pi G$, G is the gravitational constant, g is the determinant
of the metric, ${\mathcal{R}}$ represents the Ricci scalar and $\psi$
collectively denotes the matter fields. Introducing an auxiliary field $\Psi$,
one can write the dynamically equivalent action as (See Faraoni and
references therein)
${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left[f(\Psi)+f^{\prime}(\Psi)({\mathcal{R}}-\Psi)\right]+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$
(2)
where variation with respect to $\Psi$ leads to the following equation as
$f^{\prime\prime}(\Psi)({\mathcal{R}}-\Psi)=0.$ (3)
If $f^{\prime\prime}(\Psi)\neq 0$, one can then recover action (1) by setting
$\Psi={\mathcal{R}}$. Redefining the field $\Psi$ by $\phi=f^{\prime}(\Psi)$
and setting
$V(\phi)=\Psi(\phi)\phi-f(\Psi(\phi)),$ (4)
action (2) will take the following form
${\mathcal{A}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left[\phi{\mathcal{R}}-V(\phi)\right]+{\mathcal{A}}_{matter}(g_{\mu\nu},\psi),$
(5)
which corresponds to the Jordan frame representation of the action of Brans-
Dicke theory with Brans-Dicke parameter $\omega_{BD}=0$. Brans-Dicke theory
with $\omega_{BD}=0$ is sometimes called massive dilaton gravity Wands which
was originally suggested in O Hanlon in order to generate a Yukawa term in
the Newtonian limit. Extremizing the action yields the following field
equations as (we set $\kappa=8\pi G=1$ in the rest of this paper)
$G_{\mu\nu}=T^{({\rm eff})}_{\mu\nu},$ (6)
and
$3\Box\phi+2V(\phi)-\phi\frac{dV(\phi)}{d\phi}=T^{m},$ (7)
where $T^{m}$ stands for the trace of $T^{m}_{\mu\nu}$ and the subscript “$m$”
refers to the matter fields (fields other than $\phi$) and we have defined the
effective stress-energy tensor as
$T^{({\rm eff})}_{\mu\nu}=\frac{1}{\phi}\left(T^{\rm
m}_{\mu\nu}+T^{\phi}_{\mu\nu}\right),$ (8)
with
$T^{\phi}_{\mu\nu}=(\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\Box\phi)-\frac{1}{2}g_{\mu\nu}V(\phi),$ (9)
and
$T^{\mu m}_{\nu}={\rm diag}\left(\rho_{m},p_{m},p_{m},p_{m}\right),$ (10)
being the stress-energy tensors of the scalar field and a perfect fluid,
respectively.
## III Gravitational collapse of a homogeneous cloud with
$f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$
Let us now build and investigate a homogeneous class of collapsing models in
$f({\mathcal{R}})$ gravity with $m\neq 0$, where the trapping of light is
avoided till the formation of singularity, allowing the singularity to be
visible to outside observers. In order to achieve our purpose we examine a
spherically symmetric homogeneous scalar field, $\Phi=\Phi(\tau)$ originating
from geometry. Since the interior spacetime is a dynamical one, we
parameterize its line element as follows
$ds^{2}=-d\tau^{2}+a^{2}(\tau)(dr^{2}+r^{2}d\Omega^{2}),$ (11)
where $\tau$ is the proper time of a free falling observer whose geodesic
trajectories are distinguished by the comoving radial coordinate $r$ and
$d\Omega^{2}$ is the standard line element on the unit $2$-sphere. It is worth
mentioning that here we assume that starting from the homogeneous initial
data, the collapsing configuration remains homogeneous till the singularity is
formed. But as the collapse proceeds there may be some inhomogeneities
occurring throughout the collapse scenario, the existence of which can be
investigated by perturbation theory, that is, imposing inhomogeneous
perturbations on the energy density, scale factor and BD scalar field and then
see whether the terms rising from inhomogeneity are dominant in the formation
of the singularity or not. Here we do not deal with such an issue but for more
details the reader may consult per and references there in. Since the
presence of matter acting as a “seed” field prompts the collapse of the BD
scalar field, we have considered perfect fluid models obeying barotropic
equation of state as
$p_{m}=w\rho_{m}.$ (12)
Using the conservation equation for the matter
($\nabla^{\alpha}T^{m}_{\alpha\beta}=0$) together with the use of above
equation, one gets the following relations between $\rho_{m},~{}p_{m}$ and the
scale factor as
$\rho_{m}=\rho_{{}_{{}_{0}}m}a^{-3(1+w)};~{}~{}p_{m}=w\rho_{{}_{{}_{0}}m}a^{-3(1+w)},$
(13)
where $\rho_{{}_{{}_{0}}m}=\rho_{m}(a=1)$, is the initial value of energy
density of matter on the collapsing volume. Making use of equation (8) and
equation (11) one finds the following equations for the effective stress-
energy tensor as
$\rho_{{}_{({\rm eff})}}=-T^{\tau}\,_{\tau}{}^{({\rm
eff})}=\frac{1}{\phi}\left(\rho_{m}+\rho_{{}_{\phi}}\right)=\frac{1}{\phi}\left[\rho_{m}-3\frac{\dot{a}}{a}\dot{\phi}+\frac{V(\phi)}{2}\right],$
(14)
and
$p_{{}_{({\rm eff})}}=T^{r}\,_{r}{}^{({\rm
eff})}=T^{\theta}\,_{\theta}{}^{({\rm eff})}=T^{\varphi}\,_{\varphi}{}^{({\rm
eff})}=\frac{1}{\phi}\left(p_{{}_{m}}+p_{\phi}\right)=\frac{1}{\phi}\left[p_{m}+2\frac{\dot{a}}{a}\dot{\phi}+\ddot{\phi}-\frac{V(\phi)}{2}\right],$
(15)
with all other off-diagonal terms being zero and the radial and tangential
profiles of pressure are equal due to the homogeneity and isotropy.
Substituting the line element (11) into Einstein’s equation one gets the
interior solution as
$\rho_{{}_{({\rm
eff})}}=3\frac{\dot{a}^{2}}{a^{2}}=\frac{{\mathcal{M}}^{\prime}}{R^{2}R^{\prime}};~{}~{}~{}~{}p_{{}_{({\rm
eff})}}=-\left[\left(\frac{\dot{a}}{{a}}\right)^{2}+2\frac{\ddot{a}}{a}\right]=-\frac{\dot{{\mathcal{M}}}}{R^{2}\dot{R}},$
(16) $\dot{R}^{2}=\frac{{\mathcal{M}}}{R},$ (17)
where ${\mathcal{M}(\tau,r)}$ rises as a free function from the integration of
Einstein’s field equation which can be interpreted physically as the total
mass within the collapsing cloud at a coordinate radius $r$ with
${\mathcal{M}}\geq 0$, and $R(\tau,r)=ra(\tau)$ is the physical area radius
for the volume labeled by the comoving coordinate $r$. From equation (14) and
first part of equation (16), one can solve for the mass function
${\mathcal{M}}=\frac{R^{3}}{3\phi}(\rho_{{}_{\phi}}+\rho_{m}).$ (18)
Using the above equation together with equation (17) we arrive at a relation
between $\dot{a}$ and the effective energy density as follows
$\dot{a}^{2}=\frac{a^{2}}{3\phi}(\rho_{{}_{\phi}}+\rho_{m}).$ (19)
Since we are concerned with a continual collapsing scenario, the time
derivative of the scale factor should be negative $(\dot{a}<0)$ implying that
the physical area radius of the collapsing volume for constant value of $r$
decreases monotonically. The singularity arising as the final state of
collapse at $\tau=\tau_{s}$ is given by $a(\tau_{s})=0$. On the other hand
when the scale factor and physical area radius of all the collapsing shells
vanish, the collapsing cloud has reached a singularity. A point at which the
energy density and pressure blows up, the Kretschmann scalar
${\mathcal{K}}=R^{abcd}R_{abcd}$ diverges and the normal differentiability and
manifold structures break down. In order to solve the field equations we
proceed by substituting for $\rho_{{}_{({\rm eff})}}$ and $p_{{}_{({\rm
eff})}}$ from equation (16) into equations (14) and (15) and rewrite them as
follows
$3\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{\alpha
m\Phi}\left[\rho_{{}_{{}_{0}}m}a^{-3(1+w)}-3\alpha
m\frac{\dot{a}}{a}\dot{\Phi}+\frac{V(\Phi)}{2}\right],$ (20)
$-\left[\left(\frac{\dot{a}}{{a}}\right)^{2}+2\frac{\ddot{a}}{a}\right]=\frac{1}{\alpha
m\Phi}\left[w\rho_{{}_{{}_{0}}m}a^{-3(1+w)}+\alpha m\ddot{\Phi}+2\alpha
m\frac{\dot{a}}{a}\dot{\Phi}-\frac{V(\Phi)}{2}\right],$ (21)
where for later convenience we have rescaled the scalar field as $\phi=\alpha
m\Phi$ ($\alpha$ and $m$ are real constants), and by the virtue of equation
(4) the associated potential to $f({\mathcal{R}})=\alpha{\mathcal{R}}^{m}$ can
be fixed as
$V(\Phi)=\alpha(m-1)\Phi^{\frac{m}{m-1}};~{}~{}~{}~{}m\neq 1.$ (22)
Since the scalar field must diverge at the singularity we examine its behavior
by taking the following ansatz for the scalar field
$\Phi(\tau)=a^{\delta}(\tau),$ (23)
where $\delta$ is a constant whose sign decides the divergence of the scalar
field. Substituting the first and second time derivatives of the scale factor
from equations (20) and (21) into equation (7) together with the use of
equation (22) one finds
$a^{-\left[\delta+3(1+w)\right]}\left\\{\frac{2\rho_{{}_{{}_{0}}m}(\delta+3w-1)}{3\alpha
m\delta(2+\delta)}\right\\}+a^{\left[\frac{\delta}{m-1}\right]}\left\\{\frac{4+\delta(2m-1)-2m}{6m\delta+3m\delta^{2}}\right\\}=0.$
(24)
Matching the powers of scale factor in equation (24) we arrive at the
following expression for $\delta$
$\displaystyle\delta=\frac{3(1+w)(1-m)}{m},$ (25)
whence by substituting the above equation into the pair of square brackets in
equation (24) one gets the following equation to be satisfied by $m$
$\displaystyle\alpha\Big{\\{}3(1+w)+m\left[-13-9w+m(8+6w)\right]\Big{\\}}+2\rho_{{}_{{}_{0}m}}\left[4m-3(1+w)\right]=0.$
(26)
Solving the above equation, we find $m$ as a function of $\alpha$ and
$\rho_{{}_{{}_{0}m}}$ for $w$ as
$\displaystyle
m_{\pm}=\frac{13\alpha-8\rho_{{}_{{}_{0}m}}\pm\left[73\alpha^{2}-16\alpha\rho_{{}_{{}_{0}m}}+64\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{16\alpha},$
(27)
for $w=0$,
$\displaystyle
m_{\pm}=-\frac{-5\alpha+4\rho_{{}_{{}_{0}m}}\pm\left[13\alpha^{2}-16\alpha\rho_{{}_{{}_{0}m}}+16\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{6\alpha},$
(28)
for $w=-\frac{1}{3}$,
$\displaystyle
m_{\pm}=-\frac{-7\alpha+8\rho_{{}_{{}_{0}m}}\pm\left[33\alpha^{2}-80\alpha\rho_{{}_{{}_{0}m}}+64\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{8\alpha},$
(29)
for $w=-\frac{2}{3}$, and
$\displaystyle
m_{\pm}=\frac{4\alpha-2\rho_{{}_{{}_{0}m}}\pm\sqrt{2}\left[3\alpha^{2}+2\alpha\rho_{{}_{{}_{0}m}}+2\rho_{{}_{{}_{0}m}}^{2}\right]^{\frac{1}{2}}}{5\alpha},$
(30)
for $w=\frac{1}{3}$, where $\alpha\neq 0$.
## IV Time Behavior Of The Scale Factor and singular epoch
One would like to study time behavior of the scale factor as the collapse
evolves, considering matter fields. If at time $\tau=\tau^{*}$ (or
equivalently for some $a=a^{*}$) the collapse begins, then by integrating
equation (20) together with the use of equations (22) and (23) near the
singularity with respect to time one gets the time behavior of the scale
factor as
$a(\tau)=\left[a^{\ast\frac{1}{2}\left(\delta+3(1+w)\right)}-\frac{1}{2}\sqrt{\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}}\left(\delta+3(1+w)\right)(\tau-\tau_{\ast})\right]^{\frac{2}{\delta+3(1+w)}},$
(31)
and the corresponding singular epoch as
$\tau_{s}=2\sqrt{\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}}\frac{a^{\ast\frac{1}{2}\left(\delta+3(1+w)\right)}}{\delta+3(1+w)}+\tau^{\ast},$
(32)
where the time $\tau_{s}$ corresponds to a vanishing scale factor. Thus the
collapse reaches the singularity in a finite proper time. This result for the
scale factor completes the interior solution within the collapsing cloud,
providing us with the required construction.
## V The Conditions
We are now in a position to investigate the nature of singularity as the
endstate of a collapsing scenario. The singularity is called locally naked if
it is only visible to an observer being in the neighborhood of it (such a
singularity is necessarily covered by a spacetime event horizon) and is called
globally naked if there exists a family of future directed non-spacelike
geodesics reaching to the outside observers in the spacetime and terminating
at the past in the singularity. To investigate the nature of spacetime
singularity arising from the collapse procedure, we examine here whether such
a singularity could be naked, or necessarily covered within a spacetime event
horizon and if so under what conditions. Since $\alpha$ can be regarded as a
free parameter, we seek for the appropriate values of it satisfying the
following conditions
* •
The ratio ${\mathcal{M}}/R$ stays less than unity till the singular epoch
which means that the singularity has been formed earlier than the formation of
the apparent horizon or the formation of trapped surfaces have been failed
till the singular epoch.
* •
$\delta<0$ during the gravitational collapse scenario accompanied by
divergence of the scalar field.
* •
For physical reason, weak energy condition, stated as $\rho_{{}_{({\rm
eff})}}\geq 0$ and $\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm eff})}}\geq 0$ must
be satisfied during the dynamical evolution of the system.
* •
The effective energy density and effective pressure blow up in the vicinity of
the singularity, the latter being negative during the evolution of the
collapse process, since the absence of trapped surfaces is accompanied by a
negative pressure. In fact the negativeness of the effective pressure implies
that $\dot{{\mathcal{M}}}<0$, that is, the mass contained in the collapsing
volume with comoving coordinate $r$ keeps decreasing leading to an outward
energy flux during the gravitational collapse scenario.
* •
Kretschmann scalar diverges at the singular time and then converges to zero at
late times.
In order to determine whether the singularity is naked or not, one needs to
investigate the formation of trapped surfaces during the collapse procedure.
These surfaces are defined as compact two-dimensional (smooth) spacelike
surfaces such that both families of ingoing and outgoing null geodesics
orthogonal to them necessarily converge or the expansion parameter $\Theta$ of
the outgoing future-directed null geodesics is everywhere negative Frolov-
Malec . Consider a congruence of outgoing radial null geodesics having the
tangent vector $(V^{\tau},V^{r},0,0)$, where
$V^{\tau}=d\tau/dk~{}and~{}V^{r}=dr/dk$ and $k$ is an affine parameter along
the geodesics. For the spacetime metric (11), the geodesic expansion parameter
which is defined as the covariant divergence of the vector field $V^{\nu}$ is
given by Singh
$\Theta=\nabla_{\nu}V^{\nu}=\frac{2}{r}\left[1-\sqrt{\frac{{\mathcal{M}}}{R}}\right]V^{r}.$
(33)
If the null geodesics terminate at the singularity in the past with a definite
tangent, then at the singularity we have $\Theta>0$. If such family of curves
do not exist and the event horizon forms earlier than the singularity, a black
hole is formed. Utilizing equation (33), we now attempt to study the formation
of trapped surfaces during the dynamical evolution of the gravitational
collapse procedure. We show that physically, the formation of a black hole or
a naked singularity as the final state for the dynamical evolution is governed
by the rate of collapse and the presence of scalar field. It is seen that for
a specified range of variation of $\alpha$, the cosmic censorship conjecture
may be violated for all cases of matter considered below. In the following
subsections we consider first the four conditions mentioned in the beginning
of this section and postpone the last one to the next section. We begin by
calculating the ratio ${\mathcal{M}}/R$ in the general case which is
considered for the four cases of matter,
$w=\left\\{0,-\frac{1}{3},-\frac{2}{3},\frac{1}{3}\right\\}$ corresponding to
dust, cosmic strings, domain walls and radiation, respectively. By the virtue
of equation (17) we have
$\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+3w+1)}.$ (34)
The weak energy condition which states that the energy density as measured by
any local observer must be non-negative can be written for any timelike vector
$V^{\mu}$ as follows
$T_{\mu\nu}V^{\mu}V^{\nu}\geq 0,$ (35)
whereby one gets the following conditions for the effective energy density
$(\rho_{{}_{({\rm eff})}}\geq 0)$
$\rho_{{}_{({\rm eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3(w+1))}\geq 0,$ (36)
and the sum of effective energy density and pressure ($\rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm eff})}}\geq 0$) as
$\displaystyle\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm
eff})}}=\left(1+w+\frac{\delta}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-\left(\delta+3(w+1)\right)}\geq 0.$ (37)
Finally for the rate of change of mass function with respect to time one has
$\displaystyle\dot{{\mathcal{M}}}=r^{3}(\delta+3w)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+3w+1)}\mid\dot{a}\mid,$ (38)
where the minus sign has been absorbed into $\dot{a}$. At the initial epoch
where $a(\tau^{*})=1$ there should not be any trapping of light, then by
assuming that $r=r_{b}$ is the boundary of the collapsing volume one may
easily see that for a suitable initial value of energy density the ratio
${\mathcal{M}}/R$ is less than unity at the initial time. This fact is in
accordance with the regularity conditions stating that if gravitationally
collapsing massive stars are to be modeled, then the energy density, pressure,
and other physical quantities must be finite and regular at the initial
spacelike hyper-surface from which the collapse commences. For the case of
homogeneous-density collapse the resulting singularity coincides with the
curves $R(\tau_{s},0)=0$ or $R(\tau_{s},r\neq 0)=0$, corresponding to a
central or non-central singularity, respectively. In the next subsections we
first consider the simpler case of non-central singularity and investigate
formation or otherwise of trapped surfaces for different values of $w$.
### V.1 Dust ($w=0$)
For this case of matter we have the following relations (we set
$\rho_{{}_{{}_{0}m}}=1$ in the rest of this paper)
$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(1+\frac{\delta}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}\delta\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+1)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\frac{\delta}{3}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+3)},&&&&&&\\\ \end{array}\right.$ (49)
If the interval for $\alpha$ is $-3.9<\alpha<-0.1$ and by picking out $m_{-}$
in equation (27), the corresponding parameter $\delta_{-}$ is always negative,
its absolute value varies between $1<\mid\delta_{-}\mid<3$ and its minimum and
maximum values are $\delta_{-}=-2.72484$ and $\delta_{-}=-1.00569$,
respectively. It is seen that for such values of $\alpha$ and $\delta_{-}$,
the ratio ${\mathcal{M}}/R$ stays less than unity and then the expansion
parameter is always positive up to the singularity, that is the singularity is
formed earlier than the formation of apparent horizon which is the boundary of
trapped surfaces. The negativeness of effective pressure for the allowed
values of $\alpha$ ensures that $\dot{{\mathcal{M}}}$ is negative as collapse
proceeds which means that the mass contained in the collapsing volume keeps
waning. Then there exists an outward energy flux which may be visible to
outside observers (for the case of globally naked singularity) since the
trapped surfaces do not form early enough to cover the singularity. In
addition, the validity of the weak energy condition can be easily checked by
the virtue of expressions obtained for $\rho_{{}_{({\rm eff})}}$ and
$\rho_{{}_{({\rm eff})}}+p_{{}_{({\rm eff})}}$.
### V.2 Cosmic Strings ($w=-\frac{1}{3}$)
Cosmic strings are the result of hypothetical 1-dimensional (spatially)
topological defects which may have been constructed during a symmetry breaking
phase transition at the early Universe. The possibility of their existence was
first considered by Tom Kibble in 1976 TKible . A fluid of cosmic strings may
have an effective equation of state,
$p_{{}_{{}_{m}}}=-\frac{1}{3}\rho_{{}_{{}_{m}}}$, so one has the following
relations for this type of matter fluid as
$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-\delta},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+2}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta-1)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-\delta}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta-1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\\\ \end{array}\right.$ (60)
Choosing $m_{+}$ in equation (28), it can be seen that the valid range of
change for $\alpha$ is $-100<\alpha<-0.1$ upon which the absolute value of
parameter $\delta_{+}$ is restricted to vary between $0<\mid\delta_{+}\mid<2$
(in order to prevent the ratio ${\mathcal{M}}/R$ to be infinity,
$\delta_{+}=-1$ has to be excluded) and its minimum and maximum values are
$\delta_{+}=-1.86224$ and $\delta_{+}=-0.61558$, respectively. It should be
noted that one can choose the lower limit of $\alpha$ to be much less than
$-100$, but such a choice does not affect considerably the value of
$\delta_{+}$ and its magnitude remains close to $-0.6$. Taking these values
into account, the ratio ${\mathcal{M}}/R$ stays finite till the singular epoch
and causes the expansion parameter to be positive up to the singularity, and
if no trapped surfaces exist initially then none would form until the epoch
$a(\tau_{s})=0$ which is consistent with the fact that there exist families of
outgoing radial null geodesics emerging from the singularity. Also the weak
energy condition is satisfied ($\mid\delta_{+}\mid<2$) and the effective
pressure is negative (since $\delta_{+}<0$), consistent with the fact that
time derivative of mass function is negative i.e., the mass incorporated in
the region where the collapse procedure evolves keeps falling off. As a
result, there exists an outward energy flux during the collapse scenario which
may be visible to external Universe.
### V.3 Domain Walls ($w=-\frac{2}{3}$)
Domain walls are two-dimensional objects that form when a discrete symmetry is
spontaneously broken at a phase transitionWeinberg . It has been noticed that
there exist a link between domain-walls and cosmologies such as brane
cosmologyArkani . The effective equation of state for a fluid of domain walls
may be $p_{{}_{{}_{m}}}=-\frac{2}{3}\rho_{{}_{{}_{m}}}$ and the mentioned
conditions for this case can be written as
$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta-1)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta-2)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta-1)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta-2}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+1)},&&&&&&\\\ \end{array}\right.$ (71)
As long as $\alpha$ varies in the range $-100<\alpha<-0.1$ (lower limit of
$\alpha$ can be chosen much less than $-100$ but such a choice has no
noticeable influence on the maximum value of $\delta$), by choosing $m_{+}$ in
equation (29), the parameter $\delta_{+}$ is negative, its absolute value is
less than unity and its minimum and maximum values are $\delta_{+}=-0.953501$
and $\delta_{+}=-0.379572$, respectively. One then may easily see that at
initial epoch ($a(\tau^{*})=1$), the regularity condition (there should be no
trapped surfaces at the initial hyper-surface from which the collapse
commences) is satisfied and the ratio of mass function to physical area radius
of the collapsing volume is less than unity during the collapse procedure
denoting that the expansion parameter being positive up to the singularity. In
this case the collapse to a naked singularity may take place, where the
trapped surfaces do not form early enough or are avoided in the spacetime.
Also the mass contained in the collapsing ball reduces as the time advances
due to the fact that the effective pressure stays negative till the singular
epoch. It is obvious that for such values of $\alpha$ and $\delta_{+}$ the
weak energy condition is satisfied during the collapse scenario.
### V.4 Radiation ($w=\frac{1}{3}$)
Finally, for this type of matter we have following relations for the said
conditions
$\displaystyle\left\\{\begin{array}[]{ccccccc}\frac{{\mathcal{M}}}{R}=r^{2}\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+2)},&&&&&&\\\ &&&&&&\\\ \rho_{{}_{({\rm
eff})}}+p_{{}_{({\rm
eff})}}=\left(\frac{\delta+4}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},&&&&&&\left\\{\begin{array}[]{c}\rho_{{}_{({\rm
eff})}}=\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},\\\ \\\ \\\ \\\
\dot{{\mathcal{M}}}=r^{3}(\delta+1)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{6\alpha
m(1+\delta)}\right]a^{-(\delta+2)}\mid\dot{a}\mid.\\\ \end{array}\right.\\\
&&&&&&\\\ p_{{}_{({\rm
eff})}}=\left(\frac{\delta+1}{3}\right)\left[\frac{2\rho_{{}_{{}_{0}m}}+\alpha(m-1)}{2\alpha
m(1+\delta)}\right]a^{-(\delta+4)},&&&&&&\\\ \end{array}\right.$ (82)
The suitable range of variation of $\alpha$ for $m_{-}$ in equation (30) is
$-0.66<\alpha<-0.01$ which makes the corresponding values of $\delta_{-}$ to
vary in the range $2<\mid\delta_{-}\mid<4$ with $\delta_{-}=-2.01007$ and
$\delta_{-}=-3.95037$ are the maximum and minimum values of this parameter
respectively. For such values of $\alpha$ and $\delta_{-}$ the ratio
${\mathcal{M}}/R$ stays less than unity as the collapse procedure ends. Then
the expansion parameter remains positive up to the singularity denoting that
the apparent horizon is failed to form. On the other hand, weak energy
condition is satisfied for these values of $\alpha$ and $\delta_{-}$ and the
effective pressure is negative till the singular epoch. It is worth noticing
that if one chooses the parameter $\alpha$ and the power of Ricci scalar in
equations (27)-(30) other than the ones determined in the above subsections,
weak energy condition together with the conditions on effective pressure and
the ratio ${\mathcal{M}}/R$ would be violated. The central singularity
occurring at $R=0$, $r=0$ can be naked if we have any future-directed outgoing
null geodesics terminating in the past at the singularity. In order to examine
the possibility of existence of such families let us introduce a new variable
$y=r^{\gamma}$ with $\gamma>1$ defined in such away that the ratio
$R^{\prime}/r^{\gamma-1}$ is a unique finite quantity in the limit
$r\rightarrow 0$. Now consider the outgoing radial null geodesic equation
which is given by
$\displaystyle\frac{d\tau}{dr}=a(\tau).$ (83)
In terms of variables $y=r^{\gamma}$ and $R$ the above equation reads
$\displaystyle\frac{dR}{dy}=\frac{1}{\gamma
r^{\gamma-1}}\left[\dot{R}\frac{d\tau}{dr}+R^{\prime}\right],$ (84)
whence using equation (17) we have
$\displaystyle\frac{dR}{dy}=\frac{R^{\prime}}{\gamma
y^{\frac{\gamma-1}{\gamma}}}\left[1-\sqrt{\frac{{\mathcal{M}}}{R}}\right].$
(85)
If there exist outgoing radial null geodesics in the past at the central
singularity which occurs at $\tau=\tau_{s}$, then along such geodesics we have
$R\rightarrow 0$ as $r\rightarrow 0$, or in terms of the variables $y$ and
$R$, the point $y=0$, $R=0$ is a singularity of the above first order
differential equation. For such a congruence of geodesics $dR/dy$ must be
positive up to the singularity. Then as long as $\alpha$,
$\rho_{{}_{{}_{0}m}}$, and $m$ satisfy the values which was determined
previously, trapped surfaces would fail to form as the collapse evolves.
## VI nakedness of the singularity
The final fate of a continual gravitational collapse of a matter cloud ends in
either a black hole or a naked singularity where in the former there exists an
event horizon of spacetime developing earlier than the formation of the
singularity to cover it. Thus the regions of extreme curvatures and densities
are concealed from the outside observers. The event horizon or surface of a
black hole is defined as the boundary of the spacetime that is causally
connected to future null infinity Revisiting EH Finders , or in other word the
boundary between events from which light rays emitted inside this boundary
surface can not escape to future infinity while those emitted outside in a
suitable direction can. Both event and apparent horizons coincide in
stationary spacetimes, however this case is not generally true in dynamical
ones. Although the existence of an apparent horizon predicates the existence
of spacetime event horizon, the converse is not always true and the event
horizon may veil the singularity even if apparent horizon does not emerge on a
spatial slice. So far we have discussed the conditions under which the
collapse scenario ends in a locally naked singularity, that is, the
singularity is visible to an observer being in the neighborhood of it. In this
case, the trajectories coming out of the singularity do not actually come out
to a distant observer but fall back into the singularity again at a later time
without going out of the boundary of the star. Thus the locally naked
singularity could still be covered by the event horizon and only strong
version of cosmic censorship conjecture is violated but the week form of it is
intact TPSinghPSJ . However if the event horizon is delayed to form or the
singularity forms early enough before the formation of event horizon, then it
would be visible to external observers and thus a globally naked singularity
would born as the endstate of collapse rather than a black hole. Therefore in
such a situation curvature invariants namely the Kretschmann scalar should
increase near the singularity, diverge at the singular epoch and then converge
to zero at late times Kretschmann . In previous section we showed that for
suitable values of $\alpha$, trapping of light can be avoided which means that
the apparent horizon is failed to form. But since the failure of formation of
an apparent horizon does not necessarily bode the absence of an event horizon,
we investigate the nakedness of the singularity in spherically symmetric
collapse of a fluid by considering the behavior of Kretschmann scalar with
respect to time. For the line element (11) this quantity is given by
${\cal K}\equiv
R^{abcd}R_{abcd}=\frac{12}{a^{4}}\left[a^{2}\ddot{a}^{2}+\dot{a}^{4}\right].$
(86)
By the virtue of equation (31) one can easily obtain this quantity as a
function of time for
$w=\left\\{0,-\frac{1}{3},-\frac{2}{3},\frac{1}{3}\right\\}$ and the results
are sketched in Figures 1-4. It is seen that Kretschmann scalar diverges at
singular time and then tends to zero at late times. In order to better
understand the situation one should resort to critical behavior at the black
hole threshold. Such a behavior was discovered in gravitational collapse of a
spherically symmetric massless scalar field 14H , axisymmetric gravitational
waves 15H , spherical system of a radiation fluid 16H and spherical system of
a perfect fluid obeying the equation of state $p=w\rho$ 17H . Consider a type
II critical collapse in which the black hole mass is scaled as
$M_{BH}\propto\mid p-p_{*}\mid^{\gamma}$, where $p$ parameterizes a family of
initial data sets evolving through Einstein’s equations, $p_{*}$ is a critical
value and $\gamma$ is a positive constant which is called a critical exponent
Kretschmann ; Harada-LivingCarsten . For a sufficiently large value of $p$ the
collapse procedure develops to a black hole and for a sufficiently small one
it evolves to a dispersion Harada-LivingCarsten . The boundary between these
two regimes is the black hole threshold. Now consider the limit from
supercritical collapse $(p>p_{*})$ to a critical collapse, i.e., $p\rightarrow
p_{*}$. In such a limit, the black hole mass tends to zero and the maximum
value of curvature diverges just outside the event horizon. Since we have
arbitrarily strong curvature outside the event horizon by fine-tuning, the
black hole threshold can be regarded as a globally naked singularity Harada-
LivingCarsten . Let us now consider the geometry of the exterior spacetime. In
order to complete the model we need to match the interior spacetime of the
dynamical collapse to a suitable exterior geometry. The Schwarzschild solution
is a useful model to describe the spacetime outside stars but the spacetime
outside such a star may be filled with radiated energy from the star in the
form of electromagnetic radiation. The Schwarzschild solution does not
describe this properly as it deals with an empty spacetime given by
$T_{ab}=0$. The spacetime outside a spherically symmetric star surrounded by
radiation emitted from the star is described by the Vaidya metricVaidya which
can be given in the following form as
$ds^{2}_{out}=-\left[1-\frac{2M(r_{v},v)}{r_{v}}\right]dv^{2}-2dvdr_{v}+r_{v}^{2}d\Omega^{2},$
(87)
where $v$ is the retarded null coordinate, $r_{v}$ and $M(r_{v},v)$ are the
Vaidya radius and Vaidya mass, respectively. In what follows, we use the
Isreal-Darmois junction conditions to match the interior spacetime to a Vaidya
exterior geometry at the boundary hyper-surface $\Sigma$ given by $r=r_{b}$.
The spacetime metric just inside $\Sigma$ is given by
$ds^{2}_{in}=-d\tau^{2}+a^{2}(\tau)\left[dr^{2}+r_{b}^{2}d\Omega^{2}\right],$
(88)
whereby matching the area radius at the boundary one gets
$R(r_{b},\tau)=r_{v}(v).$ (89)
One then gets the interior and exterior metrics on the hyper-surface $\Sigma$
as follows
$ds^{2}_{\Sigma in}=-d\tau^{2}+a^{2}(\tau)r_{b}^{2}d\Omega^{2},$ (90)
$ds^{2}_{\Sigma
out}=-\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]dv^{2}+r_{v}^{2}d\Omega^{2},$
(91)
where matching the first fundamental form gives
$\left[\frac{dv}{d\tau}\right]_{\Sigma}=\frac{1}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}},~{}~{}~{}(r_{v})_{\Sigma}=r_{b}a(\tau).$
(92)
In order to match the second fundamental form (extrinsic curvature) for
interior and exterior spacetimes we need to find the unit vector field normal
to the hyper-surface $\Sigma$. We then proceed by taking into account the fact
that any spacetime metric can be written locally in the following form as
$ds^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)d\tau^{2}-2\beta_{i}dx^{i}d\tau+h_{ij}dx^{i}dx^{j},$
(93)
where $\alpha$, $\beta^{i}$, and $h_{ij}$ are the lapse function, shift
vector, and induced metric, respectively and $i$, $j$ run in $\\{1,2,3\\}$.
Comparing the above equation with equations (87) and (88) together with using
the following normalization condition for $n^{v}$ and $n^{r_{v}}$
$n^{v}n_{v}+n^{r_{v}}n_{r_{v}}=1,$ (94)
one gets the normal vector fields for the interior and exterior spacetimes as
$n^{a}_{in}=[0,a(\tau)^{-1},0,0],$ (95)
$n^{v}=-\frac{1}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}},~{}~{}~{}~{}~{}~{}~{}n^{r_{v}}=\frac{1-\frac{2M(r_{v},v)}{r_{v}}+\frac{dr_{v}}{dv}}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}}.$
(96)
The extrinsic curvature of the hyper-surface $\Sigma$ is defined as the Lie
derivative of the metric tensor with respect to the normal vector n, given by
the following relation as
$K_{ab}=\frac{1}{2}{\cal
L}_{\textbf{n}}g_{ab}=\frac{1}{2}\left[g_{ab,c}n^{c}+g_{cb}n^{c}_{,a}+g_{ac}n^{c}_{,b}\right],$
(97)
whereby the nonzero $\theta$ components of the extrinsic curvature read
$K^{in}_{\theta\theta}=r_{b}a(\tau),~{}~{}~{}~{}~{}~{}K^{out}_{\theta\theta}=r_{v}\frac{1-\frac{2M(r_{v},v)}{r_{v}}+\frac{dr_{v}}{dv}}{\left[1-\frac{2M(r_{v},v)}{r_{v}}+2\frac{dr_{v}}{dv}\right]^{\frac{1}{2}}}.$
(98)
Setting $\left[K^{in}_{\theta\theta}-K^{out}_{\theta\theta}\right]_{\Sigma}=0$
on the hyper-surface $\Sigma$, together by using equations (17) and (92) one
gets the following relation between the mass function and Vaidya mass on the
boundary as
${\cal M}(\tau,r_{b})=2M(r_{v},v).$ (99)
From equation (99) and equation (18) it is seen that the behavior of Vaidya
mass is decided by the allowed values of $\rho_{{}_{{}_{0}}m}$, $\alpha$ and
$m$ which prompt the gravitational collapse scenario to end in the formation
of a naked singularity. In order to get a relation describing the rate of
change of Vaidya mass with respect to $r_{v}$ one has to match the $\tau$
component of the extrinsic curvature on the hyper-surface $\Sigma$. Setting
$\left[K^{in}_{\tau\tau}-K^{out}_{\tau\tau}\right]=0$ we have
$M(r_{v},v)_{,r_{v}}=\frac{{\mathcal{M}}}{2r_{v}}+r_{b}^{2}a\ddot{a}.$ (100)
Now it can be seen that at the singular time, $\tau=\tau_{s}$ the ratio
$2M(r_{v},v)/r_{v}$ tends to zero. Thus the exterior spacetime at the singular
epoch reads
$ds^{2}=-dv^{2}-2dvdr_{v}+r_{v}^{2}d\Omega^{2},$ (101)
which describes a Minkowski spacetime in retarded null coordinates. Hence, the
exterior generalized Vaidya metric at singular time can be smoothly extended
to the Minkowski spacetime as the collapse completesPSJ . The occurrence of a
naked singularity as the final fate of a collapse scenario depends on the
existence of families of non-spacelike trajectories reaching faraway observers
and terminating in the past at the singularity. In order to show this we begin
by equation (92) and after using equation (99) we get
$\left[\frac{dv}{d\tau}\right]_{\Sigma}=\frac{1-r_{b}\dot{a}}{1-\frac{{\mathcal{M}}(\tau,r_{b})}{r_{v}}}.$
(102)
It is seen that imposing the null condition on the Vaidya metric leads to the
same relation as the above. This means that null geodesics can come out from
the singularity and reach distant observers before it evaporates into the free
space. On the other hand, since for the allowed values of $\alpha$,
$\rho_{{}_{{}_{0}m}}$ and $m$ formation of trapped surfaces in spacetime is
avoided and from another side the singularity emerges outside of the event
horizon, such a congruence of trajectories can be detected by the outside
observer.
Figure 1: The behavior of Kretschmann scalar (in units of $s^{-4}$) as a
function of proper time for $w=\frac{1}{3}$ (upper-left figure) and different
values of $\alpha$ and $\delta_{-}$. $\alpha=-0.65$, $m_{-}=2.02575$ and
$\delta_{-}=-2.02542$, Solid curve, $\alpha=-0.15$, $m_{-}=5.97355$ and
$\delta_{-}=-3.33038$, Dotted curve and $\alpha=-0.05$, $m_{-}=6.6128$ and
$\delta_{-}=-3.75922$, Dashed curve. For the initial energy density, scale
factor, and proper time we have adopted the values $\rho_{{}_{0m}}=1$,
$a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. The corresponding singular
epoches are, $\tau_{s}=2.49674$ for Solid curve, $\tau_{s}=9.44081$ for Dotted
curve, and $\tau_{s}=27.8945$ for Dashed curve. Figure 2: The behavior of
Kretschmann scalar (in units of $s^{-4}$) as a function of proper time for
$w=0$ (upper-right figure) and different values of $\alpha$ and $\delta_{-}$.
$\alpha=-3.5$, $m_{-}=1.52406$ and $\delta_{-}=-1.03175$, Solid curve,
$\alpha=-2.5$, $m_{-}=1.60424$ and $\delta_{-}=-1.12996$, Dotted curve and
$\alpha=-1.5$, $m_{-}=1.8076$ and $\delta_{-}=-1.34034$, Dashed curve. For the
initial energy density, scale factor, and proper time we have adopted the
values $\rho_{{}_{0m}}=1$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively.
The corresponding singular epoches are, $\tau_{s}=2.5083$ for Solid curve,
$\tau_{s}=2.70354$ for Dotted curve, and $\tau_{s}=3.19309$ for Dashed curve.
Figure 3: The behavior of Kretschmann scalar (in units of $s^{-4}$) as a
function of proper time for $w=-\frac{1}{3}$ (lower-left figure) and different
values of $\alpha$ and $\delta_{+}$. $\alpha=-90$, $m_{+}=1.44581$ and
$\delta_{+}=-0.616689$, Solid curve, $\alpha=-30$, $m_{+}=1.46909$ and
$\delta_{+}=-0.638609$, Dotted curve and $\alpha=-10$, $m_{+}=1.54031$ and
$\delta_{+}=-0.701562$, Dashed curve. For the initial energy density, scale
factor, and proper time we have adopted the values $\rho_{{}_{0m}}=1$,
$a^{\ast}=1$, and $\tau^{\ast}=0$, respectively. The corresponding singular
epoches are, $\tau_{s}=4.04176$ for Solid curve, $\tau_{s}=4.13327$ for Dotted
curve, and $\tau_{s}=4.38509$ for Dashed curve. Figure 4: The behavior of
Kretschmann scalar (in units of $s^{-4}$) as a function of proper time for
$w=-\frac{2}{3}$ (lower-right figure) and different values of $\alpha$ and
$\delta_{+}$. $\alpha=-100$, $m_{+}=1.61179$ and $\delta_{+}=-0.379572$, Solid
curve, $\alpha=-20$, $m_{+}=1.68699$ and $\delta_{+}=-0.407227$, Dotted curve
and $\alpha=-2$, $m_{+}=2.55425$ and $\delta_{+}=-0.608495$, Dashed curve. For
the initial energy density, scale factor, and proper time we have adopted the
values $\rho_{{}_{0m}}=1$, $a^{\ast}=1$, and $\tau^{\ast}=0$, respectively.
The corresponding singular epoches are, $\tau_{s}=10.2643$ for Solid curve,
$\tau_{s}=10.787$ for Dotted curve, and $\tau_{s}=16.808$ for Dashed curve.
## VII Conclusion and outlook
One of the physical motivations for discussing naked singularities is that
these objects provide a useful laboratory for quantum gravity, since in such
ultra-strong gravity regions the length and time scales are comparable to the
Planck length and time. In other words, quantum effects occurring in such
super-dense regimes are no longer covered by the spacetime event horizon and
the chance to observe such effects in the universe is provided. An example is
quantum particle creation due to the formation of a naked singularity, which
has been studied in the literature QPC . During the past twenty years, cosmic
censorship conjecture has been extensively investigated in spherical models of
gravitational collapse of physically reasonable matter. The simplest of those
which has been scrutinized in detail and has been shown that both black holes
and naked singularities form from generic initial conditions is gravitational
collapse of a dust fluid. Dwivedi and Joshi in Dwivedi-Joshi and Waugh and
Lake in Waugh-Lake showed that a naked strong curvature singularity can be
formed in an inhomogeneous dust collapse and a self-similar one, respectively.
Also some examples of naked singularity formation in gravitational collapse of
a scalar field is given in Christodoulou . In this work we have studied the
process of gravitational collapse of a star where the matter fluid obeys the
barotropic equation of state $p=w\rho$, in the context of $f({\mathcal{R}})$
theories of gravity. Making use of metric formalism we wrote the action of
$f({\mathcal{R}})$ gravity as the Brans-Dicke one with vanishing coupling
parameter. Having solved the resulting field equations by taking the ansatz
(23) for scalar field we arrived at the expressions (27)-(30) for the exponent
of Ricci scalar as a function of $\alpha$ and initial energy density. In
Section V we imposed five conditions on the effective energy density and
pressure, the ratio ${\mathcal{M}}/R$, parameter $\delta$, time behavior of
the mass function, and Kretschmann scalar, the validity of which depends on
determining appropriate values of $\alpha$. As long as these conditions are
fulfilled the resulting singularity can be globally naked, i.e., ultra-dense
regions are no longer covered by a spacetime event horizon and physical
effects are allowed to be shared by the external Universe. It is worth
mentioning that there are future finite-time singularities in the dark energy
universe coming from modified gravity as well as in other dark energy
theories. Considering $f(\cal{R})$ gravity models that satisfy cosmological
viability conditions (chameleon mechanism), it is possible to show that
finite-time singularities emerge in several cases. Such singularities can be
classified according to the values of the scale factor $a(t)$, the density
$\rho$, and the pressure $p$ capo .
## VIII Acknowledgments
The authors would like to express their sincere thanks to H. R. Sepangi for
useful discussions.
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|
arxiv-papers
| 2011-06-28T12:15:41 |
2024-09-04T02:49:20.138347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. H. Ziaie, K. Atazadeh, S. M. M. Rasouli",
"submitter": "Khedmat Atazadeh",
"url": "https://arxiv.org/abs/1106.5638"
}
|
1106.5644
|
# The ADS in the Information Age - Impact on Discovery
Edwin A. Henneken1, Michael J. Kurtz, Alberto Accomazzi
###### Abstract
The SAO/NASA Astrophysics Data System (ADS) grew up with and has been riding
the waves of the Information Age, closely monitoring and anticipating the
needs of its end-users. By now, all professional astronomers are using the ADS
on a daily basis, and a substantial fraction have been using it for their
entire professional career. In addition to being an indispensable tool for
professional scientists, the ADS also moved into the public domain, as a tool
for science education. In this paper we will highlight and discuss some
aspects indicative of the impact the ADS has had on research and the access to
scholarly publications.
The ADS is funded by NASA Grant NNX09AB39G.
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138
11footnotetext: Corresponding author at: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA. E-mail address:
ehenneken@cfa.harvard.edu (E. Henneken).
Keywords: Digital Libraries; SAO/NASA Astrophysics Data System; Societal
Impact
## 1\. Introduction
Why do scientists publish? First and foremost, because they want to share
their findings and further science. Essential to this process is the ability
for other people to (efficiently) discover these publications. The process of
information discovery has changed dramatically over the last two decades.
Remember the days of spending hours in a library, paging through A&A
Abstracts, catalogs and tables of contents? Sometimes a publication had to be
retrieved via Inter Library Loan, adding more time to the discovery process. A
lengthy discovery process not only means a long journey before finally
acquiring enough fuel to set the first step towards your goal, it also means
that the discovery process is a significant portion of the cost of employing a
scientist. Back then it was virtually impossible to answer the question “what
is the most popular paper on X among people interested in X?” or to find a set
of review papers on this subject (within a reasonable amount of time). Ease of
access is therefore essential for efficient information discovery. When the
digital revolution of the Information Age culminated in the birth of the
Internet, followed by the World Wide Web, the ingredients were there to take
information discovery to a new level. At this time, communication started to
change from paper to electronic, and this in turn created a fundamental change
in society. Information in electronic form resulted in a big shift in ease of
access. But just ease of access is not enough. In order to discover
information efficiently, you need tools to explore this electronic universe.
The combination of ease of access and (powerful) tools for information
discovery are central to the process of transferring knowledge. In astronomy,
the ADS has been central and pivotal in this digital revolution.
It is difficult to objectively quantify the absolute impact of the ADS (or any
online service, for that matter), but we will highlight a number of facts that
will illustrate aspects of the impact of the ADS. Firstly, the ease of access,
combined with the powerful query capabilities of the ADS, has had a direct
impact on the scientific process in the form of the amount of time gained that
researchers otherwise would have spent going to a library, physically finding
an article, Xeroxing it and returning to their office. Also, access through
the World Wide Web means that communities that historically had little or no
access to the scholarly literature, now have access to at least basic meta
data, scanned articles, Open Access literature and full text through e-prints
in the arXiv repository (with which the ADS is synchronized every night).
Thirdly, by diversifying its holdings, the ADS provides the astronomy
community not only with the essential core journals, but also with
publications from the ever-expanding periphery. Fields that seemed to have no
overlap with astronomy and astrophysics in the past, suddenly become relevant
and core journals for those fields start to have content that should be
available to astronomers and astrophysicists. Next, through its digitization
efforts, the ADS has created access to rare and historical publications.
Finally, another measure of impact is the fact that the existence of the ADS,
and other electronic resources, has had a direct influence on the publication
process itself. We will visit these various modes of impact in more detail in
the following sections.
## 2\. Impact of the ADS on astronomy
### 2.1. Efficiency of Astronomical Research
It seems reasonable to assume that an increased efficiency in discovering
information will translate into researchers being better informed because,
among other things, they will get exposed to a broader range of information
sources per search effort. To what level researchers are informed will most
certainly influence the quality of their research. The increase in efficiency,
using the ADS, is most dramatic for more complex, but still realistic queries.
For example, finding review papers on a given subject, or the most popular
papers among people also interested in a given subject is a matter of just
seconds, using the ADS “Topic Search”. Doing this type of literature research
in a traditional, “paper” library would easily take hours. But also compared
to other electronic resources, using the ADS will result in increased
efficiency and better results. The increase in efficiency becomes even more
pronounced in those cases where the ADS provides links to additional sources
of information (like SIMBAD, NED, VizieR and on-line data).
In order to quantify the increase in efficiency, Kurtz et al. (2000) developed
a metric, based on the concept of equivalent research time gained. The
physical retrieval of an article (“overhead”) was estimated to be 15 minutes
on average, in the “paper age”. Using the ADS virtually removes this overhead,
therefore resulting in a gain of roughly 15 minutes research equivalent time
when an article is downloaded. Kurtz et al. (2000) further estimate that
downloading an abstract, citation list or reference list gains one third of
the time of an entire article (5 minutes). Based on the worldwide combined ADS
logs from March 1999, Kurtz et al. (2000) found that the impact of the ADS on
astronomy is 333 FTE (Full Time Equivalent, 2000 hour) research years per
year. In a later study Kurtz et al. (2005a) found that, based on the 2002 ADS
usage logs, an increase in efficiency of astronomical research by 736 FTE
researcher equivalent years, or about 7% of all research done in astronomy. If
we apply this to the current logs (extrapolating from the Jan 2011 logs to a
full year) and just the four main astronomy journals (ApJ, A&A, MNRAS, AJ), we
arrive at a number of 985 FTE research years per year. Whatever the exact
interpretation of an FTE researcher year is, the impact of the ADS on
astronomy, and science in general, is clearly substantial.
Another example of contributions to increased efficiency are the services
myADS and myADS-arXiv, providing the scientific community with a one stop shop
for staying up-to-date. In 2003 the ADS introduced a notification service,
myADS, which uses sophisticated queries (2nd order operators) to give users a
powerful tool for staying current with the latest literature in their sub
fields of astronomy and physics. The myADS-arXiv service (“daily myADS”)
provides a powerful and unique filter on the enormous amount of bibliographic
information added to the ADS on a daily basis, as it gets synchronized with
arXiv. In essence myADS-arXiv is a tailor-made, open access, virtual journal
(see Henneken et al. (2007a)). The myADS services automate an obvious need for
most scientists: answering questions like “Who is citing my papers?” and “What
are recent, most popular and most cited papers in my field?”. Automating these
queries and providing an alert service saves time. The popularity of these
services is reflected in the steady growth of the number of myADS and myADS-
arXiv users (see figure 1).
Figure 1.: The number of myADS and myADS-arXiv users over time
### 2.2. Worldwide Access and Sociological Impact
In 2002, the Harvard-Smithsonian Center for Astrophysics Visiting Committee
reported that the ADS “empowered astronomy research in underdeveloped
countries and small institutions” (From Report of the CfA Visiting Committee,
2002). In November of 2005, the United Nations General Assembly commended the
ADS for “the mirror sites of the NASA-funded Astrophysics Data System (ADS). .
. had been enthusiastically accepted by the scientific community and had
become important assets for developing countries . . . ” (excerpt from UN
(2005)). The ADS has been instrumental in helping to bridge the “Digital
Divide” (see e.g. ITU (2007)) for astronomical research. In Henneken et al.
(2009) we showed that increased Internet access, in particular in Least
Developed Countries (UN definition), has resulted in increased ADS usage. In
that publication we examined readership in a particular region as a function
of GDP per capita (GPC), because science and technology depend heavily on
available budgets. Figure 2 (taken from that paper) shows the relation between
normalized GPC and normalized usage for a specific region. The definition for
the region “Least Developed Countries” was taken from UN (2008). The numbers
have been normalized with respect to the 1997 level, so the diagram shows a
relative growth with respect to 1997. The general evolution in this diagram is
up and to the right, as time progresses. The data used to construct figure 2
were taken from the ADS logs, the “Earth Trends Database” (WRI (2008)), the
“World Economic Outlook” (IMF (2008)) and the “World Statistics Pocketbook -
Least Developed Countries” (UN (2007)).
Figure 2.: The fraction of world usage for a number of regions as a function
of GDP per capita for that region. Both quantities have been normalized by
their value in 1997
Our logs show that growth in world usage is clearly driven by regions with the
biggest potential. High- and middle-income regions have reached a saturation
level in the density of Internet users, causing normalized ADS usage to
increase at a slower rate. Clearly, the biggest potential is in low-income
regions. There is a rapid increase in Internet user density in these regions,
and a similar rapid increase in the number of ADS users. This indicates that
the new potential is being used and in this sense there is a bridging
happening of the “Digital Divide”, at least from the ADS perspective.
Another metric for impact is the level of penetration in the scientific
community. In other words: how many people are using the ADS regularly (10 or
more times per month)? Figure 3 shows that this number of regular users is
still increasing.
Figure 3.: The number of ADS users of various types over time
In Henneken et al. (2009) we showed that usage by regular ADS users has a
median that is fairly constant at about 21 reads per month. This is an
indication for the fact that all frequent ADS users on average use the ADS on
a daily basis. Initially this meant that all professional astronomers use the
ADS daily, but by now there must be a growing group of professionals outside
astronomy (for example physicists and engineers). This is also indicated by
the fact that the current number of frequent ADS users is significantly larger
than the number of professional astronomers in the world (the IAU currently
has just over 10,000 individual members, and there were about 17,000 different
authors listed in the main astronomy journals in 2010).
Usage data also indicate another type of impact: the ADS has become a public
service. ADS usage has changed qualitatively over time. The distribution of
reads over users has changed, specifically the ratio of frequent to infrequent
users has changed considerably over time. We feel that the strong increase in
infrequent users has an impact on the science education of the general public.
Between 35% and 40% of all ADS use actually comes from links external to the
ADS. The Google, Google Scholar and Bing search engines are the largest
sources, but the ADS is also linked to by thousands of static pages. For
instance, Wikipedia has more than 17,000 links to the ADS. While some of the
page views are scientists using Google to find a reference, the vast majority
are generated by the general public.
### 2.3. Diversification and Expansion of Holdings
In order to stay relevant for its core users, the ADS holdings must accurately
reflect the complexity of the fields these people are working in. During the
lifetime of the ADS, new fields emerged in astronomy and astrophysics, and
existing fields became more complex, reflected in more diffuse boundaries with
fields that historically had a tenuous connection with astronomy at best. The
holdings of the ADS evolved accordingly. The number of journals has increased
significantly over time and we currently have over 1.8 million records in our
astronomy database, distributed over more than 4,500 journals.
Our digitization efforts are another aspect of the diversification of the ADS
holdings. All the major astronomy journals have been scanned back to volume 1,
and they have recently been re-scanned to capture grey scale and color
content. Now the ADS is focusing on scanning publications with high scientific
impact and that are not otherwise available online. We have also been
collaborating with librarians and observatories to digitize series of
historical publications that are difficult to locate and obtain. The impact of
this effort is considerable: were it not for their availability in ADS, much
of this content would be simply out of the reach of researchers, librarians,
and the general public. Consider, for instance, the content that ADS has
digitized in collaboration with the CfA library, which consists of almost
900,000 pages of historical observatory publications. During 2009 alone, more
than 1 million articles were downloaded from this collection. In other words,
many historical publications have been given a new life thanks to the
digitization efforts of the ADS. That this historical material is being read
can be illustrated by looking at the obsolescence function of reads. The ADS
logs from March and April of 2011 indicate a reads rate for historical
publications of about 1.2 reads per paper per year, which agrees with the
results found in Kurtz et al. (2005b).
### 2.4. Trends in the Electronic Publication Process
It seems almost unavoidable that the presence of the ADS would also have had
an impact on the actual products of the scientific community, specifically
publications. Ease of access is very likely to have an impact on the actual
writing process, in the sense that publications can probably be classified as
being “published before the introduction of the World Wide Web” and “published
after the introduction of the World Wide Web”. One way of finding
classifications is to represent publications as nodes in an relational
network. An obvious example is the citation network: articles A and B are
directionally connected when “A cites B”. This network turns out to have a
very specific trend: high densification over time, i.e. non-linear growth.
This is shown in figure 4, showing the relation between the number of nodes
and the number of edges in the network.
Figure 4.: Number of nodes versus the number of edges in the citation network
of the major astronomy journals in the period 1980 - 2006
One implication is that, in an average sense, bibliographies have increased in
length over time. This becomes abundantly clear in figure 5, which shows the
average number of references in bibliographies as a function of publication
year.
Figure 5.: The average number of references in bibliographies in the main
astronomy journals
When an overall linear trend is subtracted (represented by “x” in figure 5), a
deviation from this trend is observed from the mid to late 1990’s on. One
could argue that the ease of access, offered by web services like the ADS,
results in more citations (on average). Clearly, the ADS is not the only
source for bibliographic information, but since our data set consisted solely
of the major astronomy journals and the ADS has 100% penetration in the
astronomical community, it is very likely that the ADS contributed
significantly to the observed trend.
The ADS also has had impact on the study of trends in the (electronic)
publication process and in how literature is being used. Michael J. Kurtz has
contributed significantly to our understanding of e.g. usage and citation
bibliometrics (see e.g. Kurtz et al. (2000) and Kurtz et al. (2005b), Kurtz et
al. (2005c)) and the discussion of the the influence of Open Access (see e.g.
Kurtz and Henneken (2007) and Henneken et al. (2007b)).
## 3\. The Future ADS
How is the ADS moving into the future? According to some, the Information Age
is over and we are moving into the Imagination Age, where creativity and
imagination are becoming the primary creators of economic value, as opposed to
thinking and analysis (see e.g. King 2007). Whatever “age” we are in, there
will remain a desire with individuals to be able to transfer information
freely and to have instant access to information that would have been
difficult (or even impossible) to retrieve previously. But there is definitely
a new trend. It is becoming more and more common that we are faced with data
collections of such magnitude and complexity, that conventional data and
information discovery models brake. We need innovative ways to explore an
enormous, rapidly expanding data universe. Not too long ago gigabytes seemed
like a lot. The Panoramic Survey Telescope and Rapid Response System (Pan-
STARRS) project is expected to produce several terabytes of raw data per
night! The ADS holdings will never come close to these amounts of data, but in
our case it is the complexity of the data that requires our attention, and
also the more complicated demands (and perhaps expectations) of people using
the ADS. Therefore, in order to stay relevant for its end-users, the ADS has
to innovate and develop new ways to explore the literature universe. We are
doing exactly that in our test environment “ADS Labs”, where we expose our
users to new technologies and prototype services. For example, the
“streamlined search” allows users to find publications that are review papers
for the subject they are interested in. Where in the “classic” approach the
user had to have knowledge about what he/she is looking for and how to find
it, the new streamlined search in “ADS Labs” offers a means to specify
beforehand what a user is looking for. In addition to this, the results will
be displayed in a different way. We use faceted filtering allowing our users
to explore the literature by filtering collections of records by a particular
property or set of properties. This is an efficient way to quickly focus on a
particular subset of records, from the results of a broad search. In this way,
we provide the user with a custom information environment. We feel that this
approach results in an even more efficient information discovery environment
than the classic approach. In addition to this, the abstract page now also
includes recommendations. These recommendations are based on publication
similarity, in combination with article usage information from people who use
the ADS frequently. The inclusion of recommendations to the usual citations
and references links adds an element of serendipity to the usual activity of
searching and browsing the literature. In addition to the new abstract search,
ADS Labs also offers a full text search that includes current full text for
all main astronomy journals. This mode of searching add a whole new dimension
to information discovery, hitherto not available.
## 4\. Concluding Remarks
We discussed a number of metrics for the impact of the ADS. These metrics are
indicators for the impact of the ADS has on efficiency of information
retrieval (essential for both research and scholarly communication),
popularization of astronomy and the publication process itself (both in
production and understanding). On all levels, the impact of the ADS has been
significant. This impact has also been expressed in the form of recognition by
peer organizations and prizes bestowed upon ADS staff.
It has been widely recognized (Fabbiano et al. (2010)) that in this era of
data-intensive science, it is critical for researchers to be able to
seamlessly move between the description of scientific results, the data
analyzed in them, and the processes used to produce them. As observations,
derived data products, publications, and object metadata are curated by
different projects and archived in different locations, establishing the
proper linkages between these resources and describing their relationships
becomes an essential activity in their curation and preservation. The ADS, in
collaboration with the VAO, the NASA archives, and the SIMBAD project, is
leading the effort of better integrating and linking the research literature
with the body of heterogeneous astronomical resources in the VO, allowing
users as well as applications to easily cross boundaries between archives.
This endeavor has been named Semantic Interlinking of Astronomical Resources
(Accomazzi and Dave (2011)), and is funded by the Virtual Astronomical
Observatory Data Curation & Preservation project. By maintaining its
traditional role, but introducing innovations in its querying capabilities,
and by taking on this new role in the inter-linking of information sources,
the ADS intends to keep playing a central, pivotal role within the
astronomical community.
## References
* Accomazzi and Dave (2011) Accomazzi, A., Dave, R. 2011. Semantic Interlinking of Resources in the Virtual Observatory Era. arXiv:1103.5958.
* Fabbiano et al. (2010) Fabbiano, G., et al. 2010. Recommendations of the VAO-Science Council. arXiv:1006.2168
* Henneken et al. (2009) Henneken, E. A., Kurtz, M. J., Accomazzi, A., Grant, C. S., Thompson, D., Bohlen, E., Murray, S. S. 2009. Use of astronomical literature - A report on usage patterns. Journal of Infometrics 3, 1.
* Henneken et al. (2007a) Henneken, E., Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Thompson, D., Bohlen, E., Murray, S. S. 2007. myADS-arXiv – a Tailor-made, Open Access, Virtual Journal. Library and Information Services in Astronomy V 377, 106.
* Henneken et al. (2007b) Henneken, E. A., Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Thompson, D., Bohlen, E., Murray, S. S., Ginsparg, P., Warner, S. 2007. E-prints and journal articles in astronomy: a productive co-existence. Learned Publishing, 20, 16
* IMF (2008) International Monetary Fund, “World Economic Outlook” (April 2008).
* ITU (2007) International Telecommunication Union, “World Information Society Report 2007”, ITU, Geneva.
* King (2007) King, R. J. 2007. Essay for British Council (www.britishcouncil.org)
* Kurtz & Bollen (2010) Kurtz, M. J., Bollen, J. 2010. Usage Bibliometrics. Annual Review of Information Science and Technology, vol 44, p. 3-64 44, 3-64.
* Kurtz and Henneken (2007) Kurtz, M. J., Henneken, E. A. 2007. Open Access does not increase citations for research articles from The Astrophysical Journal. ArXiv e-prints arXiv:0709.0896.
* Kurtz et al. (2005a) Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Demleitner, M., Murray, S. S., Martimbeau, N., Elwell, B. 2005. The Bibliometric Properties of Article Readership Information. Journal of the American Society for Information Science and Technology 56, 111.
* Kurtz et al. (2005b) Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C., Demleitner, M., Henneken, E., Murray, S. S. 2005. The Effect of Use and Access on Citations. Information Processing and Management 41, 1395-1402.
* Kurtz et al. (2005c) Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Demleitner, M., Murray, S. S. 2005. Worldwide Use and Impact of the NASA Astrophysics Data System Digital Library. Journal of the American Society for Information Science and Technology, 56, 36
* Kurtz et al. (2000) Kurtz, M. J., Eichhorn, G., Accomazzi, A., Grant, C. S., Murray, S. S., Watson, J. M. 2000. The NASA Astrophysics Data System: Overview. Astronomy and Astrophysics Supplement Series 143, 41
* UN (2008) United nations, “About LDCs, Least Developed Countries”, UN-OHRLLS (http://www.unohrlls.org)
* UN (2007) United Nations, “World Statistics Pocketbook - Least Developed Countries”. Department of Economic and Social Affairs, Statistics Division, Series V No.31/LDC. United Nations, New York, 2007
* UN (2005) United Nations, 2005, “Report on the United Nations/European Space Agency/National Aeronautics and Space Administration of the United States of America Workshop on the International Heliophysical Year 2007”, COPUOS report A/AC.105/856.
* WRI (2008) World Resources Institute, 2008, EarthTrends database (http://earthtrends.wri.org)
|
arxiv-papers
| 2011-06-28T12:21:55 |
2024-09-04T02:49:20.147345
|
{
"license": "Public Domain",
"authors": "Edwin A. Henneken, Michael J. Kurtz, Alberto Accomazzi",
"submitter": "Edwin Henneken",
"url": "https://arxiv.org/abs/1106.5644"
}
|
1106.5674
|
# Past horizons in D-dimensional Robinson–Trautman spacetimes
Otakar Svítek Institute of Theoretical Physics, Faculty of Mathematics and
Physics, Charles University in Prague, V Holešovičkách 2, 180 00 Prague 8,
Czech Republic ota@matfyz.cz
###### Abstract
We derive the higher dimensional generalization of Penrose–Tod equation
describing past horizon in Robinson–Trautman spacetimes with a cosmological
constant and pure radiation. Existence of its solutions in $D>4$ dimensions is
proved using tools for nonlinear elliptic partial differential equations. We
show that this horizon is naturally a trapping and a dynamical horizon. The
findings generalize results from $D=4$.
higher dimensions, horizon, black holes, nonlinear PDE
###### pacs:
04.20.Gz, 04.50.Gh
## I Introduction
Robinson–Trautman spacetimes represent a class of expanding nontwisting and
nonshearing solutions RobinsonTrautman:1960 ; RobinsonTrautman:1962 ;
Stephanietal:book describing generalized black holes. Various aspects of this
family in four dimensions have been studied in the last two decades. In
particular, the existence, asymptotic behaviour and global structure of
_vacuum_ Robinson–Trautman spacetimes of type II with spherical topology were
investigated, most recently in the works of Chruściel and Singleton Chru1 ;
Chru2 ; ChruSin . In these rigorous studies, which were based on the analysis
of solutions to the nonlinear Robinson–Trautman equation for generic,
arbitrarily strong smooth initial data, the spacetimes were shown to exist
globally for all positive retarded times, and to converge asymptotically to a
corresponding Schwarzschild metric. Interestingly, extension across the
“Schwarzschild-like” future event horizon can only be made with a finite order
of smoothness. Subsequently, these results were generalized in podbic95 ;
podbic97 to the Robinson–Trautman vacuum spacetimes which admit a
nonvanishing _cosmological constant_ $\Lambda$. These cosmological solutions
settle down exponentially fast to a Schwarzschild–(anti-)de Sitter solution at
large times $u$. Finally, the Chruściel–Singleton analysis was extended to
Robinson-Trautman spacetimes including matter, namely _pure radiation_
PodSvi:2005 . It was demonstrated that these solutions with pure radiation and
a cosmological constant exist for any smooth initial data, and that they
approach the spherically symmetric Vaidya–(anti-)de Sitter metric.
In podolsky-ortaggio , Robinson–Trautman spacetimes (containing aligned pure
radiation and a cosmological constant $\Lambda$) were generalized to any
dimension. The evolution is governed by a simpler equation in higher
dimensions, contrary to the four-dimensional case where fourth order parabolic
type Robinson–Trautman equation occurs. Also, the possible algebraic types
were determined. But still several interesting features deserve attention, the
presence of horizons being among them. Similarly to four dimensions, higher-
dimensional Robinson–Trautman family of solutions contains several important
special cases, e.g. Schwarzschild–Kottler–Tangherlini black holes and
generalizations of the Vaidya metric. The study of higher-dimensional
spacetimes and their features help to comprehend which properties survive the
generalisation and which are closely tied to four dimensions, thus deepening
the understanding of General Relativity. Lately, considerable interest in
higher dimensions comes from outside of the purely relativistic community.
Our concern here is to locate the past (white hole) horizon. In general
dynamical situations this might be rather nontrivial since the obvious
candidate - event horizon - is a global characteristic and therefore the full
spacetime evolution is necessary in order to localize it. Therefore, over the
past years different quasilocal characterizations of black hole boundary were
developed. The most important ones being apparent horizon hawking-ellis ,
trapping horizon hayward and isolated or dynamical horizon ashtekar ;
krishnan . The basic local condition in the above mentioned horizon
definitions is effectively the same: these horizons are sliced by marginally
trapped hypersurfaces with vanishing expansion of outgoing (ingoing) null
congruence orthogonal to the surface. Quasilocal horizons are frequently used
in numerical relativity for locating the black holes or in black hole
thermodynamics.
For the vacuum four dimensional Robinson–Trautman solutions without
cosmological constant the location of the horizon together with its general
existence and uniqueness has been studied by Tod tod . Later, Chow and Lun
chow-lun analyzed some other useful properties of this horizon and made
numerical study of both the horizon equation and Robinson–Trautman equation.
These results were recently extended to nonvanishing cosmological constant
PodSvi:2009 .
## II Robinson–Trautman spacetime in D dimensions
Robinson–Trautman spacetimes (containing aligned pure radiation and a
cosmological constant $\Lambda$) in any dimension were obtained in podolsky-
ortaggio using the geometric conditions of the original articles about the
four-dimensional version of the spacetime RobinsonTrautman:1960 ;
RobinsonTrautman:1962 . Namely, they required the existence of a twistfree,
shearfree and expanding null geodesic congruence. They have arrived at the
following metric valid in higher dimensions
${\rm d}s^{2}=\frac{r^{2}}{P^{2}}\,\gamma_{ij}\,{\rm d}x^{i}{\rm
d}x^{j}-2\,{\rm d}u{\rm d}r-2H\,{\rm d}u^{2}$ (1)
where $2H=\frac{{\cal R}}{(D-2)(D-3)}-2\,r(\ln
P)_{,u}-\frac{2\Lambda}{(D-2)(D-1)}\,r^{2}-\frac{\mu(u)}{r^{D-3}}$. The
unimodular spatial $(D-2)$-dimensional metric $\gamma_{ij}(x)$ and the
function $P(x,u)$ must satisfy the field equation ${\cal R}_{ij}=\frac{{\cal
R}}{D-2}h_{ij}$ (with $h_{ij}=P^{-2}\gamma_{ij}$ being the rescaled metric)
and $\mu(u)$ is a “mass function” (we assume $\mu>0$). In $D=4$ the field
equation is always satisfied and ${\cal R}$ (Ricci scalar of the metric $h$)
generally depends on $x^{i}$. However, in $D>4$ the dependence on $x^{i}$ is
ruled out (${\cal R}={\cal R}(u)$). But generally, it still allows a huge
variety of possible spatial metrics $h_{ij}$ (e.g., for ${\cal R}>0$ and
$5\leq D-2\leq 9$ an infinite number of compact Einstein spaces were
classified). The dynamics is also different in $D>4$. While in four dimensions
there is a fourth order Robinson–Trautman equation, the corresponding
evolution equation is much easier in higher dimensions
$(D-1)\,\mu\,(\ln P)_{,u}-\mu_{,u}=\frac{16\pi n^{2}}{D-2}\ ,$ (2)
where function $n$ describes the aligned pure radiation.
## III Past horizon
In our case, we will be dealing only with the condition of vanishing expansion
defining the marginally trapped hypersurfaces. Concretely, we will search for
the past horizon similarly to previous studies in four dimensions and
corresponding to the form of the metric containing retarded time. As will be
clear later, we might call it trapping horizon or even dynamical horizon if it
is spacelike, assuming appropriate higher-dimensional generalization of these
notions (see Cao ). In four-dimensional case the parabolic character of
Robinson–Trautman equation makes it generally impossible to extend the
spacetime to past null infinity (the solutions of the Robinson–Trautman
equation are generally diverging when approaching $u=-\infty$) and it is
impossible to define event horizon. In higher dimensions this is no longer
truth (the evolution equation is different) but since one would like to
investigate the horizon existence generically, without prior specification of
all necessary functions (e.g. dynamics of pure radiation) and geometry of
$(D-2)$-dimensional spatial hypersurfaces, the best approach is still using
the quasilocal horizons. In figure 1, the schematic conformal picture of
Robinson–Trautman spacetime (for $D=4$ and without cosmological constant for
simplicity) is presented together with the approximate location of the
horizons (initial data are given at $u=u_{0}$).
Figure 1: Schematic conformal diagram of Robinson-Trautman spacetime in $D=4$
with $\Lambda=0$ and indicated past (trapping) horizon (PH) and event horizon
(EH).
The explicit parametrization of the past horizon hypersurface is
$r=R(u,x^{i})$ such that its intersection with each $u=u_{1}$ slice is an
outer marginally past trapped ($D-2$)-surface.
For the calculation of the expansion of an appropriate null congruence we will
use a straight-forward generalization of the tetrad formalism to arbitrary
dimension. Note that one can no longer use complex vector notation. Using two
null covectors $l_{a},n_{a}$ (with normalization $l_{a}n^{a}=-1$) and $D-2$
spatial covectors $m_{a\\{i\\}}$ ($i=1,..,D-2$) we suppose the following
decomposition of the metric
$g_{ab}=-2\,l_{(a}n_{b)}+m_{a\\{i\\}}m_{b\\{j\\}}\,\delta^{ij}$ (3)
Null D-ad (D-bein) adapted to the trapped hypersurface (using the above
mentioned parametrization) has the following form:
$\displaystyle l^{a}$ $\displaystyle=$ $\displaystyle(0,1,0,..,0)$
$\displaystyle n^{a}$ $\displaystyle=$
$\displaystyle\left(1,\,[-H+{\textstyle\frac{r^{2}}{2}}\,h({\mathbf{\nabla}}R,{\mathbf{\nabla}}R)]\,,{\mathbf{\nabla}}R\right)$
(4) $\displaystyle m^{a}_{\\{i\\}}$ $\displaystyle=$
$\displaystyle\left(0,\,Pr\,h({\mathbf{\nabla}}R,{\mathbf{w}}_{i})\,,{\textstyle\frac{1}{r}}{\mathbf{w}}_{i}\right)$
where $D-2$ vectors ${\bf w}_{i}$ diagonalize metric $h$,
${\bf\nabla}R=\\{R^{,x^{1}},..,R^{,x^{D-2}}\\}$ and $h(\cdot,\\!\cdot)$
denotes scalar product w.r.t.$\;h$. Fortunately, in subsequent calculations we
do not need the explicit form of the vectors ${\bf w}_{i}$, it is sufficient
to know their orthogonality properties.
By straight-forward computation one easily calculates the expansion associated
with the congruence generated by $l^{a}$ to be $\Theta_{l}=\frac{D-2}{r}$
meaning that the outgoing null congruence is diverging. This is exactly what
one assumes when dealing with the past trapped surface and is the additional
condition in the definition of trapping horizon hayward .
## IV Generalized Penrose–Tod equation
Ingoing null congruence expansion can be calculated using the formula
(sometimes a $(D-2)$ factor is used in the definition, but we are going to
evaluate it to zero anyway) $\Theta_{n}=n_{a;b}\,p^{ab}$, where the tensor
$p^{ab}=g^{ab}+2\,l^{(a}n^{b)}$ corresponds to the hypersurface projector.
From $\Theta_{n}=0$ (equivalent to Penrose–Tod equation in four dimensions) we
get the marginally trapped hypersurface condition
${\cal R}-{\textstyle\frac{2(D-3)}{D-1}}\Lambda
R^{2}-{\scriptstyle(D-2)(D-3)}\frac{\mu}{R^{D-3}}{\scriptstyle-{2(D-3)}}\Delta(\ln
R)-$ $-{\scriptstyle(D-4)(D-3)}\,h(\nabla\ln R,\nabla\ln R)=0$ (5)
It is a nonlinear second order partial differential equation, where both the
laplacian and scalar product in the last term correspond to the Einstein
metric $h_{ij}$. Interesting property of this equation is that for $D>4$ its
nonlinearity is much worse since the term quadratic in derivatives appears.
### IV.1 Results for $D=4$
In four-dimensional case one can no longer use the existence proof for
equation (5) given by Tod tod when the cosmological constant is present.
However, one can use the version of sub and super-solution method adapted to
Riemannian manifolds by Isenberg isenberg and valid for equations of the form
$\Delta\psi=f(x,\psi)$. For the proof of uniqueness one may use a
straightforward modification of the original Tod’s proof tod incorporating
the cosmological constant. Using Newmann-Penrose equations one can also
determine the character of the horizon as a three-dimensional hypersurface.
These results (for $\mu=2m=const.$) are derived in PodSvi:2009 and summarized
in the following table:
Table 1: $D=4$
$\begin{array}[]{||c||c|c|c||}\hline\cr\hline\cr\mbox{RESULTS}&\Lambda=0&\Lambda<0&\Lambda>0\\\
\hline\cr\hline\cr\mbox{Existence}&\mbox{Always}&\mbox{Always}&\Lambda<\frac{4}{9\mu^{2}}\\\
\hline\cr\mbox{Uniqueness}&\mbox{Always}&\mbox{Always}&R<\sqrt[3]{\frac{3\mu}{2\Lambda}}\\\
\hline\cr\mbox{Spacelike or
null}&\mbox{Always}&\mbox{Always}&R<\sqrt[3]{\frac{3\mu}{2\Lambda}}\\\
\hline\cr\hline\cr\end{array}$
The restrictions for the positive cosmological constant can be easily
understood by specializing to spherical symmetry and $\Lambda>0$
(Schwarzschild de–Sitter) :
* •
$\Lambda<\frac{4}{9\mu^{2}}=\frac{1}{9m^{2}}$ rules out an over-extreme case.
* •
$R<\sqrt[3]{\frac{3\mu}{2\Lambda}}=\sqrt[3]{\frac{3m}{\Lambda}}$ for the
extreme case ($9\Lambda m^{2}=1$) reduces to $R<3m$ which may be interpreted
as showing the uniqueness of the past black/white hole horizon (as opposed to
the cosmological one).
Both explanations are quite natural and not surprising.
### IV.2 $D>4$ : Existence of the solution
The methods used in $D=4$ are not applicable when the equation is of the form
(after the substitution $R=Ce^{-u}$ in (5), assuming $u\geq 0$ with a suitable
constant $C$)
$\Delta u=F(x,u,\nabla u)\ ,$ (6)
where $F$ is quadratic in gradient.
To prove existence of the solution to this quasilinear equation we will
proceed by combining several steps (motivated by Kuo and using results from
Besse ; Boccardo ; Gilbarg-Trudinger ).
* 1.
We will consider the differential operator $Pu=-2(D-3)\Delta u+\rho u$, with
$\rho>0$ on a Riemannian manifold $M$. By using Maximum Principle we can prove
that $\ker(P)=0$ Besse .
* 2.
The linear differential equation $Pu=f$ with $f\in C^{0,\alpha}(M)$ (Hölder
space over $M$) has unique solution $u\in C^{2,\alpha}(M)$ (this standard
result can be proven for example by Fredholm alternativeBesse and the
previous step).
* 3.
To proceed with the nonlinear problem $Pu=f(x,u,\nabla u)$, with $f$
determined from (5) as ($\|\cdot\|_{h}$ stands for the norm with respect to
the positive definite metric $h_{ij}$)
$f=-\rho u+{\cal R}-{\textstyle\frac{2(D-3)}{D-1}}\Lambda C^{2}e^{-2u}-$
$\qquad-{\scriptstyle(D-2)(D-3)}{\mu}{C^{3-D}}e^{(D-3)u}-{\scriptstyle(D-4)(D-3)}\|\nabla
u\|_{h}^{2}\ ,$
we introduce the following truncature Kuo ; Boccardo :
$f_{n}$ \- truncature of $f$ by $\pm n$.
Then the map $v\in C^{1,\beta}(M)\to f_{n}(x,v,\nabla v)$ is bounded. Using
the previous step together with results on composition of Hölder functions
there exists a unique $w\in C^{2,\alpha\beta}(M)$ solving $Pw=f_{n}(x,v,\nabla
v)$.
* 4.
The map $v\to w$ from previous step satisfies conditions of Schauder Fixed
Point theorem, namely the a priori boundedness (see Besse or Gilbarg-
Trudinger ) $\Rightarrow$ for each $n$ there is a fixed point $u_{n}\in
C^{1,\beta}(M)$ (even $u_{n}\in C^{2,\alpha\beta}(M)$) solving
$Pu_{n}=f_{n}(x,u_{n},\nabla u_{n})$ and moreover one can easily verify that
$\|u_{n}\|_{L^{\infty}}\leq\frac{n}{\rho}$ (considering
$Pu_{n}=f_{n}(x,v,\nabla v)\leq n$ and compactness for evaluation at the
maximum of $u_{n}$).
* 5.
Using results of Boccardo, Murat & Puel Boccardo , in particular their
Proposition 3.6, we can state the following corollary:
Assuming that metric $h_{ij}$ is smooth, function $F$ can be estimated like
$|F|\leq B(u)(1+|\nabla u|^{2})$ (where $B(u)$ is increasing function on
${\mathbb{R}}^{+}$), and there exist a sub- and a super-solution footnote
$u^{-}\leq u^{+}$, $u^{\pm}\in C^{1,\beta}(M)\cap L^{\infty}(M)$, then there
is a $L^{\infty}$-bounded subsequence $u_{\bar{n}}$ of the approximating
solutions from the previous step satisfying $u^{-}\leq u_{\bar{n}}\leq u^{+}$
a.e.
Indeed, inspecting the above defined function $f=\rho u-2(D-3)F$ one can
verify that function $B(u)$ might be found, namely there is no singular
behaviour at $u=0$. Also, the domain we are dealing with is compact and
therefore any dependence on $x$ can be bounded for well behaved objects we
use. For example, one may select the following bounding function
$B(u)=\max\left(W,{\textstyle\frac{D-4}{2}}\max_{x\in M}\|h_{ij}(x)\|\right)+$
$+{\textstyle\frac{(D-2)\mu C^{3-D}}{2}}e^{(D-3)u}\ ,$
where $W=\left|{\cal R}-{\textstyle\frac{\Lambda
C^{2}}{D-1}}-{\textstyle\frac{(D-2)\mu C^{3-D}}{2}}\right|$ and the matrix
norm of $h$ was used.
* 6.
Thanks to elliptic estimate $\|u_{\bar{n}}\|_{C^{2,\gamma}}\leq
K(\|u_{\bar{n}}\|_{C^{0}}+\|f_{\bar{n}}\|_{C^{0,\gamma}})\leq
K(\|u_{+}\|_{C^{0}}+N\|f\|_{C^{0,\gamma}})$ it is even $C^{1,\beta}$-bounded.
To estimate $f_{\bar{n}}$ in the last inequality one can use its
representation as $f_{\bar{n}}=fg_{\bar{n}}$, where function
$\qquad\ {\textstyle
g_{\bar{n}}=1-\Theta(f-\bar{n})\left(1-\frac{\bar{n}}{f}\right)-\Theta(-f-\bar{n})\left(1+\frac{\bar{n}}{f}\right)}$
is responsible for the truncation and $\Theta$ is the Heaviside function.
Using the results for composition (e.g. Hölder index of composed map is a
product of indices of components) and multiplication (e.g. index is a minimum
of indices of components) of Hölder continuous functions on bounded sets there
has to be a new Hölder coefficient $\gamma\leq\alpha\beta$ and a suitable
constant $N$ fulfilling the inequality. Function $f$ in the elliptic estimate
is dependent on $x$ not only explicitly but also via $u_{\bar{n}}(x)$ and
$\nabla u_{\bar{n}}(x)$, which is reflected in the constant $N$. While
$u_{\bar{n}}$ is bounded independently on $\bar{n}$ by $u^{\pm}$ we need to
bound the gradient in the same way. Using the fact that our function $F$ has
the form $F_{1}(u)+F_{2}(x)\|\nabla u\|_{h}^{2}$ (with strictly positive
$F_{2}$) and integrating over the manifold (using the Stokes theorem to
eliminate the laplacian) we get
$-\int_{M}F_{1}(u_{\bar{n}})=\int_{M}F_{2}(x)\|\nabla
u_{\bar{n}}\|_{h}^{2}\geq$ $\geq F_{2,min}\int_{M}\|\nabla
u_{\bar{n}}\|_{h}^{2}\ ,$
where the left hand side might be independently estimated. According to
previous results gradient of $u_{\bar{n}}$ is bounded and from the last
equation even independently. Therefore, the constant $N$ does not depend on
${\bar{n}}$.
* 7.
Then there is a $C^{1,\beta}$-convergent subsequence $u_{\tilde{n}}\to u^{s}$,
which proves the existence of the solution provided the sub- and super-
solutions are obtained. Moreover, using the second step with $f(x,u^{s},\nabla
u^{s})$ we must have $u^{s}\in C^{2,\alpha\beta}(M)$.
As is most common in the literature, we would be looking for constant sub- and
super-solutions, first in the case ${\cal R}>0$ (assuming $u_{min}\geq 0$
which can always be arranged by a suitable choice of $C$) :
* •
${\Lambda\leq 0}$
${\textstyle
u_{1}^{+}=\frac{1}{D-3}\ln\left[\frac{C^{D-3}}{(D-2)(D-3)\mu}{\cal R}\right]\
,}$ ${\textstyle
u_{1}^{-}=\frac{1}{D-3}\ln\left[\frac{C^{D-3}}{(D-2)(D-3)\mu}({\cal
R}-\frac{2(D-3)}{D-1}\Lambda C^{2})\right]\ ,}$
* •
${\Lambda>0}$
$u_{2}^{+}=u_{1}^{-}\ ,$ $u_{2}^{-}=u_{1}^{+}\ .$
These solutions satisfy all the conditions for any $\Lambda\leq 0$, but for
positive cosmological constant one has to demand
${\textstyle{\frac{2{\cal R}}{(D-1)(D-2)(D-3)\mu}}\left(\frac{{\cal
R}}{2\Lambda}\right)^{\frac{D-3}{2}}\geq 1\ },$ (7)
so that $u_{2}^{+}\geq 0$ is valid for such a constant $C$ which is maximizing
the value of $u_{2}^{+}$. Interestingly, this last condition reduces in the
four-dimensional case (that was not explicitly studied here, but can be
included trivially - note that then ${\cal R}$ is not a constant on $u=const.$
and $u^{\pm}$ has to be adjusted PodSvi:2009 ) to the condition from the table
1 for the existence of the solution when $\Lambda>0$. One has to remember that
for $D=4$ scalar curvature ${\cal R}$ asymptotically (as $u\to\infty$)
approaches value $2$.
One may wonder whether the condition (7) is necessary or if it might be
weakened by the choice of more suitable non-constant sub- and/or super-
solution. Let us assume we have a positive solution $R\in C^{2,\delta}(M)$ of
(5) with $\Lambda>0$. Since it represents a function on a compact manifold it
has to attain its maximum $R_{max}$ and minimum $R_{min}$. At $R_{min}$ the
gradient term in (5) vanishes while $\Delta(\ln R_{min})\geq 0$ leading to the
following inequality
$-{\cal R}+{\textstyle\frac{2(D-3)}{D-1}}\Lambda
R_{min}^{2}+{\scriptstyle(D-2)(D-3)}\mu R_{min}^{3-D}\leq 0$ (8)
The right-hand side of (8) has minimum at
$R_{min,E}^{D-1}=\frac{(D-1)(D-2)(D-3)\mu}{4\Lambda}$. The inequality (8) must
also hold for this value and after its substitution one arrives exactly at the
condition (7). Therefore, it represents not only sufficient but also necessary
condition for the existence of the horizon when positive cosmological constant
is present.
Since according to mathematical results any manifold (including compact ones)
of dimension greater than or equal to $3$ can be endowed with a complete
Riemannian metric of constant negative scalar curvature Aubin ; Lohkamp one
should also consider that ${\cal R}<0$ for our $(D-2)$-dimensional spatial
hypersurface. One can propose the following constant sub- and super-solutions
for ${\cal R}\leq 0$ (assuming $u_{min}\geq 0$)
* •
${\Lambda<0}$
$u_{3}^{+}=0$ ${\textstyle
u_{3}^{-}=\frac{1}{D-3}\ln\left[-\frac{2C^{D-1}}{(D-1)(D-2)\mu}\Lambda\right]}$
and select $C\geq C_{min}$ that is defined by
${\cal R}-{\textstyle\frac{2(D-3)}{D-1}}\Lambda
C^{2}_{min}-{\scriptstyle(D-2)(D-3)}\mu C_{min}^{-(D-3)}=0\ ,$
* •
${\Lambda\geq 0}$ : impossible to find constant $u^{+}$.
So for nonpositive scalar curvature ${\cal R}$ we can prove the existence only
for negative cosmological constant.
In the four-dimensional case one can infer some useful results from the
previous inequalities, mainly due to the fact that one can bound the scalar
curvature from the asymptotic behaviour or using Gauss-Bonnet theorem.
However, in higher-dimensional spacetime neither tool is available (the
generalizations of Gauss-Bonnet theorem are very complicated and not
immediately applicable).
### IV.3 $D>4$ : Character of the horizon
After establishing the existence of the past horizon $\mathcal{H}$ as a
hypersurface foliated by marginally trapped surfaces one is naturally
interested in whether it satisfies other conditions of recent quasilocal
horizon definitions. We will consider trapping and dynamical horizons.
The previous results tell us that $\Theta_{l}>0$ and $\Theta_{n}=0$ holds on
the past horizon. Since the Lie derivative $\mathcal{L}_{l}\Theta_{n}$ is in
general nonvanishing on the horizon its closure is a trapping horizon hayward
. Moreover, we can try to determine whether $\mathcal{L}_{l}\Theta_{n}<0$ on
the horizon which would mean that it is outer trapping horizon. After simple
manipulations one arrives at the following formula
$\mathcal{L}_{l}\Theta_{n}|_{\mathcal{H}}={\textstyle\frac{2}{D-1}\Lambda-\frac{(D-3)(D-2)}{2}\mu
R^{1-D}-\frac{1}{D-5}}\frac{\Delta R^{D-5}}{R^{D-3}}$ (9)
and we need to prove that
${\textstyle\frac{2}{D-1}\Lambda R^{D-3}-\frac{(D-3)(D-2)}{2}\mu
R^{-2}<\frac{1}{D-5}\Delta R^{D-5}}\ .$ (10)
Integrating the last equation over the $D-2$ dimensional compact subspace
spanned by coordinates $x^{i}$ (thus eliminating the right-hand side in (10))
we get the necessary condition for the horizon being outer
$\int{\textstyle\frac{2}{D-1}\Lambda R^{D-3}-\frac{(D-3)(D-2)}{2}\mu
R^{-2}}<0\ ,$ (11)
which is satisfied for any $\Lambda\leq 0$ and for positive cosmological
constant one shall demand
$R^{D-1}<{\textstyle\frac{(D-1)(D-2)(D-3)}{4}\frac{\mu}{\Lambda}}\ .$ (12)
Alternatively, one can consider (10) at the maximum of $R$, where $\Delta
R^{D-5}<0$. Notice, that for non-negative cosmological constant the right-hand
side of (10) is strictly increasing function of $R$, so it has maximal value
at maximum of $R$. If the horizon is everywhere non-degenerate
($\mathcal{L}_{l}\Theta_{n}\neq 0$) then it is really an outer trapping
horizon.
Next, we will consider a gradient of the horizon hypersurface which in our
parametrization reduces to
$\mathbf{N}={\rm d}r-R_{,u}{\rm d}u-R_{,i}{\rm d}x^{i}\ .$ (13)
We can use the sign of its norm to determine the causal character of the
horizon. Using the null D-ad (4) the corresponding vector can be expressed in
the following form
$\mathbf{N}={\textstyle\frac{1}{2}}(N^{a}N_{a})\mathbf{l}-\mathbf{n}$ (14)
and beside being normal to the horizon $\mathcal{H}$ it is also orthogonal to
its $u=const.$ $D-2$ dimensional sections $\mathcal{H}_{u}$. One can introduce
a second vector orthogonal to these sections
$\mathbf{Z}={\textstyle\frac{1}{2}}(N^{a}N_{a})\mathbf{l}+\mathbf{n}\ ,$ (15)
which satisfies $N^{a}Z_{a}=0$ and therefore is tangent to the horizon
$\mathcal{H}$. Then (inspired by hayward )
$\mathcal{L}_{Z}\Theta_{n}|_{\mathcal{H}}=0$ holds identically which gives the
following equation
${\textstyle\frac{1}{2}}(N^{a}N_{a})\mathcal{L}_{l}\Theta_{n}+\mathcal{L}_{n}\Theta_{n}=0\
.$ (16)
If the outer trapping horizon condition is satisfied
($\mathcal{L}_{l}\Theta_{n}<0$) we need to determine the sign of the second
term $\mathcal{L}_{n}\Theta_{n}$. One can consider the higher-dimensional
generalization of Raychaudhuri equation Lewandowski which in the case of
nontwisting and nonshearing solution on the horizon where $\Theta_{n}=0$
simplifies to
$\mathcal{L}_{n}\Theta_{n}=-R_{ab}n^{a}n^{b}\ .$ (17)
Since we are considering only aligned null radiation (in the direction of
$\mathbf{l}$ with radiation density $\Phi$) and cosmological constant
$\Lambda$ the Ricci tensor in (17) can be written as $R_{ab}=\Lambda
g_{ab}+\Phi l_{a}l_{b}$ and substituting into (17) we get
$\mathcal{L}_{n}\Theta_{n}=-\Phi\ .$ (18)
Therefore, for nonnegative radiation density $\Phi$ we conclude that
$\mathcal{L}_{n}\Theta_{n}\leq 0$. Since both Lie derivatives in (16) are
nonpositive it follows that $N^{a}N_{a}\leq 0$ which means that the horizon is
either null (for $\Phi=0$) or spacelike (for $\Phi>0$). In the latter case it
presents an explicit example of dynamical horizon in higher-dimensional
spacetime.
## V Conclusion and Final Remarks
We have derived the generalization of the Penrose–Tod equation to higher
dimensional Robinson-Trautman spacetimes including cosmological constant and
pure radiation. Using several mathematical tools we have proved the existence
of its solution for any $\Lambda\leq 0$ and for ${\cal R}>0$. The limitations
arising for positive $\Lambda$ (7) are shown to correspond to similar
restriction arising in four-dimensional case that are naturally related to the
more complicated horizon structure of relevant spacetimes (e.g. naked
singularities). Since the sign of scalar curvature of $(D-2)$-dimensional
spatial hypersurface does not restrict their topology as it does in $D=4$ we
have included the nonpositive case as well.
Additionally, we have proved that one can consider this horizon as being a
higher-dimensional generalization of trapping and dynamical horizon provided
additional conditions are satisfied.
The results show that in terms of the presence of the quasilocal horizons the
higher-dimensional generalization shares the same qualitative behavior as the
standard four-dimensional Robinson-Trautman spacetime. This provides support
for considering the generalization given in podolsky-ortaggio to be natural
not only mathematically but also physically.
Several important issues were not investigated here, namely uniqueness of the
horizon hypersurface and its possible topologies. The question of uniqueness
is much harder to solve for $D>4$ because of the nonlinearity in gradient. Due
to our parametrization of the horizon the issue of its topology is connected
with the topology of the underlying spatial geometry (given by $h_{ij}$) of
the $(D-2)$-dimensional manifold $M$. So the obvious starting point should be
the classification of Einstein spaces of corresponding dimension. For $D=5$
the $S^{1}\times S^{2}$ (black ring) is ruled out since it cannot be endowed
with Einstein metric Besse (the second homotopy group has to vanish
$\pi_{2}(M)=0$) and the Poincaré conjecture singles out three-sphere as the
only simply connected case. In $D=6$ topological obstructions arise for
example due to generalized Gauss-Bonnet theorem relating the Euler
characteristic $\chi(M)$ and curvature of compact oriented four-manifold. It
turns out that $\chi(M)>0$ and it is zero only in the flat case. This rules
out $S^{1}\times S^{3}$. In higher dimensions the restrictions are much weaker
(positive Ricci curvature implies finite first homotopy group).
###### Acknowledgements.
This work was supported by grant GACR 202/09/0772 and the Czech Ministry of
Education project Center of Theoretical Astrophysics LC06014.
## References
* (1) I. Robinson and A. Trautman, Phys. Rev. Lett. 4, 431 (1960).
* (2) I. Robinson and A. Trautman, Proc. Roy. Soc. Lond. A265 , 463 (1962).
* (3) H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact Solutions of the Einstein’s Field Equations, 2nd edn (CUPress Cambridge, 2002).
* (4) P.T. Chruściel, Commun. Math. Phys. 137, 289 (1991).
* (5) P.T. Chruściel, Proc. Roy. Soc. Lond. A436, 299 (1992).
* (6) P.T. Chruściel and D.B. Singleton, Commun. Math. Phys. 147, 137 (1992).
* (7) J. Bičák and J. Podolský, Phys. Rev. D 52, 887 (1995).
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* (9) J. Podolský and O. Svítek, Phys. Rev. D 71, 124001 (2005).
* (10) J. Podolský and M. Ortaggio, Class. Quant. Grav. 23, 5785 (2006).
* (11) S.W. Hawking and G.F.R. Ellis, _The large scale structure of space-time_ (CUPress Cambridge, 1975).
* (12) S.A. Hayward, Phys. Rev. D 49, 6467 (1994).
* (13) A. Ashtekar , C. Beetle and S. Fairhurst, Class. Quant. Grav. 17, 253 (2000).
* (14) A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003).
* (15) K.P. Tod, Class. Quantum Grav. 6, 1159 (1989).
* (16) E.W.M. Chow and A.W.C. Lun, J. Austr. Math. Soc. B 41, 217 (1999).
* (17) J. Podolský and O. Svítek, Phys. Rev. D 80, 124042 (2009).
* (18) L.M. Cao, JHEP 3, 112 (2011).
* (19) J. Isenberg, Class. Quantum Grav. 12, 2249 (1995).
* (20) T-H. Kuo, Nonlinear Anal. 63, e427 (2005).
* (21) A. Besse, Einstein Manifolds (Springer Berlin, 1987).
* (22) L. Boccardo, F. Murat and J.P. Puel, Annali della Scuola Normale Superiore di Pisa (Classe di Scienze) Sér. 4 vol. 11 no. 2, 213 (1984).
* (23) D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer Berlin, 2001).
* (24) E.g. supersolution satisfies $Pu^{+}\leq f(x,u^{+},\nabla u^{+})$.
* (25) T. Aubin, J. Funct. Anal. 32, 148 (1976).
* (26) J. Lohkamp, Bull. Amer. Math. Soc. (N.S.) 27 no. 2, 288 (1992).
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|
arxiv-papers
| 2011-06-28T14:12:12 |
2024-09-04T02:49:20.154703
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Otakar Svitek",
"submitter": "Otakar Svitek",
"url": "https://arxiv.org/abs/1106.5674"
}
|
1106.5697
|
# Holographic Scalar Fields Models of Dark Energy
Ahmad Sheykhi111sheykhi@uk.ac.ir Physics Department and Biruni Observatory,
Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P. O.
Box 55134-441, Maragha, Iran
###### Abstract
Many theoretical attempts toward reconstructing the potential and dynamics of
the scalar fields have been done in the literature by establishing a
connection between holographic/agegraphic energy density and a scalar field
model of dark energy. However, in most of these cases the analytical form of
the potentials in terms of the scalar field have not been reconstructed due to
the complexity of the equations involved. In this paper, by taking Hubble
radius as system’s IR cutoff, we are able to reconstruct the analytical form
of the potentials as a function of scalar field, namely $V=V(\phi)$ as well as
the dynamics of the scalar fields as a function of time, namely
$\phi=\phi(t)$, by establishing the correspondence between holographic energy
density and quintessence, tachyon, K-essence and dilaton energy density in a
flat FRW universe. The reconstructed potentials are quite reasonable and have
scaling solutions. Our study further supports the viability of the holographic
dark energy model with Hubble radius as IR cutoff.
## I Introduction
Holographic dark energy (HDE) models have got a lot of enthusiasm recently,
because they link the dark energy density to the cosmic horizon, a global
property of the universe, and have a close relationship to the spacetime foam
Ng1 ; pav2 . For a recent review on different HDE models and their consistency
check with observational data see ser . There are also a number of theoretical
motivations leading to the form of HDE, among which some are motivated by
holography and others from other principles of physics. A fairly comprehensive
motivations on HDE models can be seen in li . It is worthwhile to mention that
in the literature, various models of HDE have been investigated via
considering different system’s IR cutoff. In the presence of interaction
between dark energy and dark matter, the simple choice for IR cutoff could be
the Hubble radius, $L=H^{-1}$ which can simultaneously drive accelerated
expansion and solve the coincidence problem pav1 ; Zim . Besides, it was
argued that for an accelerating universe inside the event horizon the
generalized second law does not satisfy, while the accelerating universe
enveloped by the Hubble horizon satisfies the generalized second law Jia ;
shey1 . This implies that the event horizon in an accelerating universe might
not be a physical boundary from the thermodynamical point of view. Thus, it
looks that Hubble horizon is a convenient horizon for which satisfies all of
our accepted principles in a flat Friedmann-Robertson-Walker (FRW) universe.
There has been a lot of interest in recent years in establishing a connection
between holographic/agegraphic energy density and scalar field models of dark
energy (see e.g. XZ ; Setare ; tachADE ; quinADE ). These studies lead to
reconstruct the potential and the dynamics of the scalar fields according to
the evolution of the holographic/agegraphic energy density. Unfortunately, in
all of these cases (XZ ; Setare ; tachADE ; quinADE ) the analytical form of
the potentials have not been constructed as a function of scalar field, namely
$V=V(\phi)$, due to the complexity of the equations involved. Recently, by
implement a connection between scalar field dark energy and HDE density and
introducing a new IR cutoff, namely $L^{-2}=\alpha H^{2}+\beta\dot{H}$, the
authors of GO reconstructed explicitly the potentials and the dynamics of the
scalar fields, which describe accelerated expansion. Our work differs from GO
in that we assume the pressureless dark matter and HDE do not conserve
separately but interact with each other, while the authors of GO have
neglected the contributions from dark matter and consequently no interaction
between two dark components. Besides, our system’s IR cutoff differs from that
of Ref. GO .
In this paper, by choosing Hubble radius $L=H^{-1}$ as system’s IR cutoff, we
implement the connection between the HDE and scalar fields models including
quintessence, tachyon, K-essence and dilaton energy density in a flat FRW
universe. This simple and most natural choice for IR cutoff allows us to
reconstruct the explicit form of potentials, $V=V(\phi)$, and also the
dynamics of the scalar fields as a function of time, namely $\phi=\phi(t)$.
## II HDE with Hubble radius as an IR cut-off
For the flat FRW universe, the first Friedmann equation is
$\displaystyle H^{2}=\frac{1}{3M_{p}^{2}}\left(\rho_{m}+\rho_{D}\right),$ (1)
where $\rho_{m}$ and $\rho_{D}$ are the energy density of dark matter and dark
energy, respectively. Taking the interaction between dark matter and dark
energy into account, the continuity equation maybe written as wang1 ; pav3
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}=Q,$ (2)
$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})=-Q.$ (3)
where $w_{D}=p_{D}/\rho_{D}$ is the equation of state (EoS) parameter of HDE,
and $Q$ stands for the interaction term. It should be noted that the ideal
interaction term must be motivated from the theory of quantum gravity. In the
absence of such a theory, we rely on pure dimensional basis for choosing an
interaction $Q$. It is worth noting that the continuity equations imply that
the interaction term should be a function of a quantity with units of inverse
of time (a first and natural choice can be the Hubble factor $H$) multiplied
with the energy density. Therefore, the interaction term could be in any of
the following forms: (i) $Q\propto H\rho_{D}$, (ii) $Q\propto H\rho_{m}$, or
(iii) $Q\propto H(\rho_{m}+\rho_{D})$. We find out that for all three forms of
$Q$, the EoS parameter of HDE has a similar form, thus hereafter we consider
only the first case, namely $Q=3b^{2}H\rho_{D}$, where $b^{2}$ is a coupling
constant.
We assume the HDE density has the form
$\rho_{D}=3c^{2}M_{p}^{2}H^{2},$ (4)
where $c^{2}$ is a constant and we have set the Hubble radius $L=H^{-1}$ as
system’s IR cutoff. Inserting Eq. (4) in Eq. (1) immediately yields
$u=\frac{1-c^{2}}{c^{2}}.$ (5)
where $u={\rho_{m}}/{\rho_{D}}$ is the energy density ratio. From Eq. (5) we
see that the ratio of the energy densities is a constant; thus the coincidence
problem can be alleviated. It is worth noting that in general the term $c^{2}$
in holographic energy density can vary with time though very slowly pav4 . By
slowly varying we mean that $\dot{(c^{2})}/c^{2}$ is upper bounded by the
Hubble expansion rate, $H$, i.e., pav4
$\frac{\dot{(c^{2})}}{c^{2}}\leq H.$ (6)
Note that this condition must be fulfilled at all times; otherwise the dark
energy density would not even approximately be proportional to $L^{-2}$,
something at the core of holography pav4 . It was argued that $c^{2}$ depends
on the infrared length, $L$ pav4 . For the case of $L=H^{-1}$, it was shown
that one can take $c^{2}$ approximately constant in the late time where dark
energy dominates ($\Omega_{m}<1/3$) pav4 . Since in the present work we study
the late time cosmology, and also for later convenience, we assume the term
$c^{2}$ to be a constant. Taking the time derivative of Eq. (4), after using
Friedmann equation (1), we get
$\dot{\rho}_{D}=-3c^{2}H\rho_{D}(1+u+w_{D}).$ (7)
Substituting Eq. (7) in (3), after using relations $Q=3b^{2}H\rho_{D}$ and
(5), we obtain
$w_{D}=-\frac{b^{2}}{1-c^{2}}.$ (8)
Therefore for constant parameters $c$ and $b$ the EoS parameter becomes also a
constant. In the absence of interaction, $b^{2}=0$, we encounter dust with
$w_{D}=0$. For the choice $L=H^{-1}$ an interaction is the only way to have an
EoS different from that for dust pav1 ; Zim . Let us note that in order to
have $w_{D}<0$ we should have $c^{2}<1$. Besides, the acceleration expansion
($w_{D}<-1/3$) can be achieved provided $c^{2}>1-3b^{2}$. Thus this model can
describe the accelerated expansion if $1-3b^{2}<c^{2}<1$. Moreover, $w_{D}$
can cross the phantom line ($w_{D}<-1$) provided $b^{2}>1-c^{2}$.
## III Correspondence with scalar field models
In this section we implement a correspondence between interacting HDE and
various scalar field models, by equating the equations of state for this
models with the equations of state parameter of interacting HDE obtained in
(8).
### III.1 Reconstructing holographic quintessence model
In order to establish the correspondence between HDE and quintessence scaler
field, we assume the quintessence scalar field model of dark energy is the
effective underlying theory. The energy density and pressure of the
quintessence scalar field are given by cop
$\displaystyle\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (9)
$\displaystyle p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (10)
Thus the potential and the kinetic energy term can be written as
$\displaystyle V(\phi)=\frac{1-w_{\phi}}{2}\rho_{\phi},$ (11)
$\displaystyle\dot{\phi}^{2}=(1+w_{\phi})\rho_{\phi}.$ (12)
where $w_{\phi}=p_{\phi}/\rho_{\phi}$. In order to implement the
correspondence between HDE and quintessence scaler field, we identify
$\rho_{\phi}=\rho_{D}$ and $w_{\phi}=w_{D}$. Inserting Eqs. (4) and (8) in
(12) we reach
$\displaystyle\dot{\phi}=\sqrt{3\left(1-\frac{b^{2}}{1-c^{2}}\right)}cM_{p}\frac{\dot{a}}{a}.$
(13)
Integrating yields
$\displaystyle\phi(a)=\sqrt{3\left(1-\frac{b^{2}}{1-c^{2}}\right)}cM_{p}\ln
a,$ (14)
where we have set $\phi(a_{0}=1)=0$ for simplicity. Next we want to obtain the
scale factor as a function of $t$. Taking the time derivative of Eq. (1) and
using (8) we find
$\displaystyle\frac{\dot{H}}{H^{2}}=-\frac{3}{2}\left[1-\frac{b^{2}c^{2}}{1-c^{2}}\right]$
(15)
The fist integration gives
$\displaystyle H=\frac{da}{adt}=\frac{2}{3\rm kt},$ (16)
where $\rm k=1-\frac{b^{2}c^{2}}{1-c^{2}}$. Integrating again we find
$\displaystyle a(t)=t^{2/3\rm k}$ (17)
Hence Eq. (14) can be rewritten
$\displaystyle\phi(t)=\frac{2}{3\rm
k}cM_{p}\sqrt{3\left(1-\frac{b^{2}}{1-c^{2}}\right)}\ln t.$ (18)
Next we obtain the potential as a function of $\phi$. Combining Eq. (8) with
Eq. (11) we reach
$\displaystyle
V(\phi)=\frac{3}{2}c^{2}M_{p}^{2}\left[1+\frac{b^{2}}{1-c^{2}}\right]H^{2}.$
(19)
Using Eqs. (16) and (18) we obtain the explicit expression for potential,
namely
$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{2c^{2}M_{p}^{2}}{3\rm
k^{2}}\left[1+\frac{b^{2}}{1-c^{2}}\right]\times$
$\displaystyle\exp\left[-3\frac{\rm
k}{cM_{p}}\left(3-\frac{3b^{2}}{1-c^{2}}\right)^{-1/2}\phi\right].$
Let us discuss the condition for which the scale factor (17), and hence the
obtained potential, leads to an accelerated universe at the present time.
Requiring $\ddot{a}>0$ for the present time, leads to $\rm k<2/3$ , which can
be translated into $c^{2}>(1+3b^{2})^{-1}$. Note that the condition $\rm
k<2/3$ valid only for the late time where we have a dark energy dominated
universe. In general $\rm k$ depends on $c$, and for the matter dominated
epoch where $c$ is no longer a constant, then $\rm k$ is also not a constant
and varies with time. The obtained exponential potential here is well-known in
the literature for the quintessence scalar field cop . The cosmological
dynamics of this potential has been explored in detail cop . In addition to
the fact that exponential potentials can give rise to an accelerated
expansion, they possess cosmological scaling solutions cop ; cop2 ; sahni in
which the field energy density $\rho_{\phi}$ is proportional to the matter
energy density $\rho_{m}$.
### III.2 Reconstructing holographic tachyon model
The tachyon field has been proposed as a possible candidate for dark energy. A
rolling tachyon has an interesting EoS whose parameter smoothly interpolates
between $-1$ and $0$ Gib1 . Thus, tachyon can be realized as a suitable
candidate for the inflation at high energy Maz1 as well as a source of dark
energy depending on the form of the tachyon potential Padm . Choosing
different self-interacting potentials in the tachyon field model lead to
different consequences for the resulting DE model. These give enough
motivations us to reconstruct tachyon potential $V(\phi)$ from HDE model with
Hubble radius as the IR cutoff. The correspondence between tachyon field and
various dark energy models such as HDE Setare and agegraphic dark energy
tachADE has been already established. The extension has also been done to the
entropy corrected holographic and agegraphic dark energy models ecde .
However, in all of these cases Setare ; tachADE ; ecde the explicit form of
the tachyon potential,$V=V(\phi)$, has not been reconstructed due to the
complexity of the equations involved.
The effective lagrangian for the tachyon field is given by sen
$\displaystyle
L=-V(\phi)\sqrt{1-g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi},$ (21)
where $V(\phi)$ is the tachyon potential. The corresponding energy momentum
tensor for the tachyon field can be written in a perfect fluid form
$\displaystyle T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu},$ (22)
where $\rho$ and $p$ are, respectively, the energy density and pressure of the
tachyon and the velocity $u_{\mu}$ is
$\displaystyle
u_{\mu}=\frac{\partial_{\mu}\phi}{\sqrt{\partial_{\nu}\phi\partial^{\nu}\phi}}.$
(23)
The energy density and pressure of tachyon field are given by
$\rho=-T_{0}^{0}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^{2}}},$ (24)
$p=T_{i}^{i}=-V(\phi)\sqrt{1-\dot{\phi}^{2}}.$ (25)
Thus the EoS parameter of tachyon field is given by
$w_{T}=\frac{p}{\rho}=\dot{\phi}^{2}-1.$ (26)
To establish the correspondence between HDE and tachyon field, we equate
$w_{D}$ with $w_{T}$. From Eqs. (8) and (26) we find
$\dot{\phi}^{2}=1-\frac{b^{2}}{1-c^{2}}.$ (27)
Integrating gives
$\phi(t)=\left[1-\frac{b^{2}}{1-c^{2}}\right]^{1/2}t,$ (28)
where we set an integration constant to zero. Combining Eq. (24) with (27),
the tachyon potential is obtained as
$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle
3c^{2}M_{p}^{2}H^{2}\frac{b}{\sqrt{1-c^{2}}},$ (29)
Using Eqs. (16) and (28) we obtain tachyon potential in terms of the scalar
field
$\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle\frac{4c^{2}M_{p}^{2}}{3\rm
k^{2}}\frac{b}{\sqrt{1-c^{2}}}\left(1-\frac{b^{2}}{1-c^{2}}\right)\frac{1}{\phi^{2}},$
(30)
From Eq. (28) we see that the evolution of the tachyon is given by
$\phi(t)\propto t$. The above inverse square power-law potential corresponds
to the one in the case of scaling solutions cop ; JM ; EJ .
### III.3 Reconstructing holographic K-essence model
The scalar field model called K-essence is also employed to explain the
observed acceleration of the cosmic expansion. This model is characterized by
a scalar field with a non-canonical kinetic energy. The most general scalar-
field action which is a function of $\phi$ and $X=-\dot{\phi}^{2}/2$ is given
by arm
$S=\int d^{4}x\sqrt{-g}P(\phi,X),$ (31)
where the lagrangian density $P(\phi,X)$ corresponds to a pressure density.
According to this lagrangian the energy density and the pressure can be
written as arm ; cop
$\displaystyle\rho(\phi,X)$ $\displaystyle=$ $\displaystyle
f(\phi)(-X+3X^{2}),$ (32) $\displaystyle p(\phi,X)$ $\displaystyle=$
$\displaystyle f(\phi)(-X+X^{2}).$ (33)
Therefore the EoS parameter of the K-essence is given by
$w_{K}=\frac{X-1}{3X-1}.$ (34)
Equating $w_{K}$ with the EoS parameter of HDE (8) one finds
$X=\frac{1+b^{2}-c^{2}}{1+3b^{2}-c^{2}}.$ (35)
which implies that $X$ is a positive constant ($c^{2}<1$). Indeed the EoS
parameter in Eq. (34) diverges for $X=1/3$. Let us consider the cases with
$X>1/3$ and $X<1/3$ separately. In the first case where $X>1/3$, the condition
$w_{K}<-1/3$ leads to $X<2/3$. Thus we should have $1/3<X<2/3$ in this case.
For example we obtain the EoS of a cosmological constant ($w_{K}=-1$) for
$X=1/2$. In the second case where $X<1/3$, we have $X-1<-2/3<0$, thus
$w_{K}=\frac{X-1}{3X-1}>0$. This means that we have no acceleration at all. So
this case is ruled out. As a result in K-essence model the accelerated
universe can be achieved provided $1/3<X<2/3$, which translates into
$1-3b^{2}<c^{2}<1$. This is consistent with our previous discussions.
Combining equation (35) with $X=-\dot{\phi}^{2}/2$, one gets
$\dot{\phi}^{2}=\frac{2(1+b^{2}-c^{2})}{1+3b^{2}-c^{2}},$ (36)
and thus we obtain the expression for the scalar field in the flat FRW
background
$\phi(t)=\left[\frac{2(1+b^{2}-c^{2})}{1+3b^{2}-c^{2}}\right]^{1/2}t,$ (37)
where we have taken the integration constant $\phi_{0}$ equal to zero. Taking
the correspondence between HDE and K-essence into account, namely
$\rho_{D}=\rho(\phi,X)$, after using Eqs. (16) and (37) we find
$f(\phi)=\frac{4c^{2}M_{p}^{2}}{3\rm
k^{2}}\left[\frac{1+3b^{2}-c^{2}}{1-c^{2}}\right]\frac{1}{\phi^{2}}.$ (38)
Thus the K-essence potential $f(\phi)$ has a power law expansion. From Eq.
(37) we see that $\dot{\phi}=\rm const.$ This means that the kinitic energy of
K-essence becomes constant, though $\phi$ is not constant and evolves with
time.
### III.4 Reconstructing holographic dilaton field
The dilaton field may be used for explanation the dark energy puzzle and
avoids some quantum instabilities with respect to the phantom field models of
dark energy carroll . The lagrangian density of the dilatonic dark energy
corresponds to the pressure density of the scalar field has the following form
tsuji
$p=-X+\alpha e^{\lambda\phi}X^{2},$ (39)
where $\alpha$ and $\lambda$ are positive constants and $X=\dot{\phi}^{2}/2$.
Such a pressure (Lagrangian) leads to the following energy density tsuji
$\rho=-X+3\alpha e^{\lambda\phi}X^{2}.$ (40)
The EoS parameter of the dilaton dark energy can be written as
$w_{d}=\frac{1-\alpha e^{\lambda\phi}X}{1-3\alpha e^{\lambda\phi}X}.$ (41)
To establish the correspondence between HDE and dilaton field we equate their
EoS parameter, i.e. $w_{d}=w_{D}$. We reach
$\frac{1-\alpha e^{\lambda\phi}X}{1-3\alpha
e^{\lambda\phi}X}=-\frac{b^{2}}{1-c^{2}}.$ (42)
Using relation $X=\dot{\phi}^{2}/2$, and integrating with respect to $t$ we
find
$\phi=\frac{2}{\lambda}\ln\left[\frac{\lambda}{\sqrt{2\alpha}}\left(\frac{1+b^{2}-c^{2}}{1+3b^{2}-c^{2}}\right)^{1/2}t\right]$
(43)
The existence of scaling solutions for the dilaton was studied in tsuji and
was found that in this case the scaling solution corresponds to
$Xe^{\lambda\phi}=\rm const.$, which has the solution $\phi(t)\propto\ln t$.
The results we found here by equating the EoS parameter of HDE and dilaton
field are consistent with those obtained in tsuji .
## IV Conclusion and discussion
In this paper by choosing the Hubble radius as system’s IR cutoff for
interacting HDE model, we established a connection between the scalar field
model of dark energy including quintessence, tachyon, K-essence and dilaton
energy density and holographic energy density. As a result, we reconstructed
the analytical form of potentials namely $V=V(\phi)$ as well as the dynamics
of the scalar fields as a function of time explicitly, namely $\phi=\phi(t)$
according to the evolutionary behavior of the interacting HDE model. The
obtained expressions for the potentials are quite reasonable and lead to
scaling solutions. Our studies favor the $L=H^{-1}$ IR cutoff as a viable
phenomenological model of HDE.
Finally, I would like to mention that usually, for the sake of simplicity, the
term $c^{2}$ in holographic energy density (4) is assumed constant. However,
one should bear in mind that it is more general to consider it a slowly
varying function of time, $c^{2}(t)$ pav4 . In this case the EoS parameter
given in (8) is no longer a constant. As a result we cannot integrate easily
the resulting equations in section III and find the analytical form of the
potentials. It is important to note that, although, with implement the
correspondence between this HDE model and scalar field models, the EoS of
scalar fields are assumed to be fixed, nevertheless, neither $\phi$ nor
$V(\phi)$ are not constant and they still evolve with time. In the present
work for simplicity we have taken $c=\rm const.$ The correspondence between
HDE and scalar field models with varying $c^{2}$ term is under investigation
and will be addressed elsewhere.
###### Acknowledgements.
I am grateful to the referee for valuable comments and suggestions, which have
allowed me to improve this paper significantly. I sincerely thank Prof. Diego
Pavon for constructive comments on an earlier draft of this paper. Special
thanks go to Dr. E. Ebrahimi for many helpful discussions. This work has been
supported financially by Research Institute for Astronomy and Astrophysics of
Maragha (RIAAM) under research project No. 1/2030.
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|
arxiv-papers
| 2011-06-26T12:32:37 |
2024-09-04T02:49:20.161818
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1106.5697"
}
|
1106.5959
|
# KdV solitons in a cold quark gluon plasma
D.A. Fogaça† F.S. Navarra† and L.G. Ferreira Filho † Instituto de Física,
Universidade de São Paulo
C.P. 66318, 05315-970 São Paulo, SP, Brazil ‡ Faculdade de Tecnologia,
Universidade do Estado do Rio de Janeiro
Via Dutra km 298, CEP 27523-000, Resende, RJ, Brazil
###### Abstract
The relativistic heavy ion program developed at RHIC and now at LHC motivated
a deeper study of the properties of the quark gluon plasma (QGP) and, in
particular, the study of perturbations in this kind of plasma. We are
interested on the time evolution of perturbations in the baryon and energy
densities. If a localized pulse in baryon density could propagate throughout
the QGP for long distances preserving its shape and without loosing
localization, this could have interesting consequences for relativistic heavy
ion physics and for astrophysics. A mathematical way to proove that this can
happen is to derive (under certain conditions) from the hydrodynamical
equations of the QGP a Korteveg-de Vries (KdV) equation. The solution of this
equation describes the propagation of a KdV soliton. The derivation of the KdV
equation depends crucially on the equation of state (EOS) of the QGP. The use
of the simple MIT bag model EOS does not lead to KdV solitons. Recently we
have developed an EOS for the QGP which includes both perturbative and non-
perturbative corrections to the MIT one and is still simple enough to allow
for analitycal manipulations. With this EOS we were able to derive a KdV
equation for the cold QGP.
## I Introduction
Korteweg - de Vries solitons are very interesting non-linear waves, which may
exist in many types of fluids from ordinary water to astrophysical plasmas
drazin . In the last years we have started to produce a new kind of fluid in
laboratory: the quark gluon plasma (QGP). This is a state where quarks and
gluons, usually confined in the interior of baryons (such as the proton) and
mesons, are free to travel longer distances. With the beginning of the LHC
era, we have means to study larger and longer living samples of QGP and even
the propagation of perturbations in this new medium. In this context a natural
question is: can we have KdV solitons in the QCD plasma? In this work we give
an answer to this question.
Before the QGP there were other fluids made of strongly interacting hadronic
matter and the existence of KdV solitons in these fluids was already
investigated. The first works on the subject were published in frsw , where
the authors considered the propagation of baryon density pulses in proton-
nucleus collisions at intermediate energies. In this scenario the incoming
proton would be absorbed by the nuclear fluid generating a KdV soliton, which,
traversing the whole nucleus without distortion, would escape from the target
as a proton and would simulate an unexpected transparency. In frsw the
existence of the KdV soliton relied solely on the equation of state (EOS),
which had no deep justification. In fn1 we have reconsidered the problem,
introducing an equation of state derived from relativistic mean field models
of nuclear matter. We concluded that the homogeneous meson field approximation
was too strong and would exclude the existence of KdV solitons. We could also
trace back the derivative terms in the energy density to derivative couplings
between the nucleon and the vector meson. In fn2 we extended our analysis to
relativistic hydrodynamics and in fn3 to spherical and cylindrical
geometries. In fn4 we considered hadronic matter at finite temperature and
studied the effects of temperature on the KdV soliton. In fn5 we started the
study of perturbations in the QGP at zero and finite temperature. The
conclusion found in that work was that the existence of KdV solitons in a QGP
depends on details of the EOS and with a simple MIT bag model EOS there is no
KdV soliton! A further study of the equation of state, carried out in fn6 ,
showed that if non-perturbative effects are included in the EOS through gluon
condensates, then new terms appear in the expression of the energy density and
pressure and in the present work we show how these new terms lead to a KdV
equation, after the proper treatment of the hydrodynamical equations.
In the next section we briefly review the equations of one-dimensional
relativistic fluid dynamics. In section III we introduce the equation of
state, in section IV we derive the KdV equation and in section V we present a
numerical analysis of the obtained equation.
## II Relativistic Fluid Dynamics
Relativistic hydrodynamics is well presented in the textbooks wein ; land .
The relativistic version of the Euler equation wein ; land ; fn5 is given by:
${\frac{\partial{\vec{v}}}{\partial
t}}+(\vec{v}\cdot\vec{\nabla})\,\vec{v}=-{\frac{1}{(\varepsilon+p)\gamma^{2}}}\bigg{(}{\vec{\nabla}p+\vec{v}\,{\frac{\partial
p}{\partial t}}}\bigg{)}$ (1)
where $\vec{v}$, $\varepsilon$, $p$ and $\gamma$ are the velocity, energy
density, pressure and the Lorentz factor respectively. We employ the natural
units $c=1$ and $\hbar=1$. Space and time coordinates will be in $fm$
($1fm=10^{-15}m$). The relativistic version of the continuity equation for the
baryon density is wein :
$\partial_{\nu}{j_{B}}^{\nu}=0$ (2)
Since ${j_{B}}^{\nu}=u^{\nu}\rho_{B}$ the above equation can be rewritten as
fn5 :
${\frac{\partial\rho_{B}}{\partial
t}}+\gamma^{2}\vec{v}\,\rho_{B}\Bigg{(}{\frac{\partial\vec{v}}{\partial
t}}+\vec{v}\cdot\vec{\nabla}\vec{v}\Bigg{)}+\vec{\nabla}\cdot(\rho_{B}\,\vec{v})=0$
(3)
where $\rho_{B}$ is the baryon density. In the one dimensional Cartesian
relativistic fluid dynamics the velocity field is written as
$\vec{v}=v(x,t)\,\hat{x}$ where $\hat{x}$ is the unit vector in the $x$
direction. Equations (1) and (3) can be rewritten in the simple form:
${\frac{\partial v}{\partial t}}+v{\frac{\partial v}{\partial
x}}={\frac{(v^{2}-1)}{(\varepsilon+p)}}\bigg{(}{\frac{\partial p}{\partial
x}}+v{\frac{\partial p}{\partial t}}\bigg{)}$ (4)
and
$v\rho_{B}\bigg{(}{\frac{\partial v}{\partial t}}+v{\frac{\partial v}{\partial
x}}\bigg{)}+(1-v^{2})\bigg{(}{\frac{\partial\rho_{B}}{\partial
t}}+\rho_{B}{\frac{\partial v}{\partial x}}+v{\frac{\partial\rho_{B}}{\partial
x}}\bigg{)}=0$ (5)
## III The QGP Equation of State
In what follows we present the mean field treatment of QCD developed in fn6
(for previous works on the subject see shakin ; shakinn ) and go beyond the
homogeneous field approximation, including the terms with gradients.
The Lagrangian density of QCD is given by:
${\mathcal{L}}_{QCD}=-{\frac{1}{4}}F^{a}_{\mu\nu}F^{a\mu\nu}+\sum_{q=1}^{N_{f}}\bar{\psi}^{q}_{i}\Big{[}i\gamma^{\mu}(\delta_{ij}\partial_{\mu}-igT^{a}_{ij}G_{\mu}^{a})-\delta_{ij}m_{q}\Big{]}\psi^{q}_{j}$
(6)
where
$F^{a\mu\nu}=\partial^{\mu}G^{a\nu}-\partial^{\nu}G^{a\mu}+gf^{abc}G^{b\mu}G^{c\nu}$
(7)
The summation on $q$ runs over all quark flavors, $m_{q}$ is the mass of the
quark of flavor $q$, $i$ and $j$ are the color indices of the quarks, $T^{a}$
are the SU(3) generators and $f^{abc}$ are the SU(3) antisymmetric structure
constants. For simplicity we will consider massless quarks, i.e. $m_{q}=0$.
Moreover, we will drop the summation and consider only one flavor. At the end
of our calculation the number of flavors will be recovered. Following shakin ;
shakinn , we shall write the gluon field as:
$G^{a\mu}={A}^{a\mu}+{\alpha}^{a\mu}$ (8)
where ${A}^{a\mu}$ and ${\alpha}^{a\mu}$ are the low (“soft”) and high
(“hard”) momentum components of the gluon field respectively. We will assume
that ${A}^{a\mu}$ represents the soft modes which populate the vacuum and the
terms containing ${A}^{a\mu}$ will be replaced by their expectation values
$\langle{A}^{a\mu}\rangle$, $\langle{A}^{a\mu}{A}^{a}_{\mu}\rangle$, etc…in
the plasma. ${\alpha}^{a\mu}$ represents the modes for which the running
coupling constant is small.
In a cold quark gluon plasma the density is much larger than the ordinary
nuclear matter density. These high densities imply a very large number of
sources of the gluon field. Assuming that the coupling constant is not very
small, the existence of intense sources implies that the bosonic fields tend
to have large occupation numbers at all energy levels, and therefore they can
be treated as classical fields. This is the famous approximation for bosonic
fields used in relativistic mean field models of nuclear matter serot . It has
been applied to QCD in the past and amounts to assume that the “hard” gluon
field, ${\alpha}_{\mu}^{a}$, is simply a function of the coordinates:
${\alpha}_{\mu}^{a}(\vec{x},t)=\delta_{\mu 0}\,{\alpha}_{0}^{a}(\vec{x},t)$
(9)
with $\partial_{\nu}{\alpha}^{a}_{\mu}\neq 0$. This space and time dependence
goes beyond the standard mean field approximation serot , where
${\alpha}_{\mu}^{a}$ is constant in space and time and consequently
$\partial_{\nu}{\alpha}^{a}_{\mu}=0$. We keep assuming, as in fn6 , that the
soft gluon field ${A}^{a\mu}$ is independent of position and time and thus
$\partial^{\nu}{A}^{a\mu}=0$ . Following the same steps introduced in fn6 we
obtain the following effective Lagrangian:
$\mathcal{L}_{0}=-{\frac{1}{2}}{\alpha}^{a}_{0}\big{(}{\vec{\nabla}}^{2}{\alpha}^{a}_{0}\big{)}+{\frac{{m_{G}}^{2}}{2}}{\alpha}^{a}_{0}{\alpha}^{a}_{0}-\mathcal{B}_{QCD}+\bar{\psi}_{i}\Big{(}i\delta_{ij}\gamma^{\mu}\partial_{\mu}+g\gamma^{0}T^{a}_{ij}{\alpha}^{a}_{0}\Big{)}\psi_{j}$
(10)
where $m_{G}$ is the dynamical mass of the hard gluon $\alpha$ generated by
its interaction with the soft gluons $A^{a\,\mu}$ from the vacuum and it is
related to the dimension two $\langle A^{2}\rangle$ gluon condensate. The
constant $\mathcal{B}_{QCD}$ is related to the dimension four gluon condensate
$\langle F^{2}\rangle$ (see fn6 for details).
The effective Lagrangian (10) is quite similar to the one obtained in fn6 and
the only difference is the first term, which is new and comes from the
gradients. The equations of motion fn4 are given by:
${\frac{\partial\mathcal{L}}{\partial\eta_{i}}}-\partial_{\mu}{\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\eta_{i})}}+\partial_{\nu}\partial_{\mu}\bigg{[}{\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\partial_{\nu}\eta_{i})}}\bigg{]}=0$
(11)
Inserting (10) into (11) with $\eta_{1}={\alpha}^{a}_{0}(\vec{x},t)$ and
$\eta_{2}=\bar{\psi}(\vec{x},t)$ we find:
$-{\vec{\nabla}}^{2}{\alpha}^{a}_{0}+{m_{G}}^{2}{\alpha}^{a}_{0}=-g\rho^{a}$
(12)
$\Big{(}i\gamma^{\mu}\partial_{\mu}+g\gamma^{0}T^{a}{\alpha}^{a}_{0}\Big{)}\psi=0$
(13)
where $\rho^{a}$ is the temporal component of the color vector current given
by $j^{a\nu}=\bar{\psi}_{i}\gamma^{\nu}T^{a}_{ij}\psi_{j}$ . The energy-
momentum tensor reads fn4 :
$T^{\mu\nu}={\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\eta_{i})}}(\partial^{\nu}\eta_{i})-g^{\mu\nu}\mathcal{L}-\bigg{[}\partial_{\beta}{\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\partial_{\beta}\eta_{i})}}\bigg{]}(\partial^{\nu}\eta_{i})+{\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\partial_{\beta}\eta_{i})}}(\partial_{\beta}\partial^{\nu}\eta_{i})$
(14)
From the above expression we can obtain the energy density
($\varepsilon=<T_{00}>$) which turns out to be fn6 :
$\varepsilon={\frac{1}{2}}{\alpha}^{a}_{0}\big{(}{\vec{\nabla}}^{2}{\alpha}^{a}_{0}\big{)}-{\frac{{m_{G}}^{2}}{2}}{\alpha}^{a}_{0}{\alpha}^{a}_{0}+\mathcal{B}_{QCD}-g\rho^{a}{\alpha}^{a}_{0}+3{\frac{\gamma_{Q}}{2{\pi}^{2}}}\int_{0}^{k_{F}}dk\hskip
2.84544ptk^{2}\sqrt{{\vec{k}}^{2}+m^{2}}$ (15)
where $\gamma_{Q}$ is the quark degeneracy factor
$\gamma_{Q}=2(\mbox{spin})\times 3(\mbox{flavor})$. The sum over all the color
states was already performed and resulted in the pre-factor $3$ in the
expression above. $k_{F}$ is the Fermi momentum defined by the quark number
density $\rho$:
$\rho=\langle
N|\psi^{\dagger}_{i}\psi_{i}|N\rangle={\frac{3}{V}}\sum_{\vec{k},\lambda}\langle
N|N\rangle=3{\frac{\gamma_{Q}}{(2\pi)^{3}}}\int
d^{3}k=3{\frac{\gamma_{Q}}{2{\pi}^{2}}}\int_{0}^{k_{F}}dk\hskip
2.84544ptk^{2}={\frac{\gamma_{Q}}{2{\pi}^{2}}}{{k_{F}}}^{3}$ (16)
In the above expression $|N\rangle$ denotes a state with N quarks. In a first
approximation the field ${\alpha}^{a}_{0}$ may be estimated from (12).
Neglecting the derivative term ${\vec{\nabla^{2}}}{\alpha}^{a}_{0}$ of (12) we
have: fn4 :
${\alpha}^{a}_{0}\cong-{\frac{g}{{m_{G}}^{2}}}\rho^{a}$ (17)
Inserting (17) in the first term of (12) and then solving it for
${\alpha}^{a}_{0}$ we find:
${\alpha}^{a}_{0}=-{\frac{g}{{m_{G}}^{2}}}\rho^{a}-{\frac{g}{{m_{G}}^{4}}}{\vec{\nabla}}^{2}\rho^{a}$
(18)
We can write the color charge density $\rho^{a}$ in terms of the quark number
density $\rho$ through:
$\rho^{a}\rho^{a}=3\rho^{2}$ (19)
Analogously we have
$\rho^{a}\vec{\nabla}^{2}\rho^{a}=3\rho\vec{\nabla}^{2}\rho\hskip
2.84544pt,\hskip
8.5359pt\rho^{a}\vec{\nabla}^{4}\rho^{a}=3\rho\vec{\nabla}^{4}\rho\hskip
2.84544pt\hskip 8.5359pt$ (20)
Inserting (18), (19) and (20) into (15), performing the momentum integral and
using the baryon density, which is $\rho_{B}={\frac{1}{3}}\rho$, we arrive at
the final expression for the energy density in one spatial dimension:
$\varepsilon=\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{2}}}\bigg{)}{\rho_{B}}^{2}+\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{4}}}\bigg{)}\rho_{B}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}+\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{6}}}\bigg{)}\rho_{B}{\frac{\partial^{4}\rho_{B}}{\partial
x^{4}}}+\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{8}}}\bigg{)}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}\ {\frac{\partial^{4}\rho_{B}}{\partial x^{4}}}$
$+\mathcal{B}_{QCD}+3{\frac{\gamma_{Q}}{2{\pi}^{2}}}{\frac{{k_{F}}^{4}}{4}}$
(21)
The pressure is given by $p={\frac{1}{3}}<T_{ii}>$. Repeating the same steps
mentioned before we arrive at:
$p=\bigg{(}{\frac{27g^{2}}{2{m_{G}}^{2}}}\bigg{)}\
{\rho_{B}}^{2}+\bigg{(}{\frac{18g^{2}}{{m_{G}}^{4}}}\bigg{)}\
\rho_{B}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}-\bigg{(}{\frac{9g^{2}}{{m_{G}}^{6}}}\bigg{)}\rho_{B}{\frac{\partial^{4}\rho_{B}}{\partial
x^{4}}}-\bigg{(}{\frac{9g^{2}}{2{m_{G}}^{4}}}\bigg{)}{\frac{\partial\rho_{B}}{\partial
x}}{\frac{\partial\rho_{B}}{\partial x}}$
$+\bigg{(}{\frac{9g^{2}}{2{m_{G}}^{6}}}\bigg{)}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}-\bigg{(}{\frac{9g^{2}}{{m_{G}}^{8}}}\bigg{)}{\frac{\partial^{2}\rho_{B}}{\partial
x^{2}}}\ {\frac{\partial^{4}\rho_{B}}{\partial
x^{4}}}-\bigg{(}{\frac{9g^{2}}{2{m_{G}}^{8}}}\bigg{)}{\frac{\partial^{3}\rho_{B}}{\partial
x^{3}}}\ {\frac{\partial^{3}\rho_{B}}{\partial
x^{3}}}-\bigg{(}{\frac{9g^{2}}{{m_{G}}^{6}}}\bigg{)}{\frac{\partial\rho_{B}}{\partial
x}}\ {\frac{\partial^{3}\rho_{B}}{\partial x^{3}}}$
$-\mathcal{B}_{QCD}+{\frac{\gamma_{Q}}{2{\pi}^{2}}}{\frac{{k_{F}}^{4}}{4}}$
(22)
where now $k_{F}$ defined by (16) is given by $\rho_{B}={k_{F}}^{3}/{\pi}^{2}$
.
## IV The KdV Equation
We now combine the equations (4) and (5) to obtain the KdV equation which
governs the space-time evolution of the perturbation in the baryon density. We
first write (4) and (5) in terms of the dimensionless variables:
$\hat{\rho}={\frac{\rho_{B}}{\rho_{0}}}\hskip 5.69046pt,\hskip
14.22636pt\hat{v}={\frac{v}{c_{s}}}$ (23)
where $\rho_{0}$ is an equilibrium (or reference) density, upon which
perturbations may be gene-rated, and $c_{s}$ is the speed of sound. Next, we
introduce the $\xi$ and $\tau$ “stretched” coordinates frsw ; davidson :
$\xi=\sigma^{1/2}{\frac{(x-{c_{s}}t)}{R}}\hskip 5.69046pt,\hskip
14.22636pt\tau=\sigma^{3/2}{\frac{{c_{s}}t}{R}}$ (24)
where $\sigma$ is a small expansion parameter, $R$ is a typical size scale of
the problem. After this change of variables we expand (23) as:
$\hat{\rho}=1+\sigma\rho_{1}+\sigma^{2}\rho_{2}+\dots$ (25) $\hat{v}=\sigma
v_{1}+\sigma^{2}v_{2}+\dots$ (26)
Having rewritten (4) and (5) in the $\xi-\tau$ space and having expanded them
in powers of $\sigma$ up to $\sigma^{2}$ we organize the two equations as
series in powers of $\sigma$. After these steps (4) and (5) become:
$\sigma\Bigg{\\{}-\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{c_{s}}^{2}+3\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}\bigg{]}{\frac{\partial
v_{1}}{\partial\xi}}+\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}+\pi^{2/3}{\rho_{0}}^{4/3}\bigg{]}{\frac{\partial\rho_{1}}{\partial\xi}}\Bigg{\\}}$
$+\sigma^{2}\Bigg{\\{}\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}+\pi^{2/3}{\rho_{0}}^{4/3}\bigg{]}{\frac{\partial\rho_{2}}{\partial\xi}}-\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{c_{s}}^{2}+3\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}\bigg{]}{\frac{\partial
v_{2}}{\partial\xi}}$
$+\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{c_{s}}^{2}+3\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}\bigg{]}\bigg{(}{\frac{\partial
v_{1}}{\partial\tau}}+v_{1}{\frac{\partial
v_{1}}{\partial\xi}}\bigg{)}+\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}\rho_{1}{\frac{\partial\rho_{1}}{\partial\xi}}+\pi^{2/3}{\rho_{0}}^{4/3}{\frac{\rho_{1}}{3}}{\frac{\partial\rho_{1}}{\partial\xi}}$
$-\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}2{c_{s}}^{2}+4\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}\bigg{]}\rho_{1}{\frac{\partial
v_{1}}{\partial\xi}}-\bigg{[}\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{c_{s}}^{2}+\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}\bigg{]}v_{1}{\frac{\partial\rho_{1}}{\partial\xi}}$
$+\bigg{(}{\frac{18g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{4}R^{2}}}\bigg{)}{\frac{\partial^{3}\rho_{1}}{\partial\xi^{3}}}\Bigg{\\}}=0$
(27)
and
$\sigma\Bigg{\\{}{\frac{\partial
v_{1}}{\partial\xi}}-{\frac{\partial\rho_{1}}{\partial\xi}}\Bigg{\\}}+\sigma^{2}\Bigg{\\{}{\frac{\partial
v_{2}}{\partial\xi}}-{\frac{\partial\rho_{2}}{\partial\xi}}+{\frac{\partial\rho_{1}}{\partial\tau}}+\rho_{1}{\frac{\partial
v_{1}}{\partial\xi}}+v_{1}{\frac{\partial\rho_{1}}{\partial\xi}}-{c_{s}}^{2}v_{1}{\frac{\partial
v_{1}}{\partial\xi}}\Bigg{\\}}=0$ (28)
respectively. In the last two equations each bracket must vanish independently
and so $\\{\dots\\}=0$. From the first term of (28) we obtain
$\rho_{1}=v_{1}$. Using this identity in the first term of (27) we obtain an
equation, which solved for $c_{s}$ yields:
${c_{s}}^{2}={\frac{\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}+\pi^{2/3}{\rho_{0}}^{4/3}}{\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}+3\pi^{2/3}{\rho_{0}}^{4/3}}}$
(29)
Inserting these results into the terms proportional to $\sigma^{2}$, we find,
after some algebra the KdV equation:
${\frac{\partial\rho_{1}}{\partial\tau}}+\bigg{[}{\frac{(2-{c_{s}}^{2})}{2}}-\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{\frac{(2{c_{s}}^{2}-1)}{2A}}-{\frac{\pi^{2/3}{\rho_{0}}^{4/3}}{A}}\bigg{(}{c_{s}}^{2}-{\frac{1}{6}}\bigg{)}\bigg{]}\rho_{1}{\frac{\partial\rho_{1}}{\partial\xi}}$
$+\bigg{[}{\frac{9g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{4}R^{2}A}}\bigg{]}{\frac{\partial^{3}\rho_{1}}{\partial\xi^{3}}}=0$
(30)
where:
$A=\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{c_{s}}^{2}+3\pi^{2/3}{\rho_{0}}^{4/3}{c_{s}}^{2}=\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}+\pi^{2/3}{\rho_{0}}^{4/3}$
(31)
Returning to the $x-t$ space we obtain:
${\frac{\partial\hat{\rho}_{1}}{\partial
t}}+c_{s}{\frac{\partial\hat{\rho}_{1}}{\partial
x}}+\alpha{c_{s}}\hat{\rho}_{1}{\frac{\partial\hat{\rho}_{1}}{\partial
x}}+\beta{\frac{\partial^{3}\hat{\rho}_{1}}{\partial x^{3}}}=0$ (32)
where
$\alpha\equiv\bigg{[}{\frac{(2-{c_{s}}^{2})}{2}}-\bigg{(}{\frac{27g^{2}\,{\rho_{0}}^{2}}{{m_{G}}^{2}}}\bigg{)}{\frac{(2{c_{s}}^{2}-1)}{2A}}-{\frac{\pi^{2/3}{\rho_{0}}^{4/3}}{A}}\bigg{(}{c_{s}}^{2}-{\frac{1}{6}}\bigg{)}\bigg{]}$
(33)
and
$\beta=\bigg{[}{\frac{9g^{2}\,{\rho_{0}}^{2}{c_{s}}}{{m_{G}}^{4}A}}\bigg{]}$
(34)
which is the KdV equation at zero temperature for the small perturbation in
the baryon density $\hat{\rho}_{1}\equiv\sigma\rho_{1}$. If we neglect the
space and time derivatives in (21) and (22) and repeat the derivation sketched
above we arrive at:
${\frac{\partial\hat{\rho}_{1}}{\partial
t}}+{c_{s}}{\frac{\partial\hat{\rho}_{1}}{\partial
x}}+\alpha{c_{s}}\hat{\rho}_{1}{\frac{\partial\hat{\rho}_{1}}{\partial x}}=0$
(35)
which is a breaking wave (BW) equation for $\hat{\rho}_{1}$. We close this
section emphasizing that Eq. (32) is the main result of this work. It shows
that for suitable choices of the parameters $\alpha$ and $\beta$ we can have
KdV solitons in a quark gluon plasma.
## V Numerical analysis
The KdV equation (32) has an analytical soliton solution given by drazin :
$\hat{\rho}_{1}(x,t)={\frac{3(u-c_{s})}{\alpha c_{s}}}\
sech^{2}\Bigg{[}{\sqrt{{\frac{(u-c_{s})}{4\beta}}}}(x-ut)\Bigg{]}$ (36)
where $u$ is an arbitrary supersonic velocity. A soliton is a localized pulse
which propagates without change in shape and in this case it has the width
$\lambda$ defined by:
$\lambda={\sqrt{{\frac{4\beta}{(u-c_{s})}}}}={\sqrt{{\frac{36g^{2}\,{\rho_{0}}^{2}{c_{s}}}{(u-c_{s}){m_{G}}^{4}A}}}}$
(37)
For the numerical estimates we shall use the following values of the
parameters : $\mathcal{B}_{QCD}=\,0.0006\,\,\mbox{GeV}^{4}$, $g=0.35$,
$m_{G}=290$ MeV and $\rho_{0}=2\,fm^{-3}$ which, when sustituted in (29) yield
${c_{s}}^{2}=0.5$ $(c_{s}=0.7)$. We choose $u=0.8$. With the help of (37) we
find $\lambda=1.7\,fm$ for the soliton width and $0.6$ for its amplitude. Even
though this work is essentially qualitative the chosen numbers are well
appropriate to study a realistic situation of a perturbation traversing the
QGP.
In Fig. 1 we show the numerical solution of (32) with initial condition
$\hat{\rho}_{1}(x,t=0)$ and $\hat{\rho}_{1}(x,t)$ given by (36). We can
observe the time evolution of the initial gaussian-like pulse as a well
defined soliton, keeping its shape and form. This solution shows the behavior
expected from the analytic solution. We see that using the correct input for
the amplitude and width we obtain a pulse which propagates without distortion.
This initial condition is a very special case of little practical interest. A
real perturbation produced in the QGP will most likely have the “wrong”
amplitude and “wrong” width. For arbitrary amplitudes the solution must be
numerically calculated.
In Fig. 2 we show again the numerical solution of (32) with the initial
condition given by (36) multiplied by a factor $10$. Now we observe that the
initial pulse starts to develop secondary peaks, which are called “radiation”
in the literature. Further time evolution will increase the strength of these
peaks until the complete loss of localization.
In Fig. 3 we show the numerical solution of (35) with the initial condition
given by (36). We observe the gradual formation of a “wall” followed by the
dispersion of the initial pulse. In Fig. 4 the amplitude of the initial pulse
(36) is multiplied by a factor $10$ and used as initial condition for (35). As
expected the dispersion takes place much earlier than in Fig. 3.
Figure 1: Numerical solution of (32) with (36) as initial condition calculated
at different times. Figure 2: The same as Fig. 1 with an initial amplitude
ten times larger. Figure 3: Numerical solution of (35) with (36) as initial
condition. Figure 4: The same as Fig. 3 with an initial amplitude ten times
larger.
## VI Summary
The main conclusion of this work is that it is indeed possible to have KdV
solitons in QCD, provided that two conditions are satisified. The first
condition is that the gluon field have a dynamical mass. In this case the
equation of motion (12) can be solved in the weak inhomogeneity approximation
yielding (18). The existence of a dynamical gluon mass has been intensely
discussed in the literature during the last years and seems to be well
established (for details see the references given in fn6 ). For a massless
gluon field we can only have a breaking wave equation, as it was found in our
previous work fn5 . The second condition is the existence of second order
derivative terms in the energy density and pressure. These terms appear
naturally from the formalism, as we can see in (15). However it is necessary
to keep these derivative terms. The use of uniform field approximations
prevents us from finding KdV solitons. If we neglect the derivatives we arrive
at the breaking wave equation (35). The practical difference between
perturbations governed by the KdV and BW equations is that the former
propagate much longer keeping its localization whereas the latter loose
localization and may generate unstable “walls”. The numerical analysis of some
cases confirms the anticipated qualitative expectation. The application of the
formalism developed in this work to problems in the theory of compact stars is
in progress.
###### Acknowledgements.
This work was partially financed by the Brazilian funding agencies CAPES, CNPq
and FAPESP.
## References
* (1) P. G. Drazin and R. S. Johnson, “Solitons: An Introduction”, Cambridge University Press, 1989.
* (2) G.N. Fowler, S. Raha, N. Stelte and R.M. Weiner, Phys. Lett. B 115, 286 (1982); S. Raha, K. Wehrberger and R.M. Weiner, Nucl. Phys. A 433, 427 (1984); E.F. Hefter, S. Raha and R.M. Weiner, Phys. Rev. C 32, 2201 (1985).
* (3) D.A. Fogaça and F.S. Navarra, Phys. Lett. B 639, 629 (2006).
* (4) D.A. Fogaça and F.S. Navarra, Phys. Lett. B 645, 408 (2007).
* (5) D.A. Fogaça and F.S. Navarra, Nucl. Phys. A 790, 619c (2007); Int. J. Mod. Phys. E 16, 3019 (2007).
* (6) D.A. Fogaça, L. G. Ferreira Filho and F.S. Navarra, Nucl. Phys. A 819, 150 (2009).
* (7) D.A. Fogaça, L.G. Ferreira Filho and F.S. Navarra, Phys. Rev. C 81, 055211 (2010).
* (8) D.A. Fogaça and F.S. Navarra, Phys. Lett. B 700, 236 (2011).
* (9) S. Weinberg,“Gravitation and Cosmology”, New York: Wiley, 1972.
* (10) L. Landau and E. Lifchitz, “Fluid Mechanics”, Pergamon Press, Oxford, (1987).
* (11) L. S. Celenza and C. M. Shakin, Phys. Rev. D 34, 1591 (1986).
* (12) X. Li and C. M. Shakin, Phys. Rev. D 71, 074007 (2005).
* (13) B.D. Serot and J.D. Walecka, Advances in Nuclear Physics 16, 1 (1986).
* (14) R. C. Davidson, “Methods in Nonlinear Plasma Theory”, Academic Press, New York an London, 1972, pages 20 and 21.
|
arxiv-papers
| 2011-06-29T14:57:29 |
2024-09-04T02:49:20.172850
|
{
"license": "Public Domain",
"authors": "D.A. Foga\\c{c}a, F.S. Navarra and L.G. Ferreira Filho",
"submitter": "David Fogaca",
"url": "https://arxiv.org/abs/1106.5959"
}
|
1106.6013
|
# A survey of the regular weighted Sturm-Liouville problem - The non-definite
case 111This research is supported, in part, by grant U0167 of the Natural
Sciences and Engineering Research Council of Canada and a N.S.E.R.C.
University Research Fellowship.
PREPRINT222The original version of this article appeared with minor changes in
“A survey of the regular weighted Sturm-Liouville problem: the non-definite
case”, in Applied Differential Equations, World Scientific, Singapore and
Philadelphia, 1986: 109-137. Some typographical errors have been corrected in
this version.
Angelo B. Mingarelli333Respectfully dedicated to my wife, Leslie Jean
University of Ottawa
Current address: School of Mathematics and Statistics,
Carleton University, Ottawa, ON.,
Canada K1S 5B6
amingare@math.carleton.ca
## 1 Introduction
Let $p,q,r:\,[a,b]\;\rightarrow\;\mathbb{R}$ where $-\infty<a<b<\infty$ and
$p(x)>0$ a.e., $q,r,\frac{1}{p}\,\in\,L(a,b)$ and
$\int\limits_{a}^{b}|r(s)|\,ds\,>\,0.$ (1.1)
The weighted regular Sturm-Liouville problem consists in finding the values of
a parameter $\lambda$ (generally complex) for which the equation
$-(p(x)y^{\prime})^{\prime}+q(x)y=\lambda r(x)y\quad a\leq x\leq b,$ (1.2)
has a solution y (non-identically zero) satisfying a pair of homogeneous
separated boundary conditions,
$y(a)\cos\,\alpha\,-\,(py^{\prime})(a)\sin\,\alpha=0$ (1.3)
$y(b)\cos\,\beta\,+\,(py^{\prime})(b)\sin\,\beta=0$ (1.4)
where $0\leq\alpha,\beta\,<\,\pi.$
Let
$D\,=\,\\{y:[a,b]\,\rightarrow\,\mathcal{C}\,|\,y,py^{\prime}\,\in\,AC[a,b],r^{-1}\\{(-py^{\prime})^{\prime}+qy\\}\,\in\,L^{2}(a,b),y\;\text{satisfies}\eqref{eq3}-\eqref{eq4}\\}$,
in the case $|r(x)|>0$ a.e. on (a,b). Then associated with the problem
(1.2)-(1.4) are the quadratic forms L and R, with domain D, where, for
$y\,\in\,D$,
$(Ly,y)=|y(a)|^{2}\cot\,\alpha\,+\,(py^{\prime})(a)y^{\prime}(a)\cot\,\beta\,+\,\int\limits_{a}^{b}\\{p(x)|y^{\prime}|^{2}\,+\,q(x)|y|^{2}\\}\,dx$
(1.5)
and
$(Ry,y)=\int\limits_{a}^{b}r(x)|y|^{2}\,dx.$ (1.6)
Here (,) denotes the usual $L^{2}$-inner product. (Moreover we note, as is
usual, that the $\cot\,\alpha$ (resp. $\cot\,\beta$) term in (1.5) is absent
if $\alpha=0$ (resp. $\beta=0$) in (1.3)-(1.4)).
It was noticed at the turn of this century by Otto Haupt [27],[28] and Roland
(George Dwight) Richardson, [55], [56], [60] (and it is very likely that David
Hilbert was also aware, since Richardson went to Göttingen in 1908…) that the
nature of the boundary problem (1.2)-(1.4) is dependent upon some
”definiteness” conditions on the forms L and R.
Thus, Hilbert and his school, termed the problem (1.2)-(1.4) polar if the form
(Ly,y) is definite on D, i.e., either $(Ly,y)>0$ for each $y\neq 0$ in D or
$(Ly,y)<0$ for each $y\neq 0$ in D (modern terminology refers to this case as
the left-definite case, cf., [21]).
The problem (1.2)-(1.4) was called orthogonal (or right-definite nowadays) if
R is definite on D, (see above), whereas the general problem, i.e., when
neither L nor R is definite on D, was dubbed non-definite by Richardson (see
[60], p.285). In this respect, see also Haupt ([28], p.91). We retain
Richardson’s terminology “non-definite” in the sequel, in relation to the
“general” Sturm-Liouville boundary problem (1.2)-(1.4). Thus, in the non-
definite case, there exists functions y, z in D for which $(Ly,y)>0$ and
$(Lz,z)<0$ and also, for a possibly different set of y, z, $(Ry,y)<0$ and
$(Rz,z)>0$.
## 2 The early theory of the non-definite case
The first published results pertaining albeit indirectly, to the non-definite
Sturm-Liouville problem (1.2)-(1.4) appear to be due to Emil Hilb [30], (see
also [31]), who considered the single equation
$y^{\prime\prime}\,+\,(A\phi(x)\,+\,B)y\,=\,0,\quad 0\leq x\leq 1$ (2.1)
in the two parameters A, B and extended Klein’s oscillation theorem to this
case. In (2.1) $\phi$ is real and continuous and may change sign in (0,1).
Note that (2.1) is a special case of (1.2), with $\lambda\equiv A,\,p\equiv
1$, $r\equiv\phi,\;q\equiv-B$. In [30] Hilb noted the existence of parameter
values (A,B) (corresponding to the Dirichlet problem) which gave rise to a
“minimum oscillation number” for corresponding eigenfunctions, this, being one
of the characteristics of non-definite problems, (however this may also occur
in orthogonal problems, see [[60], p.294]). Both Haupt ([28], p.69 and p.88)
and Richardson ([58], p.289) cite Hilb’s paper [30] although it is to be noted
that Maxime Bôcher ([11], p.173) only cites Richardson, in this context, (and
the citation is by way of an oral communication of Richardson to Bôcher ([11],
p.173, footnote) in his important survey of Sturm-Liouville theory up to about
the year 1912. For a further up-date on the early developments of Sturmian
theory see Richardson’s review article ([61], pp.110-111) of Bôcher’s classic
book [13], Lichtenstein’s paper [37] and the classic book by Ince
([32],§§10-11). We note, however, that in a later paper [12], Bôcher
discovered Hilb’s article [30], (see [12], p.7, footnote).
Now it appears as if the first published theoretical investigations of the
general non-definite problem (i.e., (1.2)) with q not identically a constant
function on (a,b) are due to Haupt [27] in his dissertation, and Richardson
[57]. In each one of these works one finds the basis for a beautiful extension
of the Sturm Oscillation Theorem in relation to a non-definite problem in
which the coefficients are assumed, a priori, continuous over the finite
interval under consideration. A revised version of Haupt’s dissertation (along
with a corrected version of the said oscillation theorem) appeared much later,
in 1915, (see [28]). Richardson’s version of the “general Sturm oscillation
theorem” first appeared in [58] but the correct statement is to be found in
([58], errata) after a comment by George D. Birkhoff.
In Haupt’s paper [28] one finds many interesting results on the nature of non-
definite problems, results with which are, unfortunately, not mentioned
explicitly in Richardson [60] a paper which deals essentially with “new”
oscillation theorems for (1.2) in the non-definite case ([60], $\S$4) and for
the difficult case when q, in (1.2), is allowed to vary non-linearly with
$\lambda$, (see also McCrea-Newing [40]). However, Richardson does refer to
Haupt [28] insofar as the oscillation theorem is concerned, as Haupt’s theorem
appeared one year earlier (cf., above). Thus the first version of an
oscillation theorem (in the case q$\neq$ constant) is due to Haupt [28]. On
the other hand, a non-trivial sharpening, and the final beautiful form of the
oscillation theorem, was formulated by Richardson in ([58], errata), (see also
Richardson ([60], p.285)). Curiously enough Richardson was indeed the “father
of the non-definite case” as one can infer from his remarks in ([60], p.285),
the claim being that the first investigations of a non-definite problem
appeared in Richardson [57].
It is remarkable that Haupt’s work [28] has remained obscure, even today. For
example Ince ([32], p.248) refers to Richardson but nowhere in the book does
one find a reference to Haupt! One reason for this may be the following: As
was mentioned above Hilbert and his school referred to left-definite problems
as “polar” problems and to the right-definite as “orthogonal” problems.
However, Haupt, in [28], decided to include “non-definite” problems (in
Richardson’s terminology) under the heading of “polar” ones, Haupt ([28];
p.84, $\S$5), whereas the “orthogonal” case was included in the section
entitled “non-polar case”, Haupt ([28], p.76, $\S$3). It may then have
appeared, to those influenced by Hilbert and his school, that, at first sight,
Haupt was not doing anything radically new except for extensions, to equations
in which the parameter $\lambda$ appears non-linearly, of the Sturm
oscillation theorem, Haupt ([28], $\S$4). Haupt’s failure to emphasize the
”non-definite polar case”, Haupt ([28], p.91), as a new and distinct case may
have led to a low readership of his paper, [28].
###### Theorem 2.1.
_Haupt ([28], pp.84-85)_
In (1.2), let $p,q,r$ be continuous in $[a,b]$ along with $p$ differentiable
and $p(x)>0$ there. Then the eigenvalues of (1.2)-(1.4) are the zeros of an
entire transcendental function which is not identically zero.
###### Remark 2.2.
For results pertaining to the order of the entire transcendental function
alluded to by Haupt, see Halvorsen [26], Mingarelli ([44, 48, 42]) and
Atkinson-Mingarelli [1].
Of course, it is necessary to assume in Theorem 2.1 that there hold (1.1) or
else the entire function mentioned therein may, in fact, vanish identically on
$\mathcal{C}$ as an easy example will show. This hypothesis is not explicit in
Haupt [28] but is, however, necessary in the proof.
###### Corollary 2.3.
The eigenvalues of (1.2)-(1.4) form a discrete subset of the complex plane
(i.e., having no finite point of accumulation.)
In Haupt [28] a “normalization” is assumed for (1.2), i.e., the equation is
transformed into a “normal form” in many places (see Haupt ([28], pp.85-86)
for the transformation and Mingarelli [46] for an assumption in the same
vein).
The existence of possibly non-real eigenvalues was alluded to by Haupt ([28],
pp.94) and by Richardson ([60], p.289 and footnote) however neither author
gave an instance of such an occurrence (cf., also Mingarelli ([45], Chapter
4)). Concrete examples of non-definite problems with non-real eigenvalues were
obtained in Mingarelli ([46], pp.519-520), ([42], p.376) in which some very
early ideas of Hilb [31] were used to show that $\lambda=i$ may be turned into
an eigenvalue for a Dirichlet problem associated with (1.2), (see [42] for
details).
## 3 Terminology and notation
The theory of linear operators in spaces with an indefinite metric, Azizov and
Iohvidov [5], Bognár [14], together with their applications to quantum field
theory, Nagy [52], suggests the following terminology (much of which was due
to Werner Heisenberg).
An eigenfunction $y$ of (1.2)-(1.4) corresponding to a non-real eigenvalue
will be called a complex ghost state or complex ghost. (These necessarily
satisfy $\int\limits_{a}^{b}r(x)|y|^{2}\,dx=0$.)
If the non-real eigenvalue $\lambda$ is non-simple its corresponding
eigenfunction y will be said to be degenerate. It is non-degenerate otherwise,
i.e., $\int\limits_{a}^{b}r(x)|y|^{2}\,dx\neq 0$, (see $\S$5).
An eigenfunction corresponding to a real non-simple eigenvalue will be called
a degenerate real ghost state (or a dipole ghost, see [52]) whereas a real
eigenfunction y corresponding to the (real) eigenvalue $\lambda$ is a non-
degenerate real ghost state provided
$\text{sign}\bigg{\\{}\lambda\int\limits_{a}^{b}r(x)|y|^{2}\,dx\bigg{\\}}<0.$
The term positive eigenfunction (or ground state) will refer to a (real)
eigenfunction whose values are strictly positive in the interior of the
interval under consideration.
If $y$ is any eigenfunction the quantity
$\int\limits_{a}^{b}r(x)|y|^{2}\,dx$
will be called the $r$-Kreǐn norm of $y$. (This terminology is motivated by
the formalism whereby one may cast the setting for a spectral theory of non-
definite metric, e.g., Kreǐn space, Pontryagin space. (see, e.g., Azizov and
Iohvidov [14], Bognǎr [5] and applications in Langer [35], Daho and Langer
[17, 18], Čurgus and Langer [64], Daho [15, 16], and Mingarelli [45]). Note
that when $r(x)>0$ a.e. on $(a,b)$ the $r$-Krein-norm of $y$ is the usual norm
in the weighted Hilbert space $L_{r}^{2}(a,b)$.
As is customary we will denote by $r_{\pm}$ the positive (negative) part of r,
i.e., $r_{\pm}\,=\,max\\{\pm r(x),0\\}.$
## 4 Some motivation
Consider the boundary problem associated with Mathieu’s equation
$y^{\prime\prime}+(-\alpha+\beta p(x))y=0,\;-\infty<x<+\infty$
where $(\alpha,\beta)\,\in\,{\Re}^{2}$ are parameters and $p$ is periodic, or
more generally, locally Lebesgue integrable over $\Re$. In the usual problem,
$\beta$ is fixed and values of $\alpha$ are sought $\ldots$. However if we fix
$\alpha<<0$ and seek $\beta$’s the problem is generally non-definite. (We
recall that Hilb [30] considered the same problem, but on a finite interval.)
More generally, the equation,
$y^{\prime\prime}+(-\alpha A(x)+\beta B(x))y=0,\;-\infty<x<+\infty,$
in the two parameters $\alpha,\beta\,\in\,L_{loc}$($\Re$ ) also leads to a
non-definite problem once one of the parameters $\alpha,\beta$ is held fixed
and the other is treated as an eigenvalue parameter. The question of the
existence of positive solutions of the latter equation has received some
interest lately (e.g., the monograph of Halvorsen-Mingarelli [41]). For
applications of these to laser theory see Heading [29], also McCrea-Newing
[40] and the references therein.
Techniques from the theory of non-definite problems were recently used by
Deift-Hempel [19] with applications to the theory of color in crystals.
Various other related problems are treated, for instance, in Barkovskii-
Yudovich [6, 7] in relation to Taylor vortex formation arising from rotating
cylinders.
## 5 On the existence of eigenvalues
The problem of the actual existence of eigenvalues for the non-definite
problem (1.2)-(1.4) is treated implicitly in Haupt [28] and somewhat more
elaborately in Richardson [60], at least for the Dirichlet problem. Prüfer
angle methods have been used in this respect in Halvorsen [26] to, at least,
settle the existence of an infinite sequence of real eigenvalues under the
more general set of boundary conditions (1.3)-(1.4). A modified Prüfer angle
method has been used in Atkinson-Mingarelli [1] which also settles the
existence and, at the same time, yields their asymptotics.
###### Theorem 5.1.
_Haupt[28], Richardson [60], Halvorsen [26], Atkinson-Mingarelli [1]_.
Whenever
$\int\limits_{a}^{b}r_{+}(s)ds>0\;\;\text{and}\;\;\int\limits_{a}^{b}r_{-}(s)\,ds>0$
the problem (1.2)-(1.4) has two infinite sequences of real eigenvalues, one
positive and one negative, and each one of which has $+\infty$ and $-\infty$
for its only point of accumulation.
###### Remark 5.2.
In Haupt [28], Richardson [60] the authors deal with the case in which the
coefficients $p,q,r$ are continuous in $[a,b]$ whereas in Halvorsen [26] and
Atkinson-Mingarelli [1] the more general case referred to in the introduction
is considered.
For a non-definite problem (1.2)-(1.4), non-real eigenvalues may or may not
occur! Both these cases are possible - with regards to the former see the
comments in Haupt [28], Richardson [60], Daho and Langer [17, 18] and the
specific example in Mingarelli ([46], p.250) or Atkinson-Jabon ([4],
appendix), Jabon [33], or Marziali [39].
The question of the possible existence of non-real eigenvalues for a non-
definite problem (1.2)-(1.4) seems to have first been formulated by Haupt [28]
and then by Richardson ([60], p.100) for the case of continuous coefficients
under the assumption that the parameter is “normalized” (see Haupt [28],
pp.85-86). A result in the same vein was obtained independently by the author
in Mingarelli ([46], Theorem 2) the proof of which also includes the case of
Lebesgue integrable coefficients.
###### Theorem 5.3.
_(Haupt[28], Mingarelli [46])_
In the non-definite case and under the assumptions of $\S$, the problem
(1.2)-(1.4) has at most finitely many non-real eigenvalues, their total number
being even.
A particular case of a non-definite case of a non-definite (singular) problem
was explored in Daho-Langer [17, 18], (see also Čurgus-Langer [64] and
Mingarelli [46], Chapters 3,4).
There is indeed an intimate connection between the theory of symmetric linear
operators in a Pontryagin space (Krein space) and the theory of non-definite
Sturm-Liouville problems. We will not delve into this matter here, for the
sake of brevity, although we will refer the reader to the relevant literature
herewith: For general notions on Pontryagin spaces see the monograph by Bognár
[14] and the excellent survey paper by Azizov-Iohvidov [5]. For applications
of this theory to the problem at hand see Langer [35], Daho-Langer [17, 18],
Čurgus-Langer [64], while for interesting physico-theoretical applications of
Pontryagin spaces we refer to the monograph by Nagy [52]. We need only mention
at this point that Theorem 5.3 and other similar theorems, some of which we
will present below, may be proved with the help of the theory of symmetric
linear operators in a Pontryagin space.
Now for $\lambda$ real let n($\lambda$) denote the number of negative
eigenvalues (counting multiplicities) of the problem
$-(p(x)y^{\prime})^{\prime}+(q(x)-\lambda r(x))y=\nu y$
where $y$ is required to satisfy (1.3)-(1.4). Then the function n($\lambda$)
has an absolute minimum which will be denoted by $n_{0}$. (This construction
is due to Haupt ([28], p.85.)
###### Theorem 5.4.
_Haupt[28], Mingarelli [46]_
1. 1.
Let $p,q,r$ be continuous on $[a,b]$ and (1.2)-(1.4) non-definite. Then the
number of pairs (i.e., an eigenvalue and its complex conjugate) of non-real
eigenvalues does not exceed $n_{0}$, whenever the parameter is normalized,
([28], p.100).
2. 2.
In (i) above we may replace $n_{0}$ by the number of negative eigenvalues of
the problem
$-(p(x)y^{\prime})^{\prime}+(q(x)-\lambda r(x))y=\lambda y$ (5.2)
subject to (1.3)-(1.4) provided zero is not an eigenvalue of the said problem,
[46]. Cases of equality here may be exhibited, see [46] or Marziali [39].
We now note that, using a Green’s function argument, in the regular case of
the non-definite problem (1.2)-(1.4) the spectrum is purely discrete (i.e., it
consists only of eigenvalues of finite multiplicity, Mingarelli [50].) We now
turn our attention to the question of the “simplicity” of the eigenvalues. For
basic results regarding the nature of “simple” and “non-simple” eigenvalues we
refer to Ince ([32], $\S$ 10.72). Thus let $y(x,\lambda)$ be a non-trivial
solution of (1.2) which satisfies (1.3). Then the function
$F(\lambda)\equiv y(b,\lambda)\cos\,\beta+(py^{\prime})(b,\lambda)\sin\,\beta$
is an entire function of $\lambda\in\mathcal{C}$ whose order, generally, does
not exceed one-half (see e.g., Atkinson [2]). Actually it is now known that
its order is precisely one-half as was shown directly by Halvorsen [26] and,
as a consequence of the asymptotics, by Mingarelli [42], (cf., Mingarelli
[44]), and Atkinson-Mingarelli [1]. For some extensions of the above result on
the order of $F$ see Mingarelli [48].
Now the zeros of $F$ are in a one-to-one correspondence with the eigenvalues
of the problem (1.2)-(1.4). We say that an eigenvalue
$\lambda\,\in\,\mathcal{C}$ is simple if it is a simple zero of $F$ (i.e.,
$F^{\prime}(\lambda)\neq 0$). It is said to be non-simple otherwise, (i.e., if
$0=F(\lambda)=F^{\prime}(\lambda)$).
Note: It does not follow from the above definition that a non-simple
eigenvalue necessarily has two linearly independent eigenfunctions associated
with it, for if this were ever the case, their span would generate the space
of all solutions of (1.2) and clearly one could find solutions which do not
satisfy the first boundary condition (1.3), (cf., Ince [32], p.241).
Thus for the problem under consideration, namely (1.2)-(1.4), there is a one-
to-one correspondence between any real eigenvalue and its corresponding
eigenfunction (if it is suitably normalized).
It is known that $\lambda\,\in\,\mathcal{C}$ is non-simple if and only if it
has a corresponding (real or complex) eigenfunction which is a degenerate
ghost state. The latter result was anticipated by Richardson ([60], p.294,
Corollary) albeit without proof. For a proof and an extension of the said
result to the case of measurable coefficients, see Mingarelli [43].
It turns out that real degenerate ghost states may exist for (1.2)-(1.4), see
the example in Mingarelli [43]. Furthermore there may also exist non-
degenerate real ghost states, see, once again, an example in Mingarelli [43],
thus answering a question raised by Haupt ([28], p.100 footnote). For all of
the examples now known it appears that the complex ghost states are all non-
degenerate. There is no known example of a degenerate complex ghost state
although the author feels that these very likely do exist in some cases.
###### Theorem 5.5.
_Mingarelli[43]_
Let $\lambda>0$ and assume that zero is not an eigenvalue of (1.2)-(1.4). Then
the total number of non-degenerate and degenerate real ghost states is always
finite and bounded above by the number of negative eigenvalues of
(5.2)-(1.3)-(1.4). An analogous result holds for $\lambda<0$.
###### Remark 5.6.
Specific examples, Marziali [39], seem to indicate that the bound appearing in
Theorem 5.5 is sharp. Consolidating the above results we may formulate,
###### Theorem 5.7.
In the non-definite case of (1.2)-(1.4) the spectrum is discrete, always
consists of a doubly infinite sequence of real eigenvalues, having no finite
limit, and has at most a finite and even number of non-real eigenvalues
(necessarily occurring in complex conjugate pairs) along with at most finitely
many real non-simple eigenvalues. The totality of all such eigenvalues
comprise the spectrum of (1.2)-(1.4).
For an extension of Theorem 5.4-5.5 to an abstract setting, see Mingarelli
[47] wherein the more general operator equation $Ax=\lambda Bx$ is considered
as a generalized eigenvalue problem in a complex Hilbert space, thus allowing
for extensions of the said results to partial differential equations with
indefinite weight-functions.
Note Open Problems
1. 1.
For a given non-definite problem (1.2)-(1.4) can one find an a priori bound on
the modulus (or real/imaginary part) of the “largest” non-real eigenvalue
which might appear?
2. 2.
For a given non-definite problem (1.2)-(1.4) can one find an a priori bound on
that interval of the real axis which contains all the non-simple eigenvalues
and those eigenvalues corresponding to non-degenerate real ghost states? (Note
that this interval is finite on account of Theorem 5.5). In relation to this
question see Atkinson-Jabon [4] wherein this is done for a specific example.
3. 3.
Find a sufficient condition which will guarantee the existence of a non-real
eigenvalue for a non-definite problem (1.2)-(1.4). (Necessary conditions are
widespread: see e.g. Mingarelli ([46], p.521, Theorem 1.)
4. 4.
According to Richardson ([60], p.289) all sufficiently large real eigenvalues
may be regarded as furnishing a minimum of a calculus of variations problem
(since they all have positive r-Krein-norm?); however, no proof of this claim
is given.
(Note, however, one of Richardson’s last papers on this subject, Richardson
[62].)
## 6 A generalized Sturm Oscillation Theorem
The classical Sturm Oscillation Theorem for a left-definite Dirichlet problem
associated with (1.2) states that, if $r$ satisfies the hypothesis of Theorem
5.1, the n-th positive (negative) eigenvalue has an eigenfunction which has
precisely $n$ zeros in $(a,b)$, (see e.g. Ince [32], p.235, Theorem 3).
###### Theorem 6.1.
_Haupt[28], Richardson [60]_ In the non-definite case of (1.2)-(1.4) there
exists an integer $n_{R}\geq 0$ such that for each $n\geq n_{R}$ there are at
least two solutions of (1.2)-(1.4) having exactly $n$ zeros in $(a,b)$ while
for $n<n_{R}$ there are no real solutions having $n$ zeros in $(a,b)$.
Furthermore there exists a possibly different integer $n_{H}\geq n_{R}$ such
that for each $n\geq n_{H}$ there are precisely two solutions of (1.2)-(1.4)
having exactly $n$ zeros in $(a,b)$.
###### Remark 6.2.
We will call $n_{R}$ the Richardson Index or number and label $n_{H}$ the
Haupt Index (or number) of the problem (1.2)-(1.4) for historical reasons- The
existence of $n_{H}$ appears to have been first established in the literature
by Haupt ([28], p.86) while the existence of $n_{R}$, in general, is almost
certainly due to Richardson [57] and [60], (modulo Hilb’s special case [30]).
It now follows from the Haupt-Richardson oscillation theorem (Theorem 6.1)
that, generally speaking, a non-definite problem will tend not to have a real
ground state (positive eigenfunction)!
The existence of $n_{R}$ in the case of measurable coefficients was obtained
by the author, Mingarelli [49], via a Prüfer transformation and use of the
results in Mingarelli [43].
It was shown in Mingarelli [51] that each one of the cases $n_{H}=n_{R}$ and
$n_{H}>n_{R}$ may occur: Indeed the problem (1.2) with $p\equiv
1,\;q\equiv-9\pi^{2}/16,\;r(x)=+1$ on $[0,1]$ and $r(x)=-1$ on $(1,2]$,
$\alpha=\beta=0$ on $[a,b]=[0,2]$ has precisely one pair of non-real
eigenvalues (actually pure imaginary) situated at around $\pm 4.3$i while the
remaining real eigenvalues have eigenfunctions with at least one zero in
$(a,b)$, so that $n_{R}\geq 1$ and in fact $n_{R}=n_{H}=1$, in this case.
On the other hand if we set $q\equiv-22.206$ (a lower approximation to
$-9\pi^{2}/4$) and define the other quantities as in the above example, an
explicit calculation shows that $n_{H}=3$ while $n_{R}=2$. More recently the
calculations reported in Atkinson-Jabon [4] and Marziali [39] yield
essentially unlimited examples of such pathological behavior (see also
Richardson [60], p.298).
Armed with the Haupt-Richardson oscillation theorem one may proceed to prove
asymptotic estimates for the real eigenvalues of non-definite problems as was
done in Mingarelli [46]\- This is closely related to a conjecture of the late
Konrad Jörgens [34]. Under assumptions which, in the aftermath, are very
similar to those used by M.H. Stone [63], (e.g., if $r$ is continuous, then
$r(x)$ changes sign finitely many times in $(a,b)$), the author showed
###### Theorem 6.3.
(_Mingarelli[42]_.) If the positive eigenvalues $\lambda_{n}^{+}$ of a given
non-definite problem (1.2)-(1.4) are labeled according to the Haupt-Richardson
oscillation theorem so that $\lambda_{n}^{+}$ has an eigenfunction with
precisely $n$ zeros in $(a,b)$, then
$\lambda_{n}^{+}\sim\frac{n^{2}\pi^{2}}{\\{\int_{a}^{b}\sqrt{(\frac{{r}}{p})_{+}}\,ds\\}^{2}},\quad
n\to\infty,$ (6.3)
where $(\frac{{r}}{p})_{+}=max\\{(\frac{{r}}{p})_{+},0\\}$. An analogous
formula holds for the negative eigenvalues as well.
Let N($\lambda$) denote the number of zeros of a non-trivial solution of (1.2)
satisfying, say, (1.3). If we label our positive eigenvalues so that
$N(\lambda_{n}^{+})=n$ for $n\geq n_{H}$, then it will follow from (6.3) that
$N(\lambda)\sim\sqrt{\lambda}\pi^{-1}\int_{a}^{b}\sqrt{(\frac{{r}}{p})_{+}}\,ds,\quad\lambda\to\infty,$
(6.4)
This last relationship entails what the author calls Jörgens’ conjecture
([34], p.5.16). A proof of this conjecture, in general, is given, among other
results, in Atkinson-Mingarelli [1] thus settling the question of the validity
of (6.4), at least in the case when $p(x)>0$ a.e. on $(a,b)$. Now let
$n_{+}(\lambda$) denote the number of positive eigenvalues of (1.2)-(1.4)
which are in $(0,\lambda)$. Then it follows from (6.3) once again that
$n_{+}(\lambda)\sim\sqrt{\lambda}\pi^{-1}\int_{a}^{b}\sqrt{(\frac{{r}}{p})_{+}}\,ds,\quad\lambda\to\infty,$
(6.5)
(for a detailed calculation see Mampitiya ([38], p.28, Lemma 4.4). We mention,
in passing, that (6.5) has been recently extended to incorporate polar vector
Sturm-Liouville problems on a finite interval, see Mampitiya [38].
One of the truly remarkable results that one may extract from Richardson [60]
is the following.
###### Theorem 6.4.
(_[60], p.302, Theorem 10_.) Let $r$ be continuous and not vanish identically
in any right-neighborhood of $x=a$. If $r(x)$ changes its sign precisely once
in $(a,b)$ then the roots of the real and imaginary parts $u,v$, of any non-
real eigenfunction $y=u+iv$ corresponding to a non-real eigenvalue, separate
one another.
###### Corollary 6.5.
Under the conditions of Theorem 6.4 it follows that any non-real eigenfunction
$y$ of (1.2)-(1.4) cannot have a zero for $x$ in $(a,b)$.
Note The initial assumption on $r$ in Theorem 6.4 is not explicitly mentioned
in the proof given by Richardson [60] however it appears to be necessary for
the validity of his proof.
For an extension of Theorem 6.4 to the case when $r(x)$ changes sign its sign
finitely many times, see Mingarelli ([46], p.525, Theorem 3).
Finally we note that some interesting comparison theorems for non-definite
problems are derived in Atkinson-Jabon ([4], p.35, Proposition 3.4).
## 7 Further Results and Extensions
With regards to an eigenfunction expansion in the regular non-definite case
see Faierman [22] where the actual uniform convergence of such an expansion is
also treated. Another approach to the same problem is to be found in Binding-
Seddighi [9] with an aim towards an abstract expansion theorem which includes
the non-definite case as a particular case, but with additional boundedness
assumptions on the coefficients.
The problem of finding asymptotic expressions for the solutions of equations
of the form (1.2) with additional smoothness on the coefficients, is an old
one and much has been done in this direction- The book by Olver [53] is a good
introduction to the subject as is the book by Erdelyi [20], especially Chapter
4. Further results may be found in recent papers by Fedoryuk [23, 24] and the
paper by Olver [54], see also the references therein. Interesting results in
this connection are also buried in Birkhoff-Langer [10].
Results in the case of singular non-definite Sturm-Liouville problem (dealing
almost exclusively with a closed half-line) are more or less scattered,
although the object of recent interest. In this direction see the pioneering
papers by Atkinson-Everitt-Ong [3], and Daho-Langer [17, 18] wherein an
expansion theorem is formulated. The paper by Langer [35] sheds much insight
into the role that Krein spaces play in the study of non-definite problems.
Further expansion results (full and half-range expansions) for higher-order
scalar ordinary differential equations are given in Čurgus-Langer [64], see
also Daho [15, 16] for results dealing with the existence of a Titchmarsh-Weyl
matrix function in a higher-order singular case.
For the most up-to-date results regarding left-and right-definite problems,
which include (1.2), we refer to the paper by Everitt [21] and Bennewitz-
Everitt [8] and the references therein.
The basis for the extension of the foregoing results to partial differential
equations may be found in Fleckinger-Mingarelli [25], see also Mingarelli
[47].
At this point many questions even in the regular case, remain unanswered: For
example, questions relating to the existence/non-existence of real ground
states for (1.2)-(1.4); extensions of the preceding results to the systems of
second order differential equations and higher-dimensional analogs; an
efficient method for calculating the non-real eigenvalues of non-definite
problems (1.2)-(1.4), (current techniques of, Marziali [39], Atkinson-Jabon
[4], rely upon explicit computation of the zeros of F($\lambda$), (see §5),
which is generally inefficient; and to find a priori estimates on the Haupt
and Richardson indices for a given non-definite problem.
Acknowledgments
I am grateful to Professor F.V. Atkinson who, in 1977 introduced the author to
this fascinating field. The main reference to Richardson [60] was only
discovered in 1978 whereas the importance of Haupt’s work was only recognized
in 1984. All the other references to the early history of the non-definite
problems are basically recent finds. It is uncanny that so much work had been
done in connection with this problem at the turn of this century, work which
seemingly came to a halt around 1920. Renewed theoretical interest in this
area appears to be very recent, only after a gap of almost 60 years!
I am also grateful to Dr. Philip Hartman (formerly of Johns Hopkins
University) who, in 1978, remarked to me that Hilb may have had something to
do with this problem. This hunch proved correct as we have seen. I am grateful
to Phil Hartman for also raising the question, in 1980, of the “simplicity” of
the real eigenvalues of non-definite problems, a question which led to my
paper [43].
I wish to thank the following mathematicians for supplying me with preprints
of their work - C. Bennewitz, Paul Binding, Karim Daho, Percy Deift, Mel
Faierman, and David Jabon. I also wish to thank Derick Atkinson, Paul Binding,
Patrick Browne, Mel Faierman and Gotskalk Halvorsen for interesting
discussions. My gratitude also goes to Hans Kaper, for providing the reference
[20], and Hubert Kalf for rekindling my interest in Lichtenstein’s paper [37].
Note added June 29, 2011 The papers referred to as [1], [9], [19] and [22]
below have since appeared and the updated citations are included in the
references below. In addition, the papers by Langer [36], Richardson [59], and
Wheeler [65] were inadvertently left out of the body of the original paper as
it appeared 25 years ago. In [36] Langer makes a detailed study of the
spectral theory of a regular Sturm-Liouville problem having only one turning
point (of any order) on a finite interval. Wheeler [65] considered the
existence and completeness questions in both the orthogonal and polar cases,
while Richardson [59] considers the polar and orthogonal case of a two
dimensional elliptic problem while hinting at non definite cases as well (this
seems to be the first case of such a study for higher dimensions), cf., [[59],
p.494].
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Department of Mathematics
University of Ottawa
Ottawa, Ontario, Canada
K1N 6N5
|
arxiv-papers
| 2011-06-29T17:32:46 |
2024-09-04T02:49:20.180964
|
{
"license": "Public Domain",
"authors": "Angelo B. Mingarelli",
"submitter": "Angelo B. Mingarelli",
"url": "https://arxiv.org/abs/1106.6013"
}
|
1106.6062
|
# The Life and Death of Unwanted Bits: Towards Proactive Waste Data Management
in Digital Ecosystems
Ragib Hasan, Randal Burns
Department of Computer Science, Johns Hopkins University
3400 N. Charles Street, Baltimore, MD 21218, United States
{ragib,randal}@cs.jhu.edu
###### Abstract
Our everyday data processing activities create massive amounts of data. Like
physical waste and trash, unwanted and unused data also pollutes the digital
environment by degrading the performance and capacity of storage systems and
requiring costly disposal. In this paper, we propose using the lessons from
real life waste management in handling waste data. We show the impact of waste
data on the performance and operational costs of our computing systems. To
allow better waste data management, we define a waste hierarchy for digital
objects and provide insights into how to identify and categorize waste data.
Finally, we introduce novel ways of reusing, reducing, and recycling data and
software to minimize the impact of data wastage.
## 1 Introduction
In real life, all human activities produce unwanted, unusable, or useless by-
products. Such worthless objects are considered to be waste or trash. Waste
products impact the environment and ecosystem by using up or polluting
resources, degrading performance of physical processes, and requiring
expensive cleanup. To deal with waste in the ecosystem, various waste
management techniques have been developed over the years [10, 11]. These
techniques aim at reducing the production of waste, repurposing the waste or
its components, and efficient disposal of waste.
In today’s world, we are increasingly living virtual lives – creating,
processing and consuming data in the form of digital objects. A computing
system is similar to a real life ecosystem. In a digital ecosystem, data and
applications that consume and produce data interact and use the physical
hardware components and resources. Like real life ecosystems, a digital
ecosystem has finite resources such as storage, compute cycles, and network
bandwidth. Also, consumer applications share and compete for these resources.
No matter how much “illusion” of infinite resources various abstractions
provide, in reality resources in a digital ecosystem are not infinite.
Storing, transferring, and disposing of data consume these resources. However,
all data in a given system are not equally important or useful, and often a
significant amount of data in a system can be unwanted or unused content. Such
waste data consumes resources but provide no value to the corresponding
digital ecosystem. Unless we introduce responsible waste data management
practices, such waste data will misuse resources and make a significant impact
on the digital ecosystem.
What happens to these unwanted bits? Typical approaches to managing waste data
include compression and/or deletion of such unwanted data. However, these
processes come at a price – disposal of waste data consumes resources in the
form of energy used to delete data, tying up compute cycles, blocking I/O,
etc. Disposal via deletion also causes degradation of performance and reduces
the lifetime of storage components (such as Flash storage). We need better
waste management techniques to handle unwanted data. In this paper, we argue
about the need to examine waste data in a systematic manner. We posit that
successful real life waste management techniques can be effectively adapted to
handle waste data. The contributions of this paper are as follows:
We present a definition of waste data in digital ecosystems.
We show the impact of waste data on reduction and degradation of system
capacity.
We introduce a hierarchy for waste data management techniques.
We advocate the need for an integrated approach for managing waste data and
discuss how well known real life waste management principles can be adapted
for this. The rest of the paper is organized as follows: Section 2 presents a
definition of waste data. We discuss the impact of waste data in Section 3. We
introduce a hierarchy for waste data management and explore the use of
different real life waste management principles for managing waste data in
Section 4. Finally, we discuss related work in Section 5 and conclude in
Section 6.
## 2 Defining Waste Data
In real life, the definition of waste is somewhat subjective, as what is waste
to one system can be considered valuable resources by another system. Various
authorities and agencies have defined waste in different ways. For example,
the Basel convention and the European Union define waste as something that is
or will be discarded by the holder [1, 2]. The Organization for Economic
Cooperation and Development (OECD) defines waste as materials that are by-
products of regular processing, which have no use to the creator and which are
disposed of [9]. Pongrácz et al. [10, 11] provided a definition of waste based
on their classification of objects – an object is considered to be waste if it
is unintentionally created, or the user has used up the object, or the
object’s quality has degraded, or the object is unwanted by the user.
Similarly, providing the definition of waste data in a computing system is
difficult. Informally, a data object or software can be considered to be waste
data by a user if it has no utility for the user in the given context. To
provide a formal definition of waste data, we leverage the definition of
physical waste given by Pongrácz et al. [10, 11]. In particular, we use
Pongrácz et al.’s classification scheme to define waste data as data belonging
to any one or more of the following categories:
Unintentional data. Data unintentionally created, as a side effect or by-
product of a process, with no purpose.
Used data. Good data that has served its purpose and is no longer useful to
the user.
Degraded data. Data that has degraded in quality such that it is no longer
useful to the user.
Unwanted data. Data that was never useful to the user. Next, we discuss each
of these waste data categories with examples.
Unintentional data. Almost all data processing applications generate
unintentional by-products. We define a data object to be a by-product if it is
not included in the final set of data objects produced by the application. For
example, the goal of LaTeX compilation is to generate output .pdf or .ps files
from source text and images. However, when LaTeX is executed, it automatically
creates a number of temporary data objects and files that assist in
compilation. In the context of LaTeX compilation, these files (such as .aux,
.bbl, .log) can be considered to be unintentional by-products that assist in
the production of the final data product (e.g., .ps or .pdf).
Used data. In most cases, input data is no longer useful to the user once
computations have been performed over it. For example, an aggregation
operation can use data from many sensors. The sensor readings may be useful to
the user performing the aggregation operation only until the computation is
over. After that, the input data may become useless, and hence considered to
be waste data.
Degraded data. When data gets corrupted, it can become unusable, and
therefore be considered as waste data by a user. Also, when other changes make
data or software obsolete, it can be considered to have degraded and therefore
marked as waste data by the user. For example, newer software releases can
make old versions of the software and the related files obsolete.
Unwanted data. This class of waste data includes data that may or may not be
of high quality, but is not relevant to the user at all to begin with. For
example, software documentation in an unknown language can be of good quality
but still be irrelevant to a non-speaker.
## 3 Impact of Waste Data
In the physical reality, waste has adverse impact on the environment of the
ecosystem. The presence of waste pollutes the ecosystem, causing economic,
social, and operational impacts. We argue that in the same manner, waste data
affects a computing system by consuming resources without providing value, and
by degrading system performance and components. Storage, processing, and
transfer of data require the use of system resources such as disk space, cpu
cycles, and network or I/O bandwidth. Disposing of the waste data by deletion
also impacts the life of storage devices and incurs energy and time overheads.
Storage Consumption. Waste data consumes a lot of storage space. For example,
creating and editing a small text file in vi causes a temporary swap file to
be created. To illustrate the amount of temporary waste data created by source
code compilation, we compiled Openssl 1.0.0a on a Linux workstation.
Compilation of Openssl produces about 13.6 MB of target binary code. However,
it also produces about 44.5 MB of temporary object code that is not part of
the installation. From the viewpoint of the user, these temporary object files
produced during compilation are waste data. Such unwanted data consumes a lot
of space and needs to be deleted.
The overall amount of unused and dead data in a given system is not small. We
wrote a Perl script to determine the percentage of files that have never been
accessed since last modification. We ran the script on three different
platforms – an Apple MacBook used as a personal laptop, a Ubuntu Linux
desktop, and a student lab server running Fedora Linux. The results are show
on Table 1.
Platform | MacBook | Desktop | Server
---|---|---|---
% of files | 20.6 | 47.4 | 57.1
% of used space | 98.5 | 38.1 | 99.5
Table 1: Analysis of files in a MacBook, a desktop workstation, and a student
lab server. In all cases, a large number (20.6%–57.1%) of files have never
been accessed since last modification.
In all three cases, a large fraction of files in the system have never been
accessed since last modification, reflecting the results from previous work in
the area [4, 13]. In terms of space usage, these files amounted from 38% to as
high as 99% of the total used space on the machines. This shows that the
amount of waste data in a system is quite significant.
Reducing Device Lifetime. Disposal of waste data via deletion can impact the
lifetime of storage devices. For example, Flash based storage devices
typically have a maximum number of write cycles. Multi-Level Cell (MLC) flash
devices support a maximum of 1,000–10,000 write/erase cycles per cell while
Single-Level Cell (SLC) flash devices support up to 100,000 write-erase cycles
per cell [16]. Waste or by-product data brings no value, but uses up flash
storage write cycles, reducing the lifetime of such storage devices. As flash-
based solid state storage becomes popular, especially in mobile devices, we
need to ensure that waste data write / erase cycles do not impact the lifetime
of such storage.
Performance Degradation. Presence of unwanted waste data can degrade system
performance. For example, in a file system, the extra storage space consumed
by waste data may cause unnecessary fragmentation and use up available inodes.
If waste data can be identified and not stored by the system, we can reduce
the load and fragmentation greatly. For example, Table 1 shows that 98.5% of
the total space used by files in the laptop was actually consumed by files
that were never accessed since last modification. By storing these files
separate from the frequently accessed files, we can vastly improve system
performance. Deletion also takes up CPU cycles and consumes energy – a fact
which is significant in low-powered mobile devices.
## 4 Managing Waste Data
How do we deal with waste data? Storage and deletion of waste data is costly
in terms of energy and space consumed. Therefore, we need effective strategies
to handle waste data. To provide a guideline for waste data management, we
turned to the techniques used in real life waste management. We argue in this
paper that these lessons from real life waste management are equally effective
in managing waste data in digital environments. We start our discussion by
presenting a hierarchy of waste data management methods. Then we discuss some
specific approaches application designers and system architects can adopt to
minimize the impact of waste data.
Figure 1: The Waste Data Management Hierarchy. Processes at the top are more
preferable.
### 4.1 Waste data hierarchy
In dealing with waste in the natural environment, a waste hierarchy is widely
used to classify and organize waste management schemes according to their
usefulness and impact [17]. We propose adapting the waste hierarchy from real
life waste management to develop guidelines for choosing waste data management
schemes. Besides the “three R’s” (reduce, reuse, recycle), we use the
additional steps of recovery and disposal in our scheme, leveraging the five-
step waste management hierarchy described in [1]. We show the waste data
hierarchy in Figure 1, and describe the steps below:
Reduce. At the top of the waste data hierarchy is reduction of waste, which
refers to reducing the amount of waste data generated in the system. We opine
that this is the most favorable option, since less waste will cause the least
overhead on the system. Applications should be designed with waste-reduction
in mind and only store the desired output data in the disk. In-memory caching
of temporary values and content-based addressing can help reduce the amount of
waste data produced by applications. Operating systems and file systems can
provide incentives to applications that produce less waste data and
punishments to those that produce a lot of waste data (we discuss this later
in this section).
Reuse. In the next layer, we have reuse of waste, which refers to reusing the
waste data for other purposes. Schemes that can be classified as data reuse
include data deduplication [19], where the content of waste data objects can
be used by the deduplication scheme to achieve better compression ratios.
Another example is the reuse of translation memories in machine translation,
where the information from one translation session can be used to enrich
global translation capabilities. Google’s Translation Toolkit already allows
this type of data reuse, and it has been used successfully for translation of
English Wikipedia articles into African and South Asian languages [6]. For
degraded data, regeneration [18] or restoration [14] can be used to recover
data quality.
Recycle. Slightly less preferable than reuse is to recycle waste data, where
data objects can be broken up and used for different objects. While it is
difficult to fathom what it means to recycle application specific data for
other purposes, we can definitely recycle waste containing application code.
When an obsolete software package is going to be removed, we can extract the
usable components from it and use them for other applications.
Recover. Sometimes, the waste data cannot be recycled or reused. A
possibility of still gaining some utility from such data is to recover
information. For example, used log files can be anonymized and shared or
analyzed for getting high-level views. Obsolete data can also be mined to
gather patterns about historical trends.
Dispose. At the bottom of the hierarchy sits schemes for Disposal of data,
through deletion. This is costly in terms of time and energy spent deleting
data objects. So, we opine that deletion should be the absolute last recourse
in managing waste data.
Above the hierarchy is the ideal state of zero waste, where careful system
design results in production of no waste data. Below this hierarchy of schemes
for waste data management lies another approach not shown in the hierarchy –
physical elimination. Sometimes, the five schemes may not be enough or
feasible for managing waste data. For example, the data may be stored in
physically immutable media, and hence not subject to any of the above schemes.
Also, security issues and regulations may require physical elimination of the
storage media. This can be achieved by incinerating, degaussing, or destroying
the storage media. However, this has the worst impact on the natural
environment as any such disposal would impact the physical ecosystem.
Next, we discuss some specific strategies and best practices for waste data
management.
### 4.2 Some schemes for managing waste data
Taking a cue from successful real life waste management strategies, we propose
several schemes for managing waste data in digital ecosystems. For this, we
leverage the concepts of waste hierarchy and extended producer responsibility
[7].
Digital landfills. A digital landfill is the equivalent of real-life
landfills, where unwanted data can be disposed of without additional cost
associated with deletion. For this, we propose using a semi-volatile storage
device. Such a storage device would store data, but gradually unwanted data
objects will fade automatically and the storage space can be reclaimed. This
type of device can be implemented on a volatile storage medium using a least-
recently-used (LRU) scheme, where data which has not been used recently is
allowed to fade, while more frequently used data is refreshed.
Waste penalties for applications. A waste penalty can be imposed on
applications that create large amounts of waste data. For example, the
operating system can penalize an application that creates a lot of temporary
files by reducing its I/O bandwidth or schedule it to receive fewer CPU
cycles. This gives applications incentives to act responsibly in creating
waste data. This concept is equivalent to the Pay-as-You-Throw scheme and the
polluter-pays principle used in real life waste management [5].
Extensive system-wide Deduplication and Micro-modular software. A big problem
with recycling old or unwanted software is that software libraries are not
usually written to allow extraction of small amounts of code. Shared dynamic
link libraries do allow code sharing among multiple applications [8]. However,
they do not allow removal of unused routines to retain only the routines that
are used. When recycling a library, it is therefore not possible to extract
usable routines from it. To allow recycling old code, we propose breaking up
software code libraries into micro-modules in the level of individual routines
or algorithms, which can be extracted from the library when recycling old
code.
## 5 Related Work
While researchers have explored different storage management issues, the
systematic management of waste data has received little interest. Information
lifecycle management (ILM) has been used by the storage industry to determine
optimal management of data objects throughout their lifecycle [12]. A major
challenge in ILM is to design valuation schemes to determine the importance of
information. Chen presented such a scheme based on file access patterns [3].
We can use such schemes to identify waste data. Zadok et al. [18] advocated
for the need to reduce storage consumption through application of regeneration
and smart space reclamation policies, in order to increase device lifetimes
and available storage. Researchers have also analyzed existing systems to
identify typical usage patterns. An early work by Satyanarayanan [13]
introduced the notion of functional lifetime (f-lifetime) for files, defined
as the difference between a file’s age and the time since its last access.
Files with lower f-lifetimes are less useful, since the gap between their
creation/last modification times and last access times are short. In a study
of file systems, Douceur et al. [4] showed that 44% of the files in the
studied systems had an f-lifetime of zero (the percentage was higher, at 67%,
for technical support systems), indicating that these files have not been
accessed at all since last modification. This agrees with our findings
presented in Section 3. Vogels found that file lifetimes are often quite short
– almost 80% files are actually deleted within 4 seconds [15]. The very short
lifetime indicates that the usefulness of these files ends quickly. Deleting
these files is costly, and application designers should rethink their I/O to
prevent the creation of such waste data. Finally, researchers have developed
techniques such as software refactoring and reuse [8], and data deduplication
[19], which can be applied in different stages of the waste data hierarchy to
reduce the impact of waste data.
## 6 Conclusion
For many years, the abundance of storage space and decreasing storage costs
have allowed us to ignore the adverse impact of waste data on our digital
ecosystems. But as we start dealing with massive quantities of data, we need
to manage waste data in order to reduce overheads and energy costs, and
improve efficiency. In this paper, we defined the waste data problem and
proposed using techniques from real life waste management to minimize the
impact of waste data on our computing environment. Our waste data management
hierarchy can be used to determine the preferable option in dealing with waste
data in different applications. We also advocated the adoption of responsible
application behavior and best practices in reducing the impact of waste data.
We posit that software engineering techniques as well as hardware
architectures will need to be adapted with waste data minimization,
management, and recycling in mind, in order to build a efficient and
sustainable digital ecosystem.
## Acknowledgements
This work was supported by the National Science Foundation under Grant
#0937060 to the Computing Research Association for the CIFellows Project.
## References
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|
arxiv-papers
| 2011-06-29T20:53:02 |
2024-09-04T02:49:20.190098
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ragib Hasan and Randal Burns",
"submitter": "Ragib Hasan",
"url": "https://arxiv.org/abs/1106.6062"
}
|
1106.6105
|
# Classification of general $n$-qubit states under stochastic local operations
and classical communication in terms of the rank of coefficient matrix
Xiangrong Li1, Dafa Li2,3 1Department of Mathematics, University of California
Irvine, Irvine, California 92697, USA
2Department of Mathematical Sciences, Tsinghua University, Beijing, 100084,
China
3Center for Quantum Information Science and Technology, Tsinghua National
Laboratory for Information
Science and Technology (TNList), Beijing, 100084, China 1 Department of
Mathematics, University of California, Irvine, CA 92697-3875, USA
2 Department of mathematical sciences, Tsinghua University, Beijing 100084
CHINA
###### Abstract
We solve the entanglement classification under stochastic local operations and
classical communication (SLOCC) for general $n$-qubit states. For two
arbitrary pure $n$-qubit states connected via local operations, we establish
an equation between the two coefficient matrices associated with the states.
The rank of the coefficient matrix is preserved under SLOCC and gives rise to
a simple way of partitioning all the pure states of $n$ qubits into different
families of entanglement classes, as exemplified here. When applied to the
symmetric states, this approach reveals that all the Dicke states
$|\ell,n\rangle$ with $\ell=1,\dots,[n/2]$ are inequivalent under SLOCC.
## I I. Introduction
Entanglement, a key feature that distinguishes quantum information from
classical information, has applications in cryptography, teleportation and
quantum computation Nielsen . While the bipartite entanglement is well
understood, the task of classifying multipartite entanglement beyond two
qubits becomes increasingly difficult. To classify entangled states, some
equivalence relation has to be introduced. Of particular importance is the
equivalence under stochastic local operations and classical communication
(SLOCC) since two states belonging to the same equivalence class can perform
the same tasks of quantum information theory.
For three qubits, in terms of the local ranks of the reduced density matrices,
it has been shown that there are six inequivalent SLOCC classes Dur . For four
or more qubits, there exists an infinite number of inequivalent SLOCC classes.
It is highly desirable to partition the infinite SLOCC classes into a finite
number of families such that states belonging to the same family possess
similar properties, according to some criteria for determining which family a
given state belongs to. Considerable efforts have been undertaken over the
last decade for the SLOCC entanglement classification of four-qubit states
resulting in a finite number of families Moor2 ; Chterental ; Lamata07 ;
Borsten or classes Cao ; LDF07b ; LDFQIC ; Buniy ; Viehmann ; Zha . For more
than four qubits, a few attempts have been made for SLOCC classification for
subsets of the general $n$-qubit states such as the Greenberger-Horne-
Zeilinger (GHZ)-type, W-type, and GHZ-W-type $n$-qubit states Chen , symmetric
$n$-qubit states LDFEPL ; Bastin ; Aulbach , even $n$-qubit states LDFJPA ;
LDFJPA12 , and odd $n$-qubit states LDFQIC11 . Despite these efforts, a SLOCC
classification for general $n$-qubit states is still beyond reach.
Our aim is to solve the problem of SLOCC classification for all multipartite
pure states in the general $n$-qubit case. To this end, we demonstrate that
the rank of the coefficient matrix of a pure $n$-qubit state is invariant
under SLOCC. SLOCC invariants for subsets of $n$-qubit states have been the
subject of several recent studies LDF07a ; LDFJPA ; LDFJPA12 ; LDFQIC11 ;
Ribeiro . In LDFJPA ; LDFJPA12 , the invariant element is the determinant of
coefficient matrices of even $n$ qubits. In LDFQIC11 , the invariant element
is the rank of square matrices of size two constructed using three functions
defined on the space of odd $n$ qubits.
We construct the coefficient matrices for general $n$-qubit states by
arranging the coefficients in lexicographical order. For two states connected
via local operations, their coefficient matrices are related through an
equation. In the case where the local operations are invertible, the two
states are said to be SLOCC equivalent and the two coefficient matrices have
the same rank; i.e., the rank is preserved under SLOCC. The rank gives rise to
a simple way of partitioning all the $n$-qubit states into different SLOCC
families. For $n$-qubit symmetric Dicke states $|\ell,n\rangle$ with $\ell$
($\ell=1,\dots,[n/2]$) excitations, we show that the rank of the coefficient
matrix of $|\ell,n\rangle$ is equal to $\ell+1$ and, therefore, all these
states are inequivalent under SLOCC. Finally, composing the rank and
permutations of qubits allows us to define subfamilies by cutting each family
in pieces.
## II II. SLOCC matrix equation and the invariance of the rank
We write the state $|\psi^{\prime}\rangle$ of $n$ qubits as
$|\psi^{\prime}\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$, where $|i\rangle$
are basis states and $a_{i}$ are coefficients. We associate to an $n$-qubit
state $|\psi^{\prime}\rangle$ a $2^{[n/2]}\times 2^{[(n+1)/2]}$ coefficient
matrix $M(|\psi^{\prime}\rangle)$ whose entries are the coefficients
$a_{0},a_{1},\dots,a_{2^{n}-1}$ arranged in ascending lexicographical order.
To illustrate, we list below $M(|\psi^{\prime}\rangle)$ for $n=3$:
$M(|\psi^{\prime}\rangle)=\left(\begin{array}[]{cccc}a_{0}&a_{1}&a_{2}&a_{3}\\\
a_{4}&a_{5}&a_{6}&a_{7}\end{array}\right),$ (1)
and for $n=4$:
$M(|\psi^{\prime}\rangle)=\left(\begin{array}[]{cccc}a_{0}&a_{1}&a_{2}&a_{3}\\\
a_{4}&a_{5}&a_{6}&a_{7}\\\ a_{8}&a_{9}&a_{10}&a_{11}\\\
a_{12}&a_{13}&a_{14}&a_{15}\end{array}\right).$ (2)
We refer to the rank of the coefficient matrix $M(|\psi^{\prime}\rangle)$ as
the rank of the state $|\psi^{\prime}\rangle$, denoted as
rank$(|\psi^{\prime}\rangle)$. We exemplify with the $n$-qubit
$|\mbox{GHZ}\rangle$ state $\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
n}+|1\rangle^{\otimes n})$, and we find rank$(|\mbox{GHZ}\rangle)=2$. It is
clear that the rank of any $n$-qubit state ranges over the values 1, 2,
$\dots$, $2^{[n/2]}$.
Theorem. Let $|\psi\rangle$ be another state of $n$ qubits with
$|\psi\rangle=\sum_{i=0}^{2^{n}-1}b_{i}|i\rangle$ and $M(|\psi\rangle)$ be the
corresponding coefficient matrix constructed in the same manner as was done
for $M(|\psi^{\prime}\rangle)$. If the states $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ are related by
$|\psi^{\prime}\rangle=\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\mbox{\ldots}\otimes\mathcal{A}_{n}|\psi\rangle,$
(3)
where the local operators $\mathcal{A}_{1}$, $\mathcal{A}_{2},\dots$, and
$\mathcal{A}_{n}$ are not necessarily invertible, then the following matrix
equation holds for general $n$ qubits:
$M(|\psi^{\prime}\rangle)=(\mathcal{A}_{1}\otimes\mbox{\ldots}\otimes\mathcal{A}_{[n/2]})M(|\psi\rangle)(\mathcal{A}_{[n/2]+1}\otimes\mbox{\ldots}\otimes\mathcal{A}_{n})^{T}.$
(4)
Equation (4) holds particularly true for two SLOCC equivalent states
$|\psi\rangle$ and $|\psi^{\prime}\rangle$ which satisfy Eq. (3) along with
the local operators $\mathcal{A}_{1}$, $\mathcal{A}_{2},\dots$, and
$\mathcal{A}_{n}$ being invertible Dur . It follows from Eq. (4) that two
SLOCC equivalent states have the same rank, in other words, the rank is
invariant under SLOCC, thereby revealing that the rank is an inherent physical
property. Therefore, if two states differ in their ranks, then they belong
necessarily to different SLOCC equivalent classes. It should be noted that the
converse does not hold; i.e., two states with the same rank are not
necessarily equivalent.
We have the following two simple results: (i) The rank of a full separable
state is always 1. (ii) The rank of a genuinely entangled state is always
greater than 1.
Remark 1. Taking the determinants of both sides of Eq. (4) for even $n$ yields
LDFJPA ; LDFJPA12 :
$\det M(|\psi^{\prime}\rangle)=\det
M(|\psi\rangle)[\det(\mathcal{A}_{1})\dots\det(\mathcal{A}_{n})]^{2^{(n-2)/2}}.$
(5)
It follows from Eq. (5) that if one of $\det M(|\psi^{\prime}\rangle)$ and
$\det M(|\psi\rangle)$ vanishes while the other does not, then the state
$|\psi^{\prime}\rangle$ is not equivalent to $|\psi\rangle$ under SLOCC. In
view of the fact that the determinant of a matrix is nonvanishing if and only
if it has full rank, the SLOCC invariance of the rank is stronger than the
invariance of the determinant.
## III III. SLOCC classification in terms of the rank
We define the family $\mathcal{F}_{n,r}$ to be the set of all $n$-qubit states
with the same rank $r$. In the sequel, we will omit the subscript $n$ and
simply write $\mathcal{F}_{r}$, whenever the number of qubits is clear from
the context. Thus, there exist $2^{[n/2]}$ different SLOCC families for any
$n$ qubits. Clearly, if two states are SLOCC equivalent then they belong to
the same family. However, the converse does not hold: two states belonging to
the same family may be inequivalent under SLOCC. It is further noted that when
$n\geq 4$, at least one family contains an infinite number of SLOCC classes.
For any $n$ qubits, the following hold: (i) Family $\mathcal{F}_{1}$ contains
all the full separable states. (ii) Family $\mathcal{F}_{1}$ contains no
genuine entangled states. (iii) Family $\mathcal{F}_{1}$ contains finite SLOCC
classes. (iv) Family $\mathcal{F}_{2}$ contains the $n$-qubit
$|\mbox{GHZ}\rangle$ state. (v) Family $\mathcal{F}_{2+r}$ contains the
following state:
$\frac{1}{\sqrt{r+2}}(|0\rangle-|2^{n}-1\rangle+\sum_{k=1}^{r}|k(2^{[(n+1)/2]}+1)\rangle),$
(6)
where $1\leq r\leq 2^{[n/2]}-2$.
Now we turn to the $n$-qubit symmetric Dicke states $|\ell,n\rangle$ with
$\ell$ excitations Stockton :
$|\ell,n\rangle=\left({}_{\ell}^{n}\right)^{-1/2}\sum\limits_{k}P_{k}|1_{1},1_{2},\dots,1_{\ell},0_{\ell+1},\dots,0_{n}\rangle,$
(7)
where $\ell$ ranges from 1 to $n-1$ and $\\{P_{k}\\}$ is the set of all
distinct permutations of the spins. These states have been featured in
theoretical studies Toth ; Huber and implemented experimentally Prevedel ;
Wieczorek . The Dicke state $|1,n\rangle$ is just the $n$-qubit $|W\rangle$
state and $|\ell,n\rangle$ is equivalent to $|n-\ell,n\rangle$ under SLOCC.
As has been previously noted, all symmetric Dicke states $|\ell,n\rangle$ with
$\ell=1,\dots,[n/2]$ are SLOCC inequivalent LDFEPL ; Bastin . These states, as
demonstrated below, can also be distinguished by the rank of their coefficient
matrices which depends only on the number of excitations and is independent of
the number of qubits. Since rank$(|\ell,n\rangle)=$rank$(|n-\ell,n\rangle)$,
we only need to compute rank$(|\ell,n\rangle)$ with $1\leq\ell\leq[n/2]$. This
can be done as follows. We first construct the coefficient matrix
$M(|\psi^{\prime}\rangle)$ of state $|\psi^{\prime}\rangle$ in the same manner
as discussed above. We may write $|\psi^{\prime}\rangle$ in terms of an
orthogonal basis as $|\psi^{\prime}\rangle=\sum a_{i_{1}i_{2}\dots
i_{n}}|i_{1}i_{2}\dots i_{n}\rangle$, where $i_{1}i_{2}\dots i_{n}$ is the
$n$-bit binary form of the index $i$. Inspection of the structure of the
matrix $M(|\psi^{\prime}\rangle)$ reveals that the coefficient $a_{i_{1}\dots
i_{[n/2]}i_{[n/2]+1}\dots i_{n}}$ is the entry in the $(i_{1}\dots
i_{[n/2]})$th row and $(i_{[n/2]+1}\dots i_{n})$th column of the matrix. Here,
the $n$ bits are split into two halves, referred to as the row bits and column
bits, respectively: bits $1$ to ${[n/2]}$ are used to specify the row number,
and bits $[n/2]+1$ to $n$ are used to specify the column number. In view of
Eq. (7), the nonzero entries of the coefficient matrix $M(|\ell,n\rangle)$ are
those whose $n$-bit binary forms have $\ell$ bits equal to $1$ and the rest of
the bits equal to $0$. We further observe that the rows of $M(|\ell,n\rangle)$
with no more than $\ell$ row bits equal to 1 are nonzero rows, while the
remaining rows are identically zero. Consider the rows with $j$ ($0\leq
j\leq\ell$) row bits equal to 1. Clearly, there are $\binom{n/2}{j}$ such rows
of $M(|\ell,n\rangle)$ that are identical. Letting $j$ vary from 0 to $\ell$
gives a total of $\ell+1$ different rows. It can be verified that these
$\ell+1$ rows are independent. This yields $rank(|\ell,n\rangle)=\ell+1$.
Accordingly, for any $n$ qubits, all the Dicke states $|\ell,n\rangle$ with
$\ell=1$, $\dots$, $[n/2]$, are inequivalent under SLOCC, since they differ in
their ranks. Further, it can readily be seen that the Dicke state
$|\ell,n\rangle$ with $\ell=1$, $\dots$, $[n/2]$ belongs to the family
$\mathcal{F}_{\ell+1}$. This gives rise to a complete SLOCC classification of
all the symmetric Dicke states.
Remark 2. It follows from the discussion above that both the states
$|W\rangle$ and $|\mbox{GHZ}\rangle$ have the same rank, thereby revealing
that the two states have a similar algebraic structure. It is further noted
that both the states $|W\rangle$ and $|\mbox{GHZ}\rangle$ admit a similar
Frobenius algebra structure Coecke .
## IV IV. Ranks of coefficient matrices under permutations of qubits
In LDFJPA ; LDFJPA12 , we presented a systematic way to find all the possible
coefficient matrices for even $n$-qubit states such that the determinants of
these coefficient matrices are invariant under SLOCC. Here we extend this
construction to general $n$ qubits. Observe that to write a $2^{[n/2]}\times
2^{[(n+1)/2]}$ matrix into binary index form, we need $[n/2]$ row bits and
$[(n+1)/2]$ column bits. In the binary form of the coefficient matrix given in
Eqs. (1) and (2), bits $1$ to ${[n/2]}$ are the row bits, and bits $[n/2]+1$
to $n$ are the column bits. Alternatively, we may choose any $[n/2]$ bits as
the row bits and the remaining $[(n+1)/2]$ bits as the column bits. This
amounts to $(\frac{1}{2})^{n+1\bmod{2}}\binom{n}{[n/2]}$ different coefficient
matrices, ignoring those that end up exchanging rows or columns. The factor of
$1/2$ for even $n$ arises because exchanging the row and column bits is
equivalent to transposing the matrix. It turns out that the ranks of these
coefficient matrices are all invariant under SLOCC. To see this, we will
resort to permutations of qubits. Let $\sigma$ be a permutation of qubits
given by LDFJPA12
$\sigma=(q_{1},t_{1})(q_{2},t_{2})\dots(q_{k},t_{k}),$ (8)
where $(q_{i},t_{i})$ is the transposition of a pair of qubits $q_{i}$ and
$t_{i}$ with $q_{i}$ being a row bit and $t_{i}$ a column bit. Exhausting all
possible values of $q_{1},\dots,q_{k}$ and $t_{1},\dots,t_{k}$ such that
$1\leq q_{1}<q_{2}<\mbox{\ldots}<q_{k}<[(n+1)/2]$,
$[n/2]<t_{1}<t_{2}<\mbox{\ldots}<t_{k}\leq n$, and letting $k$ vary from 0 to
$[(n-1)/2]$ (we define $\sigma=I$ for $k=0$), yields
$(\frac{1}{2})^{n+1\bmod{2}}\binom{n}{[n/2]}$ different permutations of
qubits. Let $M^{\sigma}(|\psi^{\prime}\rangle)$ denote the coefficient matrix
of the state $|\psi^{\prime}\rangle$ under permutation $\sigma$, and let
rank${}^{\sigma}(|\psi^{\prime}\rangle)$ denote its rank. We may refer to
rank${}^{\sigma}(|\psi^{\prime}\rangle)$ as the rank of the state
$|\psi^{\prime}\rangle$ under permutation $\sigma$. Simply taking the
permutation $\sigma$ to both sides of Eq. (4) yields the following SLOCC
matrix equation:
$\displaystyle M^{\sigma}(|\psi^{\prime}\rangle)$ $\displaystyle=$
$\displaystyle(\mathcal{A}_{\sigma(1)}\otimes\mbox{\ldots}\otimes\mathcal{A}_{\sigma([n/2])})M^{\sigma}(|\psi\rangle)$
(9)
$\displaystyle\quad\quad(\mathcal{A}_{\sigma([n/2]+1)}\otimes\mbox{\ldots}\otimes\mathcal{A}_{\sigma(n)})^{T}.$
It follows immediately from Eq. (9) that two SLOCC equivalent states have the
same rank with respect to the permutation $\sigma$. That is, the rank with
respect to the permutation $\sigma$ is invariant under SLOCC. Conversely, if
two states differ in their ranks with respect to the permutation $\sigma$,
then they belong necessarily to different SLOCC classes. We define the family
$\mathcal{F}_{r}^{\sigma}$ to be the set of all $n$-qubit states with the same
rank $r$ with respect to the permutation $\sigma$, where $r$ ranges from 1 to
$2^{[n/2]}$ and we have omitted a subscript $n$. Suppose
$\sigma_{1},\sigma_{2},\dots,\sigma_{m}$ with
$m\leq(\frac{1}{2})^{n+1\bmod{2}}\binom{n}{[n/2]}$ is a sequence of
permutations of the form given in Eq. (8). In terms of the rank of
$M^{\sigma_{1}}$, the $n$-qubit states are divided into $2^{[n/2]}$ families:
$\mathcal{F}_{r_{1}}^{\sigma_{1}}$. Then, in terms of the rank of
$M^{\sigma_{2}}$, each family $\mathcal{F}_{r_{1}}^{\sigma_{1}}$ can be
further divided into $2^{[n/2]}$ subfamilies:
$\mathcal{F}_{r_{1},r_{2}}^{\sigma_{1}\sigma_{2}}=\mathcal{F}_{r_{1}}^{\sigma_{1}}\cap\mathcal{F}_{r_{2}}^{\sigma_{2}}$.
Here, each subfamily $\mathcal{F}_{r_{1},r_{2}}^{\sigma_{1}\sigma_{2}}$ is the
intersection of the families $\mathcal{F}_{r_{1}}^{\sigma_{1}}$ and
$\mathcal{F}_{r_{2}}^{\sigma_{2}}$. Assume that in terms of the ranks of
$M^{\sigma_{1}},M^{\sigma_{2}},\dots,M^{\sigma_{m-1}}$, the $n$-qubit states
are divided into $2^{(m-1)[n/2]}$ families:
$\mathcal{F}_{r_{1},r_{2},\dots,r_{m-1}}^{\sigma_{1}\sigma_{2}\dots\sigma_{m-1}}=\mathcal{F}_{r_{1}}^{\sigma_{1}}\cap\mbox{\ldots}\cap\mathcal{F}_{r_{m-1}}^{\sigma_{m-1}}$.
Then, in terms of the rank of $M^{\sigma_{m}}$, each family
$\mathcal{F}_{r_{1},r_{2},\dots,r_{m-1}}^{\sigma_{1}\sigma_{2}\dots\sigma_{m-1}}$
can be further divided into $2^{[n/2]}$ subfamilies:
$\mathcal{F}_{r_{1},r_{2},\dots,r_{m}}^{\sigma_{1}\sigma_{2}\dots\sigma_{m}}=\mathcal{F}_{r_{1},r_{2},\dots,r_{m-1}}^{\sigma_{1}\sigma_{2}\dots\sigma_{m-1}}\cap\mathcal{F}_{r_{m}}^{\sigma_{m}}=\mathcal{F}_{r_{1}}^{\sigma_{1}}\cap\mbox{\ldots}\cap\mathcal{F}_{r_{m}}^{\sigma_{m}}$.
This gives a total of $2^{m[n/2]}$ different SLOCC families.
We exemplify with the family
$L_{a_{2}b_{2}}=a(|0000\rangle+|1111\rangle)+b(|0101\rangle+|1010\rangle)+|0011\rangle+|0110\rangle$
for four qubits presented by Verstraete _et al._ Moor2 . As shown in Table 1,
the family $L_{a_{2}b_{2}}$ is further divided into four subfamilies (all
other subfamilies are empty) with respect to permutations $\sigma_{1}=I$ and
$\sigma_{2}=(1,4)$: $\mathcal{F}_{2,1}^{\sigma_{1}\sigma_{2}}$,
$\mathcal{F}_{3,3}^{\sigma_{1}\sigma_{2}}$, and
$\mathcal{F}_{4,2}^{\sigma_{1}\sigma_{2}}$ contain only a single SLOCC class,
while $\mathcal{F}_{4,3}^{\sigma_{1}\sigma_{2}}$ contains an infinite number
of SLOCC classes. In a similar fashion, we can further divide other families
presented by Verstraete _et al._ Moor2 into subfamilies.
Table 1: SLOCC classification of $L_{a_{2}b_{2}}$ $\mathcal{F}_{1}^{\sigma_{1}}$ | $\mathcal{F}_{2}^{\sigma_{1}}$ | $\mathcal{F}_{3}^{\sigma_{1}}$ | $\mathcal{F}_{4}^{\sigma_{1}}$
---|---|---|---
$\emptyset$ | $a=b=0$ | $ab=0$ & $a\neq b$ | $ab\neq 0$
$\mathcal{F}_{1}^{\sigma_{2}}$ | $\mathcal{F}_{2}^{\sigma_{2}}$ | $\mathcal{F}_{3}^{\sigma_{2}}$ | $\mathcal{F}_{4}^{\sigma_{2}}$
$a=b=0$ | $a=\pm b$ & $a\neq 0$ | $a\neq\pm b$ | $\emptyset$
$\mathcal{F}_{2,1}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{3,3}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{4,2}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{4,3}^{\sigma_{1}\sigma_{2}}$
$a=b=0$ | $ab=0$ & $a\neq b$ | $a=\pm b$ & $a\neq 0$ | $ab\neq 0$ & $a\neq\pm b$
Consider the family
span$\\{0_{k}\Psi,0_{k}\Psi\\}=|0000\rangle+|1100\rangle+\alpha|0011\rangle+\beta|1111\rangle$
for four qubits presented by Lamata _et al._ Lamata07 . As shown in Table 2,
the family $\\{0_{k}\Psi,0_{k}\Psi\\}$ is further divided into four
subfamilies (all other subfamilies are empty) with respect to permutations
$\sigma_{1}=I$ and $\sigma_{2}=(1,4)$:
$\mathcal{F}_{1,2}^{\sigma_{1}\sigma_{2}}$,
$\mathcal{F}_{1,4}^{\sigma_{1}\sigma_{2}}$, and
$\mathcal{F}_{2,3}^{\sigma_{1}\sigma_{2}}$ contain only a single SLOCC class,
while $\mathcal{F}_{2,4}^{\sigma_{1}\sigma_{2}}$ contains an infinite number
of SLOCC classes. In a similar way, other families presented by Lamata _et
al._ Lamata07 can also be further divided into subfamilies.
Table 2: SLOCC classification of span$\\{0_{k}\Psi,0_{k}\Psi\\}$ $\mathcal{F}_{1}^{\sigma_{1}}$ | $\mathcal{F}_{2}^{\sigma_{1}}$ | $\mathcal{F}_{3}^{\sigma_{1}}$ | $\mathcal{F}_{4}^{\sigma_{1}}$
---|---|---|---
$\alpha=\beta$ | $\alpha\neq\beta$ | $\emptyset$ | $\emptyset$
$\mathcal{F}_{1}^{\sigma_{2}}$ | $\mathcal{F}_{2}^{\sigma_{2}}$ | $\mathcal{F}_{3}^{\sigma_{2}}$ | $\mathcal{F}_{4}^{\sigma_{2}}$
$\emptyset$ | $\alpha=\beta=0$ | $\alpha\beta=0$ & $\alpha\neq\beta$ | $\alpha\beta\neq 0$
$\mathcal{F}_{1,2}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{1,4}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{2,3}^{\sigma_{1}\sigma_{2}}$ | $\mathcal{F}_{2,4}^{\sigma_{1}\sigma_{2}}$
$\alpha=\beta=0$ | $\alpha=\beta\neq 0$ | $\alpha\beta=0$ & $\alpha\neq\beta$ | $\alpha\beta\neq 0$ & $\alpha\neq\beta$
By using the filters, it has been shown that four five-qubit states
$|\Psi_{2}\rangle$, $|\Psi_{4}\rangle$, $|\Psi_{5}\rangle$, and
$|\Psi_{6}\rangle$ are in different orbits Osterloh . Letting $\sigma_{1}=I$,
$\sigma_{2}=(1,5)$, and $\sigma_{3}=(1,3)$, it can be shown that the above
four states belong to the families
$\mathcal{F}_{2,2,2}^{\sigma_{1}\sigma_{2}\sigma_{3}}$,
$\mathcal{F}_{3,3,3}^{\sigma_{1}\sigma_{2}\sigma_{3}}$,
$\mathcal{F}_{2,4,2}^{\sigma_{1}\sigma_{2}\sigma_{3}}$ and
$\mathcal{F}_{2,4,4}^{\sigma_{1}\sigma_{2}\sigma_{3}}$, respectively.
Therefore, these five-qubit states can also be distinguished by ranks.
Furthermore, it has been shown that five six-qubit states $|\Xi_{2}\rangle$,
$|\Xi_{4}\rangle$, $|\Xi_{5}\rangle$, $|\Xi_{6}\rangle$, and $|\Xi_{7}\rangle$
are distinguished by the six-qubit filters Osterloh . Letting $\sigma_{1}=I$,
$\sigma_{2}=(1,4)$, and $\sigma_{3}=(1,5)$, it can be shown that the above
five states belong to the families
$\mathcal{F}_{2,2,2}^{\sigma_{1}\sigma_{2}\sigma_{3}}$,
$\mathcal{F}_{2,2,4}^{\sigma_{1}\sigma_{2}\sigma_{3}}$,
$\mathcal{F}_{2,4,4}^{\sigma_{1}\sigma_{2}\sigma_{3}}$,
$\mathcal{F}_{3,4,4}^{\sigma_{1}\sigma_{2}\sigma_{3}}$, and
$\mathcal{F}_{3,3,3}^{\sigma_{1}\sigma_{2}\sigma_{3}}$, respectively.
Therefore, these six-qubit states can also be distinguished by ranks.
## V V. Discussion and summary
Chterental _et al._ (see Remark 3.5 in Chterental ) stated that the family
$L_{ab_{3}}$ is equivalent to a subfamily of $L_{abc_{2}}$ obtained by setting
$a=c$, where $L_{ab_{3}}$ and $L_{abc_{2}}$ are given by Moor2
$\displaystyle L_{ab_{3}}$ $\displaystyle=$ $\displaystyle
a(|0000\rangle+|1111\rangle)+\frac{a+b}{2}(|0101\rangle+|1010\rangle)$ (10)
$\displaystyle+\frac{a-b}{2}(|0110\rangle+|1001\rangle)+\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle$
$\displaystyle+|0111\rangle+|1011\rangle),$ $\displaystyle L_{abc_{2}}$
$\displaystyle=$
$\displaystyle\frac{a+b}{2}(|0000\rangle+|1111\rangle)+\frac{a-b}{2}(|0011\rangle+|1100\rangle)$
(11) $\displaystyle+c(|0101\rangle+|1010\rangle)+|0110\rangle.$
In terms of the rank, the families $L_{ab_{3}}$ and $L_{abc_{2}}$ with $a=c$
are both divided into four subfamilies, see Table 3. As can be seen, the
subfamily $\mathcal{F}_{2}$ of $L_{ab_{3}}$ is a single class with
representative
$\frac{i}{\sqrt{2}}(|0001\rangle+|0010\rangle+|0111\rangle+|1011\rangle)$,
whereas the subfamily $\mathcal{F}_{2}$ of $L_{abc_{2}}(a=c)$ is a single
class with representative
$\frac{b}{2}(|0000\rangle+|1111\rangle-|0011\rangle-|1100\rangle)+|0110\rangle$.
In light of Theorem 1 in LDF07a , the two representative states are not
equivalent to each other. This reveals that $L_{ab_{3}}$ is not equivalent to
a subfamily of $L_{abc_{2}}$ obtained by setting $a=c$.
Table 3: SLOCC classifications of $L_{ab_{3}}$ and $L_{abc_{2}}$ $L_{ab3}$ | $\mathcal{F}_{1}$ | $\mathcal{F}_{2}$ | $\mathcal{F}_{3}$ | $\mathcal{F}_{4}$
---|---|---|---|---
| $\emptyset$ | $a=b=0$ | $ab=0$ & $a\neq b$ | $ab\neq 0$
$L_{abc_{2}}$ | $\mathcal{F}_{1}$ | $\mathcal{F}_{2}$ | $\mathcal{F}_{3}$ | $\mathcal{F}_{4}$
$(a=c)$ | $a=b=0$ | $a=0$ & $b\neq 0$ | $a\neq 0$ & $b=0$ | $ab\neq 0$
To determine if a four-qubit state belongs to a family according to the
criteria given by Verstraete _et al._ Moor2 and Lamata _et al._ Lamata07 ,
one needs to check if the state is equivalent to the representative state of
that family. For the classification scheme proposed in this Letter, to
determine if an $n$-qubit state belongs to a family, one needs only to
calculate the rank of the coefficient matrix of the state.
In summary, we have studied SLOCC classification for general $n$-qubit states
via the invariance of the rank of the coefficient matrix and given several
examples for $n$ up to six. We have also characterized full separable states
and genuinely entangled states in terms of the rank. We expect that the
proposed entanglement classification for general $n$-qubit states may find
further experimental consequences.
This work was supported by NSFC (Grant No. 10875061) and Tsinghua National
Laboratory for Information Science and Technology.
## References
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|
arxiv-papers
| 2011-06-30T02:14:25 |
2024-09-04T02:49:20.197893
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiangrong Li, Dafa Li",
"submitter": "Dafa Li",
"url": "https://arxiv.org/abs/1106.6105"
}
|
1106.6130
|
# Top Physics at CDF
Chang-Seong Moon111csmoon@fnal.gov, Speaker on behalf of the CDF collaboration
###### Abstract
We present the recent results of top-quark physics using up to 6 fb-1 of
$p\bar{p}$ collisions at a center of mass energy of $\sqrt{s}$ = 1.96 TeV
analyzed by the CDF collaboration. Thanks to this large data sample, precision
top quark measurements are now a reality at the Tevatron. Further, several new
physics signals could appear in this large dataset. We will present the latest
measurements of top quark intrinsic properties as well as direct searches for
new physics in the top sector.
###### Keywords:
Tevatron, CDF, Standard Model, Top quark
###### :
14.65.Ha
## 1 Introduction
Since the top quark has been discovered in 1995 by CDF top_cdf and D0 top_d0
experiments at the Fermilab Tevatron collider, the studies of top quark
physics have been continued. Up to now, the great performance of the Tevatron
accelerator and the excellent operation of the CDF experiment allows to
measure the top quark properties with improved precision and also explore
several new physics signals in the top sector using large data samples. The
representative recent results on top quark physics from the CDF experiments
with 6 fb-1 are described due to the limited available space cdf_top .
## 2 Top Physics
The top quark which completes the third fermion generation in the Standard
Model (SM), is the most massive of the known elementary particles. This has
led many physicists to believe that the top quark may shed light on the path
to new physics. Due to the large mass of the top quark, its properties allow
predictions to be made of the mass of the Higgs boson under certain extensions
of the SM. At the Tavatron $p\bar{p}$ collider, top quark is mainly produced
in pairs through quark and anti-quark (85%) or gluon-gluon fusion (15%) and
decays through the weak force almost exclusively into a $W$ boson and a $b$
quark. The $t\bar{t}$ decays are labeled as dilepton, lepton+jets and all
hadronic depending on whether a leptonic decay has occurred in both, one only
or none of the two W bosons respectively. The dilepton final state is only 5
percent of top pairs decay but it is the easiest to identify because not very
many other processes can produce such a striking final state. The lepton+jets
decays are considered as the golden channel because of a sizable branching
fraction combined with manageable background levels. The all hadronic decay
channel has the largest cross section but also a huge background from QCD
multijet production.
## 3 Top Pair Production Cross Section Measurement
The SM predicts the $t\bar{t}$ production cross section at next-to-leading
order (NLO) of $\sigma_{t\bar{t}}$ = 7.45${}^{+0.72}_{-0.63}$ pb for the top
quark mass of 172.5 GeV/$c^{2}$ xsec . A measurement of the $t\bar{t}$
production cross section provides a test of the QCD calculations and can
probes new physics signals beyond the SM. Most precise measurements come from
the lepton+jets signature with two techniques xsec_lj . First one used a
topological selection based on a Neural Network (NN) exploiting the
kinematical properties of the event and the other on $b$-tagging. The latest
analysis is performed in the dilepton signature xsec_dil . In 5 fb-1 of CDF
data in the dilepton channel 343 signal candidates are found, the cross
section is measured to be $\sigma_{t\bar{t}}$ = 7.4 $\pm$ 0.6 (stat) $\pm$ 0.6
(syst) $\pm$ 0.5 (lumi) pb. After the requirement of at least one of the jets
originating from a $b$ quark by the SecVtx algorithm, 137 candidate events are
found, for a measured cross section value of $\sigma_{t\bar{t}}$ = 7.3 $\pm$
0.7 (stat) $\pm$ 0.5 (syst) $\pm$ 0.4 (lumi) pb. CDF also measured the
$\sigma_{t\bar{t}}$ with lower precision in all hadronic channel xsec_had .
Figure 1 (left) shows a summary of CDF $t\bar{t}$ cross section measurements
which agree well with the SM calculations.
Figure 1: Left: Summary of CDF $t\bar{t}$ cross section measurements in
different decay channels, Right: Summary of CDF top mass measurements in each
decay channel
## 4 Measurement of Top Quark Mass
The top quark mass is fundamentally important parameter of the Standard Model
because of measurement of the top quark mass constraint the mass of the Higgs
boson. CDF present two new preliminary measurements using events in jets plus
missing transverse energy and all hadronic final states channel using 6 fb-1.
The measured top masses are respectively $m_{t}$ = 172.3 $\pm$ 2.4 (stat+JES)
$\pm$ 1.0 (syst) GeV/$c^{2}$ mass_met and $m_{t}$ = 172.5 $\pm$ 1.4 (stat)
$\pm$ 1.0 (JES) $\pm$ 1.2 (syst) GeV/$c^{2}$. mass_had Figure 1 (right) shows
the CDF top mass measurements and the combination. The precision is reached to
$\Delta M_{top}/M_{top}$ = 0.63% and the uncertainty of top mass measurement
is only 1.09 GeV/$c^{2}$. And the top mass different between top and anti-top
quark is measured using 6 fb-1 to be $\delta$Mtop = -3.3 $\pm$ 1.7 GeV/$c^{2}$
mass_diff . It is consistent with CPT symmetry at a 2$\sigma$ level.
Figure 2: Left: The cos$\theta*$ distribution between data and the SM
expectation in dilepton channel, Right: The $\Delta_{y}$ distribution of the
top and anti-top in dilepton channel
## 5 Study of Top Quark Properties
Recent CDF results for the top quark properties are presented. First, CDF have
reported a measurement of the $W$ boson polarization in top quark decay. The
$V$-$A$ structure of the weak interaction of the SM predicts that the $W^{+}$
bosons from the top quark decay $t\rightarrow W^{+}b$ are dominantly either
longitudinally polarized ($\sim$70%) or left-handed ($\sim$30%), while right-
handed $W$ bosons are heavily suppressed and are forbidden in the limit of
massless b quarks. The latest CDF measurements using 5 fb-1 in dilepton
channel present longitudinal fraction, $f_{0}=0.73^{+0.18}_{-0.17}$ (stat)
$\pm$ 0.06 (syst) and right-handed fraction, $f_{+}$ = -0.08 $\pm$ 0.09 (stat)
$\pm$ 0.03 (syst). Figure 2 (left) shows the distribution of observable for
$W$ boson polarization and good agreement between data and the SM predictions.
Next measurement is the forward-backward asymmetry, $A_{fb}$ of the top quark
pair production. The asymmetry can arise only in the next to leading order and
is predicted to be about 0.078 afb . In the lepton+jets channel, the corrected
parton level asymmetry is 0.158 $\pm$ 0.074. And the recent analysis using the
dilepton events shows that at the parton level the measured $A_{fb}$ is 0.42
$\pm$ 0.15(stat) $\pm$ 0.05(syst) This result confirms the deviation from the
theoretical predictions. The rapidity difference between the top and anti-top
quarks in the dileptonic decay mode is shown in Figure 2 (right).
## 6 Searches in the Top Sector
CDF has searched for a heavy top quark $t^{\prime}\rightarrow Wb$ in the
lepton+jets events using 6 fb-1 tprime1 . We exclude the SM fourth-generation
$t^{\prime}$ quark with mass below 358 GeV/$c^{2}$ at 95% C.L. (see Figure
3(left)). Another result is search for dark matter through the production of
exotic $4^{th}$ generation quarks $t^{\prime}$ decaying via
$t^{\prime}\rightarrow t+X$, where $X$ is dark matter tprime2 . We find that
using 5 fb-1 of data will allow for exclusion of $m_{T^{\prime}}$ ¡ 360 GeV at
95% C.L. for $m_{X}$ ¡ 100 GeV. The Figure 3 (right) describes the expected
exclusion with $\pm 1\sigma$ vs observed exclusion.
Figure 3: Left: Upper limit, at 95% CL, on the production rate for
$t^{\prime}$ as a function of $t^{\prime}$ mass (red), Right: Observed versus
expected exclusion in ($m_{T^{\prime}}$, $m_{X}$)
## 7 Conclusion
We have presented recent results of top physics from CDF experiment All the
measurements are consistent with the SM prediction so far and new physics is
not found yet in top sector. More than 1000 reconstructed $t\bar{t}$ events in
6 fb-1 of dataset allow precision measurements of top quark properties. New
results will be updated with 7$\sim$8 fb-1 of data. CDF top physics program
and understanding of systematic effects will continue to play a significant
role for years to come using the expected more than 10 fb-1 of CDF full
dataset at the end of Run II of the Tevatron.
I would like to thank my colleagues of the CDF collaboration for their hard
work and dedication to provide these rich physics results. I also thank the
organizers of DIS 2011 in Jefferson Lab for an interesting and successful
conference.
## References
* (1) F. Abe et al. (CDF Collaboration), _Phys. Rev. Lett._ 74 2626 (1995).
* (2) S. Abachi et al. (D0 Collaboration), _Phys. Rev. Lett._ 74, 2632 (1995).
* (3) Public web-pages and archive of conference notes from the Top Quark Physics Groups of CDF http://www-cdf.fnal.gov/physics/new/top/top.html
* (4) S. Moch, P. Uwer, _Nucl. Phys. Suppl._ 183 75 (2008).
* (5) T. Aaltonen et al. (CDF Collaboration), _Phys. Rev. Lett_ 105, 012001 (2010).
* (6) T. Aaltonen et al. (CDF Collaboration), CDF Conference Note 10163.
* (7) T. Aaltonen et al. (CDF Collaboration), _Phys. Rev. D_ 81, 052011 (2010).
* (8) T. Aaltonen et al. (CDF Collaboration), CDF Conference Note 10433.
* (9) T. Aaltonen et al. (CDF Collaboration), CDF Conference Note 10456.
* (10) T. Aaltonen el al. (CDF Collaboration), _Phys. Rev. Lett._ , 106 152001 (2011).
* (11) J.H.Kuhn and G.Rodrigo, _Phys. Rev. Lett._ 81, 49 (1998);
J.H.Kuhn and G.Rodrigo, _Phys. Rev. D_ 59, 054017 (1999);
O.Antunano, J.H.KuhnandG.Rodrigo, _Phys. Rev. D_ 77, 014003 (2008).
* (12) T. Aaltonen et al. (CDF Collaboration), CDF Conference Note 10395.
* (13) T. Aaltonen el al. (CDF Collaboration), _Phys. Rev. Lett._ 106 191801 (2011).
|
arxiv-papers
| 2011-06-30T07:22:02 |
2024-09-04T02:49:20.204177
|
{
"license": "Public Domain",
"authors": "Chang-Seong Moon",
"submitter": "Chang-Seong Moon",
"url": "https://arxiv.org/abs/1106.6130"
}
|
1106.6150
|
# Quantum effects in energy and charge transfer in an artificial
photosynthetic complex
Pulak Kumar Ghosh1111Author to whom correspondence should be addressed.
Electronic mail: pulak@riken.jp, Anatoly Yu. Smirnov1,2, and Franco Nori1,2 1
Advanced Science Institute, RIKEN, Wako, Saitama, 351-0198, Japan
2 Physics Department, The University of Michigan, Ann Arbor, MI 48109-1040,
USA
###### Abstract
We investigate the quantum dynamics of energy and charge transfer in a wheel-
shaped artificial photosynthetic antenna-reaction center complex. This complex
consists of six light-harvesting chromophores and an electron-acceptor
fullerene. To describe quantum effects on a femtosecond time scale, we derive
the set of exact non-Markovian equations for the Heisenberg operators of this
photosynthetic complex in contact with a Gaussian heat bath. With these
equations we can analyze the regime of strong system-bath interactions, where
reorganization energies are of the order of the intersite exciton couplings.
We show that the energy of the initially-excited antenna chromophores is
efficiently funneled to the porphyrin-fullerene reaction center, where a
charge-separated state is set up in a few picoseconds, with a quantum yield of
the order of 95%. In the single-exciton regime, with one antenna chromophore
being initially excited, we observe quantum beatings of energy between two
resonant antenna chromophores with a decoherence time of $\sim$ 100 fs. We
also analyze the double-exciton regime, when two porphyrin molecules involved
in the reaction center are initially excited. In this regime we obtain
pronounced quantum oscillations of the charge on the fullerene molecule with a
decoherence time of about 20 fs (at liquid nitrogen temperatures). These
results show a way to directly detect quantum effects in artificial
photosynthetic systems.
## I Introduction
The multistep energy-transduction process in natural photosystems begins with
capturing sunlight photons by light-absorbing antenna chromophores surrounding
a reaction center Blankenship02 ; Amerongen00 . The antenna chromophores
transfer radiation energy to the reaction center directly or through a series
of accessory chromophores. The reaction center harnesses the excitation energy
to create a stable charge-separated state.
Energy transfer in natural and artificial photosynthetic structures has been
an intriguing issue in quantum biophysics due to the conspicuous presence of
long-lived quantum coherence observed with two-dimensional Fourier transform
electronic spectroscopy Engel07 ; Panitch10 . These experimental achievements
have motivated researchers to investigate the role of quantum coherence in
very efficient energy transmission, which takes place in natural photosystems
Guzik ; Plenio ; IshizakiPNAS ; ChengFleming09 ; Sarovar . Quantum coherent
effects surviving up to room temperatures have also been observed in
artificial polymers Collini09 . Artificial photosynthetic elements, mimicking
natural photosystems, might serve as building blocks for efficient and
powerful sources of energy Barber09 ; Larkum10 . Some of these elements have
been created and studied experimentally in Refs. gali1 ; gali2 ; gust1 ; gust2
; gust3 ; Imahori . The theoretical modelling of artificial reaction centers
has been recently performed in Refs. GhoshJCP09 ; SmirnovJPC09 .
Figure 1: (Color online) Schematic diagram of the wheel-shaped artificial
antenna-reaction center complex reported in ref gust3 . We use the short
notation, _BPF Complex_ , to denote this photosynthetic device. The antenna-
reaction center complex contains six light-harvesting pigmets: (i) two bis
(phenylethynyl)anthracene chromophores, BPEAa and BPEAb, (ii) two
borondipyrromethene chromophores, BDPYa and BDPYb, and (iii) two zinc
tetraarylporphyrin chromophores, ZnPya and ZnPyb. All the chromophores are
attached to a rigid hexaphenyl benzene core. In addition to the antenna
components, the photosystem contains a fullerene derivative (F) containing two
pyridyl groups, acting as an electron acceptor. The fullerene derivative F is
attached to the both ZnPy chromophores via the coordination of the pyridyl
nitrogens with the zinc atoms. For structural details of the BPF Complex we
refer gust3 ; gust4 .
Here we study energy transfer and charge separation in a wheel-shaped
molecular complex (BPF complex, see Fig. 1) mimicking a natural photosynthetic
system. This complex has been synthesized and experimentally investigated in
Ref. gust3 . It has four antennas - two bis(phenylethynyl)anthracene (BPEA)
molecules and two borondipyrromethene (BDPY) chromophores, as well as two zinc
porphyrins (${\rm ZnPy_{a}}$ and ${\rm ZnPy_{b}}$). These six light-absorbing
chromophores are attached to a central hexaphenylbenzene core. Electrons can
tunnel from the zinc porphyrin molecules to a fullerene F (electron acceptor).
Thus, two porphyrins and the fullerene molecule form an artificial reaction
center (${\rm ZnPy_{a}-F-ZnPy_{b}}$). The BPEA chromophores strongly absorb
around 450 nm (the blue region), while the BDPY moieties have good absorptions
around 513 nm (green region). Porphyrins have absorption peaks at both red and
orange wavelengths. Therefore, the BPF complex can utilize most of the rainbow
of sunlight – from blue to red photons. It is shown in gust3 that the
absorption of photons results in the formation of a porphyrin-fullerene
charge-separated state with a lifetime of 230 ps; in doing so, excitations
from the BPEA and BDPY antenna chromophores are transferred to the porphyrins
with a subsequent donation of an electron from the excited states of the
porphyrins to the fullerene moiety. This process takes a few picoseconds,
suggesting that the excitonic coupling between chromophores is sufficiently
strong. The electronic coupling between the porphyrins and the fullerene
controlling tunneling of electrons in the artificial reaction center also
should be quite strong. It should be noted, however, that spectroscopic data
gust1 ; gust2 ; gust3 show that the absorption spectrum of the BPF complex is
approximately represented as a superposition of contributions from the
individual chromophores with almost no perturbations due to the links between
the chromophores. This means that the chromophores comprising the light-
harvesting complex can be considered as individual interacting units, but not
as an extended single chromophore. We can expect that, at these conditions,
quantum coherence is able to play an important role in energy and charge
transfer dynamics, manifesting itself in quantum beatings of chromophore
populations as well as in quantum oscillations of the charge accumulated on
the fullerene molecule. In principle, these oscillations could be measured by
a sensitive single-electron transistor, thus providing a direct proof of
quantum behavior in the artificial photosynthetic complex. Since these
phenomena occurs at very short time scales (a few femtoseconds), these could
be within the reach of femtosecond spectroscopy in the near future. The main
goal of this study is to explore quantum features of the energy and charge
transfer in a wheel-shaped antenna-reaction center complex at subpicosecond
timescales.
## II Model and Methods
### II.1 Hamiltonian
Each chromophore has one ground and one excited state, whereas the electron
acceptor fullerene F has just one energy level with energy $E_{F}$. We
introduce creation (annihilation) operators, $a_{k}^{\dagger}$ ($a_{k}$), of
an electron on the $k$th site. The electron population operators are defined
as $n_{k}=a_{k}^{\dagger}a_{k}$. We assume that each electron state can be
occupied by a single electron, as spin degrees of freedom are neglected. The
basic Hamiltonian of the system has the form:
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\sum_{k}(E_{k}n_{k}+E_{k^{*}}n_{k^{*}})+E_{F}n_{F}+H_{C}+\sum_{k\neq
l}V_{kl}a^{\dagger}_{k^{*}}a_{k}\;a^{\dagger}_{l}a_{l^{*}}-\sum_{\sigma\sigma^{\prime}}\Delta_{\sigma\sigma^{\prime}}a^{\dagger}_{\sigma}a_{\sigma^{\prime}},$
(1)
where the first part incorporates the energies of the electron states
(hereafter $k,l$ = BPEAa, BPEAb, BDPYa, BDPYb, ZnPya, ZnPyb), and the second
term is related to a fullerene energy level $E_{F}$ with a population operator
$n_{F}=a_{F}^{\dagger}a_{F}$. The pair ($k,k^{*}$) denotes a ground ($k$) and
an excited ($k^{*}$) state of an electron located on the site $k$ with the
corresponding energy $E_{k}\;(E_{k^{*}})$. The term $H_{C}$ represents the
contribution of Coulomb interactions between electron-binding sites. This term
is given in Appendix A. The fourth term of Eq. (1) describes excitonic
couplings between the chromophores $k$ and $l$. The matrix element $V_{kl}$ is
a measure of an interchromophoric coupling strength. The last term in Eq. (1)
describes the electron tunneling from excited states of the porphyrin
molecules ZnPya, ZnPyb to the electron acceptor F characterized by the
tunneling amplitudes $\Delta_{\sigma\sigma^{\prime}}$, where
$\sigma,\;\sigma^{\prime}$ = ZnPy${}_{a}^{*}$, ZnPy${}_{b}^{*}$, F.
The interaction of the system with the environment (heat bath), represented
here by a sum of independent oscillators with Hamiltonian
$\displaystyle
H_{\rm{env}}=\sum_{j}\left(\frac{p_{j}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2}x_{j}^{2}}{2}\right),$
(2)
is given by the term
$\displaystyle H_{e-{\rm ph}}=-\sum_{jk}m_{j}\omega_{j}^{2}x_{jk}x_{j}n_{k},$
(3)
where $x_{j}$ and $p_{j}$ are the position and momentum of the $j$th
oscillator having an effective mass $m_{j}$ and a frequency $\omega_{j}$. The
coefficients $x_{jk}$ define the strength of the coupling between the electron
subsystem and the environment.
The contribution of the energy-quenching mechanisms responsible for the
recombination processes in the system is given by the Hamiltonian
$H_{\rm
quen}=-\sum_{l}(q_{l}^{\dagger}a_{l^{*}}^{\dagger}a_{l}+q_{l}a_{l}^{\dagger}a_{l^{*}}).$
(4)
For the sake of simplicity, we include the radiation damping of the excited
states into the energy-quenching operator $q_{l}$. The first term in the
Hermitian Hamiltonian $H_{\rm quen}$ is related to the excitation of the
$l-$chromophore by the quenching bath, whereas the second term corresponds to
the reverse process, namely, to the absorption of chromophore energy by the
bath. Both processes are necessary to provide correct conditions for the
thermodynamic equilibrium between the system and the bath.
The total Hamiltonian of the system is
$\displaystyle H=H_{0}+H_{e-{\rm ph}}+H_{\rm env}+H_{\rm quen}.$ (5)
We omit here the Hamiltonian of the quenching (radiation) heat bath.
### II.2 Diagonalization of $H_{0}$
We choose 160 basis states $|M\rangle$ of the complex including a vacuum
state, where all chromophores are in the ground state and the F site is empty.
We diagonalize the Hamiltonian $H_{0}$ (1) to consider the case where the
excitonic coupling between chromophores, described by coefficients $V_{lm},$
and the porphyrin-fullerene tunneling, which is determined by amplitudes
$\Delta_{\sigma\sigma^{\prime}}$, cannot be analyzed within perturbation
theory. In the new basis, $|\mu\rangle=\sum_{M}|M\rangle\langle M|\mu\rangle$,
the Hamiltonian $H_{0}$ is diagonal with the energy spectrum {$E_{\mu}$}, so
that the total Hamiltonian of the system $H$ has the form
$\displaystyle H=\sum_{\mu}E_{\mu}|\mu\rangle\langle\mu|-\sum_{\mu\nu}{\cal
A}_{\mu\nu}|\mu\rangle\langle\nu|+H_{\rm env}.$ (6)
Here
${\cal A}_{\mu\nu}=Q_{\mu\nu}+q_{\mu\nu}$ (7)
is the combined operator for both heat baths with fluctuating in time
variables
$\displaystyle
Q_{\mu\nu}=\sum_{j}m_{j}\omega_{j}^{2}x_{j}[x_{jF}\langle\mu|n_{F}|\nu\rangle+\sum_{k}(x_{jk}\langle\mu|n_{k}|\nu\rangle+x_{jk^{*}}\langle\mu|n_{k^{*}}|\nu\rangle)],$
(8)
and
$q_{\mu\nu}=\sum_{l}\,\langle\mu|a^{{\dagger}}_{l}a_{l^{*}}|\nu\rangle\,q_{l}+H.c.$
(9)
To distinguish the processes of energy transfer, where the number of electrons
on each chromophore remains constant, from the processes of charge transfer,
where the total population of the site changes, we introduce the following
operators
$\displaystyle S_{l}=n_{l}+n_{l^{*}},\;\;M_{l}=n_{l}-n_{l^{*}},$ (10)
together with coefficients
$\bar{x}_{jl}=\frac{x_{jl}+x_{jl^{*}}}{2},\;\;\;\;\;\;\tilde{x}_{jl}=\frac{x_{jl}-x_{jl^{*}}}{2}.$
(11)
Thus, the environment operator $Q_{\mu\nu}$ can be rewritten as
$\displaystyle Q_{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{j}m_{j}\omega_{j}^{2}x_{j}\Lambda^{\mu\nu}_{j}$ (12)
with
$\displaystyle\Lambda^{\mu\nu}_{j}=\sum_{l}\left\\{\bar{x}_{jl}\langle\mu|S_{l}|\nu\rangle+\tilde{x}_{j{l}}\langle\mu|M_{l}|\nu\rangle\right\\}+x_{jF}\langle\mu|n_{F}|\nu\rangle$
(13)
### II.3 Non-Markovian equations for the system operators
An arbitrary electron operator $W$ can be expressed in terms of the basic
operators $\rho_{\mu\nu}~{}=~{}|\mu\rangle\langle\nu|$; with
$W=\sum_{\mu\nu}W_{\mu\nu}\,\rho_{\mu\nu},$ and
$W_{\mu\nu}=\langle\mu|W|\nu\rangle.$ The operator $\rho_{\mu\nu}$ denotes a
matrix with zero elements, with the exception of the single element at the
crossing of the $\mu-$row and the $\nu-$column. The matrix elements
$W_{\mu\nu}$ of any electron operator can be easily calculated (see, e.g.,
Eqs. (S10) and (S11) in the Supporting Information for Ref. SmirnovJPC09 ).
For example, an electron localized in a two-well potential Leggett87 , with
the right and left states $|1\rangle$ and $|2\rangle$, is described by the
Pauli matrices
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}:\;\sigma_{z}=|1\rangle\langle
1|-|2\rangle\langle 2|,\,\sigma_{x}=|1\rangle\langle 2|+|2\rangle\langle
1|,\,$ and $\sigma_{y}=i(|2\rangle\langle 1|-|1\rangle\langle 2|)$, which are
expressed in terms of the basic operators $|\mu\rangle\langle\nu|$ with
$\mu,\,\nu=1,2$.
In the Heisenberg picture, the operator $W$ evolves in time according to the
equation: $i\left(\partial W/\partial t\right)=[W,H]_{-}\;.$ This evolution
can be described with the time-evolving operators,
$\rho_{\mu\nu}(t)~{}=~{}(|\mu\rangle\langle\nu|)(t)$, which satisfy the
Heisenberg equation:
$i\frac{\partial{\rho}_{\mu\nu}}{\partial
t}=[\rho_{\mu\nu},H]_{-}\;=\;-\,\omega_{\mu\nu}\rho_{\mu\nu}-\sum_{\alpha}({\cal
A}_{\nu\alpha}\rho_{\mu\alpha}-{\cal A}_{\alpha\mu}\rho_{\alpha\nu}),$ (14)
where $\omega_{\mu\nu}=E_{\mu}-E_{\nu},$ and the heat bath operator ${\cal
A}_{\mu\nu}$ is defined in Eq. (7). Here, we use the fact that the Hamiltonian
$H$ Eq. (6) is also expressed in terms of the operators $\rho_{\mu\nu}$ taken
at the same moment of time $t$. For two of these operators, $\rho_{\mu\nu}(t)$
and $\rho_{\alpha\beta}(t)$, we have simple multiplication rules:
$\rho_{\mu\nu}\rho_{\alpha\beta}=\delta_{\nu\alpha}\rho_{\mu\beta}.$ These
rules allow to calculate commutators of basic operators taken at the same
moment of time. We note that at the initial moment of time the operator,
$\rho_{\mu\nu}(0)~{}\equiv~{}|\mu\rangle\langle\nu|$, is represented by the
above-mentioned zero matrix with a single unit at the $\mu$-$\nu$
intersection. The matrix elements of the electron operators in Eqs. (9,13) are
taken over the time-independent eigenstates of the Hamiltonian $H_{0}$. The
bath operators ${\cal A}_{\mu\nu}$ fluctuate in time since they depend on the
environmental variables, $\\{x_{j}(t)\\}$, and on the variables
$\\{q_{l}(t)\\}$ of the quenching bath.
It is known that the dissipative evolution of the two-state system can be
described by the Heisenberg equations for the Pauli matrices
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ with the spin-boson Hamiltonian [see
Eq. (1.4) in Ref. Leggett87 ], which includes environmental degrees of
freedom. The artificial photosynthetic complex analyzed in the present paper
has 160 states. A dissipative evolution of this complex is described by the
Hamiltonian $H$ in Eq. (6), written in terms of the Heisenberg operators
$\rho_{\mu\nu}(t)~{}=~{}(|\mu\rangle\langle\nu|)(t)$ taken at the moment of
time $t$. Instead of the time-dependent Pauli matrices, the time evolution of
the two-state dissipative system can be described by the basic operators
$|1\rangle\langle 1|,\,|1\rangle\langle 2|,\,|2\rangle\langle
1|,\,|2\rangle\langle 2|,$ evolving in time. In a similar manner, the
evolution of the multi-state photosynthetic complex is described by the set of
the time-dependent Heisenberg operators $\rho_{\mu\nu}(t)$, which obey the
equation (14). As its spin-boson counterpart, the Hamiltonian $H$ in Eq. (6)
contains the Hamiltonian, $H_{\rm env}$, of the heat bath as well as the
system-bath interaction terms. Here, we generalize the spin-boson model from
the case of two states to the case of 160 states. With a knowledge of the
operators $\rho_{\mu\nu}(t)$, it is possible to find the time evolution of any
Heisenberg operator of the system. Only at the initial moment of time, $t=0$,
the operators $\rho_{\mu\nu}(0)$ form the basis of the Liouville space. Note
that we work in the Heisenberg representation, without using the description
based on the von Neumann equations for the density matrix.
To obtain functions that can be measured in experiments, we have to average
the operator $\rho_{\mu\nu}(t)$ and the equation (14) over the initial state
$|\Psi^{0}\rangle$ of the electron subsystem as well as over the Gaussian
distribution, $\rho_{T}=\exp(-H_{\rm bath}^{(0)}/T),$ of the equilibrium bath,
$\langle\ldots\rangle_{T},$ with temperature $T$ and with a free Hamiltonian
$H_{\rm bath}^{(0)}$, which is comprised of the free environment Hamiltonian
and the free Hamiltonian of the quenching bath. The notation
$\langle\ldots\rangle$ means double averaging:
$\langle\ldots\rangle=\langle\langle\Psi^{0}|\ldots|\Psi^{0}\rangle\rangle_{T}.$
(15)
The quantum-mechanical average value of the initial basic matrix,
$\langle\Psi^{0}|\rho_{\mu\nu}(0)|\Psi^{0}\rangle=\langle\Psi^{0}|\mu\rangle\langle\nu|\Psi^{0}\rangle,$
is determined by the product of amplitudes to find the electron subsystem at
the initial moment of time in the eigenstates $|\mu\rangle$ and $|\nu\rangle$
of the Hamiltonian $H_{0}$.
A standard density matrix, $\bar{\rho}=\\{\bar{\rho}_{\mu\nu}\\}$, of the
electron subsystem is a deterministic function which allows to calculate the
average value of an arbitrary operator $W$ with the formula:
$\langle
W(t)\rangle=Tr[\bar{\rho}(t)\,W]=\sum_{\mu\nu}W_{\mu\nu}\,\bar{\rho}_{\nu\mu}(t).$
(16)
The same average value can be written as $\langle
W(t)\rangle=\sum_{\mu\nu}W_{\mu\nu}\,\langle\rho_{\mu\nu}(t)\rangle,$ which
means that the average matrix,
$\langle\rho_{\mu\nu}(t)\rangle=\bar{\rho}_{\nu\mu}(t)$, has matrix elements
related to the transposed density matrix $\bar{\rho}(t)$.
It should be emphasized that the time evolution of the heat-bath operators
$\\{x_{j},\,p_{j}\\}$ and $\\{q_{l}\\}$, as well as their linear combinations
$Q_{\mu\nu},\,q_{\mu\nu},$ and ${\cal A}_{\mu\nu}$, are determined by the
total Hamiltonian $H$ in Eq. (6). In the absence of an interaction with the
dynamical system (the electron-binding sites), the free-phonon operators
$Q_{\mu\nu}^{(0)}$, as well as the free operators of the other baths,
$q_{\mu\nu}^{(0)}$, are described by Gaussian statistics Tanimura06 , as in
the case of an environment comprised of independent linear oscillators with
the Hamiltonian $H_{\rm env}$ (2). Using the Gaussian property, Efremov and
coauthors Efremov81 derived non-Markovian Heisenberg-Langevin equations,
without using perturbation theory, that assumes a weak system-bath
interaction. Recently, a similar non-perturbative approach has been developed
by Ishizaki and Fleming in Ref. IshizakiJCP09 . Due to Gaussian properties of
the free bath, the total operator ${\cal A}_{\mu\nu}$ of the combined
dissipative environment is a linear functional of the operators
$\rho_{\mu\nu}$,
${\cal A}_{\mu\nu}(t)={\cal
A}_{\mu\nu}^{(0)}(t)+\sum_{\bar{\mu}\bar{\nu}}\int\langle i[{\cal
A}_{\mu\nu}^{(0)}(t),{\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})]_{-}\rangle\theta(t-t_{1})\rho_{\bar{\mu}\bar{\nu}}(t_{1}),$
(17)
where $\theta(\tau)$ is the Heaviside step function. We note that this
expansion directly follows from the solution of the Heisenberg equations for
the positions $\\{x_{j}\\}$ and $\\{q_{l}\\}$ of the bath oscillators. It is
shown in Ref. Efremov81 that the average value of the free operator ${\cal
A}_{\mu\nu}^{(0)}(t)$ multiplied by an arbitrary operator ${\cal B}(t)$ is
proportional to the functional derivative of the operator ${\cal B}$ over the
variable ${\cal A}_{\mu\nu}^{(0)}(t)$:
$\langle{\cal A}_{\mu\nu}^{(0)}(t){\cal
B}(t)\rangle=\sum_{\bar{\mu}\bar{\nu}}\int dt_{1}\;\langle{\cal
A}_{\mu\nu}^{(0)}(t){\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\rangle\times\left\langle\frac{\delta{\cal
B}(t)}{\delta{\cal A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})}\right\rangle,$ (18)
with
$\frac{\delta{\cal B}(t)}{\delta{\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})}=i\;[{\cal
B}(t),\rho_{\bar{\mu}\bar{\nu}}(t_{1})]_{-}\;\theta(t-t_{1}).$ (19)
Substituting Eqs. (17,18,19) into Eq. (14) we derive the exact non-Markovian
equation for the Heisenberg operators $\rho_{\mu\nu}$ of the dynamical system
(chromomorphic sites + fullerene) interacting with a Gaussian heat bath,
$\displaystyle\langle\dot{\rho}_{\mu\nu}\rangle-i\,\omega_{\mu\nu}\langle\rho_{\mu\nu}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\alpha\bar{\mu}\bar{\nu}}\int_{0}^{t}dt_{1}\left\\{\langle{\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1}){\cal
A}_{\nu\alpha}^{(0)}(t)\rangle\langle\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rho_{\mu\alpha}(t)\rangle\right.$
(20) $\displaystyle-$ $\displaystyle\left.\langle{\cal
A}_{\nu\alpha}^{(0)}(t){\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\rangle\langle\rho_{\mu\alpha}(t)\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rangle+\langle{\cal
A}_{\alpha\mu}^{(0)}(t){\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\rangle\langle\rho_{\alpha\nu}(t)\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rangle\right.$
$\displaystyle-$ $\displaystyle\left.\langle{\cal
A}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1}){\cal
A}_{\alpha\mu}^{(0)}(t)\rangle\langle\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rho_{\alpha\nu}(t)\rangle\right.\\}.$
The time evolution of the average operator $\langle\rho_{\mu\nu}\rangle$ is
determined by the second-order correlation functions of the system operators
as well as by the correlation functions of the free dissipative environment.
Here we do not impose any restrictions on the spectrum of the environment. It
should be emphasized that the exact non-Markovian equation (20) goes far
beyond the von Neumann equation, $i\dot{\bar{\rho}}=[\bar{\rho},H]_{-},$ for
the density matrix $\bar{\rho}$ of the electron subsystem.
### II.4 Beyond the system-bath perturbation theory.
We assume that the coupling of the system to the quenching heat bath
determined by the Hamiltonian $H_{\rm quen}$ (4) is weak enough to be analyzed
perturbatively. However, an interaction of the chromophores with the protein
environment cannot be treated entirely within perturbation theory since the
reorganization energies are of the order of the intersite couplings. As in the
theory of modified Redfield equations Zhang98 ; YangFleming02 , the phonon
operator $Q_{\mu\nu}$ in Eq. (12) can be represented as a sum of diagonal
$Q_{\mu}=Q_{\mu\mu}$ and off-diagonal $\tilde{Q}_{\mu\nu}$ parts:
$Q_{\mu\nu}=Q_{\mu}\delta_{\mu\nu}+(1-\delta_{\mu\nu})\tilde{Q}_{\mu\nu}.$
(21)
We derive equations for diagonal and off-diagonal elements of the matrix
$\langle\rho_{\mu\nu}(t)\rangle$ (see Appendix B for details about the
derivation), where the interaction with the off-diagonal elements of the
environment operators $\tilde{Q}_{\mu\nu}$ are considered within perturbation
theory, and the effects of the diagonal elements $Q_{\mu}$ are treated
exactly.
The time dependence of the electron distribution $\langle\rho_{\mu}\rangle$
(diagonal elements) over eigenstates of the Hamiltonian $H_{0}$ is governed by
the equation
$\langle\dot{\rho}_{\mu}\rangle+\gamma_{\mu}\langle\rho_{\mu}\rangle=\sum_{\alpha}\gamma_{\mu\alpha}\langle\rho_{\alpha}\rangle,$
(22)
where the relaxation matrix $\gamma_{\mu\alpha}$ contains a contribution,
$\tilde{\gamma}_{\mu\alpha}$, from the non-diagonal environment operators [see
Eq. (46)] as well as a contribution from the quenching processes,
$\gamma_{\mu\alpha}^{\rm quen}$ [see Eq. (54)],
$\gamma_{\mu\alpha}=\tilde{\gamma}_{\mu\alpha}+\gamma_{\mu\alpha}^{\rm quen},$
(23)
with the total relaxation rate $\gamma_{\mu}=\sum_{\alpha}\gamma_{\alpha\mu}.$
The time evolution of the off-diagonal elements are given by Eq. (55) in
Appendix B.
Equations (22,55) allow us to determine the time evolution of an average value
for an arbitrary operator $W$ of the system: $\langle
W(t)\rangle=\sum_{\mu\nu}\langle\mu|W|\nu\rangle\langle\rho_{\mu\nu}(t)\rangle$.
## III Energies and other parameters
### III.1 Energy levels and electrochemical potentials
The energies of the excited states of chromophores BPEA, BDPY, and ZnPy, in
the BPF complex are estimated from an average between the longest wavelength
absorption band and the shortest wavelength emission band of the chromophores.
The average excited state energies of the chromophores BPEA, BDPY and ZnPy are
2610 meV, 2370 meV, and 2030 meV, respectively, if we count from the
corresponding ground energy levels gust2 ; gust3 . Cyclic voltammetric studies
gust3 of reduction potentials with respect to the standard calomel electrode
show that the first reduction potential of the fullerene derivative, F, is
about – 0.62 V and the first oxidation potential of ZnPy is about 0.75 V. From
these data we calculate that the energy of the charge separated state
ZnPy${}^{+}-$F- is about 1370 meV. This energy is a sum of the energy of an
electron on site F and a Coulomb interaction energy between a positive charge
on ZnPy and a negative charge on F. The Coulomb energy can be calculated with
the formula $u=e^{2}/4\pi\epsilon_{0}\epsilon r$, where $\epsilon_{0}$ is the
vacuum dielectric constant. The dielectric constant $\epsilon$ of 1,2
diflurobenzene (a solvent used in all experimental measurements of Ref. gust3
) is about 13.8. If the distance $r$ between porphyrin ZnPy and fullerene F is
about 1 nm, the Coulomb interaction energy is about 105 meV. Thus, the
estimated energy of the electron on F can be of the order of 1475 meV.
Table 1: This table presents the chosen values of the excitonic couplings ($V$) and reorganization energies for energy transfer ($\Lambda$) of the six antenna chromophores. We choose two sets of parameters, one set (denoted by I) corresponds to $V>\Lambda$ and the other set (II) to the opposite limit $V<\Lambda$. The calculated values of the time constants using both sets of parameters agree with the experimental values. Chromophores | | Set I
---
Coupling (V)
| Set I
---
Reorganization
energy ($\Lambda$)
| Set II
---
Coupling (V)
| Set II
---
Reorganization
energy ($\Lambda$)
| BPEAa $\leftrightarrow$ BPEAb,
---
BPEAb $\leftrightarrow$ BPEAa
50 meV | | $\Lambda_{{\rm BPEA}_{a}}=20$ meV
---
$\Lambda_{{\rm BPEA}_{b}}=20$ meV
30 meV | | $\Lambda_{{\rm BPEA}_{a}}=40$ meV
---
$\Lambda_{{\rm BPEA}_{b}}=40$ meV
| BPEAa $\leftrightarrow$ BDPYa,
---
BPEAb $\leftrightarrow$ BDPYb
30 meV | | $\Lambda_{{\rm BDPY}_{a}}=15$ meV
---
$\Lambda_{{\rm BDPY}_{b}}=15$ meV
17 meV | | $\Lambda_{{\rm BDPY}_{a}}=30$ meV
---
$\Lambda_{{\rm BDPY}_{b}}=30$ meV
| BDPYa $\leftrightarrow$ ZnPya,
---
BDPYb $\leftrightarrow$ ZnPyb
60 meV | | $\Lambda_{{\rm ZnPy}_{a}}=20$ meV
---
$\Lambda_{{\rm ZnPy}_{b}}=20$ meV
25 meV | | $\Lambda_{{\rm ZnPy}_{a}}=40$ meV
---
$\Lambda_{{\rm ZnPy}_{b}}=40$ meV
| BPEAa $\leftrightarrow$ ZnPya,
---
BPEAb $\leftrightarrow$ ZnPyb
50 meV | - | 40 meV | -
| BPEAb $\leftrightarrow$ ZnPya,
---
BPEAa $\leftrightarrow$ ZnPyb
60 meV | - | 40 meV | -
### III.2 Reorganization energies and coupling strengths
The reorganization energies for exciton and electron transfer processes and
electronic coupling strengths between the chromophores depend on the mutual
distances and orientations of the components, strengths of chemical bonds,
solvent polarity and other structural details of the system. Precise values of
these parameters are not available. However, time constants for energy
transfer between different chromophores in the BPF complex, as well as rates
for transitions of electrons between the fullerene F and porphyrin
chromophores ZnPy, have been reported in Ref. gust3 . We fit the experimental
values of these time constants with the rates following from our equations
with the goal of extracting reasonable values for the reorganization energies
and the electronic and excitonic couplings. In principle, many combinations of
reorganization energies and coupling constants could be possible. For the sake
of simplicity, we consider two sets of parameters, for two limiting
situations. One parameter set (denoted by I in Table I) corresponds to a
larger excitonic couplings, $V$, compared to the reorganization energies,
$\Lambda$, whereas another set of parameters (denoted by II in Table I)
considers the opposite case: where the reorganization energies are larger than
the excitonic couplings. These two sets of parameters are presented in Table
I. In addition to the parameters listed in Table I, we consider the following
values for the charge-transfer reorganization energies (set I):
$\lambda_{F}=200$ meV, $\lambda_{lM}=100$ meV, and $\lambda_{F}=230$ meV,
$\lambda_{lM}=120$ meV (set II), where $l={\rm ZnPy}_{a},{\rm ZnPy}_{b}$. The
values of the reorganization energies for energy-transfer processes are much
smaller than those for charge transfer.
Table 2: This table presents a comparison between the calculated values of the time constants (using the parameters sets I and II) to the experimental values reported in Ref. gust3 . Process | $\tau$ (Set I) | $\tau$ (Set II) | $\tau$ (Experimental)
---|---|---|---
| BPEAa $\rightarrow$ BPEAb,
---
BPEAb $\rightarrow$ BPEAa
$\sim$ 0.4 ps | $\sim$ 0.2 ps | 0.4 ps
| BPEAa $\rightarrow$ BDPYa,
---
BPEAb $\rightarrow$ BDPYb
$\sim$ 5 ps | $\sim$ 5.4 ps | 5-13 ps
| BDPYa $\rightarrow$ ZnPya,
---
BDPYb $\rightarrow$ ZnPyb
$\sim$ 5 ps | $\sim$ 3.9 ps | 2-15 ps
| BPEAa $\rightarrow$ ZnPya,
---
BPEAb $\rightarrow$ ZnPyb
$\sim$ 12 ps | $\sim$ 12 ps | 7 ps
| BPEAb $\rightarrow$ ZnPya,
---
BPEAa $\rightarrow$ ZnPyb
$\sim$ 10 ps | $\sim$ 12 ps | 6 ps
| ZnPyb $\rightarrow$ F,
---
ZnPya $\rightarrow$ F
$\sim$ 3 ps | $\sim$ 3 ps | 3 ps
References gust3 ; gust4 reported a very fast electron transfer (with a time
constant $\tau\sim$ 3 ps) between excited states of zincporphyrins
(ZnPya,ZnPyb) and the fullerene derivative F. This fact indicates a good
porphyrin-fullerene electronic coupling, which is due to the short covalent
linkage and close spatial arrangement of the components gust4 . Hereafter, we
assume that the ZnPy-F tunneling amplitudes $\Delta$ are about 100 meV
(parameter set I) and 80 meV (parameter set II). These parameters provide a
quite fast electron transfer, despite of a significant energy gap between the
ZnPy excited states and the fullerene energy level.
To describe recombination processes, we introduce a coupling of the $l$-th
chromophore to a quenching heat-bath characterized for simplicity by the Ohmic
spectral density: $\chi^{\prime\prime}_{l}(\omega)=\alpha_{l}\,\omega$ with a
dimensionless constant $\alpha_{l}$. We assume that the shifts of the energy
levels caused by the quenching bath are included into the renormalized
parameters of the electron subsystem. The experimental values gust3 ; gust4
of the lifetimes $\tau^{e}_{l}$ for excited states of chromophores BPEA, BDPY
and ZnPy: $\tau^{e}_{\rm BPEA}=2.82\;{\rm ns},\;\tau^{e}_{\rm BDPY}=0.26\;{\rm
ns},\;$ and $\tau^{e}_{\rm ZnPY}=0.45\;{\rm ns},$ can be achieved with the
following set of coupling constants: $\alpha_{\rm BPEA}\sim
10^{-7},\;\alpha_{\rm BDPY}\sim 10^{-6},\;\;{\rm and}\;\;\alpha_{\rm ZnPy}\sim
7\times 10^{-7}.$
## IV Results and discussions
Using Eqs. (55,22) and two sets of parameters discussed in Sec. III, here we
study electron and energy transfer kinetics in the BPF complex with special
emphasis on the femtosecond time range, where the effects of quantum coherence
can play an important role. We consider both single- and double-exciton
regimes.
### IV.1 Evolution of a single exciton in the BPF complex
Figure 2: (Color online) Site populations as a function of time for the
parameter set I. The inset plots depict the features of site populations for
short times, at two different temperatures: $T$ = 300 K and 77 K. The site
populations of the BPEA moieties oscillate with a considerably large
amplitude, while the oscillations of the other site populations are hardly
observable.
In Fig. 2 we show the time evolution of the excited states populations
provided that only the BPEAa chromophore is excited at $t=0$ (single-exciton
regime). We use here the parameter set I, where excitonic couplings are larger
than reorganization energies (see Sec. III). The process starts with quantum
beatings between the resonant BPEAa and BPEAb chromophores, with a decoherence
time of the order of 100 fs (at $T$ = 300 K). In a few picoseconds, the
excitation energy is subsequently transferred to the adjacent BDPY moieties
and to the ZnPy chromophores. Later on, an electron moves from the excited
energy level of the porphyrins to the fullerene moiety; thus, producing a
charge-separated state, ZnPy${}^{+}-$F-, with a quantum yield 95%, which is in
agreement with experimental results gust2 .
Figure 3: (Color online) This figure presents site populations as a function
of time for the parameter set II. The inset plots show the site populations
for short times, at two different temperatures: $T$ = 300 K and 77 K. The
amplitudes of the site-population oscillations are much smaller and die out
earlier, compared to Fig. 2. This figure indicates that even for $\Lambda>V$,
the energy transfer between BPEA chromophores is dominated by wave-like
coherent motion.
It is evident from Fig. 2 that excited state populations of the BDPY
chromophores oscillate with much lower amplitudes and die out within a very
short time, $t<10$ fs, at both temperatures: $T$ = 300 K and 77 K. The
populations of the other sites of the BPF complex do not exhibit any
oscillatory behavior. This can be ascribed to incoherent hopping becoming
dominant because of significant energy mismatch between these chromophores.
Figure 4: (Color online) Site populations as a function of time for the
parameter set I, when the ZnPya chromophore is in the excited state and all
the other chromophores are in the ground state at $t=0$. The inset plots
depict the site populations at short times for two temperatures: $T$ = 300 and
77. Lowering the temperature enhances the oscillations of the charge density
on the fullerene moiety. Despite the huge energy difference between
ZnPy${}^{*}-$F and ZnPy${}^{+}-$F-, the charge of the fullerene site exhibits
oscillatory behavior for short times, specially at lower temperatures.
Figure 3 shows the time-dependence of the excited state populations of
chromophores for the parameter set II, where the reorganization energies are
larger than the excitonic couplings between chromophores. At $t=0$ the BPEAa
chromophore is excited (single-exciton regime). Then, after a few picoseconds,
the charge-separated state is formed with a quantum yield of the order of 97%.
However, owing to a stronger system-environment coupling, quantum beats
between the BPEAa and BPEAb chromophores have a lower amplitude and shorter
decoherence time ($\sim$50 fs) than in the previous case when we use the
parameter set I. We note that no quantum oscillations of the fullerene
population (site F) are visible in Figs. 2 and 3.
No significant oscillations of the site populations were observed (not shown
here) when the BDPY chromophores were initially (at $t=0$) excited. In this
case, due to the considerable energy gaps between the BDPY and the adjacent
BPEA and ZnPy chromophores, incoherent hopping dominates over the coherent
transfer of excitons. Furthermore, the structure of the BPF complex gust1 ;
gust4 does not allow direct energy transfer between two BDPY chromophores.
Figure 5: (Color online) Time evolution of the site populations for the
parameter set II, starting with an exciton on the chromophore ZnPya at $t=0$.
The inset plots depict the features of the site populations for a shorter time
regime and at two temperatures: $T$ = 300 K and 77 K. Lowering the temperature
enhances oscillations of the charge density on the fullerene derivative. These
results indicate that the population of the site F oscillates for short times,
even for $\Lambda>V$. These oscillations are more pronounced at lower
temperatures.
Figures 4 and 5 demonstrate charge- and energy-transfer dynamics for two
parameter sets, I and II, for the case when one of the porphyrin chromophores
(ZnPya) is excited. Here we do not show the time evolution of the BPEA and
BDPY chromophores since these moieties have higher excitation energies than
the ZnPy chromophore and they are not excited in the process. As evident from
Figs. 4 and 5, the excited porphyrin molecule rapidly transfers an electron to
fullerene, thus, producing a charge-separated state ZnPy${}^{+}-$F- with a
quantum yield of about 98%. The most important feature here is that the
population and charge of the fullerene molecule oscillates in time due to a
quantum superposition of the porphyrin excited state and the state of an
electron on the fullerene. The amplitude of these quantum beats is very small
and the decoherence time is quite short ($\sim$10 fs at T = 77 K). This fact
can be explained by the significant energy mismatch between the ZnPy${}^{*}-$F
and ZnPy${}^{+}-$F- states as well as by the strong influence of the
environment on the electron dynamics.
### IV.2 Evolution of double excitons in the BPF complex
Figure 6: (Color online) Time evolution of the populations on the site F, for
both sets of parameters, I and II, comparing the double-exciton case (the two
ZnPy chromophores are excited) with the single-exciton case. (a) Time
evolution of the populations on the site F for the parameter set I. (b) Time
evolution of the populations on the site F for the parameter set II. Note that
the double-excitation significantly enhances the amplitude of the charge
oscillations at the fullerene site for both sets of parameters, either at low
or high temperatures.
In the previous subsection, we consider a single exciton case with just one
chromophore initially being in the upper energy state. Here we analyze a
situation where two porphyrin molecules (ZnPya and ZnPyb) are excited at
$t=0$.
Figure 7: (Color online) Time evolution of the population on the site F for
the parameters set II when both ZnPy chromophores are excited at $t=0$. (a)
Effects of the coupling $\Delta$ on the time evolution of the populations on
the site F. (b) Effects of the energy gap between an excited state of a ZnPy
chromophore and the charge-separated state, $E_{\rm ch}$, on the time
evolution of populations on the site F. (c) Effects of the reorganization
energy $\lambda$ on the time evolution of populations on the site F. As can be
seen from these plots, the contribution of wave-like coherent motion to
electron-transfer dynamics is significantly enhanced when strengthening the
coupling between fullerene and porphyrin, lowering the energy gap between the
fullerene and porphyrin sites, and decreasing the reorganization energy.
Figures 6a and 6b show the coherent dynamics of the fullerene population (and
the fullerene charge) for the parameter sets I (Fig. 6a) and II (Fig. 6b) at
two different temperatures, $T=77$ K and $T=300$ K. We also compare the
double-exciton case with the previously analyzed single-exciton case. It is
apparent from Fig. 6, that the double excitation significantly enhances the
amplitude of quantum oscillations of the fullerene charge for both sets of
parameters. As one might expect, the frequency of the quantum beatings and the
decoherence time are not affected by the number of excitons.
### IV.3 Amplification of charge oscillations
In the previous discussion we observed that lowering the temperature and the
simultaneous excitation of both porphyrins significantly enhances quantum
oscillations of the fullerene charge. In this subsection we show that these
oscillations can also be controlled by tuning the following parameters:
#### IV.3.1 Electron tunneling amplitude $\Delta\,.$
The electronic coupling between the fullerene electron acceptor and zinc
porphyrins has a strong effect on the quantum oscillations of the fullerene
charge. To explore this effect, in Fig. 7a we plot the electron population of
the fullerene as a function of time, for different values of the coupling
$\Delta$. Figure 7a clearly shows that, with increasing $\Delta$, the
amplitude of the charge oscillations is significantly enhanced. This coupling
can be increased by attaching the fullerene to porphyrins with better ligands
which form much stronger covalent bonds.
#### IV.3.2 Energy of the charge-separated state $E_{\rm ch}\,.$
The energy $E_{\rm ch}\sim$ 1370 meV, of the charge separated state,
ZnPy${}^{+}-$F- is much lower than the energy of the zinc porphyrin excited
state, $E_{{\rm ZnPy}^{*}}\sim$ 2030 meV. It is evident from Fig. 7b that
increasing the energy $E_{\rm ch}$, which leads to a decrease of the porpyrin-
fullerene energy mismatch, results in a pronounced amplification of the
quantum oscillations of the fullerene charge. The energy of the fullerene can
be changed by placing nearby a charge residue, electrostatically coupled to
the fullerene.
#### IV.3.3 Reorganization energy $\lambda_{F}\,.$
In Fig. 7c we present the time evolution of the fullerene population for
different values of charge transfer reorganization energy $\lambda_{F}$. This
parameter can be decreased by replacing the polar solvent with another one
which has a much lower polarity. As can be seen from Fig. 7c, the quantum
oscillations of the fullerene charge survive much longer times for smaller
values of the reorganization energy, which correspond to weaker system-
environment couplings. A similar effect is expected when the porphyrin
reorganization energy is changed.
## V Conclusions
We theoretically studied the energy and electron-transfer dynamics in a wheel-
shaped artificial antenna-reaction center complex. This complex gust3 ,
mimicking a natural photosystem, contains six chromophores (BPEAa, BPEAb,
BDPYa, BDPYb, ZnPya, ZnPyb) and an electron acceptor (fullerene, F). Using
methods of dissipative quantum mechanics we derive and solve a set of
equations for both the diagonal and off-diagonal elements of the density
matrix, which describe quantum coherent effects in energy and charge transfer.
We consider two sets of parameters, one corresponding to the case where the
energy-transfer reorganization energy $\Lambda$ is less than the resonant
coupling $V$ between the chromophores, $\Lambda<V$, and another regime where
$\Lambda>V$. For these two sets of parameters we examine the electron and
exciton dynamics, with special emphasis on the short-time regime ($\sim$
femtoseconds). We demonstrate that, in agreement with experiments performed in
Ref. gust3 , the excitation energy of the BPEA antenna chromophores is
efficiently funneled to porphyrins (ZnPy). The excited ZnPy molecules rapidly
donate an electron to the fullerene electron acceptor, thus creating a charge-
separated state, ZnPy${}^{+}-$F-, with a quantum yield of the order of 95%.
There is no observable difference in energy transduction efficiency for these
two sets of parameters. In the limit of strong interchromophoric coupling,
coherent dynamics dominates over incoherent-hopping motion. In the single-
exciton regime, when one of the BPEA chromophores is initially excited,
quantum beatings between two resonant BPEA chromophores occur with decoherence
times of the order of 100 fs. However, here the electron transfer process is
dominated by incoherent hopping. For the case where one porphyrin molecule is
excited at the beginning, we obtain small quantum oscillations of the
fullerene charge characterized by a short decay time scale ($\sim$ 10 fs).
More pronounced quantum oscillations of the fullerene charge (with an
amplitude $\sim$ 0.1 electron charge and decoherence time of about 20 fs at
$T$ = 77 K) are predicted for the double-exciton regime, when both porphyrin
molecules are initially excited. We also show that the contribution of wave-
like coherent motion to electron-transfer dynamics could be enhanced by
lowering the temperature, strengthening the fullerene-porphyrin bonds,
shrinking the energy gap between the zinc porphyrin and fullerene moieties
(e.g., by attaching a charged residue to the fullerene), as well as by
decreasing the reorganization energy (by tuning the solvent polarity).
Acknowledgements. FN acknowledges partial support from the Laboratory of
Physical Sciences, National Security Agency, Army Research Office, DARPA, Air
Force Office of Scientific Research, National Science Foundation grant No.
0726909, JSPS-RFBR contract No. 09-02-92114, Grant-in-Aid for Scientific
Research (S), MEXT Kakenhi on Quantum Cybernetics, and Funding Program for
Innovative Research and Development on Science and Technology (FIRST).
## Appendix A Coulomb interaction energies
The Coulomb interactions between the electron states are,
$\displaystyle H_{\rm C}$ $\displaystyle=$ $\displaystyle-u_{\rm
F}\left[(1-\bar{n}_{\rm ZnPy_{a}})n_{\rm F}+(1-\bar{n}_{\rm ZnPy_{b}})n_{\rm
F}\right]+u_{\rm Py}(1-\bar{n}_{\rm ZnPy_{a}})(1-\bar{n}_{\rm ZnPy_{b}})$ (24)
$\displaystyle+$ $\displaystyle u_{\rm ZnPy_{a}}n_{\rm ZnPy_{a}}n_{\rm
ZnPy_{a}^{*}}+u_{\rm ZnPy_{b}}n_{\rm ZnPy_{b}}n_{\rm ZnPy_{b}^{*}},$
where,
$\bar{n}_{\rm ZnPy_{a}}=n_{\rm ZnPy_{a}}+n_{\rm ZnPy_{a}^{*}}\;\;\;\;\;{\rm
and}\;\;\;\;\;\bar{n}_{\rm ZnPy_{b}}=n_{\rm ZnPy_{b}}+n_{\rm ZnPy_{b}^{*}}.$
The first term of (24) represents the electrostatic attraction (so the minus
sign) between the positively charged ZnPy chromophores and the negatively-
charged fullerene. The second term is due to the Coulomb repulsion (so the
plus sign) between two ZnPy chromophores. The last two terms are the repulsive
interaction energies when both the excited and ground states of the ZnPy
chromophores are occupied by electrons. The coefficients $u_{\rm F},u_{\rm
Py},u_{\rm ZnPy_{a}},\;{\rm and}\;u_{\rm ZnPy_{a}}$ represent the magnitude of
the electrostatic interactions and these are calculated using the Coulomb
formula. We have assumed that the empty ZnPy chromophores ($n_{\rm
ZnPy}+n_{\rm ZnPy^{*}}=0$) have positive charges and the acceptor state F
becomes negatively-charged when it is occupied by an electron.
## Appendix B Derivation of equations for the matrix
$\langle\rho_{\mu\nu}\rangle$
Our derivation of the equations for the matrix $\langle\rho_{\mu\nu}\rangle$
is based on the exact solution for the operator
$\rho_{\mu\nu}=(|\mu\rangle\langle\nu|)(t)$ of the system influenced only by
diagonal fluctuations of the bath. In this case the “system + bath”
Hamiltonian has the form
$H_{\rm
diag}=\sum_{\mu}E_{\mu}|\mu\rangle\langle\mu|+\sum_{j}\left(\frac{p_{j}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2}x_{j}^{2}}{2}\right)-\sum_{\mu}\sum_{j}m_{j}\omega_{j}^{2}\Lambda_{j}^{\mu}x_{j}|\mu\rangle\langle\mu|,$
(25)
where $\Lambda_{j}^{\mu}=\Lambda_{j}^{\mu\mu}$ [see Eq. (13)]. The time
evolution of the exciton operators $\rho_{\mu\nu}$ is governed by the
Heisenberg equation
$i\dot{\rho}_{\mu\nu}=-\,\omega_{\mu\nu}\rho_{\mu\nu}+\sum_{j}m_{j}\omega_{j}^{2}(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})x_{j}\rho_{\mu\nu}.$
(26)
It is possible to verify that the solution of Eq. (26) is given by the
equation
$\displaystyle\rho_{\mu\nu}(t)=\exp[i\Omega_{\mu\nu}(t-t_{0})]\times\exp\left[i\sum_{j}p_{j}(t)(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})\right]\times$
$\displaystyle\exp\left[-i\sum_{j}p_{j}(t_{0})(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})\right]\rho_{\mu\nu}(t_{0}),$
(27)
where
$\Omega_{\mu\nu}=\omega_{\mu\nu}-\sum_{j}\frac{m_{j}\omega_{j}^{2}}{2}\left[(\Lambda_{j}^{\mu})^{2}-(\Lambda_{j}^{\nu})^{2}\right],$
(28)
and $p_{j}$ is the Heisenberg operator of the dissipative environment. The
evolution begins at time $t=t_{0}$. The diagonal operators
$\rho_{\mu}=\rho_{\mu\mu}$ are constant, $\rho_{\mu}(t)=\rho_{\mu}(t_{0})$, in
the presence of a strong interaction with the diagonal operators of the
protein environment.
For uncorrelated diagonal and off-diagonal environment operators, when
$\langle Q^{(0)}_{\alpha}(t)\tilde{Q}^{(0)}_{\mu\nu}(t^{\prime})\rangle=0$,
the contribution of the environment to the non-Markovian equation (20)
consists of two parts:
$\displaystyle\langle-i[\rho_{\mu\nu},H_{e-{\rm
ph}}]_{-}\rangle=\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
diag}]_{-}\rangle+\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
n-diag}]_{-}\rangle.$ (29)
The diagonal elements, $Q_{\mu}$, of the environment contribute to the first
part,
$\displaystyle\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
diag}]_{-}\rangle=\int_{0}^{t}dt_{1}\langle(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)Q_{\bar{\nu}}^{(0)}(t_{1})\rangle\langle\rho_{\mu\nu}(t)\rho_{\bar{\nu}}(t_{1})\rangle-$
$\displaystyle\int_{0}^{t}dt_{1}\langle
Q_{\bar{\nu}}^{(0)}(t_{1})(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)\rangle\langle\rho_{\bar{\nu}}(t_{1})\rho_{\mu\nu}(t)\rangle,$
(30)
whereas the second part is due to a contribution of the non-diagonal
(abbreviated as n-diag in the super-index) operators, $\tilde{Q}_{\mu\nu}$,
$\displaystyle\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
n-diag}]_{-}\rangle=-\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\nu\alpha}^{(0)}(t)\tilde{Q}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\rangle\langle\rho_{\mu\alpha}(t)\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rangle+$
$\displaystyle\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\tilde{Q}_{\nu\alpha}^{(0)}(t)\rangle\langle\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rho_{\mu\alpha}(t)\rangle+$
$\displaystyle\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\alpha\mu}^{(0)}(t)\tilde{Q}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\rangle\langle\rho_{\alpha\nu}(t)\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rangle-$
$\displaystyle\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\bar{\mu}\bar{\nu}}^{(0)}(t_{1})\tilde{Q}_{\alpha\mu}^{(0)}(t)\rangle\langle\rho_{\bar{\mu}\bar{\nu}}(t_{1})\rho_{\alpha\nu}(t)\rangle.$
(31)
We note that the time evolution of the diagonal elements of the system
operator, $\rho_{\mu}=\rho_{\mu\mu},$ is determined by the non-diagonal
operators $\tilde{Q}_{\mu\nu}$ as well as by quenching terms. Strong diagonal
fluctuations of the environment have no effect on the evolution of the
diagonal elements of the matrix. Thus, in Eq. (30) we assume that
$\rho_{\bar{\nu}}(t_{1})=\rho_{\bar{\nu}}(t),$ so that Eq. (30) can be
rewritten as
$\displaystyle\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
diag}]_{-}\rangle=-(\Gamma_{\mu\nu}^{\rm diag}+i\delta\Omega_{\mu\nu}^{\rm
diag})(t)\langle\rho_{\mu\nu}(t)\rangle,$ (32)
where the time-dependent rate, $\Gamma_{\mu\nu}^{\rm diag}(t)$, and the
frequency shift, $\delta\Omega_{\mu\nu}^{\rm diag}$, can be found from the
following expression
$\displaystyle\Gamma_{\mu\nu}^{\rm diag}(t)+i\delta\Omega_{\mu\nu}^{\rm
diag}(t)=\int_{0}^{t}dt_{1}\left\\{\left\langle(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)Q_{\nu}^{(0)}(t_{1})\right\rangle-\left\langle
Q_{\mu}^{(0)}(t_{1})(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)\right\rangle\right\\}.$
(33)
The rate $\Gamma_{\mu\nu}^{\rm diag}(t)$ determines the fast decay of quantum
coherence in our system. For an environment composed of independent
oscillators we obtain
$\displaystyle\langle(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)Q_{\nu}^{(0)}(t_{1})\rangle-\langle
Q_{\mu}^{(0)}(t_{1})(Q_{\mu}^{(0)}-Q_{\nu}^{(0)})(t)\rangle=$
$\displaystyle-\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})^{2}\coth\left(\frac{\omega_{j}}{2T}\right)\cos\omega_{j}(t-t_{1})-$
$\displaystyle
i\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}\left[(\Lambda_{j}^{\mu})^{2}-(\Lambda_{j}^{\nu})^{2}\right]\sin\omega_{j}(t-t_{1}).$
(34)
The fluctuations of the diagonal operators of the environment can be described
by the set of spectral functions,
$\displaystyle
J_{\mu}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}(\Lambda_{j}^{\mu})^{2}\delta(\omega-\omega_{j}),$
$\displaystyle\bar{J}_{\mu\nu}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})^{2}\delta(\omega-\omega_{j}),$
(35)
together with the corresponding reorganization energies,
$\displaystyle\lambda_{\mu}=\int_{0}^{\infty}\frac{d\omega}{\omega}J_{\mu}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{2}}{2}(\Lambda_{j}^{\mu})^{2},$
$\displaystyle\bar{\lambda}_{\mu\nu}=\int_{0}^{\infty}\frac{d\omega}{\omega}\bar{J}_{\mu\nu}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{2}}{2}(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})^{2}.$
(36)
We also introduce a spectral function, $\tilde{J}_{\mu\nu}(\omega)$, which
characterizes the non-diagonal ($\mu\neq\nu$) environment fluctuations,
$\tilde{J}_{\mu\nu}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}|\tilde{\Lambda}_{j}^{\mu\nu}|^{2}\delta(\omega-\omega_{j}),$
(37)
where $\tilde{\Lambda}_{j}^{\mu\nu}=\Lambda_{j}^{\mu\nu}$ (13) taken at
$\mu\neq\nu$. With Eq. (34) we calculate the contributions of the diagonal
environment fluctuations into the decoherence rate and the frequency shift of
the off-diagonal elements of the system matrix $\langle\rho_{\mu\nu}\rangle$
in (32),
$\displaystyle\Gamma_{\mu\nu}^{\rm
diag}(t)=\int_{0}^{\infty}\frac{d\omega}{\omega}\bar{J}_{\mu\nu}(\omega)\coth\left(\frac{\omega}{2T}\right)\sin\omega
t,$ $\displaystyle\delta\Omega_{\mu\nu}^{\rm
diag}(t)=\int_{0}^{\infty}\frac{d\omega}{\omega}[J_{\mu}(\omega)-J_{\nu}(\omega)](1-\cos\omega
t).$ (38)
The contribution of the non-diagonal fluctuations of the environment to the
evolution of the electron operators $\langle\rho_{\mu\nu}\rangle$ is defined
by Eq. (31). To calculate the products of exciton variables taken at different
moments of time, for example,
$\rho_{\mu\alpha}(t)\rho_{\bar{\mu}\bar{\nu}}(t_{1})$, we use Eq. (27), which
describes the evolution of exciton operators in the presence of strong
coupling to the diagonal operators, $Q_{\mu}$, of the environment. We assume
that the interaction with the non-diagonal environment operators,
$\tilde{Q}_{\mu\nu}$, is weak. With Eq. (27) we express the operators at time
$t_{1}$ in terms of operators taken at time $t$:
$\displaystyle\rho_{\bar{\mu}\bar{\nu}}(t_{1})=\exp\left[-i\Omega_{\bar{\mu}\bar{\nu}}\tau\right]\exp\left[iu_{\bar{\mu}\bar{\nu}}(\tau)\right]\exp\left[-iv_{\bar{\mu}\bar{\nu}}(t,t_{1})\right]\rho_{\mu\nu}(t),$
$\displaystyle\rho_{\bar{\mu}\bar{\nu}}(t_{1})=\rho_{\mu\nu}(t)\exp\left[-i\Omega_{\bar{\mu}\bar{\nu}}\tau\right]\exp\left[-iu_{\bar{\mu}\bar{\nu}}(\tau)\right]\exp\left[-iv_{\bar{\mu}\bar{\nu}}(t,t_{1})\right],$
(39)
where $\tau=t-t_{1}$, and
$\displaystyle
u_{\mu\nu}(\tau)=\int_{0}^{\infty}\frac{d\omega}{\omega}\bar{J}_{\mu\nu}(\omega)\sin\omega\tau,$
$\displaystyle
v_{\mu\nu}(t,t_{1})=\sum_{j}(\Lambda_{j}^{\mu}-\Lambda_{j}^{\nu})[p_{j}(t)-p_{j}(t_{1})].$
(40)
Here we assume that $p_{j}(t),p_{j}(t_{1})$ are free-evolving momentum
operators of the environment, which are described by Gaussian statistics with
a correlation function
$\left\langle\frac{1}{2}\left[\;p_{j}(t),p_{j}(t_{1})\right]_{+}\right\rangle=\frac{\hbar
m_{j}\omega_{j}}{2}\coth\left(\frac{\hbar\omega_{j}}{2T}\right)\cos\omega_{j}(t-t_{1}).$
(41)
The operator function $v_{\mu\nu}(t,t_{1})$ does not commute with the exciton
matrix $\rho_{\mu\nu}(t)$, and, therefore, we need two expressions for the
operator $\rho_{\mu\nu}(t_{1})$, which are distinguished by the order of the
operators $\rho_{\mu\nu}(t)$ and $\exp\left[-iv_{\mu\nu}(t,t_{1})\right].$ For
the average value of the operator $\exp\left[-iv_{\mu\nu}(t,t_{1})\right]$ we
obtain
$\displaystyle\langle\exp\left[-iv_{\mu\nu}(t,t_{1})\right]\rangle=\exp\left\\{-\int_{0}^{\infty}\frac{d\omega}{\omega^{2}}\bar{J}_{\mu\nu}(\omega)\coth\left(\frac{\hbar\omega}{2T}\right)[1-\cos\omega(t-t_{1})]\right\\}.$
(42)
Substituting Eqs. (39) to Eq. (31) and using the secular approximation we
obtain a contribution of the non-diagonal environment operators,
$\tilde{Q}_{\mu\nu}$, to the evolution of diagonal exciton operators
$\langle\rho_{\mu}\rangle$,
$\displaystyle\langle-i[\rho_{\mu},H_{e-{\rm ph}}^{\rm
n-diag}]_{-}\rangle=-\sum_{\alpha}\tilde{\gamma}_{\alpha\mu}(t)\langle\rho_{\mu}\rangle+\sum_{\alpha}\tilde{\gamma}_{\mu\alpha}(t)\langle\rho_{\alpha}\rangle,$
(43)
characterized by the following relaxation matrix,
$\displaystyle\tilde{\gamma}_{\mu\alpha}(t)=\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\alpha\mu}^{(0)}(t)\tilde{Q}_{\mu\alpha}^{(0)})(t_{1})\rangle
e^{-i\Omega_{\mu\alpha}(t-t_{1})}e^{-iu_{\mu\alpha}(t-t_{1})}\langle
e^{-iv_{\mu\alpha}(t,t_{1})}\rangle+$
$\displaystyle\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\alpha\mu}^{(0)}(t_{1})\tilde{Q}_{\mu\alpha}^{(0)})(t)\rangle
e^{-i\Omega_{\alpha\mu}(t-t_{1})}e^{iu_{\alpha\mu}(t-t_{1})}\langle
e^{-iv_{\alpha\mu}(t,t_{1})}\rangle,$ (44)
where
$\displaystyle\langle\tilde{Q}_{\alpha\mu}^{(0)}(t)\tilde{Q}_{\mu\alpha}^{(0)})(t_{1})\rangle=(1/2)\int_{0}^{\infty}\tilde{J}_{\alpha\mu}(\omega)\times$
$\displaystyle\left\\{\left[\coth\left(\frac{\omega}{2T}\right)-1\right]e^{i\omega(t-t_{1})}+\left[\coth\left(\frac{\omega}{2T}\right)+1\right]e^{-i\omega(t-t_{1})}\right\\}.$
(45)
When the environment is at high temperatures ($2T\gg\omega$) and at low
frequencies of the diagonal fluctuations ($\omega\tau\ll 1$) we have:
$u_{\mu\nu}(\tau)\simeq\bar{\lambda}_{\mu\nu}\tau,$
and
$\langle\exp[-iv_{\mu\nu}(t,t_{1})]\rangle\simeq\exp[-\bar{\lambda}_{\mu\nu}T(t-t_{1})^{2}].$
With these assumptions the relaxation matrix has a simple form
$\displaystyle\tilde{\gamma}_{\mu\alpha}=\sqrt{\frac{\pi}{\bar{\lambda}_{\alpha\mu}}}\int_{0}^{\infty}d\omega\,\tilde{J}_{\alpha\mu}(\omega)\,n(\omega)\times$
$\displaystyle\left\\{\exp\left[-\frac{(\omega+\Omega_{\alpha\mu}-\bar{\lambda}_{\alpha\mu})^{2}}{4\bar{\lambda}_{\alpha\mu}T}\right]+\exp\left(\frac{\omega}{T}\right)\exp\left[-\frac{(\omega-\Omega_{\alpha\mu}+\bar{\lambda}_{\alpha\mu})^{2}}{4\bar{\lambda}_{\alpha\mu}T}\right]\right\\},$
(46)
where $n(\omega)=[\exp(\omega/T)-1]^{-1}$ is the Bose distribution function at
the temperature $T$. The moment of time $t$ in the expression (44) for the
relaxation matrix is usually higher than the effective retardation time,
$\tau_{c}\sim(\bar{\lambda}_{\alpha\mu}T)^{-1/2}$, of the integrand in Eq.
(44): $t\gg\tau_{c}$. Therefore, we assume that $t\simeq\infty$, so that
$\tilde{\gamma}_{\mu\alpha}(t)\simeq\tilde{\gamma}_{\mu\alpha}(\infty)=\tilde{\gamma}_{\mu\alpha}.$
It follows from Eq. (31) that a contribution of the non-diagonal environment
operators $\tilde{Q}_{\mu\nu}$ to the evolution of the off-diagonal elements
$\rho_{\mu\nu}$ is given by the formula
$\displaystyle\langle-i[\rho_{\mu\nu},H_{e-{\rm ph}}^{\rm
n-diag}]_{-}\rangle=-(\tilde{\Gamma}_{\mu\nu}+i\delta\tilde{\Omega}_{\mu\nu})(t)\langle\rho_{\mu\nu}(t)\rangle,$
(47)
where
$\displaystyle\tilde{\Gamma}_{\mu\nu}(t)+i\delta\tilde{\Omega}_{\mu\nu}(t)=\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\nu\alpha}^{(0)}(t)\tilde{Q}_{\alpha\nu}^{(0)}(t_{1})\rangle
e^{-i\Omega_{\alpha\nu}(t-t_{1})}e^{-iu_{\alpha\nu}(t-t_{1})}\langle
e^{-iv_{\alpha\nu}(t,t_{1})}\rangle+$
$\displaystyle\int_{0}^{t}dt_{1}\langle\tilde{Q}_{\mu\alpha}^{(0)}(t)\tilde{Q}_{\alpha\mu}^{(0)}(t_{1})\rangle
e^{-i\Omega_{\mu\alpha}(t-t_{1})}e^{iu_{\mu\alpha}(t-t_{1})}\langle
e^{-iv_{\mu\alpha}(t,t_{1})}\rangle.$ (48)
A small frequency shift, $\delta\tilde{\Omega}_{\mu\nu},$ can be hereafter
ignored. The dephasing rate, $\tilde{\Gamma}_{\mu\nu}$, has two parts,
$\tilde{\Gamma}_{\mu\nu}=\tilde{\Gamma}_{\mu}+\tilde{\Gamma}_{\nu},$ where
$\displaystyle\tilde{\Gamma}_{\mu}=\frac{1}{2}\sum_{\alpha}\sqrt{\frac{\pi}{\bar{\lambda}_{\mu\alpha}T}}\int_{0}^{\infty}d\omega\tilde{J}_{\mu\alpha}(\omega)n(\omega)\times$
$\displaystyle\left\\{\exp\left[-\frac{(\omega+\Omega_{\mu\alpha}-\bar{\lambda}_{\mu\alpha})^{2}}{4\bar{\lambda}_{\mu\alpha}T}\right]+\exp\left(\frac{\omega}{T}\right)\exp\left[-\frac{(\omega-\Omega_{\mu\alpha}+\bar{\lambda}_{\mu\alpha})^{2}}{4\bar{\lambda}_{\mu\alpha}T}\right]\right\\}.$
(49)
We note that
$\tilde{\Gamma}_{\mu}=(1/2)\sum_{\alpha}\tilde{\gamma}_{\alpha\mu},$ and
$\Omega_{\mu\nu}=\omega_{\mu\nu}-\lambda_{\mu}+\lambda_{\nu}$ from Eq.
(28),(36).
Assuming that the environment fluctuations acting on each electron-binding
site are independent and using Eq. (13) for the coefficients
$\Lambda_{j}^{\mu\nu}$, we obtain
$\displaystyle\tilde{J}_{\mu\nu}(\omega)=\sum_{l}\left[J_{lS}(\omega)|\langle\mu|S_{l}|\nu\rangle|^{2}+J_{lM}(\omega)|\langle\mu|M_{l}|\nu\rangle|^{2}\right]+J_{F}(\omega)|\langle\mu|n_{F}|\nu\rangle|^{2},$
(50)
where
$\displaystyle
J_{lS}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}\bar{x}^{2}_{jl}\delta(\omega-\omega_{j}),$
$\displaystyle
J_{lM}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}\tilde{x}^{2}_{jl}\delta(\omega-\omega_{j}),$
$\displaystyle
J_{F}(\omega)=\sum_{j}\frac{m_{j}\omega_{j}^{3}}{2}x_{jF}^{2}\delta(\omega-\omega_{j}).$
(51)
The results obtained above are valid for an arbitrary frequency dependence of
the spectral densities $J_{lS}(\omega),J_{lM}(\omega),J_{F}(\omega)$.
Hereafter we assume that these functions are described by the Lorentz-Drude
formula characterized by a common inverse correlation time,
$\gamma_{c}=\tau_{c}^{-1}$, and by a corresponding reorganization energy
$\lambda_{lS},\lambda_{lM},$ or $\lambda_{F}$, e.g.
$J_{lS}(\omega)=2\frac{\lambda_{lS}}{\pi}\frac{\omega\gamma_{c}}{\omega^{2}+\gamma_{c}^{2}}.$
(52)
Quenching processes also contribute to the decay of the off-diagonal elements,
$\langle\rho_{\mu\nu}\rangle$, with the following decoherence rates:
$\Gamma_{\mu\nu}^{\rm quen}=\Gamma_{\mu}^{\rm quen}+\Gamma_{\nu}^{\rm quen},$
where
$\displaystyle\Gamma_{\mu}^{\rm
quen}=\sum_{l\alpha}|\langle\mu|a_{l}^{\dagger}a_{l^{*}}|\alpha\rangle|^{2}\chi^{\prime\prime}_{l}(\omega_{\mu\alpha})\left[\coth\left(\frac{\omega_{\mu\alpha}}{2T}\right)+1\right].$
(53)
Here we consider an Ohmic quenching heat-bath with the spectral density
$\chi^{\prime\prime}_{l}(\omega)=\alpha_{l}\omega$, which is determined by a
set of site-dependent dimensionless coupling constants $\alpha_{l}\ll 1$. The
contribution of quenching to the relaxation of the diagonal elements of the
electron matrix, $\langle\rho_{\mu}\rangle$, is determined by the standard
Redfield term
$\displaystyle\gamma_{\mu\nu}^{\rm
quen}=\sum_{l}(|\langle\mu|a_{l}^{\dagger}a_{l^{*}}|\nu\rangle|^{2}+|\langle\nu|a_{l}^{\dagger}a_{l^{*}}|\mu\rangle|^{2})\chi^{\prime\prime}_{l}(\omega_{\mu\nu})\left[\coth\left(\frac{\omega_{\mu\nu}}{2T}\right)-1\right].$
(54)
As a result, we find that the time evolution of the off-diagonal elements of
the electron matrix is determined by the expression
$\displaystyle\langle\rho_{\mu\nu}\rangle(t)=\exp\,(\,i\,\omega_{\mu\nu}\,t-\bar{\lambda}_{\mu\nu}\,T\,t^{2}\,)\times\exp\,(\,-\Gamma_{\mu\nu}\,t\,)\;\rho_{\mu\nu}(0),$
(55)
with the decoherence rates $\Gamma_{\mu\nu}=\Gamma_{\mu}+\Gamma_{\nu}$, where
the coefficient $\Gamma_{\mu}$ contains contributions of the off-diagonal
fluctuations of the environment (49) as well as quenching processes
$\Gamma_{\mu}^{\rm quen}$ (53):
$\Gamma_{\mu}=\tilde{\Gamma}_{\mu}+\Gamma_{\mu}^{\rm quen}$. The evolution
starts at the moment $t=0$ with the initial matrix $\rho_{\mu\nu}(0)$. An
effect of diagonal environment fluctuations is determined by the rate
$\sqrt{\bar{\lambda}_{\mu\nu}\,T}$, where $\bar{\lambda}_{\mu\nu}$ is the
reorganization energy defined by Eq. (36) and $T$ is the temperature of the
environment.
## References
* (1) R.E. Blankenship _Molecular mechanisms of photosynthesis_ (Blackwell Science, Oxford, UK, 2002).
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* (3) G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mančal, Y. C. Cheng, R. E. Blankenship and G. R. Fleming, Nature 446, 782 (2007).
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|
arxiv-papers
| 2011-06-30T09:06:46 |
2024-09-04T02:49:20.210447
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pulak Kumar Ghosh, Anatoly Yu. Smirnov, and Franco Nori",
"submitter": "Pulak Kumar Ghosh Dr.",
"url": "https://arxiv.org/abs/1106.6150"
}
|
1106.6331
|
# Thermodynamic properties of tunneling quasiparticles in graphene-based
structures
Dima Bolmatova***e-mail: d.bolmatov@qmul.ac.uk a School of Physics,Queen Mary
University of London, Mile End Road, London, E1 4NS, UK
###### Abstract
Thermodynamic properties of quasiparticles in a graphene-based structures are
investigated. Two graphene superconducting layers (one superconducting
component is placed on the top layered-graphene structure and the other
component in the bottom) separated by oxide dielectric layers and one normal
graphene layer in the middle. The quasiparticle flow emerged due to external
gate voltage, we considered it as a gas of electron-hole pairs whose
components belong to different layers. This is a striking result in view of
the complexity of these systems: we have established that specific heat
exhibits universal (-$T^{3}$) behaviour at low $T$, independent from the gate
voltage and the superconducting gap. The experimental observation of this
theoretical prediction would be an important step towards our understanding of
critical massless matter.
###### pacs:
74.25.Jb, 81.05.ue,74.50.+r,74.45.+c
## I Introduction
Graphene is a unique system in many ways Novosel-1 ; Topsakal-1 ; Sahin-1 . It
is truly two-dimensional systems, has unusual electronic excitations described
in terms of Dirac fermions that move in a curved space, is an interesting mix
of a semiconductor (zero density of states) and a metal (gaplessness), and has
properties of soft matter. The electrons in graphene seem to be almost
insensitive to disorder and electron-electron interactions and have very long
mean free paths Denis-1 . Hence, graphene’s properties are different from what
is found in usual metals and semiconductors GN-1 ; Sahin-2 . Graphene has also
a robust but flexible structure with unusual phonon modes that do not exist in
ordinary three-dimensional solids. In some sense, graphene brings together
issues in quantum gravity and particle physics, and also from soft and hard
condensed matter. Interestingly enough, these properties can be easily
modified with the application of electric and magnetic fields, addition of
layers, control of its geometry, and chemical doping Luis-1 ; Jun-1 .
Moreover, graphene can be directly and relatively easily probed by various
scanning probe techniques from mesoscopic down to atomic scales, because it is
not buried inside a 3D structure Pan-1 . This makes graphene one of the most
versatile systems in condensed-matter research.
Figure 1: Graphene multilayer device tunneling structure. Two sheets of
superconducting graphene are separated by thin dielectric oxide layers and one
normal graphene monolayer in the middle. Separate gate electrodes make it
possible to vary independently the carrier concentration in the normal and
superconducting graphene layers.
Besides the unusual basic properties, graphene has the potential for a large
number of applications Geim-1 , from chemical sensors Chen-1 to transistors
Nel-1 ; Oos-1 . Graphene can be chemically and/or structurally modified in
order to change its functionality and henceforth its potential applications.
Moreover, graphene can be easily obtained from graphite, a material that is
abundant on the Earth s surface. This particular characteristic makes graphene
one of the most readily available materials for basic research. Whereas many
papers have been written on monolayer graphene in the past few years, only a
small fraction actually deal with multilayers Haas-1 ; Yan-1 . The majority of
the theoretical and experimental efforts have concentrated on the single
layer, perhaps because of its simplicity and the natural attraction that a one
atom thick material, which can be produced by simple methods in almost any
laboratory, creates. Nevertheless, few-layer graphene is equally interesting
and unusual with a technological potential, perhaps larger than the single
layer Bol-1 . Indeed, the theoretical understanding and experimental
exploration of multilayers is far behind the single layer.
Graphene can be considered as a semiconductor with zero band gap. Electron
energy spectrum of graphene contains two Dirac points that separate the
electron and the hole subband. In a multilayer structure the Fermi levels of
the hybrid system layers can be adjusted independently by the gate voltage.
The electron-hole symmetry near the Dirac points ensures perfect nesting
between the electron and the hole Fermi surfaces. A flow of electron-hole
pairs in the graphene-layered structures is equivalent to two oppositely
directed electrical currents in the layers. Therefore, the flow of such pairs
is a kind of superconductivity Bol-2 . It is believed that electron-hole pairs
may demonstrate superfluid behaviour Hon-1 ; Fil-1 . In this letter we
consider multilayer graphene structure and claim: flow of electron-hole pairs
consisting of an electron from the top (bottom) graphene layer and a hole from
the bottom (top) graphene layer behave as superconductive one. This phenomenon
can be obtained in a graphene-layered structure with dielectric layers,
pairing of electrons in one top (bottom) graphene layer with holes in the
bottom (top) graphene layer occurs.
In the weak-coupling limit, exciton condensation is a consequence of the
Cooper instability Kel-1 of solids with occupied conduction-band states and
empty valence-band states inside identical Fermi surfaces. Each layer has two
Dirac-cone bands centered at inequivalent points in its Brillouin zone. The
particle-hole symmetry of the Dirac equation ensures perfect nesting: the
nesting condition requires only that the Fermi surfaces be identical in area
and shape and not that layers have aligned honeycomb lattices and hence
aligned Brillouin zones, between the electron Fermi spheres in one layer and
its hole counterparts in the opposite layer, thereby driving the Cooper
instability. Global wave vector mismatches can be removed by gauge
transformations. When weak inter-valley electron-electron scattering processes
are included only simultaneous momentum shifts of both valleys in a layer are
allowed.
## II Graphene-based Structures
Multilayered structures are the building blocks of many of the most advanced
devices presently being developed and produced. They are essential elements of
the highest-performance optical sources and detectors, and are being employed
increasingly in high-speed and high-frequency digital and analog devices. The
usefulness of such structures is that they offer precise control over the
states and motions of charge carriers.
In Fig.1 we illustrate a device geometry in which flow of electron-hole pairs
may be observed and in Fig.2 we sketchily represent the dispersion relation of
elementary excitations in graphene multilayer device tunneling structure. We
have two superconducting graphene monolayers which are separated by thin oxide
barriers and one normal graphene monolayer in the middle. Top and bottom gates
($V_{tg}$ and $V_{bg}$) are used to electrostatically. The top and bottom
gates are separated manipulate the quasiparticle concentrations in the top and
bottom layers. The top and bottom gates are separated from the graphene layers
by gate oxides which are several nanometers thick. As the phenomenon of
superfluidity of electron-hole pairs has not yet been experimentally observed
in graphene multilayer structures.
A dissipationless flow of electron-hole pairs in equilibrium through the
graphene mono-layer and thin oxide layers, depending on the phase difference
between the two superconducting graphene layers. The single-particle
Hamiltonian in graphene is the two-dimensional Dirac Hamiltonian
$H=\left(\begin{array}[]{c}\begin{array}[]{cc}H_{+}&0\\\
0&H_{-}\end{array}\end{array}\right)$
where
$H_{\pm}=-i\hbar\upsilon_{F}(\sigma_{x}\partial_{x}\pm\sigma_{y}\partial_{y}+U)$
acting on a four-dimensional spinor
($\Psi_{A+}$,$\Psi_{B+}$,$\Psi_{A-}$,$\Psi_{B-}$). The indices $A,B$ label the
two sublattices of the honeycomb lattice of carbon atoms, while the indices
$\pm$ label the two valleys of the band structure. There is an additional spin
degree of freedom, which plays no role here. The 2$\times$2 Pauli matrices
$\sigma_{i}$ act on the sublattice index.
The time-reversal operator interchanges the valleys
$\Upsilon=\left(\begin{array}[]{c}\begin{array}[]{cc}0&\sigma_{z}\\\
\sigma_{z}&0\end{array}\end{array}\right)\wp=\Upsilon^{-1}$
with $\wp$ the operator of complex conjugation. We consider a sheet of
graphene in the $x-y$ plane. Electron and hole excitations are described by
the Bogoliubov-de Gennes equation
$\left(\begin{array}[]{c}\begin{array}[]{cc}H-E_{F}&\Delta\\\
\Delta^{*}&E_{F}-\Upsilon
H\Upsilon^{-1}\end{array}\end{array}\right)\left(\begin{array}[]{c}\begin{array}[]{cc}u\\\
v\end{array}\end{array}\right)=\varepsilon\left(\begin{array}[]{c}\begin{array}[]{cc}u\\\
v\end{array}\end{array}\right)$
with $u$ and $v$ the electron and hole wave functions, $\varepsilon>0$ the
excitation energy (relative to the Fermi energy $E_{F}$), $H$ the single-
particle Hamiltonian, and $\Upsilon$ the time-reversal operator. The pair
potential $\Delta(\mathbb{r})$ couples time-reversed electron and hole states.
In the absence of a magnetic field, the Hamiltonian is time-reversal
invariant, $\Upsilon H\Upsilon^{-1}=H$ and we yield two decoupled sets of four
equations each, of the form
$\left(\begin{array}[]{c}\begin{array}[]{cc}H_{\pm}-E_{F}&\Delta\\\
\Delta^{*}&E_{F}-H_{\pm}\end{array}\end{array}\right)\left(\begin{array}[]{c}\begin{array}[]{cc}u\\\
v\end{array}\end{array}\right)=\varepsilon\left(\begin{array}[]{c}\begin{array}[]{cc}u\\\
v\end{array}\end{array}\right)$
Figure 2: Scheme of dispersion relation of elementary excitations in graphene
multilayer device tunneling structure. The spectrum of elementary excitations
of quasiparticles in graphene-based structure was proposed in the He4-like
manner: the linear portion near $k=0$ represents ”phonons” (due to single
normal graphene layer in the middle of structure) and the portion near
$k=k_{0}$ corresponds to ”rotons”, which requires a minimal energy
$\Delta_{0}$ for its creation.
Separate gate electrodes make it possible to vary independently the carrier
concentration of electron-hole pairs in the normal direction ($z$-direction)
to graphene mono-layer, thin oxide layers and superconducting graphene layers.
For $-d<z<d$, the pair potential vanishes identically, disregarding any
intrinsic superconductivity of graphene, where $d$ is a total width of
graphene mono-layer and thin oxide layers. For $z<-d$ and $z>d$ the
superconducting graphene layers will induce a nonzero pair potential
$\Delta(z)$ via the proximity effect similarly to what happens in a planar
junction between a two-dimensional electron gas and a superconductorVolkov-1 .
The bulk value $\Delta_{0}e^{i\phi}$ for $z<-d$ and $\Delta_{0}e^{-i\phi}$ for
$z>d$ (with $\phi$ the superconducting phase) is reached at a distance from
the interface which becomes negligibly small if the Fermi wavelength
$\lambda_{F}^{{}^{\prime}}$ in superconducting layers is much smaller than the
value $\lambda_{F}$ in graphene mono-layer and thin oxide layers. We assume
that the electrostatic potential $U$ in graphene mono-layer, thin oxide and
superconducting graphene layers may be adjusted independently by a gate
voltage or by doping. Since the zero of potential is arbitrary, we may take:
$U(\mathbb{r})=0$ for $-d<z<d$ and $U(\mathbb{r})=-U_{0}$ otherwise. For
$U_{0}$ large positive, and $E_{F}\geq 0$, the Fermi wave vector
$k_{F}^{{}^{\prime}}\equiv
2\pi/\lambda_{F}^{{}^{\prime}}=(E_{F}+U_{0})/\hbar\upsilon_{F}$ in graphene
superconducting layers is large compared to the value $k_{F}\equiv
2\pi/\lambda_{F}=E_{F}/\hbar\upsilon_{F}$ in graphene mono-layer and thin
oxide layers (with $\upsilon_{F}$ the energy-independent velocity in
graphene).
## III Specific heat
For the graphene mono-layer and oxide thin layers (where $\Delta=0=U$) we
assume that Fermi level is tuned to the point of zero carrier concentration.
In the superconducting graphene layers there is a gap in the spectrum of
magnitude $|\Delta|=\Delta_{0}$.
The canonical partition function for this graphene-based structure can be
interpreted as a collection of normal mode oscillators, the oscillator
labelled by $\mathbb{k}$ containing $n_{\mathbb{k}}$ quanta of energy
$\varepsilon(\mathbb{k})$. First we calculate the Helmholtz free energy for
normal graphene mono-layer ($phonon$-like dependence) and thin oxide layers
(its make dispersion relation between linear and quadratic dependence smooth,
see Fig.2) which can be written as
$\displaystyle
F_{n}(T,V)=-2Nk_{B}TV\int_{-k_{d}}^{k_{d}}\frac{dk}{2\pi}k\ln{\left[1+\exp{\left(-\frac{\hbar\upsilon_{F}k}{k_{B}T}\right)}\right]}$
(1)
where $k_{d}$ ($-k_{d}$) is the largest (lowest) wavevector for which the
linear dispersion is a reasonable approximation. Therefore, the contribution
to free energy from the $normal$ region of graphene-based structure is
$\displaystyle
F_{n}(T,V)\simeq-\frac{3\zeta(3)Nk_{B}T^{3}V}{2\pi\hbar^{2}\upsilon_{F}^{2}}$
(2)
where $N$ is the number of the $2$ component Dirac flavors, $N=4$ in the
single layer graphene and $\zeta(n)$ is the Riemann zeta function.
Contribution to the specific heat per unit volume is
$\displaystyle C_{V}^{n}=-\frac{T}{V}\left(\frac{\partial^{2}F_{n}}{\partial
T^{2}}\right)_{V}=\frac{43.2k_{B}T^{2}}{\pi\hbar^{2}\upsilon_{F}^{2}}$ (3)
### III.1 Specific heat of superconducting layers: graphene-based layers
The grand potential of the system ($roton$-like dependence) is given by
$\displaystyle q(V,T)\equiv-\beta
F_{s}(V,T)=-\sum_{\mathbb{k}}\ln{[1-\exp{(-\beta\varepsilon(\mathbb{k}))}]}$
$\displaystyle\simeq\sum_{\mathbb{k}}\exp{(-\beta\varepsilon(\mathbb{k}))}\simeq\bar{N}$
(4)
where $\bar{N}$ is the ”equilibrium” number of $rotons$ in the superconducting
graphene layers. The summation over $\mathbb{k}$ may be replaced by an
integration, with the result
$\displaystyle F_{s}$ $\displaystyle=$ $\displaystyle-4k_{B}T\bar{N}$ (5)
$\displaystyle=$
$\displaystyle-8k_{B}TV\int_{k_{1}}^{k_{2}}\frac{dk}{2\pi}ke^{-\beta(\Delta+U+\hbar\upsilon_{F}(k-k_{0}))}$
where $k_{1}$ and $k_{2}$ are the lowest and highest values respectively of
$k$ for which the quadratic approximation to the dispersion curve is
reasonable (the limits of integration may be extended to $\pm\infty$). The
free energy of the $roton$-like superfluid gas is given by
$\displaystyle
F_{s}\simeq-\frac{4(k_{B}T)^{2}V}{\pi\hbar\upsilon_{F}}\left(\frac{k_{B}T}{\hbar\upsilon_{F}}-k_{0}\right)e^{-(2\Delta_{0}\cos{\phi}+U)/k_{B}T}$
(6)
The $roton$-like contribution to the specific heat we estimate as:
$C_{V}^{s}=C_{V,1}^{s}+C_{V,2}^{s}$, where
$\displaystyle
C_{V,1}^{s}=\frac{4k_{B}(k_{B}T)^{2}}{\pi(\hbar\upsilon_{F})^{2}}e^{-(2\Delta_{0}\cos{\phi}+U)/k_{B}T}\times$
$\displaystyle\left[6+4\cdot\frac{2\Delta_{0}\cos{\phi}+U}{k_{B}T}+\left(\frac{2\Delta_{0}\cos{\phi}+U}{k_{B}T}\right)^{2}\right]$
(7) $\displaystyle
C_{V,2}^{s}=-\frac{4k_{B}k_{0}(k_{B}T)}{\pi\hbar\upsilon_{F}}e^{-(2\Delta_{0}\cos{\phi}+U)/k_{B}T}\times$
$\displaystyle\left[2+2\cdot\frac{2\Delta_{0}\cos{\phi}+U}{k_{B}T}+\left(\frac{2\Delta_{0}\cos{\phi}+U}{k_{B}T}\right)^{2}\right]$
(8)
Figure 3: Dependence of specific heat versus temperature exhibits Debye-like
behaviour ($T^{3}$-like). This is a striking result in view of all four curves
sit on top of each other.
where $k_{0}=n/d$, $d=0.4$ nm is the thickness of graphene sheet, n is the
number of oxide dielectric layers (the thickness of each of them is presumed
equivalent to graphene one) between normal and superconducting graphene
layers. Rough low-temperature estimates often quote the result
$\displaystyle S\propto C_{V}^{n}+C_{V}^{s}$ (9)
where $S$ is thermopower Scar-1 . In Fig.3 is depicted the specific heat
against temperature. It is remarkable that in graphene-based superconducting
structure the phonon-assisted drag effect is suppressed. The carrier
concentration can be tuned varying different parameters: the thickness of
oxide layers, phase difference in superconducting layers, pair and
electrostatic potentials on one hand and on the another proposed structure
exhibits Debye-like behaviour.
## IV Conclusion
The practical significance of this investigation rests on the expectation that
high-quality contacts between a superconducting graphene and normal graphene
sheets can be realized. This expectation is supported by the experience with
carbon nanotubes (rolled up sheets of graphene), which have been contacted
succesfully by superconducting electrodes. Graphene-based structures provides
a unique opportunity to explore the physics of the ”relativistic Josephson
effect”, which had remained unexplored in earlier work on relativistic effects
in high-temperature and heavy-fermion superconductors.
We believe that this device structure can also lead to greater understanding
in another condensed matter systems such as high-temperature superconductors
and exciton gases. The low cost experimental techniques allow for easy
verification of the proposed hypothesis.
## V ACKNOWLEDGMENT
Authors are very indebted to C. Y. Mou, E. Babaev, M. M. Parish, M. Baxendale
and D. J. Dunstan for stimulating discussions and fruitful suggestions. We
acknowledge Myerscough Bequest and School of Physics at Queen Mary University
for financial support.
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|
arxiv-papers
| 2011-06-30T18:35:05 |
2024-09-04T02:49:20.223513
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dima Bolmatov",
"submitter": "Dima Bolmatov",
"url": "https://arxiv.org/abs/1106.6331"
}
|
1107.0080
|
# Elliptic and triangular flow of identified particles measured with ALICE
detector at the LHC.
Mikołaj Krzewicki for the ALICE Collaboration NIKHEF, Science Park 105, 1098
XG Amsterdam, The Netherlands. mikolaj.krzewicki@cern.ch
###### Abstract
We report on the first measurements of elliptic and triangular flow for
charged pions, kaons and anti-protons in lead-lead collisions at
$\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$ measured with the ALICE detector at the
LHC. We compare the observed mass splitting of differential elliptic flow at
LHC energies to RHIC measurements at lower energies and theory predictions. We
test the quark coalescence picture with the quark number scaling of elliptic
and triangular flow.
## 1 Introduction
Anisotropic flow [1], described by the coefficients in the Fourier expansion
of the azimuthal particle distribution [2], is an important probe of
collectivity in the system created in heavy ion collisions. The first
measurement of elliptic flow ($v_{2}$) at the LHC shows an increase (with
respect to RHIC energies) of the integrated value by about 30% and no
significant increase of the $p_{t}$ differential flow [3]. Hydrodynamical
models [4] predict a rise of the radial expansion velocity from RHIC to the
LHC energies which might explain this behaviour. Elliptic flow of identified
hadrons is sensitive to the hydrodynamical radial expansion of the medium.
Since flow develops into a common velocity field a mass-dependent shift
towards higher momenta occurs giving rise to mass splitting and ordering of
flow for different particle species. Triangular flow ($v_{3}$) of identified
particles in hydrodynamics is expected to exhibit similar mass scaling and
additionally it is expected to be a sensitive probe for the viscosity to
entropy ratio $\eta/s$ of the created medium [5]. The measurement of
identified particle $v_{3}$ can therefore provide a constraint on $\eta/s$
after more detailed theoretical calculations become available for LHC
energies.
We present the measurement of identified particle $v_{2}$, compare it to the
RHIC measurement and test the quark number scaling in terms of transverse
kinetic energy ($KE_{t}$ scaling). For identified particle $v_{3}$ we present
the first preliminary results.
## 2 Data analysis
For this analysis the standard ALICE minimum bias event selection was used
together with an additional requirement on the primary vertex position
$|z|<7\,\mathrm{cm}$ yielding a sample of around 4 million events. The
collision centrality was determined using the forward VZERO scintillator
arrays [6]. Particle tracking was done using the time projection chamber (TPC)
and the silicon inner tracking system with full azimuth coverage for
$|\eta|<0.8$. The particle identification was done by combining the time-of-
flight (TOF) measurement with the energy loss measurement in the TPC. The
purity was estimated to be better than $95\%$ at $p<3\,\mathrm{GeV}/c$ for
pions and kaons and at $p<5\,\mathrm{GeV}/c$ for protons. In order to reduce
the contamination from non-primary particles the reconstructed particles were
required to have a distance of closest approach to the primary vertex of less
than 1 mm. Main sources of systematic uncertainty on the flow values
considered in this analysis are non-flow, feed-down and centrality
determination. In the following figures, uncertainty bands indicate systematic
and statistical uncertainties added in quadrature. Elliptic flow is measured
using the two-particle scalar product method [7] with a large $\eta$ gap
($|\Delta\eta|>1$) to reduce the contribution from short range non-flow
correlations.
## 3 Results
Figure 1: Identified particle elliptic flow in two centrality bins compared to
a hydrodynamic prediction [4]. Theory describes the data well at low to
intermediate $p_{t}$ except for the anti-proton $v_{2}$ in the more central
collisions (10%-20%).
Figure 1 shows elliptic flow measured for more central ($10\%-20\%$) and more
peripheral ($40\%-50\%$) collisions overlaying a theoretical prediction from
ideal hydrodynamics [4]. As can be seen, the theory is in agreement with the
measurement for pions and kaons, but in the case of the more central events
this model does not describe anti-protons. The measured $v_{2}$ is compared to
the published results from the PHENIX [8] and STAR [9] collaborations in
figure 2. Since PHENIX reports a combined result for pions and kaons we only
compare the anti-proton results which show lower values in ALICE data
consistent with larger radial flow at $2.76\,\mathrm{TeV}$. In the comparison
with STAR data we observe a larger mass splitting in both pion and anti-proton
comparisons.
We report on the results at LHC energies of $KE_{t}$ scaling, introduced and
described in [10] and [11]. From figure 3 it can be seen that, within errors,
the flow of pions and kaons follows the scaling while the flow of anti-protons
deviates for the more central and the more peripheral events.
Figure 2: Elliptic flow for Pb-Pb at $\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$
compared to RHIC results shows a larger mass splitting than at RHIC energies.
Figure 3: Elliptic flow per constituent quark vs. transverse kinetic energy
per quark (the $KE_{t}$ scaling) for more central ($10\%-20\%$) and more
peripheral ($40\%-50\%$) Pb-Pb collisions at
$\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$.
Figure 4: Triangular flow for more central ($10\%-20\%$) and more peripheral
($40\%-50\%$) Pb-Pb collisions at $\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$.
Triangular flow, depicted in figure 4, qualitatively exhibits the same
features as elliptic flow, i.e. the mass splitting and mass ordering as
expected from hydrodynamic models and a crossing point between pion and proton
flow at intermediate $p_{t}$ as expected from the quark coalescence picture.
Similarly to elliptic flow, triangular flow shows deviations from $KE_{t}$
scaling (see figure 5).
Figure 5: $KE_{t}$ scaling of triangular flow for more central ($10\%-20\%$)
and more peripheral ($40\%-50\%$) Pb-Pb collisions at
$\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$.
## 4 Summary
We presented the $p_{t}$ differential elliptic flow of identified particles
for Pb-Pb collisions at $\sqrt{s_{NN}}=2.76\,\mathrm{TeV}$ measured with ALICE
and compared it to measurements at RHIC energies and a hydrodynamic model. The
model correctly describes elliptic flow of pions and kaons, but overpredicts
the flow of protons for more central collisions. Compared to the RHIC data we
observed a larger mass splitting, mostly apparent in the proton flow. We also
showed deviations of elliptic flow from $KE_{t}$ scaling. Additionally we
presented the first measurement of pt differential triangular flow of
identified particles at the LHC. We observed that $v_{3}$ has features similar
to $v_{2}$, i.e. mass scaling and a crossing point for pions and protons at
intermediate $p_{t}$.
## References
## References
* [1] J.Y.Ollitrault, Phys. Rev. D 46, 229 (1992).
* [2] S.Voloshin, Y.Zhang, Z. Phys. C 70, 665 (1996).
* [3] K.Aamodt _et al._ (ALICE Collaboration), Phys. Rev. Lett. 105, 252302 (2010).
* [4] Chun Shen, Ulrich W.Heinz, Pasi Huovinen, Huichao Song,c arXiv:1105.3226v1.
* [5] B.H.Alver, C.Gombeaud, M.Luzum, J.Y.Ollitrault, Phys. Rev. C 82, 034913 (2010).
* [6] A.Toia, these proceedings.
* [7] C.Adler _et al._ (STAR Collaboration), Phys. Rev. C 66, 034904 (2002).
* [8] S.S.Adler _et al._ (PHENIX Collaboration), Phys. Rev. Lett. 91, 182301 (2003).
* [9] B.I.Abelev _et al._ (STAR Collaboration), Phys. Rev. C 77, 054901 (2008).
* [10] A.Adare _et al._ (PHENIX collaboration), Phys. Rev. Lett. 98, 162301 (2007).
* [11] A.Taranenko (for the PHENIX Collaboration), J. Phys. G 34, S1069-1072 (2007).
|
arxiv-papers
| 2011-06-30T22:40:18 |
2024-09-04T02:49:20.234712
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.Krzewicki (for the ALICE Collaboration)",
"submitter": "Miko{\\l}aj Krzewicki",
"url": "https://arxiv.org/abs/1107.0080"
}
|
1107.0101
|
# Existence of Dyons in Minimally Gauged
Skyrme Model via Constrained Minimization
Zhifeng Gao111Email address: gzf@henu.edu.cn
Institute of Contemporary Mathematics
School of Mathematics
Henan University
Kaifeng, Henan 475001, PR China
Yisong Yang222Email address: yyang@math.poly.edu
Department of Mathematics
Polytechnic Institute of New York University
Brooklyn, New York 11201, USA
###### Abstract
We prove the existence of electrically and magnetically charged particle-like
static solutions, known as dyons, in the minimally gauged Skyrme model
developed by Brihaye, Hartmann, and Tchrakian. The solutions are spherically
symmetric, depend on two continuous parameters, and carry unit monopole and
magnetic charges but continuous Skyrme charge and non-quantized electric
charge induced from the ’t Hooft electromagnetism. The problem amounts to
obtaining a finite-energy critical point of an indefinite action functional,
arising from the presence of electricity and the Minkowski spacetime
signature. The difficulty with the absence of the Higgs field is overcome by
achieving suitable strong convergence and obtaining uniform decay estimates at
singular boundary points so that the negative sector of the action functional
becomes tractable.
Key words and phrases: Skyrme model, gauge fields, electromagnetism,
monopoles, dyons, topological invariants, calculus of variations for
indefinite action functional, constraints, weak convergence.
## 1 Introduction
It was Dirac [14] who first explored the electromagnetic duality in the
Maxwell equations and came up with a mathematical formalism of magnetic
monopoles, which was initially conceptualized by P. Curie [12]. Motivated by
the search of a quark model, Schwinger [35] extended the study of Dirac [14]
to obtain a new class of particle-like solutions of the Maxwell equations
carrying both electric and magnetic charges, called dyons, and derived an
elegant charge-quantization formula for dyons, generalizing that of Dirac for
monopoles. However, both the Dirac monopoles and Schwinger dyons are of
infinite energy and deemed unphysical. In the seminal works of Polyakov [30]
and ’t Hooft [40], finite-energy smooth monopole solutions were obtained in
non-Abelian gauge field theory. Later, Julia and Zee [23] extended the works
of Polyakov and ’t Hooft and obtained finite-energy smooth dyon solutions in
the same non-Abelian gauge field theory framework. See Manton and Sutcliffe
[28] for a review of monopoles and dyons in the context of a research
monograph on topological solitons. See also [1, 18, 32] for some earlier
reviews on the subject. In contemporary physics, monopoles and dyons are
relevant theoretical constructs for an interpretation of quark confinement
[19, 27, 36].
Mathematically, the existence of monopole and dyons is a sophisticated and
highly challenging problem. In fact, the construction of monopoles and dyons
was first made possible in the critical Bogomol’nyi [6] and Prasad–Sommerfeld
[31] (BPS) limit, although an analytic proof of existence of spherically
symmetric unit-charge monopoles was also obtained roughly at the same time
[4]. A few years later, the BPS monopoles of multiple charges were obtained by
Taubes [22, 39] using a gluing technique to patch a distribution of widely
separated unit-charge BPS monopoles together. Technically, the existence of
dyons is a more difficult problem even for spherically symmetric solutions of
unit charges. The reason is that the presence of electricity requires a non-
vanishing temporal component of gauge field as a consequence of the ’t Hooft
construction [42] of electromagnetism so that the action functional governing
the equations of motion becomes indefinite due to the Minkowski spacetime
signature. In fact, the original derivation of the BPS dyons is based on an
internal-space rotation of the BPS monopoles, also called the Julia–Zee
correspondence [1]. An analytic proof for the existence of the Julia–Zee dyons
[23], away from the BPS limit, was obtained by Schechter and Weder [34] using
a constrained minimization method. Developing this method, existence theorems
have been established for dyons in the Weinberg–Salam electroweak theory [11,
44, 45], and in the Georgi–Glashow–Skyrme model [8, 25], as well as for the
Chern–Simons vortex equations [10, 24].
It is well known that the Skyrme model [37, 38] is important for baryon
physics [17, 20, 26, 46] and soliton-like solutions in the Skyrme model,
called Skyrmions, are used to model elementary particles. Thus, in order to
investigate inter-particle forces among Skyrmions, gauge fields have been
introduced into the formalism [2, 7, 8, 9, 13, 15, 29, 43]. Here, we are
interested in the minimally gauged Skyrme model studied by Brihaye, Hartmann,
and Tchrakian [7], where the Skyrme (baryon) charge may be prescribed
explicitly in a continuous interval. The Skyrme map is hedgehog and the
presence of gauge fields makes the static solutions carry both electric and
magnetic charges. In other words, these gauged Skyrmions are dyons. In [7],
numerical solutions are obtained which convincingly support the existence of
such solutions. The purpose of this paper is to give an analytic proof for the
existence of these solutions, extending the methods developed in the earlier
studies [25, 34, 44, 45] for the dyon solutions in other models in field
theory described above. See also [5, 16]. Note that, since here we are
interested in the minimally gauged Skyrme model where no Higgs field is
present, we lose the control over the negative terms in the indefinite action
functional which can otherwise be controlled if a Higgs field is present [8,
25, 34, 44, 45]. In order to overcome this difficulty, we need to obtain
suitable uniform estimates for a minimizing sequence at singular boundary
points and to achieve strong convergence results, for the sequences of the
negative terms.
The contents of the rest of the paper are outlined as follows. In Section 2,
we review the minimally gauged Skyrme model of Brihaye–Hartmann–Tchrakian [7]
and then state our main existence theorem for dyon solutions. It is
interesting that the solutions obtained are of unit monopole and magnetic
charges but continuous Skyrme charge and non-quantized electric charge. In the
subsequent three sections, we establish this existence theorem. In Section 3,
we prove the existence of a finite-energy critical point of the indefinite
action functional by formulating and solving a constrained minimization
problem. In Section 4, we show that the critical point obtained in the
previous section for the constrained minimization problem solves the original
equations of motion by proving that the constraint does not give rise to a
Lagrange multiplier problem. In Section 5, we study the properties of the
solutions. In particular, we obtain some uniform decay estimates which allow
us to describe the dependence of the (’t Hooft) electric charge on the
asymptotic value of the electric potential function at infinity.
## 2 Dyons in the minimally gauged Skyrme model
As in the classical Skyrme model [37, 38], the minimally gauged Skyrme model
of Brihaye–Hartmann–Tchrakian [7] is built around a wave map,
$\phi=(\phi^{a})$ ($a=1,2,3,4$), from the Minkowski spacetime
${\mathbb{R}}^{3,1}$ of signature $(+---)$ into the unit sphere, $S^{3}$, in
${\mathbb{R}}^{4}$, so that $\phi$ is subject to the constraint
$|\phi|^{2}=(\phi^{a})^{2}=1$, where and in the sequel, summation convention
is implemented over repeated indices. Finite-energy condition implies that
$\phi$ approaches a fixed vector in $S^{3}$, at spatial infinity. Thus, at any
time $t=x^{0}$, $\phi$ may be viewed as a map from $S^{3}$, which is a one-
point compactification of ${\mathbb{R}}^{3}$, into $S^{3}$. Hence, $\phi$ is
naturally characterized by an integral class, say $[\phi]$, in the homotopy
group $\pi_{3}(S^{3})={\mathbb{Z}}$. The integer $[\phi]$, also identified as
the Brouwer degree of $\phi$, may be represented as a volume integral of the
form
$B_{\phi}=[\phi]=\frac{1}{12\pi^{2}}\int_{{\mathbb{R}}^{3}}\varepsilon_{ijk}\varepsilon^{abcd}\partial_{i}\phi^{a}\partial_{j}\phi^{b}\partial_{k}\phi^{c}\phi^{d}\,\mbox{d}x,$
(2.1)
where $i,j,k=1,2,3$ denote the spatial coordinate indices and $\varepsilon$ is
the Kronecker skewsymmetric tensor. This topological invariant is also
referred to as the Skyrme charge or baryon charge. Let
$(\eta_{\mu\nu})=\mbox{diag}\\{1,-1,-1,-1\\}$ ($\mu,\nu=0,1,2,3$) be the
Minkowski metric tensor and $(\eta^{\mu\nu})$ its inverse. We use
$|A_{\mu}|^{2}=\eta^{\mu\nu}A_{\mu}A_{\nu}$ to denote the squared Minkowski
norm of a 4-vector $A_{\mu}$ and
$A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$ to denote the skewsymmetric
tensor product of $A_{\mu}$ and $B_{\mu}$. The Lagrangian action density of
the Skyrme model [37, 38] is of the form
${\cal
L}=\frac{1}{2}\kappa_{1}^{2}|\partial_{\mu}\phi^{a}|^{2}-\frac{1}{2}k_{2}^{4}|\partial_{[\mu}\phi^{a}\partial_{\nu]}\phi^{b}|^{2},$
(2.2)
where $\kappa_{1},\kappa_{2}>0$ are coupling constants. The model is invariant
under any internal space rotation. That is, the model enjoys a global $O(4)$
symmetry. Such a symmetry is broken down to $SO(3)$ by suppressing the vacuum
manifold to a fixed point, say ${\bf n}=(0,0,0,1)$, which may be specified by
inserting a potential term of the form
$V=\lambda(1-\phi^{4})=\lambda(1-{\bf n}\cdot\phi)^{4},\quad\lambda>0,$ (2.3)
into the Skyrme Lagrangian density.
The ‘residual’ $SO(3)$ symmetry is now to be gauged. In order to do so, we
follow [7] to set $\phi=(\phi^{a})=(\phi^{\alpha},\phi^{4})$ and replace the
common derivative by the $SO(3)$ gauge-covariant derivative
$D_{\mu}\phi^{\alpha}=\partial_{\mu}\phi^{\alpha}+\varepsilon^{\alpha\beta\gamma}A^{\beta}_{\mu}\phi^{\gamma},\quad\alpha,\beta,\gamma=1,2,3,\quad
D_{\mu}\phi^{4}=\partial_{\mu}\phi^{4},$ (2.4)
where $A^{\alpha}_{\mu}$ is the $\alpha$-component of the $SO(3)$-gauge field
${\bf A}_{\mu}$ in the standard isovector representation ${\bf
A}_{\mu}=(A_{\mu}^{\alpha})$, which induces the gauge field strength tensor
${\bf F}_{\mu\nu}=\partial_{\mu}{\bf A}_{\nu}-\partial_{\nu}{\bf A}_{\mu}+{\bf
A}_{\mu}\times{\bf A}_{\nu}=(F^{\alpha}_{\mu\nu}).$ (2.5)
As a result, the $SO(3)$ gauged Skyrme model is then defined by the Lagrangian
density [7]
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\kappa^{4}_{0}|F^{\alpha}_{\mu\nu}|^{2}+\frac{1}{2}\kappa^{2}_{1}|D_{\mu}\phi^{a}|^{2}-\frac{1}{2}\kappa^{4}_{2}|D_{[\mu}\phi^{a}D_{\nu]}\phi^{b}|^{2}-V_{\omega}(\phi),$
(2.6)
where $\kappa_{0}>0$ and the potential function $V_{\omega}$ is taken to be
$V_{\omega}(\phi)=\lambda(\cos\omega-\phi^{4})^{2},\quad 0\leq\omega\leq\pi,$
(2.7)
with $\omega$ an additional free parameter which is used to generate a rich
vacuum manifold defined by
$|\phi^{\alpha}|=\sin\omega,\quad\phi^{4}=\cos\omega.$ (2.8)
In order to stay within the context of minimal coupling, we shall follow [7]
to set $\lambda=0$ to suppress the potential term (2.7) but maintain the
vacuum manifold (2.8) by imposing appropriate boundary condition at spatial
infinity.
Besides, since the topological integral (2.1) is not gauge-invariant, we need
to replace it by the quantity [3, 7]
$Q_{S}=B_{\phi,A}=\frac{1}{12\pi^{2}}\int_{{\mathbb{R}}^{3}}\left(\varepsilon_{ijk}\varepsilon^{abcd}D_{i}\phi^{a}D_{j}\phi^{b}D_{k}\phi^{c}\phi^{d}-3\varepsilon_{ijk}\phi^{4}F_{ij}^{\alpha}D_{k}\phi^{\alpha}\right)\,\mbox{d}x,$
(2.9)
as the Skyrme charge or baryon charge. On the other hand, following [18, 33],
the monopole charge $Q_{M}$ is given by
$Q_{M}=\frac{1}{16\pi}\int_{{\mathbb{R}}^{3}}\varepsilon_{ijk}F^{\alpha}_{ij}D_{k}\phi^{\alpha}\,\mbox{d}x,$
(2.10)
which defines the homotopy class of $\phi$ viewed as a map from a 2-sphere
near the infinity of ${\mathbb{R}}^{3}$ into the vacuum manifold described in
(2.8) which happens to be a 2-sphere as well when $\omega\in(0,\pi)$.
Following [7], we will look for solutions under the spherically symmetric
ansatz
$\displaystyle A^{\alpha}_{0}$ $\displaystyle=$ $\displaystyle
g(r)\left(\frac{x^{\alpha}}{r}\right),\quad
A^{\alpha}_{i}=\frac{a(r)-1}{r}\varepsilon_{i\alpha\beta}\left(\frac{x^{\beta}}{r}\right),$
(2.11) $\displaystyle\phi^{\alpha}$ $\displaystyle=$ $\displaystyle\sin
f(r)\left(\frac{x^{\alpha}}{r}\right),\ \ \ \phi^{4}=\cos f(r),$ (2.12)
where $r=|x|$ ($x\in{\mathbb{R}}^{3}$). Since the presence of the function $g$
gives rise to a nonvanishing temporal component of the gauge field, $g$ may be
regarded as an electric potential. With (2.12), the Skyrme charge $Q_{S}$ can
be shown to be given by [3, 7, 25]
$Q_{S}=-\frac{2}{\pi}\int_{0}^{\infty}\sin^{2}f(r)f^{\prime}(r)\,\mbox{d}r.$
(2.13)
Recall also that, with the notation $\vec{\phi}=(\phi^{\alpha})$ and the
updated gauge-covariant derivative
$D_{\mu}\vec{\phi}=\partial_{\mu}{\vec{\phi}}+{\bf
A}_{\mu}\times{\vec{\phi}},$ (2.14)
we may express the ’t Hooft electromagnetic field $F_{\mu\nu}$ by the formula
[23, 41, 42]
$F_{\mu\nu}=\frac{1}{|\vec{\phi}|}{\vec{\phi}}\cdot{\bf
F}_{\mu\nu}-\frac{1}{|\vec{\phi}|^{3}}{\vec{\phi}}\cdot(D_{\mu}{\vec{\phi}}\times
D_{\nu}{\vec{\phi}}).$ (2.15)
Inserting (2.11) and (2.12) into (2.15), we see that the electric and magnetic
fields, ${\bf E}=(E^{i})$ and ${\bf B}=(B^{i})$, are given by [18, 23, 31]
$\displaystyle E^{i}$ $\displaystyle=$
$\displaystyle-F^{0i}=\frac{x^{i}}{r}\frac{\mbox{d}g}{\mbox{d}r},$ (2.16)
$\displaystyle B^{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\epsilon_{ijk}F^{jk}=\frac{x^{i}}{r^{3}}.$ (2.17)
Therefore the magnetic charge $Q_{m}$ may be calculated immediately to give us
$Q_{m}=\frac{1}{4\pi}\,\lim_{r\to\infty}\oint_{S^{2}_{r}}{\bf
B}\cdot\mbox{d}{\bf S}=1,$ (2.18)
where $S^{2}_{r}$ denotes the 2-sphere of radius $r$, centered at the origin
in the 3-space. Similarly, the monopole charge $Q_{M}$ may be shown to be 1 as
well [25].
Within the ansatz (2.11)–(2.12), using suitable rescaling, and denoting
$\kappa^{4}_{2}\equiv\kappa$, it is shown [7] that the Lagrangian density
(2.6) may be reduced into the following one-dimensional one, after suppressing
the potential term,
${\cal L}={\cal E}_{1}-{\cal E}_{2},$ (2.19)
where
$\displaystyle{\cal E}_{1}$ $\displaystyle=$ $\displaystyle
2\left(2(a^{\prime})^{2}+\frac{(a^{2}-1)^{2}}{r^{2}}\right)+{1\over
2}\left(r^{2}(f^{\prime})^{2}+2a^{2}\sin^{2}f\right)$ (2.20)
$\displaystyle+2\kappa a^{2}\sin^{2}f\bigg{(}2f^{\prime
2}+\frac{a^{2}\sin^{2}f}{r^{2}}\bigg{)},$ $\displaystyle{\cal E}_{2}$
$\displaystyle=$ $\displaystyle r^{2}(g^{\prime})^{2}+2a^{2}g^{2},$ (2.21)
and ′ denotes the differentiation $\frac{\small\mbox{d}}{{\small\mbox{d}}r}$,
such that the associated Hamiltonian (energy) density is given by
${\cal E}={\cal E}_{1}+{\cal E}_{2}.$ (2.22)
The equations of motion of the original Lagrangian density (2.6) now become
the variational equation
$\delta{L}=0,$ (2.23)
of the static action
${L}(a,f,g)=\int_{0}^{\infty}{\cal L}\,\mbox{d}r=\int_{0}^{\infty}({\cal
E}_{1}-{\cal E}_{2})\,\mbox{d}r,$ (2.24)
which is indefinite. Explicitly, the equation (2.23) may be expressed in terms
of the unknowns $a,f,g$ as
$\displaystyle a^{\prime\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}a(a^{2}-1)+{1\over 4}a\sin^{2}f+\kappa
a\sin^{2}f(f^{\prime})^{2}$ (2.25) $\displaystyle\quad+{1\over{r^{2}}}\kappa
a^{3}\sin^{4}f-\frac{ag^{2}}{2},$ $\displaystyle
8\kappa(a^{2}\sin^{2}ff^{\prime})^{\prime}+(r^{2}f^{\prime})^{\prime}$
$\displaystyle=$ $\displaystyle 2a^{2}\sin f\cos f+8\kappa a^{2}\sin f\cos
f(f^{\prime})^{2}$ (2.26) $\displaystyle\quad+\frac{8\kappa a^{4}\sin^{3}f\cos
f}{r^{2}},$ $\displaystyle(r^{2}g^{\prime})^{\prime}$ $\displaystyle=$
$\displaystyle 2a^{2}g.$ (2.27)
We are to solve these equations under suitable boundary conditions. First we
observe in view of the ansatz (2.11)–(2.12) that the regularity of the fields
$\phi$ and $A_{\mu}$ imposes at $r=0$ the boundary condition
$a(0)=1,\quad f(0)=\pi,\quad g(0)=0.$ (2.28)
Furthermore, the finite-energy condition
$E(a,f,g)=\int_{0}^{\infty}{\cal E}\,\mbox{d}r=\int_{0}^{\infty}({\cal
E}_{1}+{\cal E}_{2})\,\mbox{d}r<\infty,$ (2.29)
the definition of the vacuum manifold (2.8), and the non-triviality of the
$g$-sector lead us to the boundary condition at $r=\infty$, given as
$a(\infty)=0,\quad f(\infty)=\omega,\quad g(\infty)=q,$ (2.30)
where $q>0$ (say) is a parameter, to be specified later, which defined the
asymptotic value of the electric potential at infinity.
Applying the boundary conditions (2.28) and (2.30) in (2.13), we obtain
$Q_{S}=Q_{S}(\omega)=1+\frac{1}{\pi}\left(\frac{1}{2}\sin(2\omega)-\omega\right),$
(2.31)
which is strictly decreasing for $\omega\in[0,\pi]$ with
$Q_{S}(0)=1,Q_{S}(\frac{\pi}{2})=\frac{1}{2},Q_{S}(\pi)=0$, and the range of
$Q_{S}(\omega)$ over $[0,\pi]$ is the entire interval $[0,1]$.
We now evaluate the electric charge. Using (2.16) and the equation (2.27), we
see that the electric charge $Q_{e}$ is given by
$\displaystyle Q_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\lim_{r\to\infty}\oint_{S^{2}_{r}}{\bf
E}\cdot\,\mbox{d}{\bf
S}=\frac{1}{4\pi}\,\lim_{r\to\infty}\int_{|x|<r}\nabla\cdot{\bf
E}\,\mbox{d}x=\frac{1}{4\pi}\,\lim_{r\to\infty}\int_{|x|<r}\partial_{i}\left(\frac{x^{i}}{r}\frac{\mbox{d}g}{\mbox{d}r}\right)\,\mbox{d}x$
(2.32) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{\mbox{d}}{\mbox{d}r}\left(r^{2}\frac{\mbox{d}g}{\mbox{d}r}\right)\,\mbox{d}r=2\int_{0}^{\infty}a^{2}(r)g(r)\,\mbox{d}r.$
With the above preparation, we can state our main result regarding the
existence of dyon solitons in the minimally gauged Skyrme model [7] as
follows.
###### Theorem 2.1
. For any parameters $\omega$ and $q$ satisfying
$\frac{\pi}{2}<\omega<\pi,\quad
0<q<\min\left\\{\frac{1}{\sqrt{2}}\sin\omega,\sqrt{2}\left(1-\frac{\omega}{\pi}\right)\right\\},$
(2.33)
the equations of motion of the minimally gauged Skyrme model defined by the
Lagrangian density (2.6), with $\lambda=0$, have a static finite-energy
spherically symmetric solution described by the ansatz (2.11)–(2.12) so that
$(a,f,g)$ satisfies the boundary conditions (2.28) and (2.30),
$a(r)>0,\omega<f(r)<\pi,0<g(r)<q$ for all $r>0$, and $a,f,g$ are strictly
monotone functions of $r$. Moreover, $a(r)$ vanishes at infinity exponentially
fast and $f(r),g(r)$ approach their limiting values at the rate O$(r^{-1})$ as
$r\to\infty$. The solution carries a unit monopole charge, $Q_{M}=1$, a
continuous Skyrme charge $Q_{S}$ given as function of $\omega$ by
$Q_{S}(\omega)=1+\frac{1}{\pi}\left(\frac{1}{2}\sin(2\omega)-\omega\right),\quad\frac{\pi}{2}<\omega<\pi,$
(2.34)
which may assume any value in the interval $(0,\frac{1}{2})$, a unit magnetic
charge $Q_{m}=1$, and an electric charge $Q_{e}$ given by the integral
$Q_{e}=2\int_{0}^{\infty}a^{2}(r)g(r)\,{\rm\mbox{d}}r>0,$ (2.35)
which depends on $q$ and approaches zero as $q\to 0$.
It is interesting that the ’t Hooft electric charge $Q_{e}$ cannot be
quantized as stated in the Dirac quantization formula, which reads in
normalized units [33],
$q_{e}q_{m}=\frac{n}{2},\quad n\in{\mathbb{Z}},$ (2.36)
where $q_{e}$ and $q_{e}$ are electric and magnetic charges, respectively.
Indeed, according to Theorem 2.1, $Q_{e}=0$ is an accumulation point of the
set of electric charges of the model. On the other hand, the formula (2.36)
says that, for $q_{m}>0$, the smallest positive value of $q_{e}$ is
$(2q_{m})^{-1}$.
We note that the expression (2.35) suggests that $Q_{e}$ should depend on $q$
continuously, although a proof of this statement is yet to be worked out.
The above theorem will be established in the subsequent sections.
## 3 Constrained minimization problem
We first observe that the action density (2.19) is invariant under the
transformation $f\mapsto\pi-f$. Hence we may ‘normalize’ the boundary
conditions (2.28) and (2.30) into
$\displaystyle a(0)$ $\displaystyle=$ $\displaystyle 1,\quad f(0)=0,\quad
g(0)=0,$ (3.1) $\displaystyle a(\infty)$ $\displaystyle=$ $\displaystyle
0,\quad f(\infty)=\pi-\omega,\quad g(\infty)=q,$ (3.2)
The proof of our main existence theorem, Theorem 2.1, for the dyon solutions
in the minimally gauged Skyrme model amounts to establishing the following.
###### Theorem 3.1
. Given $\omega$ satisfying
$\frac{\pi}{2}<\omega<\pi,$ (3.3)
set
$q_{\omega}=\frac{\sqrt{2}}{\pi}(\pi-\omega).$ (3.4)
For any constant $q$ satisfying $0<q<q_{\omega}$ where $\omega$ lies in the
interval (3.3), and
$q<\frac{1}{\sqrt{2}}\sin\omega,$ (3.5)
the functional (2.24) has a finite-energy critical point $(a,f,g)$ which
satisfies the equations (2.25)–(2.27) and the boundary conditions (3.1)–(3.2).
Furthermore, such a solution enjoys the property that $f(r),g(r)$ are strictly
increasing, and $a(r)>0,0<f(r)<\pi-\omega,0<g(r)<q$, for $r>0$.
The proof of the theorem will be carried out through establishing a series of
lemmas. In this section, we concentrate on formulating and solving a
constrained minimization problem put forth to overcome the difficulty arising
from the negative terms in the action functional (2.24). In the next section,
we show that the solution obtained in this section is indeed a critical point
of (2.24) so that the constraint does not give rise to a Lagrangian multiplier
problem.
To proceed, we begin by defining the admissible space of our one-dimensional
variational problem to be
$\displaystyle{\cal A}$ $\displaystyle=$
$\displaystyle\left\\{(a,f,g)|a,f,g\mbox{ are continuous functions over
$[0,\infty)$ which are absolutely}\right.$ continuous on any compact
subinterval of $(0,\infty)$, satisfy the boundary conditions
$a(0)=1,a(\infty)=0,f(0)=0,f(\infty)=\pi-\omega,$ $\displaystyle
g(\infty)=q,\left.\mbox{ and of finite-energy $E(a,f,g)<\infty$}\right\\}.$
Note that in the admissible space $\cal A$ we only implement partially the
boundary conditions (3.1)–(3.2) to ensure the compatibility with the
minimization process. The full set of the boundary conditions will eventually
be recovered in the solution process.
In order to tackle the problem arising from the negative terms involving the
function $g$ in the action (2.24), we use the methods developed in [25, 34,
44] by imposing the constraint
$\int_{0}^{\infty}(r^{2}g^{\prime}G^{\prime}+2a^{2}gG)\,\mbox{d}r=0,$ (3.6)
to ‘freeze’ the troublesome $g$-sector, where $G$ is an arbitrary test
function satisfying $G(\infty)=0$ and
$E_{2}(a,G)=\int_{0}^{\infty}(r^{2}[G^{\prime}]^{2}+2a^{2}G^{2})\,\mbox{d}r<\infty.$
(3.7)
That is, for given $a$, the function $g$ is taken to be a critical point of
the energy functional $E_{2}(a,\cdot)$ subject to the boundary condition
$g(\infty)=q$.
We now define our constrained class $\mathcal{C}$ to be
${\cal C}=\\{(a,f,g)\in{\cal A}|\,(a,f,g)\mbox{ satisfies (\ref{8})}\\}.$
(3.8)
In the rest of this section, we shall study the following constrained
minimization problem
$\min\left\\{L(a,f,g)|(a,f,g)\in\mathcal{C}\right\\}.$ (3.9)
###### Lemma 3.2
. Assume (3.3). For the problem (3.9), we may always restrict our attention to
functions $f$ satisfying $0\leq f\leq\pi/2$.
Proof.Since the action (2.24) is even in $f$, it is clear that
$L(a,f,g)=L(a,|f|,g)$. Hence we may assume $f\geq 0$ in the minimization
problem. Besides, since $f(\infty)=\pi-\omega<\frac{\pi}{2}$, if there is some
$r_{0}>0$ such that $f(r_{0})>\frac{\pi}{2}$, then there is an interval
$(r_{1},r_{2})$ with $0\leq r_{1}<r_{0}<r_{2}<\infty$ such that
$f(r)>\frac{\pi}{2}$ ($r\in(r_{1},r_{2})$) and
$f(r_{1})=f(r_{2})=\frac{\pi}{2}$. We now modify $f$ by reflecting $f$ over
the interval $[r_{1},r_{2}]$ with respect to the level $\frac{\pi}{2}$ to get
a new function $\tilde{f}$ satisfying $\tilde{f}(r)=\pi-f(r)$
($r\in[r_{1},r_{2}]$) and $\tilde{f}(r)=f(r)$ ($r\not\in[r_{1},r_{2}]$). We
have $L(a,f,g)\geq L(a,\tilde{f},g)$ again.
###### Lemma 3.3
. The constrained admissible class $\cal C$ defined in (3.8) is non-empty.
Furthermore, if $q>0$ and $(a,f,g)\in{\cal C}$, we have $0<g(r)<q$ for all
$r>0$ and that $g$ is the unique solution to the minimization problem
$\min\left\\{E_{2}(a,G)\,\bigg{|}\,G(\infty)=q\right\\}.$ (3.10)
Proof.Consider the problem (3.10). Then the Schwartz inequality gives us the
asymptotic estimate
$|G(r)-q|\leq\int_{r}^{\infty}\left|G^{\prime}(\rho)\right|\,\mbox{d}\rho\leq
r^{-\frac{1}{2}}\left(\int_{r}^{\infty}\rho^{2}(G^{\prime}(\rho))^{2}\,\mbox{d}\rho\right)^{1\over
2}\leq r^{-{1\over 2}}E_{2}^{\frac{1}{2}}(a,G),$ (3.11)
which indicates that the limiting behavior $G(\infty)=q$ can be preserved for
any minimizing sequence of the problem (3.10). Hence (3.10) is solvable. In
fact, it has a unique solution, say $g$, for any given function $a$, since the
functional $E_{2}(a,\cdot)$ is strictly convex. Since $E_{2}(a,\cdot)$ is
even, we have $g\geq 0$. Applying the maximum principle in (2.27), we conclude
with $0<g(r)<q$ for all $r>0$. The uniqueness of the solution to (3.10), for
given $a$, is obvious.
###### Lemma 3.4
. For any $(a,f,g)\in\mathcal{C}$, $g(r)$ is nondecreasing for $r>0$ and
$g(0)=0.$
Proof.To proceed, we first claim that
$\liminf\limits_{r\rightarrow 0}r^{2}|g^{\prime}(r)|=0.$ (3.12)
Indeed, if (3.12) is false, then there are $\epsilon_{0},\delta>0$, such that
$r^{2}|g^{\prime}(r)|\geq\epsilon_{0}$ for $0<r<\delta$, which contradicts the
convergence of the integral $\int_{0}^{\infty}r^{2}(g^{\prime})^{2}\mbox{d}r$.
As an immediate consequence, (3.12) implies that there is a sequence
$\\{r_{k}\\}$, such that $r_{k}\rightarrow 0$ and
$r_{k}^{2}|g^{\prime}(r_{k})|\rightarrow 0$, as $k\rightarrow\infty$. In view
of this fact and (2.27), we have
$\displaystyle r^{2}g^{\prime}(r)$ $\displaystyle=$ $\displaystyle
r^{2}g^{\prime}(r)-\lim\limits_{k\rightarrow\infty}r_{k}^{2}g^{\prime}(r_{k})$
(3.13) $\displaystyle=$
$\displaystyle\lim\limits_{k\rightarrow\infty}\int_{r_{k}}^{r}(\rho^{2}g^{\prime}(\rho))^{\prime}\,\mbox{d}\rho=\int_{0}^{r}(\rho^{2}g^{\prime}(\rho))^{\prime}\,\mbox{d}\rho$
$\displaystyle=$
$\displaystyle\int_{0}^{r}2a^{2}(\rho)g(\rho)\,\mbox{d}\rho\geq 0,\quad r>0.$
Hence $g^{\prime}(r)\geq 0$ and $g(r)$ is nondecreasing. In particular, we
conclude that there is number $g_{0}\geq 0$ such that
$\lim\limits_{r\rightarrow 0}g(r)=g_{0}.$ (3.14)
We will need to show $g_{0}=0$. Otherwise, if $g_{0}>0$, we can use
$a(0)=1,r^{2}g^{\prime}(r)\rightarrow 0$ ($r\rightarrow 0$) (this latter
result follows from (3.13)), and L’Hopital’s rule to get
$2g_{0}=2\lim\limits_{r\rightarrow 0}a^{2}(r)g(r)=\lim\limits_{r\rightarrow
0}(r^{2}g^{\prime})^{\prime}=\lim\limits_{r\rightarrow
0}\frac{r^{2}g^{\prime}(r)}{r}=\lim\limits_{r\rightarrow 0}rg^{\prime}(r).$
Hence, there is a $\delta>0$, such that
$g^{\prime}(r)\geq\frac{g_{0}}{r},\quad 0<r<\delta.$ (3.15)
Integrating (3.15), we obtain
$\left|g(r_{2})-g(r_{1})\right|\geq g_{0}\left|\ln\frac{r_{2}}{r_{1}}\right|,$
which contradicts the existence of limit stated in (3.14). So $g_{0}=0,$ and
the lemma follows.
###### Lemma 3.5
. With (3.3) and (3.4), for any $0<q<q_{\omega}$,
$q<\frac{1}{\sqrt{2}}(\pi-\omega),$ (3.16)
and $(a,f,g)\in{\cal C}$, we have the following partial coercive lower
estimate
$\displaystyle L(a,f,g)$ $\displaystyle\geq$
$\displaystyle\int_{0}^{\infty}{\rm\mbox{d}}r\left\\{2\left(2(a^{\prime})^{2}+\frac{(a^{2}-1)^{2}}{r^{2}}\right)+C_{1}r^{2}(f^{\prime})^{2}\right.$
(3.17) $\displaystyle\left.+2\kappa
a^{2}\sin^{2}f\bigg{(}2(f^{\prime})^{2}+\frac{a^{2}\sin^{2}f}{r^{2}}\bigg{)}+C_{2}a^{2}f^{2}\right\\},$
where $C_{1},C_{2}>0$ are constants depending on $\omega$ and $q$ only.
Proof.For any $(a,f,g)\in\mathcal{C}$, set $g_{1}={q}{(\pi-\omega)^{-1}}f$.
Then $g_{1}$ satisfies $g_{1}(\infty)=q$. As a consequence, we have
$E_{2}(a,g_{1})\geq E_{2}(a,g),$ (3.18)
and thus,
$\displaystyle L(a,f,g)=E_{1}(a,f)-E_{2}(a,g)\geq E_{1}(a,f)-E_{2}(a,g_{1})$
$\displaystyle=\int_{0}^{\infty}\mbox{d}r\left\\{2\left(2(a^{\prime})^{2}+\frac{(a^{2}-1)^{2}}{r^{2}}\right)+\left({1\over
2}-\frac{q^{2}}{(\pi-\omega)^{2}}\right)r^{2}(f^{\prime})^{2}\right.$
$\displaystyle+2\kappa
a^{2}\sin^{2}f\bigg{(}2(f^{\prime})^{2}+\frac{a^{2}\sin^{2}f}{r^{2}}\bigg{)}\left.+\left(\frac{\sin^{2}f}{f^{2}}-\frac{2q^{2}}{(\pi-\omega)^{2}}\right)a^{2}f^{2}\right\\}.$
(3.19)
Using the elementary inequality $\frac{\sin t}{t}\geq\frac{2}{\pi}$
($0<t\leq\frac{\pi}{2}$) and Lemma 3.2, we have
$\displaystyle\left(\frac{\sin^{2}f}{f^{2}}-\frac{2q^{2}}{(\pi-\omega)^{2}}\right)f^{2}$
$\displaystyle\geq$ $\displaystyle
2\left(\frac{2}{\pi^{2}}-\frac{q^{2}}{(\pi-\omega)^{2}}\right)f^{2}$ (3.20)
$\displaystyle=$
$\displaystyle\frac{2}{(\pi-\omega)^{2}}(q_{\omega}^{2}-q^{2})f^{2}\equiv
C_{2}f^{2}.$
Inserting (3.20) into (3) and setting in (3) the quantity
$C_{1}\equiv{1\over
2}-\frac{q^{2}}{(\pi-\omega)^{2}}=\frac{1}{(\pi-\omega)^{2}}\left(\frac{1}{2}(\pi-\omega)^{2}-q^{2}\right)>0,$
(3.21)
in view of (3.16), we see that the lower estimate (3.17) is established.
###### Lemma 3.6
. Under the conditions stated in Theorem 3.1, the constrained minimization
problem (3.9) has a solution.
Proof.We start by observing that the condition (3.16) is implied by the
condition (3.5). So Lemma 3.5 is valid. Hence, applying Lemma 3.5, we see that
$\eta=\inf\\{L(a,f,g)\,|\,(a,f,g)\in{\cal C}\\}$ (3.22)
is well defined. Let $\\{(a_{n},f_{n},g_{n})\\}$ denote any minimizing
sequence of (3.9). That is, $(a_{n},f_{n},g_{n})\in{\cal C}$ and
$L(a_{n},f_{n},g_{n})\to\eta$ as $n\to\infty$. Without loss of generality, we
may assume $L(a_{n},f_{n},g_{n})\leq\eta+1$ (say) for all $n$. In view of
(3.17) and the Schwartz inequality, we have
$|a_{n}(r)-1|\leq\int_{0}^{r}\left|a^{\prime}_{n}(\rho)\right|\,\mbox{d}\rho\leq
r^{\frac{1}{2}}\left(\int_{0}^{r}(a_{n}^{\prime}(\rho))^{2}\,\mbox{d}\rho\right)^{\frac{1}{2}}\leq
Cr^{\frac{1}{2}}(\eta+1)^{\frac{1}{2}},$ (3.23)
$\left|f_{n}(r)-(\pi-\omega)\right|\leq\int_{r}^{\infty}\left|f^{\prime}_{n}(\rho)\right|\,\mbox{d}\rho\leq
r^{-\frac{1}{2}}\left(\int_{r}^{\infty}\rho^{2}(f^{\prime}_{n}(\rho))^{2}\,\mbox{d}\rho\right)^{\frac{1}{2}}\leq
Cr^{-\frac{1}{2}}(\eta+1)^{\frac{1}{2}},$ (3.24)
where $C>0$ is a constant independent of $n$. In particular,
$a_{n}(r)\rightarrow 1$ and $f_{n}(r)\rightarrow(\pi-\omega)$ uniformly as
$r\rightarrow 0$ and $r\rightarrow\infty$, respectively.
For any $(a_{n},f_{n},g_{n})$, the function
$G_{n}=\frac{{q}}{{(\pi-\omega)}}f_{n}$ satisfies $G_{n}(\infty)=q$. Thus, by
virtue of the definition of $g_{n}$ and (3.17), we have
$E_{2}(a_{n},g_{n})\leq
E_{2}(a_{n},G_{n})=\frac{q^{2}}{(\pi-\omega)^{2}}\int_{0}^{\infty}(r^{2}(f^{\prime}_{n})^{2}+a_{n}^{2}f_{n}^{2})\,\mbox{d}r\leq
CL(a_{n},f_{n},g_{n}),$ (3.25)
where $C>0$ is a constant, which shows that $E_{2}(a_{n},f_{n})$ is bounded as
well.
With the above preparation, we are now ready to investigate the limit of the
sequence $\\{(a_{n},f_{n},g_{n})\\}$.
Consider the Hilbert space $(X,(\cdot,\cdot))$, where the functions in $X$ are
all continuously defined in $r\geq 0$ and vanish at $r=0$ and the inner
product $(\cdot,\cdot)$ is defined by
$(h_{1},h_{2})=\int_{0}^{\infty}h^{\prime}_{1}(r)h^{\prime}_{2}(r)\,\mbox{d}r,\
\ h_{1},h_{2}\in X.$
Since $\\{a_{n}-1\\}$ is bounded in $(X,(\cdot,\cdot))$, we may assume without
loss of generality that $\\{a_{n}\\}$ has a weak limit, say, $a$, in the same
space,
$\int_{0}^{\infty}a^{\prime}_{n}h^{\prime}\,\mbox{d}r\rightarrow\int_{0}^{\infty}a^{\prime}h^{\prime}\,\mbox{d}r,\
\ \ \forall h\in X,$
as $n\rightarrow\infty$.
Similarly, for the Hilbert space $(Y,(\cdot,\cdot))$ where the functions in
$Y$ are all continuously defined in $r>0$ and vanish at infinity and the inner
product $(\cdot,\cdot)$ is defined by
$(h_{1},h_{2})=\int_{0}^{\infty}r^{2}h_{1}^{\prime}h_{2}^{\prime}\,\mbox{d}r,\
\ h_{1},h_{2}\in Y.$
Since $\\{f_{n}-(\pi-\omega)\\}$, $\\{g_{n}-q\\}$ are bounded in
$(Y,(\cdot,\cdot))$, we may assume without loss of generality that there are
functions $f,g$ with $f(\infty)=\pi-\omega,g(\infty)=q$, and
$f-(\pi-\omega),g-q\in(Y,(\cdot,\cdot))$, such that
$\int_{0}^{\infty}r^{2}H_{n}^{\prime}h^{\prime}\,\mbox{d}r\rightarrow\int_{0}^{\infty}r^{2}H^{\prime}h^{\prime}\,\mbox{d}r,\quad\forall
h\in Y,$ (3.26)
as $n\to\infty$, for $H_{n}=f_{n}-(\pi-\omega),\ H=f-(\pi-\omega)$, and
$H_{n}=g_{n}-q,\ H=g-q$, respectively.
Next, we need to show that the weak limit $(a,f,g)$ of the minimizing sequence
$\\{(a_{n},f_{n},g_{n})\\}$ obtained above actually lies in $\mathcal{C}$.
There are two things to be verified for $(a,f,g)$: the boundary conditions and
the constraint (3.6). From the uniform estimates (3.11), (3.23), and (3.24),
we easily deduce that $a(0)=1,f(\infty)=\pi-\omega,g(\infty)=q$. Moreover,
applying Lemma 3.5, we get $a\in W^{1,2}(0,\infty)$. Hence $a(\infty)=0$. To
verify $f(0)=0$, we use (3.23) to get a $\delta>0$ such that
$|a_{n}(r)|\geq\frac{1}{2},\quad r\in[0,\delta].$ (3.27)
Then, using (3.27), we have
$\displaystyle\sin^{2}f_{n}(r)$ $\displaystyle\leq$ $\displaystyle
2\int_{0}^{r}|\sin f_{n}(\rho)f^{\prime}_{n}(\rho)|\,\mbox{d}\rho$ (3.28)
$\displaystyle\leq$ $\displaystyle
4r^{\frac{1}{2}}\left(\int_{0}^{r}a_{n}^{2}(\rho)\sin^{2}f_{n}(\rho)(f^{\prime}_{n}(\rho))^{2}\,\mbox{d}\rho\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
2\kappa^{-\frac{1}{2}}r^{\frac{1}{2}}L^{\frac{1}{2}}(a_{n},f_{n},g_{n}),\quad
r\in[0,\delta].$
Since $0\leq f_{n}\leq\frac{\pi}{2}$, we can invert (3.28) to obtain the
uniform estimate
$0\leq f_{n}(r)\leq Cr^{\frac{1}{4}},\quad r\in[0,\delta],$ (3.29)
where $C>0$ is independent of $n$. Letting $n\to\infty$ in (3.29), we see that
$f(0)=0$ as anticipated.
Thus, it remains to verify (3.6). For this purpose, it suffices to establish
the following results,
$\displaystyle\int_{0}^{\infty}(a^{2}_{n}g_{n}-a^{2}g)G\,\mbox{d}r$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (3.30) $\displaystyle\
\int_{0}^{\infty}(r^{2}g^{\prime}_{n}-r^{2}g^{\prime})G^{\prime}\,\mbox{d}r$
$\displaystyle\rightarrow$ $\displaystyle 0,$ (3.31)
for any test function $G$ satisfying (3.7) and $G(\infty)=0$, as $n\to\infty$.
From the fact $G\in Y$ and (3.26), we immediately see that (3.31) is valid.
To establish (3.30), we rewrite
$\int_{0}^{\infty}(a^{2}_{n}g_{n}-a^{2}g)G\,\mbox{d}r=\int_{0}^{\delta_{1}}+\int_{\delta_{1}}^{\delta_{2}}+\int_{\delta_{2}}^{\infty}\equiv
I_{1}+I_{2}+I_{3},$ (3.32)
for some positive constants $0<\delta_{1}<\delta_{2}<\infty$, and we begin
with
$I_{1}=\int_{0}^{\delta_{1}}(a_{n}^{2}-a^{2})g_{n}G\,\mbox{d}r+\int_{0}^{\delta_{1}}a^{2}(g_{n}-g)G\,\mbox{d}r\equiv
I_{11}+I_{12}.$ (3.33)
In view of (3.23) and (3.25), we see that there is a small $\delta>0$ such
that $g_{n}\in L^{2}(0,\delta)$ and there holds the uniform bound
$\|g_{n}\|_{L^{2}(0,\delta)}\leq K,$ (3.34)
for some constant $K>0$. Thus, we may assume $g_{n}\to g$ weakly in
$L^{2}(0,\delta)$ as $n\to\infty$. In particular, $g\in L^{2}(0,\delta)$ and
$\|g\|_{L^{2}(0,\delta)}\leq K$. Besides, since in (3.7), the function $a$
satisfies $a(0)=1$, we have $G\in L^{2}(0,\delta)$ when $\delta>0$ is chosen
small enough. Thus, using (3.23) and taking $\delta_{1}\leq\delta$, we get
$\displaystyle|I_{11}|$ $\displaystyle\leq$
$\displaystyle\int_{0}^{\delta_{1}}|a^{2}_{n}-a^{2}||g_{n}G|\,\mbox{d}r\leq\int_{0}^{\delta_{1}}\left(|a^{2}_{n}-1|+|a^{2}-1|\right)|g_{n}G|\,\mbox{d}r$
(3.35) $\displaystyle\leq$ $\displaystyle
CK\delta^{\frac{1}{2}}\|G\|_{L^{2}(0,\delta)},$
where $C>0$ is a constant independent of $n$. Thus, for any $\varepsilon>0$,
we can choose $\delta_{1}>0$ sufficiently small to get $|I_{11}|<\varepsilon$.
On the other hand, since $g_{n}\to g$ weakly in $L^{2}(0,\delta)$ and $G\in
L^{2}(0,\delta)$, we have $I_{12}\to 0$ as $n\to\infty$.
Since $\\{a_{n}\\}$ and $\\{g_{n}\\}$ are bounded sequences in
$W^{1,2}(\delta_{1},\delta_{2})$, using the compact embedding
$W^{1,2}(\delta_{1},\delta_{2})\mapsto C[\delta_{1},\delta_{2}]$, we see that
$a_{n}\to a$ and $g_{n}\to g$ uniformly over $[\delta_{1},\delta_{2}]$ as
$n\to\infty$. Thus $I_{2}\to 0$ as $n\to\infty$.
To estimate $I_{3}$, we recall that $\\{E_{2}(a_{n},g_{n})\\}$ is bounded by
(3.25), $g_{n}(r)\rightarrow q$ uniformly as $n\to\infty$ by (3.11), and
$G(r)=\mbox{O}(r^{-\frac{1}{2}})$ as $r\to\infty$ by (3.7). In particular,
since $q>0$, we may choose $r_{0}>0$ sufficiently large so that
$|g(r)|\geq\frac{q}{2},\quad\inf_{n}|g_{n}(r)|\geq\frac{q}{2},\quad r\geq
r_{0}.$ (3.36)
Combining the above facts, we arrive at
$|I_{3}|\leq\int_{r}^{\infty}\left(|a^{2}_{n}g_{n}|+|a^{2}g|\right)|G|\,\mbox{d}\rho\leq
Cr^{-\frac{1}{2}}\int^{\infty}_{r}\frac{2}{q}(a^{2}_{n}g_{n}^{2}+a^{2}g^{2})\,\mbox{d}\rho,$
(3.37)
where $r\geq r_{0}$ (cf. (3.36)) and $C>0$ is a constant. Using (3.25) in
(3.37), we see that for any $\varepsilon>0$ we may choose $\delta_{2}$ large
enough to get $|I_{3}|<\varepsilon$.
Summarizing the above discussion, we obtain
$\limsup_{n\to\infty}\left|\int_{0}^{\infty}(a^{2}_{n}g_{n}-a^{2}g)G\,\mbox{d}r\right|\leq
2\varepsilon,$ (3.38)
which proves the desired conclusion (3.30). Thus, the claim $(a,f,g)\in{\cal
C}$ follows.
To show that $(a,f,g)$ solves (3.9), we need to establish
$\eta=\liminf_{n\to\infty}L(a_{n},f_{n},g_{n})\geq L(a,f,g).$ (3.39)
This fact is not automatically valid and extra caution is to be exerted
because the functional $L$ contains negative terms.
With
$\Omega=\pi-\omega,\quad 0<\Omega<\frac{\pi}{2},$ (3.40)
we may rewrite the Lagrange density (2.19) as
${\mathcal{L}}(a,f,g)={\cal L}_{0}(a,f)-{\cal E}_{0}(a,g),$ (3.41)
where
$\displaystyle{\cal L}_{0}(a,f)$ $\displaystyle=$ $\displaystyle
2\bigg{(}2(a^{\prime})^{2}+\frac{(a^{2}-1)^{2}}{r^{2}}\bigg{)}+{1\over
2}r^{2}(f^{\prime})^{2}+2\kappa
a^{2}\sin^{2}f\bigg{(}2(f^{\prime})^{2}+\frac{a^{2}\sin^{2}f}{r^{2}}\bigg{)}$
(3.42)
$\displaystyle+a^{2}(\sin^{2}\Omega-2q^{2})+a^{2}\sin^{2}\Omega\left(\cos^{2}(f-\Omega)-1\right)$
$\displaystyle+2a^{2}\sin\Omega\cos\Omega\sin(f-\Omega)\cos(f-\Omega)+a^{2}\cos^{2}\Omega\sin^{2}(f-\Omega),$
$\displaystyle{\cal E}_{0}(a,g)$ $\displaystyle=$ $\displaystyle
r^{2}(g^{\prime})^{2}+2a^{2}(g-q)^{2}+4a^{2}(g-q)q,$ (3.43)
Thus, in order to establish (3.39), it suffices to show that
$\liminf_{n\to\infty}\int_{0}^{\infty}{\cal
L}_{0}(a_{n},f_{n})\,\mbox{d}r\geq\int_{0}^{\infty}{\cal
L}_{0}(a,f)\,\mbox{d}r,$ (3.44) $\lim_{n\to\infty}\int_{0}^{\infty}{\cal
E}_{0}(a_{n},g_{n})\,\mbox{d}r=\int_{0}^{\infty}{\cal E}_{0}(a,f)\,\mbox{d}r.$
(3.45)
We first show (3.45). To this end, we observe that, since both $(a_{n},g_{n})$
and $(a,g)$ satisfy (3.6), i.e.,
$\int_{0}^{\infty}(r^{2}g^{\prime}_{n}G^{\prime}+2a^{2}_{n}g_{n}G)\,\mbox{d}r=0,\quad\int_{0}^{\infty}(r^{2}g^{\prime}G^{\prime}+2a^{2}gG)\,\mbox{d}r=0,$
(3.46)
we can set $G=g-g_{n}$ in the above equations and subtract them to get
$\displaystyle\int_{0}^{\infty}r^{2}(g^{\prime}_{n}-g^{\prime})^{2}\mbox{d}r$
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}(a^{2}_{n}g_{n}-a^{2}g)(g-g_{n})\mbox{d}r$ (3.47)
$\displaystyle=$
$\displaystyle\int_{0}^{\delta_{1}}+\int_{\delta_{1}}^{\delta_{2}}+\int_{\delta_{2}}^{\infty}\equiv
I_{1}+I_{2}+I_{3},$
where $0<\delta_{1}<\delta_{2}<\infty$.
To study $I_{1}$, we need to get some uniform estimate for the sequence
$\\{g_{n}\\}$ near $r=0$. From (3.23), we see that for any
$0<\gamma<\frac{1}{2}$ (say) there is a $\delta>0$ such that
$2a^{2}_{n}(r)\geq(2-\gamma),\quad r\in[0,\delta].$ (3.48)
Consider the comparison function
$\sigma(r)=Cr^{1-\gamma},\quad r\in[0,\delta],\quad C>0.$ (3.49)
Then
$(r^{2}\sigma^{\prime})^{\prime}=(1-\gamma)(2-\gamma)\sigma<2a^{2}_{n}(r)\sigma,\quad
r\in[0,\delta].$ (3.50)
Consequently, we have
$(r^{2}(g_{n}-\sigma)^{\prime})^{\prime}>2a^{2}_{n}(r)(g_{n}-\sigma),\quad
r\in[0,\delta].$ (3.51)
Choose $C>0$ in (3.49) large enough so that $C\delta^{1-\gamma}\geq q$. Since
$g_{n}<q$ (Lemma 3.3), we have $(g_{n}-\sigma)(\delta)<0$ and
$(g_{n}-\sigma)(0)=0$. In view of these boundary conditions and applying the
maximum principle to (3.51), we obtain $g_{n}(r)<\sigma(r)$ for all
$r\in(0,\delta)$. Or, more precisely, we have
$0<g_{n}(r)<\left(\frac{q}{\delta^{1-\gamma}}\right)r^{1-\gamma},\quad
0<r<\delta.$ (3.52)
Of course, the weak limit $g$ of $\\{g_{n}\\}$ satisfies the same estimate.
Therefore, using the uniform estimates (3.23) and (3.52), we see that for any
$\varepsilon>0$ there is some $\delta_{1}>0$ ($\delta_{1}<\delta$) such that
$|I_{1}|<\varepsilon$.
Moreover, in view of the uniform estimate (3.11) and (3.36), we have
$\displaystyle|I_{3}|$ $\displaystyle\leq$ $\displaystyle
2\int_{\delta_{2}}^{\infty}(a_{n}^{2}g_{n}+a^{2}g)(|g_{n}-q|+|g-q|)\,\mbox{d}r$
(3.53) $\displaystyle\leq$
$\displaystyle\frac{4}{q}(|g_{n}(\delta_{2})-q|+|g(\delta_{2})-q|)\int_{0}^{\infty}(a_{n}^{2}g_{n}^{2}+a^{2}g^{2})\,\mbox{d}r$
$\displaystyle\leq$
$\displaystyle\frac{2}{q}\delta_{2}^{-\frac{1}{2}}\left(E_{2}^{\frac{1}{2}}(a_{n},g_{n})+E_{2}^{\frac{1}{2}}(a,g)\right)\left(E_{2}(a_{n},g_{n})+E_{2}(a,g)\right),$
which may be made small than $\varepsilon$ when $\delta_{2}>0$ is large enough
due to (3.25).
Furthermore, since $a_{n}\to a$ and $g_{n}\to g$ in
$C[\delta_{1},\delta_{2}]$, we see that $I_{2}\to 0$ as $n\to\infty$.
In view of the above results regarding $I_{1},I_{2},I_{3}$ in (3.47), we
obtain the strong convergence
$\lim\limits_{n\rightarrow\infty}\int_{0}^{\infty}r^{2}(g^{\prime}_{n}-g^{\prime})^{2}\,\mbox{d}r=0.$
(3.54)
In particular, we have
$\lim_{n\to\infty}\int_{0}^{\infty}r^{2}(g^{\prime}_{n})^{2}\,\mbox{d}r=\int_{0}^{\infty}r^{2}(g^{\prime})^{2}\,\mbox{d}r.$
(3.55)
We can also show that
$\lim_{n\to\infty}\int_{0}^{\infty}\left(a_{n}^{2}(g_{n}-q)^{2}+2a_{n}^{2}(g_{n}-q)q\right)\,\mbox{d}r=\int_{0}^{\infty}\left(a^{2}(g-q)^{2}+2a^{2}(g-q)q\right)\,\mbox{d}r.$
(3.56)
In fact, we have seen that $\\{(a_{n},f_{n},g_{n})\\}$ is bounded in
$W^{1,2}_{\mbox{loc}}(0,\infty)$. Thus, the sequence is convergent in
$C[\alpha,\beta]$ for any pair of numbers, $0<\alpha<\beta<\infty$. Since we
have shown that $a_{n}(r)\to 1$ and $g_{n}(r)\to 0$ as $r\to 0$ uniformly,
with respect to $n=1,2,\cdots$, we conclude that $a_{n}\to a$ and $g_{n}\to g$
uniformly over any interval $[0,\beta]$ ($0<\beta<\infty$). Thus, combining
this result with the uniform estimate (3.11), we see that (3.56) is proved.
In view of (3.55) and (3.56), we see that (3.45) follows.
On the other hand, applying the uniform estimate (3.24), we also have
$\displaystyle\lim_{n\to\infty}\int_{0}^{\infty}a_{n}^{2}(\cos^{2}(f_{n}-\Omega)-1)\,\mbox{d}r$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}a^{2}(\cos^{2}(f-\Omega)-1)\,\mbox{d}r,$ (3.57)
$\displaystyle\lim_{n\to\infty}\int_{0}^{\infty}a_{n}^{2}\sin(f_{n}-\Omega)\cos(f_{n}-\Omega)\,\mbox{d}r$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}a^{2}\sin(f-\Omega)\cos(f-\Omega)\,\mbox{d}r.$
Finally, using (3.57), (LABEL:lim2), and the condition (3.5), i.e.,
$\sin^{2}\Omega-2q^{2}>0,$ (3.59)
we see that (3.44) is established and the proof of the lemma is complete.
## 4 Fulfillment of the governing equations
Let $(a,f,g)$ be the solution of (3.9) obtained in the previous section. We
need to show that it satisfies the governing equations (2.25)–(2.27) for
dyons. Since we have solved a constrained minimization problem, we need to
prove that the Lagrange multiplier problem does not arise as a result of the
constraint, which would otherwise alter the original equations of motion. In
fact, since the constraint (3.6) involves $a$ and $g$ only and (3.6)
immediately gives rise to (2.27), we see that all we have to do is to verify
the validity of (2.25) because (2.26) is the $f$-equation and (3.6) does not
involve $f$ explicitly.
To proceed, we take $\widetilde{a}\in C^{1}_{0}$. For any $t\in{\mathbb{R}}$,
there is a unique corresponding function $g_{t}$ such that
$(a+t\widetilde{a},f,g_{t})\in\mathcal{C}$ and that $g_{t}$ smoothly depends
on $t$. Set
$g_{t}=g+\widetilde{g}_{t},\quad\widetilde{G}=\left.\left(\frac{\mbox{d}}{\mbox{d}t}\widetilde{g}_{t}\right)\right|_{t=0}.$
(4.1)
Since $(a+t\widetilde{a},f,g_{t})|_{t=0}=(a,f,g)$ is a minimizing solution of
(3.9), we have
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left.\frac{\mbox{d}}{\mbox{d}t}L(a+t\widetilde{a},f,g_{t})\right|_{t=0}$
(4.2) $\displaystyle=$ $\displaystyle
8\int_{0}^{\infty}\mbox{d}r\bigg{\\{}a^{\prime}\widetilde{a}^{\prime}+\frac{(a^{2}-1)a\widetilde{a}}{r^{2}}+\frac{1}{4}\sin^{2}f\,a\widetilde{a}$
$\displaystyle+\kappa\sin^{2}f\bigg{(}(f^{\prime})^{2}a\widetilde{a}+\frac{\sin^{2}f}{r^{2}}a^{3}\widetilde{a}\bigg{)}-\frac{1}{2}g^{2}a\widetilde{a}\bigg{\\}}-2\int_{0}^{\infty}\mbox{d}r\bigg{\\{}r^{2}g^{\prime}\widetilde{G}^{\prime}+2a^{2}g\widetilde{G}\bigg{\\}}$
$\displaystyle\equiv$ $\displaystyle 8I_{1}-2I_{2}.$
It is clear that the vanishing of $I_{1}$ implies (2.25) so that it suffices
to prove that $I_{2}$ vanishes. To this end and in view of (3.6), we only need
to show that $\widetilde{G}$ satisfies the same conditions required of $G$ in
(3.6).
In (3.6), when we make the replacements $a\mapsto a+t\widetilde{a},g\mapsto
g_{t},G\mapsto\widetilde{g}_{t}$, we have
$\int_{0}^{\infty}\left(r^{2}g^{\prime}_{t}\widetilde{g}^{\prime}_{t}+2(a+t\widetilde{a})^{2}g_{t}\widetilde{g}_{t}\right)\,\mbox{d}r=0.$
(4.3)
Or, with $g_{t}=g+\widetilde{g}_{t}$, we have
$\int_{0}^{\infty}\left(r^{2}(g^{\prime}+\widetilde{g}^{\prime}_{t})\widetilde{g}^{\prime}_{t}+2a^{2}(g+\widetilde{g}_{t})\widetilde{g}_{t}+2t^{2}\widetilde{a}^{2}g_{t}\widetilde{g}_{t}+4ta\widetilde{a}g_{t}\widetilde{g}_{t}\right)\,\mbox{d}r=0.$
(4.4)
Recall that
$\int_{0}^{\infty}\left(r^{2}g^{\prime}\widetilde{g}^{\prime}_{t}+2a^{2}g\widetilde{g}_{t}\right)\,\mbox{d}r=0$.
Thus (4.4) and the Schwartz inequality give us
$\displaystyle\int_{0}^{\infty}(r^{2}(\widetilde{g}^{\prime}_{t})^{2}+2a^{2}\widetilde{g}^{2}_{t})\,\mbox{d}r=\left|2t\int_{0}^{\infty}(t\widetilde{a}^{2}+2a\widetilde{a})g_{t}\widetilde{g}_{t}\,\mbox{d}r\right|$
(4.5)
$\displaystyle\leq|2t|\left(|2t|\int_{0}^{\infty}\widetilde{a}^{2}g_{t}^{2}\mbox{d}r+\frac{1}{|2t|}\int_{0}^{\infty}a^{2}\widetilde{g}^{2}_{t}\,\mbox{d}r\right)+2t^{2}\int_{0}^{\infty}\widetilde{a}^{2}|g_{t}|\,|\widetilde{g}_{t}|\,\mbox{d}r$
$\displaystyle=$ $\displaystyle
4t^{2}\int_{0}^{\infty}\widetilde{a}^{2}g^{2}_{t}\,\mbox{d}r+\int_{0}^{\infty}a^{2}\widetilde{g}^{2}_{t}\,\mbox{d}r+2t^{2}\int_{0}^{\infty}\widetilde{a}^{2}|g_{t}|\,|\widetilde{g}_{t}|\,\mbox{d}r.$
Applying the bounds $0\leq g,g_{t}\leq q$ and the relation
$\widetilde{g}_{t}=g_{t}-g$ in (4.5), we have
$\displaystyle\int_{0}^{\infty}(r^{2}(\widetilde{g}^{\prime}_{t})^{2}+a^{2}\widetilde{g}^{2}_{t})\,\mbox{d}r$
$\displaystyle\leq$ $\displaystyle
4t^{2}\int_{0}^{\infty}\widetilde{a}^{2}g^{2}_{t}\,\mbox{d}r+2t^{2}\int_{0}^{\infty}\widetilde{a}^{2}|g_{t}|\,|\widetilde{g}_{t}|\,\mbox{d}r$
(4.6) $\displaystyle\leq$ $\displaystyle
8q^{2}t^{2}\int_{0}^{\infty}\widetilde{a}^{2}\,\mbox{d}r.$
As a consequence, we have
$\int_{0}^{\infty}\left(r^{2}\left(\frac{\widetilde{g}^{\prime}_{t}}{t}\right)^{2}+a^{2}\left(\frac{\widetilde{g}_{t}}{t}\right)^{2}\right)\,\mbox{d}r\leq
8q^{2}\int_{0}^{\infty}\widetilde{a}^{2}\,\mbox{d}r,\quad t\neq 0.$ (4.7)
Using $\widetilde{g}_{t}(\infty)=0$, the Schwartz inequality and (4.7), we
have for $t\neq 0$ the estimate
$\left|\frac{\widetilde{g}_{t}}{t}(r)\right|\leq\int_{r}^{\infty}\left|\frac{\widetilde{g}_{t}^{\prime}(\rho)}{t}\right|\,\mbox{d}\rho\\\
\leq
r^{-\frac{1}{2}}\left(\int_{r}^{\infty}\rho^{2}\left(\frac{\widetilde{g}^{\prime}_{t}}{t}\right)^{2}\,\mbox{d}\rho\right)^{\frac{1}{2}}\leq
2\sqrt{2}q\|\widetilde{a}\|_{L^{2}(0,\infty)}.$
Letting $t\to 0$ in (4.7) and (4), we obtain $E_{2}(a,\widetilde{G})<\infty$
and $\widetilde{G}(r)=\mbox{O}(r^{-\frac{1}{2}})$ (for $r$ large). In
particular, $\widetilde{G}(\infty)=0$ and $\widetilde{G}$ indeed satisfies all
conditions required in (3.6) for $G$. Hence $I_{2}$ vanishes in (4.2).
Consequently, the equation (2.25) has been verified.
## 5 Properties of the solution obtained
In this section, we study the properties of the solution, say $(a,f,g)$, of
the equations (2.25)–(2.27) obtained as a solution of the constrained
minimization problem (3.9). We split the investigation over a few steps.
###### Lemma 5.1
. The solution $(a,f,g)$ enjoys the properties $a(r)>0$, $0<g(r)<q$,
$0<f(r)<\pi-\omega$, and both $f(r)$ and $g(r)$ are strictly increasing, for
any $r>0$.
Proof.We have $0\leq g\leq q$ and $0\leq f\leq\frac{\pi}{2}$ from Lemmas 3.2
and 3.3. Besides, it is clear that $a\geq 0$ since both (2.20) and (2.21) are
even in $a$.
If $a(r_{0})=0$ for some $r_{0}>0$, then $r_{0}$ is a minimizing point and
$a^{\prime}(r_{0})=0$. Using the uniqueness of the solution to the initial
value problem consisting of (2.25) and $a(r_{0})=a^{\prime}(r_{0})=0$, we get
$a\equiv 0$ which contradicts $a(0)=1$. Thus, $a(r)>0$ for all $r>0$. The same
argument shows that $f(r)>0,g(r)>0$ for all $r>0$. Since (3.13) is valid, we
see that $g(r)$ is strictly increasing. In particular, $g(r)<q$ for all $r>0$.
Lemma 3.2 already gives us $f\leq\frac{\pi}{2}$. We now strengthen it to
$f<\pi-\omega$. First it is easy to see that $f\leq\pi-\omega$. Otherwise
there is a point $r_{0}>0$ such that $f(r_{0})>\pi-\omega$. Thus, we can find
two points $r_{1},r_{2}$, with $0\leq r_{1}<r_{0}<r_{2}$, such that
$f(r_{1})=f(r_{2})$ and $f(r)\geq f(r_{1})$ for all $r\in(r_{1},r_{2})$.
Modify $f$ to $\tilde{f}$ by setting $\tilde{f}(r)=f(r_{1})$,
$r\in(r_{1},r_{2})$; $\tilde{f}=f$, elsewhere. Then $(a,\tilde{f},g)\in{\cal
C}$ and $L(a,\tilde{f},g)<L(a,f,g)$ because $f$ cannot be constant-valued over
$(r_{1},r_{2})$ by virtue of the equation (2.26) and the energy density ${\cal
E}_{1}$ defined in (2.20) increases for $f\in[0,\frac{\pi}{2}]$. This
contradiction establishes the result $f\leq\pi-\omega$. Next, we assert that
$f<\pi-\omega$. Otherwise, if $f(r_{0})=\pi-\omega$ for some $r_{0}>0$, then
$r_{0}$ is a maximum point of $f$ such that $f^{\prime}(r_{0})=0$ and
$f^{\prime\prime}(r_{0})\leq 0$. Inserting these results into (2.26), we
arrive at a contradiction since $0<\pi-\omega<\frac{\pi}{2}$.
To see that $f$ is non-decreasing, we assume otherwise that there are
$0<r_{1}<r_{2}$ such that $f(r_{1})>f(r_{2})$. Since $f(0)=0$, we see that $f$
has a local maximum point $r_{0}$ below $r_{2}$, which is known to be false.
Thus $f$ is non-decreasing. To see that $f$ is strictly increasing, we assume
otherwise that there are $0<r_{1}<r_{2}$ such that $f(r_{1})=f(r_{2})$. Hence
$f$ is constant-valued over $[r_{1},r_{2}]$ which is impossible.
The proof of the lemma is complete.
###### Lemma 5.2
. For the solution $(a,f,g)$, we have the asymptotic estimates
$a(r)={\rm\mbox{O}}\left({\rm\mbox{e}}^{-\gamma(1-\varepsilon)r}\right),\quad
g(r)=q+{\rm\mbox{O}}\left(r^{-1}\right),\quad
f(r)=(\pi-\omega)+{\rm\mbox{O}}\left(r^{-1}\right),$ (5.1)
as $r\to\infty$, where $\varepsilon\in(0,1)$ is arbitrarily small and
$\gamma=\frac{1}{2}\sqrt{\sin^{2}\omega-2{q^{2}}}.$ (5.2)
Moreover, the exponential decay rate for $a(r)$ stated in (5.1) is uniform
with respect to the parameter $q$ when $q$ is restricted to any allowed
interval $[0,q_{0}]$ where $q_{0}$ satisfies
$0<q_{0}<\min\left\\{\frac{1}{\sqrt{2}}\sin\omega,\sqrt{2}\left(1-\frac{\omega}{\pi}\right)\right\\}.$
(5.3)
Proof.From (3) and (3.24), we see that $f(r)\to\pi-\omega$ uniformly fast for
$q\in[0,q_{0}]$. Applying this and the other properties derived in Lemma 5.1
for $a,f,g$ in the equation (2.25), we have
$a^{\prime\prime}\geq\left(\frac{1}{4}\sin^{2}\omega(1-\delta)-\frac{1}{2}q^{2}\right)a,\quad
r>R_{\delta},$ (5.4)
where $R_{\delta}>0$ is sufficiently large but independent of $q\in[0,q_{0}]$
and $\delta>0$ is arbitrarily small. Write
$\frac{1}{4}\sin^{2}\omega(1-\delta)-\frac{1}{2}q^{2}=\frac{1}{4}\left(\sin^{2}\omega-2q^{2}\right)(1-\varepsilon)^{2},$
(5.5)
and $r_{\varepsilon}=R_{\delta}$. Then (5.4) gives us
$a^{\prime\prime}\geq\gamma^{2}(1-\varepsilon)^{2}a$, $r>r_{\varepsilon}$.
Using the comparison function
$\sigma(r)=C_{\varepsilon}\mbox{e}^{-\gamma(1-\varepsilon)r}$, we have that
$(a-\sigma)^{\prime\prime}\geq\gamma^{2}(1-\varepsilon)^{2}(a-\sigma)$,
$r>r_{\varepsilon}$. Thus, by virtue of the maximum principle, we have
$(a-\sigma)(r)<0$ for all $r>r_{\varepsilon}$ when the constant
$C_{\varepsilon}$ is chosen large enough so that
$a(r_{\varepsilon})\leq\sigma(r_{\varepsilon})$. This establishes the uniform
exponential decay estimate for $a(r)$ as $r\to\infty$ with respect to
$q\in[0,q_{0}]$.
To get the estimate for $g$, we note from (3.13) that
$g^{\prime}(r)=\frac{1}{r^{2}}\int_{0}^{r}2a^{2}(\rho)g(\rho)\,\mbox{d}\rho,\quad
r>0,$ (5.6)
which leads to
$q-g(r)=\int_{r}^{\infty}\frac{1}{\rho^{2}}\int_{0}^{\rho}2a^{2}(\rho^{\prime})g(\rho^{\prime})\,\mbox{d}\rho^{\prime}\,\mbox{d}\rho=\mbox{O}(r^{-1}),$
(5.7)
for $r>0$ large, since $a(r)$ vanishes exponentially fast at $r=\infty$.
To study the asymptotic behavior of $f$, we integrate (2.26) over the interval
$(r_{0},r)$ ($0<r_{0}<r<\infty$) to get
$8\kappa\,a^{2}(r)\sin^{2}f(r)f^{\prime}(r)+r^{2}f^{\prime}(r)=C_{0}+\int_{r_{0}}^{r}F(\rho)\,\mbox{d}r,$
(5.8)
where $C_{0}$ and $F(r)$ are given by
$\displaystyle C_{0}$ $\displaystyle=$ $\displaystyle
8\kappa\,a^{2}(r_{0})\sin^{2}f(r_{0})f^{\prime}(r_{0})+r_{0}^{2}f^{\prime}(r_{0}),$
(5.9) $\displaystyle F$ $\displaystyle=$ $\displaystyle 2a^{2}\sin f\cos
f+8\kappa a^{2}\sin f\cos f(f^{\prime})^{2}+\frac{8\kappa a^{4}\sin^{3}f\cos
f}{r^{2}}.$ (5.10)
Take $r_{0}>0$ large enough so that $\frac{1}{2}(\pi-\omega)\leq
f(r)<\pi-\omega$ for $r\geq r_{0}$. Thus
$0<\sin\frac{1}{2}(\pi-\omega)\leq\sin f(r),\quad r\geq r_{0}.$ (5.11)
Using (5.11) and recalling the definition of ${\cal E}_{1}$, we see that the
integral
$\int^{\infty}_{r_{0}}a^{2}\sin f\cos f(f^{\prime})^{2}\,\mbox{d}r$ (5.12)
is convergent. Applying this result and the exponential decay estimate of
$a(r)$ as $r\to\infty$, we obtain
$f^{\prime}(r)=\frac{C_{0}+\int_{r_{0}}^{r}F(\rho)\,\mbox{d}\rho}{8\kappa\,a^{2}(r)\sin^{2}f(r)+r^{2}}=\mbox{O}(r^{-2}),\quad
r>r_{0}.$ (5.13)
Integrating (5.13) over $(r,\infty)$ ($r>r_{0}$), we arrive at
$(\pi-\omega)-f(r)=\mbox{O}(r^{-1}).$ (5.14)
The proof of the lemma is complete.
###### Lemma 5.3
. For the solution $(a,f,g)$ with fixed $\omega\in(\frac{\pi}{2},\pi)$, the
electric charge
$Q_{e}(q)=2\int_{0}^{\infty}a^{2}(r)g(r)\,\mbox{d}r$ (5.15)
enjoys the property $Q_{e}(q)\to 0$ as $q\to 0$.
Proof.For fixed $\omega$, let $q_{0}$ satisfy (5.3). Since $a$ vanishes
exponentially fast at infinity uniformly with respect to $q\in(0,q_{0}]$ and
$0<g(r)<q$ for all $r>0$, we see that we can apply the dominated convergence
theorem to (5.15) to conclude that $Q_{e}(q)\to 0$ as $q\to 0$.
It should be noted that the special case $\kappa=0$ is to be treated
separately since the study above relies on the condition $\kappa>0$ (see
(3.28)). In fact, when $\kappa=0$, the Skyrme term in (2.6) is absent and the
model becomes the gauged sigma model which is easier. However, technically,
the boundary condition $f(0)=0$ has to be removed from the definition of the
admissible space $\cal A$ but recovered later for the obtained solution to the
constrained minimization problem as is done for the function $g$ in the proof
of Lemma 3.4. The details are omitted here.
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|
arxiv-papers
| 2011-07-01T04:41:55 |
2024-09-04T02:49:20.240259
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhifeng Gao, Yisong Yang",
"submitter": "Zhifeng Gao",
"url": "https://arxiv.org/abs/1107.0101"
}
|
1107.0262
|
# Numerical Bifurcation Analysis of Conformal
Formulations of the Einstein Constraints
M. Holst mholst@math.ucsd.edu and V. Kungurtsev vkungurt@math.ucsd.edu
Department of Mathematics
University of California San Diego
La Jolla CA 92093
###### Abstract.
The Einstein constraint equations have been the subject of study for more than
fifty years. The introduction of the conformal method in the 1970’s as a
parameterization of initial data for the Einstein equations led to increased
interest in the development of a complete solution theory for the constraints,
with the theory for constant mean curvature (CMC) spatial slices and closed
manifolds completely developed by 1995. The first general non-CMC existence
result was establish by Holst et al. in 2008, with extensions to rough data by
Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC
theory remains mostly open; moreover, recent work of Maxwell on specific
symmetry models sheds light on fundamental non-uniqueness problems with the
conformal method as a parameterization in non-CMC settings. In parallel with
these mathematical developments, computational physicists have uncovered
surprising behavior in numerical solutions to the extended conformal thin
sandwich formulation of the Einstein constraints. In particular, numerical
evidence suggests the existence of multiple solutions with a quadratic fold,
and a recent analysis of a simplified model supports this conclusion. In this
article, we examine this apparent bifurcation phenomena in a methodical way,
using modern techniques in bifurcation theory and in numerical homotopy
methods. We first review the evidence for the presence of bifurcation in the
Hamiltonian constraint in the time-symmetric case. We give a brief
introduction to the mathematical framework for analyzing bifurcation
phenomena, and then develop the main ideas behind the construction of
numerical homotopy, or path-following, methods in the analysis of bifurcation
phenomena. We then apply the continuation software package AUTO to this
problem, and verify the presence of the fold with homotopy-based numerical
methods. We discuss these results and their physical significance, which lead
to some interesting remaining questions to investigate further.
###### Key words and phrases:
bifurcation theory, nonlinear PDE, non-uniqueness, solution folds, pitchfork
bifurcation, compact operators, Fredholm operators, Fredholm index, homotopy
methods, continuation methods, pseudo-arclength continuation, Einstein
constraint equations
PACS Numbers: 04.20.Ex, 02.30.Jr, 02.30.Sa, 04.25D-
MH was supported in part by NSF Awards 0715146 and 0915220, and by DOD/DTRA
Award HDTRA-09-1-0036. VK was supported in part by NSF Awards 0715146 and
0915220.
###### Contents
1. 1 Introduction
2. 2 Conformal Thin Sandwich Decomposition
3. 3 Nonlinear Operators on Banach Spaces
4. 4 Bifurcation Theory for Nonlinear Operators Equations
5. 5 Numerical Bifurcation Theory
6. 6 Setup for Hamiltonian Constraint Bifurcation
7. 7 Numerical Results
8. 8 Discussion
9. 9 Conclusion
10. 10 Acknowledgments
## 1\. Introduction
Einstein’s gravitational field equations for relating the space curvature at a
time slice to the stress-energy can be split into a set of evolution and
constraint equations. The four constraint equations, known as the (scalar)
Hamiltonian constraint and the (3-vector) momentum constraint, constrain the
induced spatial metric $g_{ij}$ and extrinsic curvature $K_{ij}$. The Einstein
constraint equations have been the subject of study for more than fifty years
(cf. [2]). The introduction of the conformal method in the 1970’s as a way of
parameterizing initial data to the Einstein equations led to increased
interest in the development of a complete solution theory for the constraints,
with the theory for constant mean curvature (CMC) spatial slices and closed
manifolds completely developed by 1995. The CMC theory on closed manifolds is
particular satisfying, with nearly all physically interesting cases exhibiting
both existence and uniqueness of solutions.
However, other than the near-CMC result of Isenberg and Moncrief in 1996 [10],
the theory for non-CMC solutions remained completely open until the first far-
from-CMC existence result was establish by Holst et al. [8] in 2008, with
extensions to rough data by Holst et al. in 2009 [9], and to vacuum spacetimes
by Maxwell in 2009 [13]. However, the non-CMC theory remains mostly open, and
what is known is much less satisfying than the CMC case; the new non-CMC
results of Holst et al. and Maxwell are based on new types of topological
fixed point arguments, and while they establish existence, these arguments do
not give uniqueness. Moreover, more recent work of Maxwell on specific
symmetry models show in fact that uniqueness is lost, and also sheds light on
some fundamental problems with the conformal method itself as a
parameterization of initial data on manifolds that are not very close to CMC.
Several decompositions of the equations have been formulated in addition to
the conformal method, although the solution theory for only the conformal
method is well-developed (cf. [3]). In particular, in the extended conformal
thin sandwich (XCTS) decomposition of the constraints, a conformal factor
$\psi$, the lapse $N$, and the shift $\beta_{i}$, are solved for in five
coupled elliptic equations with supplied background data. In parallel with the
mathematical developments in the theory for the conformal method,
computational physicists have uncovered surprising behavior in numerical
solutions to the XCTS formulation. In particular, numerical evidence suggests
the existence of multiple solutions; Pfeiffer and York in [14] numerically
construct two solutions for a specific choice of free data.
In addition, there have been more theoretical results suggesting non-
uniqueness. Under additional assumptions of conformal flatness and time-
symmetry, the Hamiltonian constraint becomes a decoupled equation for the
conformal factor. Baumgarte, Murchadha and Pfeiffer [5] solve this form of the
Hamiltonian constraints using Sobolev functions, to find that an equation
parameter has two admissible values consistent with the constraints. Walsh
[16] performs a Lyapunov-Schmidt analysis (a standard technique in bifurcation
theory) of this form of the Hamiltonian constraint, and shows that at a
critical point with two expected solution branches, the solution branches
should take the form of a quadratic fold. In this article, we examine this
apparent bifurcation phenomena in a methodical way, using modern techniques in
bifurcation theory and in numerical homotopy methods.
Outline of the paper. The remainder of the paper is structured as follows. In
§2, give an overview of the constraint equations in the Einstein system and in
particular the XCTS formulation thereof. In §3, we give a brief introduction
to the mathematical framework for analyzing nonlinear operators in general. We
build on this in §4 where we expound on the theoretical framework underlying
bifurcation analysis. In §5, we develop the main ideas behind the construction
of numerical homotopy, or path-following, methods in the numerical treatment
of bifurcation phenomena, and subsequently in §6 present the methodology of
the numerical techniques we perform for this problem. In §7, we apply the
continuation software package AUTO to the constraint problem, and verify the
presence of the fold with homotopy-based numerical methods. We confirm the
earlier results, as well as provide a framework for a more careful exploration
of the solution theory for various parameterizations of the constraint
equations. Analyzing the Hamiltonian constraint for time-symmetric conformally
flat initial data, as in [16, 5], we demonstrate the existence and location of
a critical point, evidence that the solution branch at the critical point
forms a one-dimensional fold, and the form of the solution as continued past
the critical point.
## 2\. Conformal Thin Sandwich Decomposition
In General Relativity it is common to look at the curvature of spatial
hypersurfaces taken at time-slices of space-time. The Einstein constraint
equations are conditions on the induced spatial metric $g_{ij}$ and the second
fundamental form $K_{ij}$ for being a spatial slice. In addition, there are
six evolution equations that govern how this geometric data evolves in a full
spacetime. In solving the equations, a variety of decompositions have been
proposed. In the XCTS formulation, proposed by York [11], the shift vector
$\beta_{i}$, the lapse $N$ and a conformal factor $\psi$ are solved for given
a set of supplied data that includes
$(\tilde{g}_{ij},\tilde{u}_{ij},K,\partial_{t}K)$, where $\tilde{g}_{ij}$ is
the conformally related induced spatial metric, $\tilde{u}_{ij}$ its time
derivative, and $K$ the trace of the extrinsic curvature. The lapse and shift
are formed from taking a level set of $t$ and looking at the normal to the
hypersurface $n=N^{-1}(\partial_{t}-\beta^{i}\partial_{i})$ where $i$ indexes
over spatial coordinates [4]. The Hamiltonian constraint of the XCTS equations
can be written as
$\tilde{\nabla}^{2}\psi-\frac{1}{8}\tilde{R}\psi-\frac{1}{12}K^{2}\psi^{5}+\frac{1}{8}\psi^{-7}\tilde{A}^{ij}\tilde{A}_{ij}+2\pi\psi^{5}\rho=0$
(2.1)
Here $\tilde{\nabla}$ represents the covariant derivative, $\tilde{R}$ the
trace of the Ricci tensor and $\tilde{A}$ the trace free part of the extrinsic
curvature, all associated with the conformally scaled metric $\tilde{g}_{ij}$.
Following [5, 16] we assume time-symmetry (so $K_{ij}=0$) and conformally flat
initial data. This constraint then reduces to
$\nabla^{2}\psi+2\pi\rho\psi^{5}=0$ (2.2)
Following [5] we let $\rho$ be the constant mass-density of a star. Without
loss of generality, we take $\rho=0$ outside $r=1$ and look for solutions to
(2.2) with boundary conditions
$\displaystyle\frac{\partial\psi}{\partial r}$ $\displaystyle=0,\quad r=0$
(2.3) $\displaystyle\psi$ $\displaystyle=1,\quad r=1$ (2.4)
We can note that the maximum principle does not apply for this equation and
hence uniqueness is not guaranteed. We will see that this equation has two,
one, or no solutions, depending on the value of $\rho$.
## 3\. Nonlinear Operators on Banach Spaces
Let $X$ and $Y$ be Banach spaces, and let $X^{\prime}$ and $Y^{\prime}$ be
their respective dual spaces. Given a (generally nonlinear) map $F\colon X\to
Y$, we are interested in the following general problem:
$\mbox{Find}~{}u\in X~{}\mbox{such~{}that}~{}F(u)=0\in Y.$ (3.1)
We will give a brief overview of techniques for analyzing solutions to (3.1),
and for characterizing their behavior with respect to parameters.
We say that the problem ${F(u)=0}$ is well-posed if there is (a) existence,
(b) uniqueness, and (c) continuous dependence of the solution on the data of
the problem. Recall that if $F$ is both one-to-one (injective) and onto
(surjective), it is called a bijection, in which case the inverse mapping
$F^{-1}$ exists, and we would have both existence and uniqueness of solutions
to the problem ${F(u)=0}$. Recall that $F\colon X\to Y$ is a continuous map
from the normed space $X$ to the normed space $Y$ if
$\lim_{j\to\infty}u_{j}=u$ implies that $\lim_{j\to\infty}F(u_{j})=F(u)$,
where $\\{u_{j}\\}$ is a sequence, $u_{j}\in X$. If both $F$ and $F^{-1}$ are
continuous, then $F$ is called a homeomorphism. If both $F$ and $F^{-1}$ are
differentiable (see the below for the definition of differentiation of
abstract operators in Banach spaces), then $F$ is called a diffeomorphism. If
both $F$ and $F^{-1}$ are $k$-times continuously differentiable, then $F$ is
called a $C^{k}$-diffeomorphism. A linear map between two vector spaces is a
type of homomorphism (structure-preserving map); a linear bijection is called
an isomorphism.
We will need to assemble just a few basic concepts involving general nonlinear
maps in Banach spaces, to provide the mathematical framework for the
discussions in the next section. In particular, the following notion of
differentiation of maps on Banach spaces will be required (see [15, 17] for
more complete discussions).
###### Definition 3.1.
Let $X$ and $Y$ be Banach spaces, let $F\colon X\to Y$, and let $D\subset X$
be an open set. Then the map $F$ is called Fréchet- or F-differentiable at
$u\in D$ if there exists a bounded linear operator $F_{u}(u)\colon X\to Y$
such that:
$\lim_{\|h\|_{X}\to 0}\frac{1}{\|h\|_{X}}\|F(u+h)-F(x)-F_{u}(u)(h)\|_{Y}=0.$
The bounded linear operator $F_{u}(u)$ is called the F-derivative of $F$ at
$u$. The F-derivative of $F$ at $u$ can again be shown to be unique. If the
F-derivative of $F$ exists at all points $u\in D$, then we say that $F$ is
F-differentiable on $D$. If in fact $D=X$, then we simply say that $F$ is
F-differentiable, and the derivative $F_{u}(\cdot)$ defines a map from $X$
into the space of bounded linear maps, $F_{u}\colon X\to\mathcal{L}(X,Y)$. In
this case, we say that $F\in C^{1}(X;Y)$. Many of the properties of the
derivative of smooth functions over domains in $\mathbb{R}^{n}$ carry over to
this abstract setting, including the chain rule: If $X$, $Y$, and $Z$ are
Banach spaces, and if the maps $F\colon X\to Y$ and $G\colon Y\to Z$ are
differentiable, then the derivative of the composition map $H=G\circ F$ also
exists, and takes the form
$H_{u}(u)=(G\circ F)_{u}(u)=G_{F}(F(u))\circ F_{u}(u),$
where $H_{u}\colon X\to\mathcal{L}(X,Z)$, $F_{u}\colon X\to\mathcal{L}(X,Y)$,
and $G_{F}\colon Y\to\mathcal{L}(Y,Z)$. Higher order Fréchet (and Gâteaux)
derivatives can be defined in the obvious way, giving rise to multilinear maps
and giving meaning to the notation $F\in C^{k}(X;Y)$. Note that below we will
often encounter functions of two variables $F(u,\lambda)$, and will be
interested in the derivatives of such functions with respect to each variable;
we will denote these using the consistent notation $F_{u}$ and $F_{\lambda}$.
See [1, 15] for more complete discussions of general maps and differentiation
in Banach spaces.
A fundamental concept concerning linear operators that we will need for the
discussions below is that of a Fredholm operator. Let $X$ and $Y$ be Banach
spaces, and let $A\in\mathcal{L}(X,Y)$, or in other words, $A$ is a bounded
linear operator from $X$ to $Y$. In the case that $X$ and $Y$ have additional
Hilbert space structure, wherein there is an inner-product, the Riesz
Representation Theorem implies that the adjoint operator
$A^{*}\in\mathcal{L}(Y,X)$, defined as the operator for which
$(Ax,y)=(x,A^{*}y)$, exists uniquely. There are four fundamental subspaces of
$X$ and $Y$ associated with $A$ and $A^{*}$, namely:
1. (i)
$\mathcal{N}(A):$ null space (or kernel) of $A$
2. (ii)
$\mathcal{R}(A):$ range space of $A$
3. (iii)
$\mathcal{N}(A^{*}):$ null space (or kernel) of $A^{*}$
4. (iv)
$\mathcal{R}(A^{*}):$ range space of $A^{*}$
One can consider the dimension (dim) and co-dimension (codim) of each of these
four spaces, where co-dimension is taken to be relative to the larger spaces
that they are subspaces of. The operator $A\in\mathcal{L}(X,Y)$ is called a
Fredholm operator if and only if:
1. (i)
$\mathrm{dim}(\mathcal{N}(A))<\infty$
2. (ii)
$\mathrm{dim}(\mathcal{R}(A))<\infty$
The difference of these two dimensions, namely
$\mathrm{ind}(A)=\mathrm{dim}(\mathcal{N}(A))-\mathrm{dim}(\mathcal{R}(A))$
(3.2)
is called the Fredholm index of $A$. A basic result about Fredholm operators
is the following.
###### Theorem 3.2.
If $A\in\mathcal{L}(X,Y)$ is a Fredholm operator, then
1. (i)
$\mathcal{R}(A)$ is closed.
2. (ii)
$A^{*}$ is Fredholm with index $\mathrm{ind}(A^{*})=-\mathrm{ind}(A)$.
3. (iii)
$K$ is a compact operator, then $A+K$ is Fredholm with
$\mathrm{ind}(A+K)=\mathrm{ind}(A)$.
###### Proof.
See [17]. ∎
## 4\. Bifurcation Theory for Nonlinear Operators Equations
Bifurcation Theory studies the branching of solutions as governed by
parameter(s). Throughout this section we will refer to a ”solution curve”
which, for $F(u,\lambda)$ plots points in $(||u||,\lambda)$ space at which
$(u,\lambda)$ solves $F(u,\lambda)=0$, and $||u||$ denotes the norm of $u$.
Specifically, we are interested in the local behavior of the solution curve in
a neighborhood of a known solution $(u_{0},\lambda_{0})$. This is because, in
order to explore the solution space to a certain operator equation
$F(u,\lambda)$, we often solve $F(u,\lambda_{0})$ for $u$, then obtain another
$(u_{1},\lambda_{1})$ from the original solution data we obtain, and similarly
continue to subsequent solutions.
Bifurcation theory has its foundation in the implicit function theorem,
###### Theorem 4.1 (Implicit Function Theorem).
If it holds that, for an operator $F:X\times\mathbb{R}\rightarrow Y$, if
$F(u,\lambda)$ with $F(u_{0},\lambda_{0})=0$, $F$ and $F_{u}$ are continuous
on some region $U\times V$ with $(x_{0},\lambda_{0})\in U\times V$, and
$F_{u}(u_{0},\lambda_{0})$ is nonsingular with a bounded inverse, then there
is a unique branch of solutions $(u(\lambda),\lambda))$ in for $\lambda\in V$,
e.g. $F(u(\lambda),\lambda)=0$. Furthermore $u(\lambda)$ is continuous with
respect to $\lambda$ in $V$
This theorem states that if the operator $F_{u}$ is nonsingular at a certain
point $(x_{0},\lambda_{0})$ there is a unique solution $u$ for each $\lambda$
close to $\lambda_{0}$ on either side of $\lambda_{0}$ and we can plot a one-
dimensional curve in $(||u||,\lambda)$ space through
$(||u_{0}||,\lambda_{0})$. With a singular $F_{u}$, however, such a branch is
not guaranteed, suggesting the possibility of two or more such $u(\lambda)$
branches or no solutions for some $\lambda$ in every neighborhood around
$\lambda_{0}$. The exact form of the branching depends on the dimensions of
$F_{u}(x_{0},\lambda_{0})$, $F_{\lambda}(x_{0},\lambda_{0})$ and whether
$F_{\lambda}(x_{0},\lambda_{0})\in\mathcal{R}(F_{u}(x_{0},\lambda_{0}))$.
In the case of a ”fold”, wherein there is still a one-dimensional path through
$(u_{0},\lambda_{0})$ but the path exists solely on one side of $\lambda_{0}$
for $\lambda$, we have
1. (i)
$\mathrm{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})))=1$
2. (ii)
$\mathrm{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})^{*}))=1$, where
$F_{u}(u_{0},\lambda_{0})^{*}$ is the adjoint operator of
$F_{u}(u_{0},\lambda_{0})$
3. (iii)
$F_{\lambda}(u_{0},\lambda_{0})\notin\mathcal{R}(F_{u}(u_{0},\lambda_{0})$.
It can be shown, under these circumstances, near $(u_{0},\lambda_{0})$ the
continuation of the solution will be of the form
$\displaystyle u(\epsilon)$
$\displaystyle=u_{0}+\epsilon\phi+C_{1}\epsilon^{2}+O(\epsilon^{3})$ (4.1)
$\displaystyle\lambda(\epsilon)$
$\displaystyle=\lambda_{0}+C_{2}\epsilon^{2}+O(\epsilon^{3})$ (4.2)
where $\phi$ is a basis for the null-space of $F_{u}(u_{0},\lambda_{0})$ (see
[6]). Note that there is no linear $\lambda(\epsilon)$ term, so the solution
curve, at first approximation, stays at a constant value of $\lambda$ and
changes in $u$. The specific values of $C_{1}$ and $C_{2}$ are based on the
Hessian of the operator $F$. With a fold the $C_{2}$ will be negative, so the
solution curve shows $\lambda$ increase up to a certain critical point
$\lambda_{c}$ then decreases. With $C_{1}<0$ the actual solution $u$ changes
at first-order along $\phi$, the change then dampens in magnitude. Hence with
$C_{2}\neq 0$, the resulting solution curve appears locally as a sideways
parabola, and hence called a ”simple quadratic fold”. If, on the other hand,
$C_{2}=0$, we have at $(x_{0},\lambda_{0})$ a ”fold of order m” if
$\lambda^{(k)}(\epsilon)=0$ for $k<m$ [12]. In the case of a simple singular
point, we have either
1. (i)
$\text{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})))=1$, and
2. (ii)
$F_{\lambda}(u_{0},\lambda_{0})\in\mathcal{R}(F_{u}(u_{0},\lambda_{0}))$,
or we have
1. (i)
$\text{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})))=2$, and
2. (ii)
$F_{\lambda}(u_{0},\lambda_{0})\notin\mathcal{R}(F_{u}(u_{0},\lambda_{0}))$.
In this case, we have a situation of branch-switching, in which there are are
two branches of solutions crossing the point $(u_{0},\lambda_{0})$ [12]. With
high-dimensional null-spaces, the situation becomes increasingly complicated,
with multiple branching solutions of various forms. We illustrate the three
representative cases as they would appear in $(||u||,\lambda)$ space in Figure
1.
Figure 1. Common Locally Bifurcating Solution Paths.
We conclude, for completeness, this section with an exposition of the
generalized form of the bifurcation analysis method of Lyapunov-Schmidt, as
used for this problem by Walsh [16]. The exposition follows Zeidler [17]. We
assume that
1. (i)
$F_{u}(u_{0},\lambda_{0})$ is a Fredholm operator of index $k$
2. (ii)
$\text{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})))=n$
Then we define projection operators $P:X\rightarrow X$ and $Q:X\rightarrow X$
with $P(X)=\mathcal{N}(F_{u}(u_{0},\lambda_{0}))$ and
$(I-Q)(Y)=\mathcal{R}(F_{u}(u_{0},\lambda_{0}))$. Now the equation
$F(u,\lambda)=0$ is equivalent to the pair
$\displaystyle(I-Q)F(y+z,\lambda)$ $\displaystyle=0$ (4.3) $\displaystyle
QF(y+z,\lambda)$ $\displaystyle=0$ (4.4)
with $y=(I-P)u$ and $z=Pu$. Now the first equation satisfies the assumptions
of the Implicit Function Theorem, and so we can get a unique solution branch
$y(z,\lambda)$, substitute the solution in the second equation to obtain the
branching equation
$QF(y(z,\lambda)+z,\lambda)=0$ (4.5)
then solve for $z(\lambda)$ to get the branch
$u=y(z(\lambda),\lambda)+z(\lambda)$ Note that, in practice, one solves (4.3)
by expanding the operators in the bases of
$\mathcal{N}(F_{u}(u_{0},\lambda_{0}))$ and
$\mathcal{N}(F_{u}(u_{0},\lambda_{0})^{*})$. Hence, to be constructive, one
must already have an estimation of the dimension of the null-space of
$F_{u}(u_{0},\lambda_{0})$. In particular, Walsh [16] performs the analysis
with the starting assumption that
$\text{dim}(\mathcal{N}(F_{u}(u_{0},\lambda_{0})))=1$. So while the technique
is appropriate once this is known, it is not a standalone method of
bifurcation analysis.
## 5\. Numerical Bifurcation Theory
In the procedure of continuation, we seek to find a solution $u$ to a problem
$F(u,\lambda)=0$ as we move along $\lambda$. In the case of a nonsingular
Jacobian of the operator ($F_{u}(u_{0},\lambda_{0})$), it is standard to apply
Newton’s method on a discretization of the solution space. So we perform
continuation by repeatedly iterating $\lambda=\lambda+\Delta\lambda$ then
resolving $F(u,\lambda)=0$ for $u$. However, the procedure is invalid in the
case of a singular Jacobian of the equation operator, necessitating
alternatives for traversing a solution as the parameter varies. Depending on
the form of the bifurcation, various procedures exist. Considering the finite-
dimensional case (e.g. for a discretization of $u$), we have that
$F(u,\lambda)$ maps $\mathbb{R}^{N}\times\mathbb{R}$ to $\mathbb{R}^{N}$. In
this case, the rank of $[F_{u},F_{\lambda}]=N$ if either $F_{u}$ is
nonsingular, which is the case above, or
$F_{\lambda}(u_{0},\lambda_{0})\notin\mathcal{R}(F_{u}(u_{0},\lambda_{0}))$.
In this case $\text{dim}(\mathcal{N}([F_{u},F_{\lambda}]))=1$. Let’s say we
have a solution $F(u_{0},\lambda_{0})=0$. In the latter case, we cannot just
set $\lambda_{1}=\lambda_{0}+\Delta$ (with a constant $\Delta$) and solve for
$u_{1}$, but we can solve for $F(u_{1},\lambda_{1})=0$ together with one more
scalar equation, which we can set to be a constraint on the total magnitude in
the change of $(u,\lambda)$. With pseudo-arclength continuation, at a certain
parameter $\lambda_{0}$ and solution vector $u_{0}$, and a direction vector
$(\dot{u}_{0},\dot{\lambda}_{0})$ of the solution branch determined thus far,
you run Newton’s method on the two equations $F(u_{1},\lambda_{1})=0$ and
$(u_{1}-u_{0})^{*}\dot{u}_{0}+(\lambda_{1}-\lambda_{0})\dot{\lambda}-\Delta
s=0$, where $\Delta s$ is a constant ”arclength” term. It can be shown that
the Newton’s method Jacobian matrix is nonsingular if the point is either one
at which $F_{u}$ is nonsingular or a fold [6].
If $\text{dim}(\mathcal{N}(F_{u}))>1$ or
$F_{\lambda}(u_{0},\lambda_{0})\in\mathcal{R}(F_{u}(u_{0},\lambda_{0}))$ then
the Jacobian of Newton’s method in the pseudo-arclength continuation equations
is singular as well, and other procedures must be used to continue the
solution. Methods of continuation usually involve constructive techniques,
wherein the nullspace vectors for $F_{u}$ and the solution
$F_{u}\phi_{r}=F_{\lambda}$ are explicitly found and coefficients constructed
(for a list of relevant algorithms, see [12]). With even higher-dimensional
null-spaces, this framework is generalized.
## 6\. Setup for Hamiltonian Constraint Bifurcation
We reprint the equations (2.2)–(2.3) here
$\displaystyle\nabla^{2}\psi+2\pi\rho\psi^{5}$ $\displaystyle=0$ (6.1)
$\displaystyle\frac{\partial\psi}{\partial r}$ $\displaystyle=0,\quad r=0$
(6.2) $\displaystyle\psi$ $\displaystyle=1,\quad r=1$ (6.3)
and perform a reduction for bifurcation analysis following [6].
The goal is to transform this equation into a form wherein its linearization
becomes an eigenvalue problem. This can be done by parameterizing an extra
boundary condition. The equation can be rewritten in an equivalent form as
$\displaystyle\nabla^{2}\psi+2\pi\rho\psi^{5}$ $\displaystyle=0$ (6.4)
$\displaystyle\psi(0)$ $\displaystyle=p$ (6.5)
$\displaystyle\frac{\partial\psi}{\partial r}(0)$ $\displaystyle=0$ (6.6)
Now we seek to solve $F(p,\rho)\equiv\psi(1,p,\rho)-1=0$. Writing
$\psi_{p}(t,p,\rho)=\frac{d\psi}{dp}(t,p,\rho)$,
$F_{p}(p,\rho)=\psi_{p}(1,p,\rho)$ and similarly
$F_{\rho}(p,\rho)=\psi_{\rho}(1,p,\rho)$ Now $\psi_{p}$ satisfies
$\displaystyle\nabla^{2}\psi_{p}+10\pi\rho\psi^{4}\psi_{p}$ $\displaystyle=0$
(6.7) $\displaystyle\psi_{p}(0)$ $\displaystyle=1$ (6.8)
$\displaystyle\frac{\partial\psi_{p}}{\partial r}(0)$ $\displaystyle=0$ (6.9)
Which is an eigenvalue problem with a distinct solution. Define
$a(p,\lambda)=\psi_{p}(1,p,\rho)=F_{p}(p,\rho)$. Similarly $\psi_{\rho}$
satisfies
$\displaystyle\nabla^{2}\psi_{\rho}+2\pi\psi^{5}+10\pi\rho\psi^{4}\psi_{\rho}$
$\displaystyle=0$ (6.10) $\displaystyle\psi_{\rho}(0)$ $\displaystyle=0$
(6.11) $\displaystyle\frac{\partial\psi_{\rho}}{\partial r}(0)$
$\displaystyle=0$ (6.12)
and likewise define $b(p,\rho)=\psi_{\rho}(1,p,\rho)=F_{\rho}(p,\rho)$. Now we
have $F_{(p,\rho)}=(a,b)$, So $N(F_{(p,\rho)}))$ can be $\begin{pmatrix}1\\\
0\end{pmatrix}$, $\begin{pmatrix}0\\\ 1\end{pmatrix}$, both, or neither. Let
us discuss each of these cases separately. If the null-space is empty, then we
have a nonsingular Jacobian of the operator, so continuation can proceed with
Newton’s method as standard. If the null-space is $\begin{pmatrix}0\\\
1\end{pmatrix}$, so $b(p,\rho)=0$, then this is a situation where the same
solution exists for a differential increase in $\rho$, which is also an
uninteresting case. If the null-space is $\begin{pmatrix}1\\\ 0\end{pmatrix}$,
so $a(p,\rho)=0$ and $b(p,\rho)\neq 0$, we now have a case where $N(F_{p})=1$
but $F_{\rho}\notin F_{p}$, so we have a fold. Pseudo-arclength continuation
must be performed, and the form of the continuation gleamed from the
coefficients. Finally, in the case of a full two-dimensional null-space, we
have reached another solution branch. Here, there are two solution branches
intersecting, and so we have a choice between three directions in the
continuation of the solution. We now seek to perform this analysis numerically
to locate any critical points for (6.1) and see one of these scenarios occurs.
## 7\. Numerical Results
Recall that a continuation procedure attempts to solve $F(u,\lambda)$ for
varying $\lambda$ by a series of successive iterations along discrete steps of
$\lambda$. The procedure begins at some $\lambda_{0}$, finds the solution
$F(u_{0},\lambda_{0})$, finds the nullspace information of
$F_{u}(u_{0},\lambda_{0})$ and $F_{\lambda}(u_{0},\lambda_{0})$, then using
this information performs a suitable continuation step to find the next
solution $F(u_{1},\lambda_{1})$. We used the continuation software AUTO to
trace the solutions to (2.2) with boundary conditions (2.3). AUTO ([7])
performs numerical continuation on ODEs (notice that due to symmetry the
differential equation becomes an ODE with just variable $r$). AUTO calculates
the dimension of the null-space of the Jacobian of the differential operator
and performs, as required, Newton’s method, pseudo-arclength continuation or
explicit calculation of the null-space directions if it identifies higher-
order bifurcations, as discussed in (5). It uses the bordering algorithm for
the case of pseudo-arclength continuation ([6]) and otherwise more intricate
linear algebraic algorithms for higher-dimensional bifurcations.
Figure 2. Solution curve for (6.1).
Continuation reveals a quadratic fold at a value of $\rho$ around
$\rho_{c}\approx 0.35$. For $\rho<\rho_{c}$ there are two solutions to (2.2),
at $\rho=\rho_{c}$, there is exactly one solution, and at $\rho>\rho_{c}$
there are no solutions. AUTO finds that at $\rho=\rho_{c}$ the nullspace of
the discrete Jacobian differential operator has a dimension of one. In
addition, it finds that the 2nd derivative information indicates that the
continuation has a quadratic form. At all other points along the solution
curve, the Jacobian is nonsingular. Now we let the exponent vary. Writing the
equation as $\nabla^{2}\psi+2\pi\rho\psi^{a}=0$, we investigate the
continuation for different values of $a$. Knowing that the Laplacian operator
on its own has an invertible Jacobian, we expect a fold to appear for some
value of $a$. We find that indeed, with $a>1$ a fold appears, the curvature of
the solution continuation becoming sharper with increasing $a$. In Figure 3 we
see the bifurcation diagram for $a=1,1.25,5,10$.
Figure 3. Bifurcation diagram for $a=1,1.25,5,10$.
We’d like to note that, in contrast to the previous numerical evidence for
non-uniqueness [14], rather than looking at the form of the solution curve and
making an educated guess at the presence of a second solution curve, we
confirm that there is one and only one additional solution curve which
connects to the primary solution set in a quadratic curve and there are no
additional branches or any solutions past $\rho_{c}$.
Now we look at two representative solutions from the two branches. Taking the
two solutions at $\rho=0.2$ for the original problem $a=5$, we see
Figure 4. Solutions for $a=5$, $\rho=0.2$.
We note that this confirms the results of [5], as the solutions have the form
of a Sobolev function and furthermore the conformal factor is considerably
larger, suggesting a higher ADM energy, for the upper branch solutions.
## 8\. Discussion
The Conformal Thin Sandwich approach has been a promising formulation of the
Einstein constraints. The XCTS method appears to have the benefit of more
physically intuitive and less computationally onerous specification of free
data In contrast to its namesake predecessors, the conformal method and thin
sandwich approach. In particular, you save the task of having to specify a
divergence-free part of a symmetric tracefree tensor in the conformal method.
As such it has been used extensively in the construction of the geometry of
binary neutron stars and black holes (see the bibliography 5-12 in [14]). As
such this paper provides a point of caution for numerical relativists.
As Baumgarte et al. [5] noted, the upper branch corresponds to a higher level
of ADM energy than the lower branch in this model of a constant density black
hole. As such, in most applications with this set of asymptotically flat,
spherically symmetric free data, the lower branch should be chosen as the
correct, physically representative solution, but numerical relativists should
be aware that a solver could reach either one and be able to identify the need
to switch to the other branch. In addition, as $\rho\rightarrow\rho_{c}$
standard numerical procedures will become increasingly ill-conditioned.
Pinning the solution to the expected branch is recommended as a solution
strategy.
Finally, it should be noted that since this bifurcation analysis is complete,
rather than simply constructive, and so demonstrates conclusively that 1)
there are exactly two solutions for $\rho<\rho_{c}$ and 2) there are no
solutions for $\rho>\rho_{c}$ for this choice of free data. The fact that the
same choice of free data is consistent with two qualitatively disparate
geometries suggests future investigation in the relation between the structure
in the free and dependent set of data. In addition, it would be an interesting
investigation as to why a higher density is inconsistent with time-symmetry,
conformal flatness and spherical symmetry.
## 9\. Conclusion
In this article, we have examined the apparent bifurcation phenomena in the
XTCS formulation of the Einstein constraints in a methodical way, using modern
techniques in bifurcation theory and in numerical homotopy methods. We first
gave an overview of the Einstein constraints in §2, and followed this in §3
with a brief introduction to the mathematical foundations for, and in §4 the
framework for analyzing bifurcation phenomena in nonlinear operators
equations. In §5, we developed the main ideas behind the construction of
numerical homotopy, or path-following, methods in the numerical treatment of
bifurcation phenomena, and in §6 we presented the set up of and in §7 we
applied the continuation software package AUTO to the constraint problem. We
verified the presence of the fold with homotopy-based numerical methods,
confirming the earlier results. Analyzing the Hamiltonian constraint for time-
symmetric conformally flat initial data, as in [16, 5], we demonstrated the
existence and location of a critical point, evidence that the solution branch
at the critical point is one-dimensional, and the form of the solution as
continued past the critical point is a simple quadratic fold. We confirm
Walsh’s [16] constructive Lyapunov-Schmidt analysis by showing numerically
that there is indeed a solution point at which there is a one-dimensional
kernel for the linearization of the differential equation operator, justifying
the assumption made in his analysis, as well as confirming numerically that
indeed the form of the solution continuation expansion coefficients are such
that this equation exhibits a quadratic fold rather than branching or higher-
order folds.
The techniques presented here can be viewed as providing a framework for a
more careful exploration of the solution theory for various parameterizations
of the constraint equations, as well as the geometric relationship between the
free data in the model problem investigated.
## 10\. Acknowledgments
MH would like to express his appreciation to Herb Keller for introducing him
to the continuation methodology and techniques used in the paper, while MH was
a postdoc with Professor Keller at Caltech. Professor Keller was very
enthusiastic about eventually doing a careful numerical bifurcation analysis
of the Einstein constraint equations. MH would also like to thank Kip Thorne
for the many conversations about mathematical and numerical general
relativity.
MH was supported in part by the NSF through Awards 0715146 and 0915220, and by
DOD/DTRA through Award HDTRA-09-1-0036. VK was supported in part by NSF Award
0715146 and 0915220.
## References
* [1] R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer-Verlag, New York, NY, 1988.
* [2] R. Arnowitt, S. Deser, and C. Misner. The dynamics of general relativity. In L. Witten, editor, Gravitation, pages 227–265. Wiley, New York, NY, 1962.
* [3] R. Bartnik and J. Isenberg. The constraint equations. In P. Chruściel and H. Friedrich, editors, 50 Years of the Cauchy Problem, in honour of Y. Choquet-Bruhat, 2002 Cargese Meeting, 2003\. to appear.
* [4] R. Bartnik and J. Isenberg. The Einstein equations and the large scale behavior of gravitational fields, chapter The Einstein Constraints. Birkhauser Verlag, 2004.
* [5] T. W. Baumgarte, N. O. Murchadha, and H. P. Pfeiffer. Einstein constraints: Uniqueness and nonuniqueness in the conformal thin sandwich approach. Phys. Rev. D, 75(044009), 2007.
* [6] E. J. Doedel. Numerical analysis of bifurcation problems. Spring School on Numerical Software, 1997.
* [7] T. F. F. Y. A. K. B. S. X.-J. W. E. J. Doedel, A. R. Champneys. Auto97: Continuation and bifurcation software for ordinary differential equations (with homcont), 1997. Description available at: http://cmvl.cs.concordia.ca/publications/auto97.ps.gz.
* [8] M. Holst, G. Nagy, and G. Tsogtgerel. Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics. Phys. Rev. Lett., 100(16):161101.1–161101.4, 2008. Available as arXiv:0802.1031 [gr-qc].
* [9] M. Holst, G. Nagy, and G. Tsogtgerel. Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions. Comm. Math. Phys., 288(2):547–613, 2009. Available as arXiv:0712.0798 [gr-qc].
* [10] J. Isenberg and V. Moncrief. A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds. Class. Quantum Grav., 13:1819–1847, 1996.
* [11] J. James W. York. Conformal ’thin-sandwich’ data for the initial-value problem of general relativity. Phys. Rev. Lett., 82(7), 1999.
* [12] H. B. Keller. Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research, Bombay, India, 1987.
* [13] D. Maxwell. A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature. Math. Res. Lett., 16(4):627–645, 2009.
* [14] H. P. Pfeiffer and J. James W. York. Uniqueness and nonuniqueness in the einstein constraints. Phys. Rev. Lett., 95(091101), 2005.
* [15] I. Stakgold and M. Holst. Green’s Functions and Boundary Value Problems. John Wiley & Sons, Inc., New York, NY, third edition, 888 pages, February 2011. The preface and table of contents of the book are available at: http://ccom.ucsd.edu/~mholst/pubs/dist/StHo2011a-preview.pdf.
* [16] D. M. Walsh. Non-uniqueness in conformal formulations of the einstein constraints. Class. Quantum Grav., 24(8), 2007.
* [17] E. Zeidler. Nonlinear Functional Analysis and its Applications, volume I: Fixed Point Theorems. Springer-Verlag, New York, NY, 1991.
|
arxiv-papers
| 2011-07-01T16:06:02 |
2024-09-04T02:49:20.253355
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Holst and V. Kungurtsev",
"submitter": "Michael Holst",
"url": "https://arxiv.org/abs/1107.0262"
}
|
1107.0363
|
# Beyond the Standard Model Higgs Boson Searches at the Tevatron
Tim Scanlon on behalf of the D0 and CDF Collaborations
Results are presented for beyond the Standard Model Higgs boson searches using
up to 8.2 fb-1 of data from Run II at the Tevatron. No significant excess is
observed in any of the channels so 95% confidence level limits are presented.
## 1 Introduction
The search for the Higgs boson is one of the main goals in High Energy Physics
and one of the highest priorities at Run II of the Tevatron. There are many
alternative Higgs boson models beyond the SM, including Supersymmetry (SUSY)
$\\!{}^{{\bf?}}$, Hidden Valley (HV) $\\!{}^{{\bf?},{\bf?}}$ and Fermiophobic
Higgs bosons $\\!{}^{{\bf?}}$, which can actively be probed at the Tevatron,
and in the absence of an excess constrained. The latest limits for several
SUSY Higgs boson searches are presented in Section 2, for HV Higgs boson
searches in Section 3 and for Fermiophobic Higgs boson searches in Section 4.
More information on all these searches, along with the latest results, can be
found on the CDF and D0 public results webpages $\\!{}^{{\bf?},{\bf?}}$.
## 2 Minimal Supersymmetric Standard Model Higgs Boson Searches
The Minimal Supersymmetric extension of the SM (MSSM) $\\!{}^{{\bf?}}$
introduces two Higgs doublets which results in five physical Higgs bosons
after electroweak symmetry breaking. Three of the Higgs bosons are neutral,
the CP-odd scalar, $A$, and the CP-even scalars, $h$ and $H$ ($h$ is the
lighter and SM like), and two are charged, $H^{\pm}$.
At tree level only two free parameters are needed for all couplings and masses
to be calculated. These are chosen as the mass of the CP-odd pseudoscalar
($m_{A}$) and tan$\beta$, the ratio of the two vacuum expectation values of
the Higgs doublets.
The Higgs boson production cross section in the MSSM is proportional to the
square of tan$\beta$. Large values of tan$\beta$ thus result in significantly
increased production cross sections compared to the SM. Moreover, one of the
CP-even scalars and the CP-odd scalar are degenerate in mass, leading to a
further approximate doubling of the cross section.
The main production mechanisms for the neutral Higgs bosons are the
$gg,b\bar{b}\rightarrow\phi$ and $gg,q\bar{q}\rightarrow\phi+b\bar{b}$
processes, where $\phi=h,H,A$. The branching ratio of $\phi\rightarrow
b\bar{b}$ is around 90% and $\phi\rightarrow\tau^{+}\tau^{-}$ is around 10%.
This results in three channels of interest: $\phi\rightarrow\tau^{+}\tau^{-}$,
$\phi b\rightarrow b\bar{b}b$ and $\phi b\rightarrow\tau^{+}\tau^{-}b$. The
overall experimental sensitivity of the three channels is similar due to the
lower background from the more unique signature of the $\tau$ decays.
### 2.1 Higgs $\rightarrow\tau^{+}\tau^{-}$
D0’s most recent search is in the $\tau_{\mu}\tau_{had}$ final state using 1.2
fb-1 of Run II data, where $\tau_{had}$ refers to a hadronic decay and
$\tau_{\mu}$ to a leptonic decay (to a $\mu$) of the $\tau$. This result is an
extension to, and combined with, the published 1 fb-1 result which also
included the $\tau_{\mu}\tau_{e}$, and $\tau_{e}\tau_{had}$ channels
$\\!{}^{{\bf?}}$. CDF have published a search combining the
$\tau_{\mu}\tau_{e}$, $\tau_{\mu}\tau_{had}$ and $\tau_{e}\tau_{had}$ final
states using 1.8 fb-1 of RunII data $\\!{}^{{\bf?}}$.
Both searches require events to have an isolated $\mu$ ($e$), separated from
an opposite signed $\tau_{had}$ (or $e$ for the $\tau_{\mu}\tau_{e}$ channel).
Hadronic $\tau$ candidates are identified at D0 by neural networks designed to
distinguish $\tau_{had}$ from multi-jet events and at CDF by using a variable
size isolation cone. To minimise the $W$+jets background events are removed
which have a large $W$ transverse mass (D0) or by placing a cut on the
relative direction of the visible $\tau$ decay products and the missing
$E_{T}$ (CDF).
In both analyses the $Z/\gamma\rightarrow\tau\tau$ and $W$+jets backgrounds
are modelled using PYTHIA $\\!{}^{{\bf?}}$, with the $W$+jets normalisation
and the multi-jet contribution modelled using data. Limits are set using the
visible mass distribution ($m_{vis}$), which is the invariant mass of the
visible $\tau$ products and the missing $E_{T}$. The CDF model independent 95%
CL upper limit on the branching ratio multiplied by cross section is shown in
Fig. 1.
Figure 1: Model independent $95\%$ CL upper limit on the branching ratio
multiplied by cross section from the 1.8 fb-1 CDF publication. The dark
(light) grey bands show the 1 (2) standard deviation bands around the expected
limit.
### 2.2 Higgs $+~{}b\rightarrow b\bar{b}b$
This channel has a signature of at least three $b$ jets, with the background
consequentially dominated by heavy flavour multi-jet events. D0 have recently
published a search in this channel using 5.2 fb-1 of data $\\!{}^{{\bf?}}$ and
CDF have a preliminary result using 2.2 fb-1 of data.
Both searches require three $b$-tagged jets, D0 uses its standard neural
network $b$-tagging algorithm $\\!{}^{{\bf?}}$ and CDF uses its standard
secondary vertex algorithm. Due to the difficultly of simulating the heavy
flavour multi-jet background both analyses use data driven approaches. D0 uses
a fit to data over several different $b$-tagging criteria whereas CDF uses
fits to dijet invariant and secondary vertex mass templates to determine the
heavy flavour sample composition.
To increase the sensitivity of the analysis, D0 splits it into exclusive three
and four-jet channels, training a likelihood to distinguish the signal from
background in each. Limits are set by both CDF and D0 on the Higgs boson
production cross section times branching ratio using the dijet invariant mass
as the discriminating variable. The D0 model independent $95\%$ CL upper limit
on the branching ratio multiplied by cross section is shown in Fig. 2.
Figure 2: Model independent $95\%$ CL upper limit on the branching ratio
multiplied by cross section from the 5.2 fb-1 D0 publication. The yellow
(green) bands show the 1 (2) standard deviation bands around the expected
limit.
### 2.3 Higgs $+~{}b\rightarrow\tau^{+}\tau^{-}b$
D0 has performed a search for both the $\tau_{\mu}\tau_{had}$ and
$\tau_{e}\tau_{had}$ signatures using 4.3 fb-1 and 3.7 fb-1 of Run II data
respectively. Events are selected by requiring an isolated muon or electron
separated from an opposite sign $\tau_{had}$ candidate, along with a
$b$-tagged jet. The $\tau_{had}$ decays are identified using the standard D0
neural networks and $b$ jets using the neural network $b$-tagging algorithm.
The dominant backgrounds are $t\bar{t}$, $W$+jets, multi-jet and $Z$+jet
events. The multi-jet and $W$+jets backgrounds are estimated from data with
$t\bar{t}$ modelled using ALPGEN $\\!{}^{{\bf?}}$ interfaced with PYTHIA.
To improve the sensitivity of the analysis discriminants are trained which
differentiate the signal from the $t\bar{t}$, multi-jet and $Z$+light parton
(muon channel only) events respectively. The discriminants are combined to
form a final discriminant which is used to set limits. Figure 3 shows the
model independent 95% CL upper limit for the D0 $\tau_{\mu}\tau_{had}$
channel.
Figure 3: Model independent $95\%$ CL upper limit on the branching ratio
multiplied by cross section from the 4.2 fb-1 D0 $\tau_{\mu}\tau_{had}$
channel. The green (yellow) bands show the 1 (2) standard deviation bands
around the expected limit.
### 2.4 Combined Limits
The channels described in Sections 2.1–2.3 are complementary and can be
combined to increase the reach of the MSSM Higgs boson searches at the
Tevatron. D0 has combined its three neutral Higgs boson channels (using an
earlier version of the $\tau^{+}\tau^{-}b$ analysis based on only 1.2 fb-1 of
RunII data and not including the $\tau_{e}\tau_{had}$ channel) and interpreted
the limits in the standard MSSM scenarios $\\!{}^{{\bf?}}$. A combined
Tevatron limit on the MSSM Higgs sector has also been produced from D0 and
CDF’s Higgs $\rightarrow\tau^{+}\tau^{-}$ channels. The combined Higgs
$\rightarrow\tau^{+}\tau^{-}$ result has been interpreted in a quasi-model
independent limit, as well as in the standard scenarios. Both the D0 and
Tevatron combined 95% CL limits for one of the scenarios are shown in Fig. 4
along with the limit from LEP $\\!{}^{{\bf?}}$.
Figure 4: The combined D0 (left) and Tevatron (right) 95% CL limits on
tan$\beta$ versus ma for the $\mu<0$, no mixing scenario. The green area is
the region excluded by LEP.
### 2.5 Next-to-MSSM Higgs Bosons Searches
In the next-to-MSSM (nMSSM) $\\!{}^{{\bf?}}$ the branching ratio of
Higgs$\rightarrow b\bar{b}$ is greatly reduced. Instead the Higgs boson
predominantly decays to a pair of lighter neutral pseudoscalor Higgs bosons,
$a$. The nMSSM scheme is interesting as it allows the LEP limit on the $h$
boson to be naturally lowered to the general Higgs boson search limit from LEP
of $M_{h}>82$ GeV $\\!{}^{{\bf?}}$.
CDF has conducted a search for a light nMSSM Higgs boson using 2.7 fb-1 of
data in top quark decays, where $t\rightarrow W^{\pm(*)}ab$ and
$a\rightarrow\tau\tau$. The $\tau$ particles are identified by the presence of
additional isolated tracks in the event due to their low $p_{T}$. The dominant
background is from soft parton interactions and is modelled using data. Upper
limits are set at the 95% CL on the branching ratio of a top quark decaying to
a charged Higgs boson from a fit to the $p_{T}$ spectrum of the lead isolated
track and are shown in Fig. 5.
Figure 5: The 95% CL upper limits on branching ratio of top decaying to
$H^{+}b$ for various Higgs bosons masses.
## 3 Hidden Valley Higgs
CDF has conducted Higgs boson searches in Hidden Valley (HV) models, which
contain long-lived particles which travel a macroscopic distance before
decaying into two jets. The signature of this search is a Higgs boson decaying
to two HV particles, which travel for $\sim 1$ cm before decaying to two
b-quarks. Although there are four b-jets present in the decay, to increase the
efficiency only three are required, two of which must be $b$-tagged and not
back-to-back in the detector. A specially adapted version of CDF’s secondary
vertex $b$-tagging tool is used to reconstruct the displaced secondary
vertices and the reconstructed HV decay points are required to have a large
decay length.
Due to the difficulty of usinge Monte Carlo to model background events with
large decay lengths, a data driven approach is used. The predicted number of
background events is compared to the number seen in data and in the absence of
a significant excess, limits are set on the production cross section times
branching ratio on the benchmark HV model. Figure 6 shows the 95% CL upper
limit for a Higgs mass of 130 GeV and a HV particle mass of 40 GeV as a
function of the HV particle lifetime.
Figure 6: The 95% CL upper limit on $\sigma\times BR$ as a function of the
Hidden Valley particle’s lifetime.
## 4 Fermiophobic Higgs Boson Searches
The Standard Model Higgs boson branching ratio to a pair of photons is small.
There are however several models where the decay of the Higgs boson to
fermions is suppressed. In these models the decay of the Higgs boson to
photons is greatly enhanced. Both D0 and CDF $\\!{}^{{\bf?}}$ have carried out
searches for the Fermiophobic Higgs boson using 8.2 and 4.2 fb-1 respectively.
D0 requires two photon candidates in the central calorimeter, with jets
misidentified as photons rejected by use of a neutral network. Electrons are
suppressed by requiring that the photon candidates are not matched to activity
in the tracking detectors. A decision tree is trained using five variables to
distinguish signal from background events. The three main background sources
are estimated separately: the jet and diphoton backgrounds are estimated from
data and the Drell-Yan contribution is estimated using PYTHIA.
CDF’s search also requires two photons, with only one of them required to be
in the central region of the calorimeter. This looser photon requirement
approximately doubles the acceptance compared to requiring both photons in the
central region. In addition a cut is placed on the transverse momentum of the
two photons which significantly reduces the background, which is estimated
using a purely data-based approach.
Upper limits are set on the Higgs boson production cross section times
branching ratio using the decision tree output (D0) or diphoton mass (CDF) as
the discriminating variable. The 95% CL upper limit are shown in Fig. 7 for
the D0 search.
Figure 7: The 95% CL upper limit on $\sigma\times BR$ as a function of the
Fermiophobic Higgs boson mass for D0.
## 5 Conclusions
CDF and D0 have a wide variety of beyond the Standard Model Higgs boson
searches, presented here using up to 8.2 fb-1 of data. These searches are
already powerful, and have set some of the best limits in the world. No signal
has been observed yet, but with their rapidly improving sensitivity, due to
both improved analysis techniques and the addition of between 2–5 times more
data (which has already been recorded), these analyses will continue to probe
extremely interesting regions of parameter space, promising many exciting
results in the near future.
## Acknowledgments
I would like to thank all the staff at Fermilab, the Tevatron accelerator
division along with the CDF and the D0 Collaborations.
## References
## References
* [1] S. Dimopoulos, H. Georgi, Nucl. Phys. B 193, 150 (1981).
* [2] M. Strassler, K. Zurek, Phys. Lett. B 651, 374 (2007).
* [3] M. Strassler, K. Zurek, Phys. Lett. B 661, 263 (2008).
* [4] H. E. Haber, G. L. Kane, T. Sterling, Nucl. Phys. B 161, 493 (1979).
* [5] `http://www-cdf.fnal.gov`
* [6] `http://www-d0.fnal.gov`
* [7] V.M. Abazov _et al._ (D0 Collaboration), Phys. Rev. Lett. 101, 071804 (2008).
* [8] T. Aaltonen _et al._ (CDF Collaboration,) Phys. Rev. Lett. 103, 201801 (2009).
* [9] Sjöstrand T, Lönnblad L, Mrenna S and Skands P, J. High Energy Phys. 05, 026 (2006).
* [10] V.M. Abazov _et al._ (D0 Collaboration), Phys. Lett. B 698, 97 (2011).
* [11] V.M. Abazov _et al._ (D0 Collaboration), Nucl. Instrum. Methods A 620, 490 (2010).
* [12] M. L. Mangano, M. Moretti, F. Piccinini, R. Pittau, A. D. Polosa, J. High Energy Phys. 307, 001 (2003).
* [13] M. Carena, S. Heinemeyer, C. E. M. Wagner and G. Weiglein, Eur. Phys. J. C 45, 797 (2006).
* [14] Schael S et al., Eur. Phys. J. C 47, 547-587 (2006).
* [15] U. Ellwanger, M. Rausch de Traubenberg, C. A. Savoy, Nucl. Phys. B 492, 21 (1997).
* [16] G. Abbiendi et al. (OPAL Collaboration), Eur. Phys. J. C 27, 311 (2003).
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* [18] T. Aaltonen _et al._ (CDF Collaboration), Phys. Rev. Lett. 103, 061803 (2009).
|
arxiv-papers
| 2011-07-02T06:45:58 |
2024-09-04T02:49:20.262569
|
{
"license": "Public Domain",
"authors": "Tim Scanlon",
"submitter": "Tim Scanlon",
"url": "https://arxiv.org/abs/1107.0363"
}
|
1107.0532
|
# $J/\psi$ production and correlation in $p+p$ and Au+Au collisions at STAR
Zebo Tang (for the STAR collaboration) Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui, China, 230026
zbtang@mail.ustc.edu.cn
###### Abstract
The results on $J/\psi$ $p_{T}$ spectra in 200 GeV $p+p$ and Au+Au collisions
at STAR with $p_{T}$ in the range of 3-10 GeV/$c$ are presented. Nuclear
modification factor of high-$p_{T}$ $J/\psi$ is found to be consistent with no
suppression in peripheral Au+Au collisions and significantly smaller than
unity in central Au+Au collisions. The $J/\psi$ elliptic flow is measured to
be consistent with no flow at $p_{T}<10$ GeV/$c$ in 20-60% Au+Au collisions.
## 1 Introduction
$J/\psi$ suppression in heavy-ion collisions due to color-screening of its
constituent quarks was proposed as the signature for the formation of quark-
gluon plasma by T. Masui and H. Satz 25 years ago [1]. But results from SPS
and RHIC showed some other effects such as cold nuclear matter (CNM) effect
and recombination of charm quarks may play an important role in the observed
$J/\psi$ suppression in relativistic heavy-ion collisions [2, 3]. It is
believed that high-$p_{T}$ $J/\psi$ is less affected by CNM effect and charm
quark recombination effect, thus providing a cleaner probe to search for
evidence of color-screening effect in relativistic heavy-ion collisions [4, 5,
6]. STAR’s previous measurements showed no suppression for high-$p_{T}$
$J/\psi$ in Cu+Cu collisions at 200 GeV, but with limited statistics [7]. And
system size for Cu+Cu collisions may be too small. The same measurement in
Au+Au collisions with higher statistics is needed for better understanding. On
the other hand, the $J/\psi$ collective flow measurement is crucial for the
test of charm quark recombination effect. It is also a clean probe to the
charm quark flow in case of coalescence hadronization.
The interpretation of $J/\psi$ modification by the medium created in heavy-ion
collisions also requires understanding of the quarkonium production mechanism
in hadronic collisions, but no model at present fully explains the $J/\psi$
systematic observed in elementary collisions. The $J/\psi$ spectrum
measurement at intermediate and high-$p_{T}$ range and high-$p_{T}$
$J/\psi$-hadron correlation measurement may provide additional insights into
the basic processes underlying quarkonium production.
In this paper, we present the measurement of $J/\psi$ $p_{T}$ spectra at mid-
rapidity with the STAR experiment in $p+p$ and Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV in RHIC year 2009 and 2010 high luminosity
runs. We also present the measurement of $J/\psi$ elliptic flow $v_{2}$ from
low to high-$p_{T}$ range in 20-60% Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV. The high-$p_{T}$ $J/\psi$-hadron
correlation in $p+p$ collisions has been discussed in Ref. [8].
## 2 Results and Discussions
The $J/\psi$ reconstruction method is similar to what we used in year 2005 and
year 2006 data [7, 9]. The integrated luminosity used for this analysis is 1.8
$pb^{-1}$ (1.4 $nb^{-1}$) with transverse energy threshold $E_{T}>$ 2.6 (4.3)
GeV in $p+p$ (Au+Au ) collisions. Since year 2009, STAR installed a large area
Time-Of-Flight (TOF), consist of 72% and 100% full barrel system at mid-
rapidity ($|\eta|<0.9$) in year 2009 and 2010 run respectively. Including TOF
in the analysis increases the signal-to-background ratio of $J/\psi$ and
reduces the statistical uncertainties.
Figure 1: $J/\psi$ $p_{T}$ spectra in $p+p$ (left) and Au+Au (right)
collisions.
The left panel of Fig. 1 shows the fully corrected $J/\psi$ $p_{T}$ spectra in
$p+p$ collisions at $\sqrt{s}$ = 200 GeV. STAR new measurements are consistent
with previous STAR and PHENIX measurements in the overlapping $p_{T}$ region.
The solid line represents a Tsallis statistics based Blast-Wave (TBW) model
[10, 11] fit to all of the data points. The dashed line and gray band depict
theoretical calculations of NRQCD from color-coctet (CO) and color-singlet
(CS) transitions [12] and NNLO⋆ CS result [13] for direct $J/\psi$ in $p+p$
collisions respectively. The CS+CO calculation leaves no room for feeddown
from $\psi^{\prime}$, $\chi_{c}$ and $B$, estimated to be a factor of $\sim$
0.5 of direct $J/\psi$. NNLO⋆ CS predicts a steeper $p_{T}$ dependence. The
dot-dashed line shows the calculation from color evaporation model (CEM) for
inclusive $J/\psi$, which can reasonably well explain the $p_{T}$ spectra at
$p_{T}>$ 1 GeV/c [14].
The right panel of Fig. 1 shows the fully corrected $J/\psi$ $p_{T}$ spectra
in Au+Au collisions with different centralities. STAR and PHENIX measurements
are consistent with each other at the overlapped $p_{T}$ range. The solid
lines present TBW fits to STAR and PHENIX data points simultaneously with
radial flow velocity $\beta$ fixed to 0, which can describe the data points
very well. The dashed lines show the TBW predictions assuming $J/\psi$ has the
same radial flow and freeze-out condition as light hadrons, and they are much
harder than the measurements [10, 11]. These indicate 1) $J/\psi$ has very
small (or 0) radial flow; and/or 2) there are significant contribution from
charm quark recombination at low $p_{T}$.
Figure 2: Left: $J/\psi$ $R_{AA}$ vs. $p_{T}$ in 0-20% and 40-60% Au+Au
collisions. Right: $R_{AA}$ vs. $N_{part}$ for low-$p_{T}$, high-$p_{T}$
$J/\psi$s and high-$p_{T}$ charged pion in Au+Au collisions.
The $J/\psi$ nuclear modification factor $R_{AA}$ as a function of $p_{T}$ in
Au+Au collisions at different centralities measured by STAR at high $p_{T}$
are shown in the left panel of Fig. 2 and compared to PHENIX measurements at
low $p_{T}$ [3]. There is a increasing trend from low to high $p_{T}$, maybe
due to CNM or $J/\psi$ formation time effect. The high-$p_{T}$ $J/\psi$
$R_{AA}$ is consistent with no suppression in 40-60% centrality, but
systematically smaller than unity in 0-20% centrality. The solid and dashed
lines show two theoretical calculations including both primordial $J/\psi$ and
statistical charm quark regeneration $J/\psi$ [4, 5].
The high-$p_{T}$ ($p_{T}>5$ GeV/$c$) $J/\psi$ $R_{AA}$ as a function of number
of participants ($N_{part}$) in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ =
200 GeV are shown in the right panel of Fig. 2. In peripheral collisions
(20-60%), high-$p_{T}$ $J/\psi$ has no suppression, consistent with STAR
previous measurements in Cu+Cu collisions at 200 GeV. In central collisions
(0-20%), high-$p_{T}$ $J/\psi$ is significantly suppressed, which may be due
to color-screening effect. The $R_{AA}$ of low-$p_{T}$ ($0<p_{T}<5$ GeV/$c$)
$J/\psi$ measured by PHENIX and high-$p_{T}$ ($p_{T}>5$ GeV/$c$) charged pion
measured by STAR are also shown for comparison. The high-$p_{T}$ $J/\psi$
$R_{AA}$ is systematic higher than that for low-$p_{T}$ $J/\psi$, and has
different trend from high-$p_{T}$ charged pion.
Figure 3: Left: $v_{2}$ vs. $p_{T}$ for $J/\psi$, $\phi$ and charge hadrons in
Au+Au collisions. Right: $J/\psi$ $v_{2}$ vs. $p_{T}$ from model calculations,
see text for detail.
For $J/\psi$ elliptic flow $v_{2}$ analysis [15], we use all of the available
data because this analysis does not need to correct for absolute normalization
and efficiency. The event planes are reconstructed by using charged particles
at mid-rapidity measured by TPC. The $v_{2}$ results of inclusive charged
hadrons using these event planes are consistent with previous STAR
measurement. $J/\psi$ $v_{2}$ as a function of $p_{T}$ in 20-60% Au+Au
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV are shown in the left panel
of Fig. 3, with the vertical lines (caps) representing statistical
(systematic) uncertainties, and the boxes depict uncertainties from non-flow
effect. $J/\psi$ $v_{2}$ is consistent with zero in all of the measured
$p_{T}$ range within uncertainties, and significantly lower than $\phi$ and
inclusive charged hadron $v_{2}$. Several model calculations [16, 17, 18, 19,
20, 21] with slightly different centralities are shown in the right panel of
Fig. 3. The picture that $J/\psi$ production is dominated by charm quark
recombination with significant charm quark flow is disfavored by STAR
measurements, but models can describe STAR measurements assuming that charm
quark recombination is dominant at low $p_{T}$ and primordial production is
dominant at high $p_{T}$.
The author is supported in part by the National Natural Science Foundation of
China under Grant No. 11005103 and the China Fundamental Research Funds for
the Central Universities.
## References
## References
* [1] T. Matsui, H. Satz, Phys. Lett.T. Matsui, H. Satz. B178 (1986) 416.
* [2] M. C. Abreu, et al., Phys. Lett.M. C. Abreu, et al. B499 (2001) 85–96.
* [3] A. Adare, et al., Phys. Rev. Lett.A. Adare, et al. 98 (2007) 232301.
* [4] Y.-p. Liu, Z. Qu, N. Xu, P.-f. Zhuang, Phys. Lett. B678 (2009) 72–76. arXiv:0901.2757.
* [5] X. Zhao, R. Rapp, Phys. Rev. C82 (2010) 064905.
* [6] H. Liu, K. Rajagopal, U.A.Wiedemann, Phys. Rev. Lett. 98 (2007) 182301.
* [7] B. I. Abelev, et al., Phys. Rev. C80 (2009) 041902. B. I. Abelev, et al., arXiv:0904.0439.
* [8] Z. Tang, Nucl. Phys. A855 (2011) 396–399. Z. Tang, arXiv:arXiv:1012.0233.
* [9] Z. Tang, Ph.D. thesis, University of Science and Technology and China (2009).
* [10] Z. Tang, et al., Phys. Rev. C79 (2009) 051901. Z. Tang, et al., arXiv:0812.1609.
* [11] Z. Tang, et al.arXiv:1101.1912.
* [12] G. C. Nayak, M. X. Liu, F. Cooper, Phys. Rev. D68 (2003) 034003.
* [13] P. Artoisenet, et al., Phys. Rev. Lett. 101 (2008) 152001, and J.P. Lansberg private communication.
* [14] M. Bedjidian, et al., hep-ph/0311048 and R. Vogt private communication (2004).
* [15] H. Qiu, Quark Matter 2011, poster.
* [16] V. Greco, C. M. Ko, R. Rapp, Phys. Lett. B595 (2004) 202–208.
* [17] L. Ravagli, R. Rapp, Phys. Lett. B655 (2007) 126–131. L. Ravagli, R. Rapp, arXiv:0705.0021.
* [18] L. Yan, P. Zhuang, N. Xu, Phys. Rev. Lett. 97 (2006) 232301.
* [19] X. Zhao, R. RapparXiv:0806.1239.
* [20] Y. Liu, N. Xu, P. Zhuang, Nucl. Phys. A834 (2010) 317c–319c.
* [21] U. W. Heinz, C. Shen private communication.
|
arxiv-papers
| 2011-07-04T05:58:49 |
2024-09-04T02:49:20.273161
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zebo Tang (for the STAR Collaboration)",
"submitter": "Zebo Tang",
"url": "https://arxiv.org/abs/1107.0532"
}
|
1107.0548
|
# On the statistical description of classical open systems with integer
variables by the Lindblad equation
E. D. Vol vol@ilt.kharkov.ua B. Verkin Institute for Low Temperature Physics
and Engineering of the National Academy of Sciences of Ukraine 47, Lenin Ave.,
Kharkov 61103, Ukraine.
###### Abstract
We propose the consistent statistical approach to consider a wide class of
classical open systems whose states are specified by a set of positive
integers(occupation numbers).Such systems are often encountered in physics,
chemistry, ecology, economics and other sciences.Our statistical method based
on ideas of quantum theory of open systems takes into account both
discreteness of the system variables and their time fluctuations - two effects
which are ignored in usual mean field dynamical approach.The method let one to
calculate the distribution function and (or)all moments of the system of
interest at any instant.As descriptive examples illustrating the effectiveness
of the method we consider some simple models:one relating to nonlinear
mechanics,and others taken from population biology .In all this examples the
results obtained by the method for large occupation numbers coincide with
results of purely dynamical approach but for small numbers interesting
differences and new effects arise.The possible observable effects connected
with discreteness and fluctuations in such systems are discussed.
###### pacs:
03.65.Ta, 05.40.-a
## I Introduction
Among the vast number of classical open dynamical systems under consideration
in physics, chemistry, biology, economics and other sciences there are many
such whose states in accordance with the sense of the problem are specified by
a set of integer variables $\left(\left\\{n_{i}\right\\}\right)$ , where $i$ =
0,1,2,..N (N-number of degrees of freedom). For example in physics $n_{i}$\-
are occupation numbers of cell states in phase space, in chemistry - numbers
of molecules of reactive elements, in ecology -numbers of individuals in
populations which live in the area and interact with other populations, in
economics the number of companies operating on the market. Since all these
systems are classical their dynamics as a rule is described by a system of
differential equations of the form
$\frac{dn_{i}}{dt}=F_{i}\left(\left\\{n_{\alpha}\right\\}\right),$ (1)
where $F_{i}(\left\\{n_{\alpha}\right\\})$ -some nonlinear functions,
depending on concrete problem. Obviously notation Eq. (1) implies that
variables $n_{i}$ in system Eq. (1) -are considered as continuous. In the case
when all $n_{i}\gg 1$ such approach can be easy justified. From the physical
point of view the system of equations (1) corresponds to mean field
approximation and $n_{i}$ are occupation numbers averaged over some
appropriate statistical ensemble. In the case when all $n_{i}\gtrsim 1$,
dynamical description becomes inadequate and the question naturally arises: is
there consistent statistical approach which takes into account both
discreteness of variables $n_{i\text{ }}$ and their time fluctuations that may
be not small. In addition it is naturally to demand that such approach gave
the same results as dynamical description in the large $n_{i\text{ }}$ limit.
In this paper we propose such approach based on the ideas of quantum theory of
open systems (QTOS) and consider some examples that demonstrate its
effectiveness. The rest of the article organized as follows. In the Sect.2 we
briefly describe minimal information from (QTOS) which is necessary for
understanding of the method used and present main steps of our method. In
Sect.3 we consider simple model of nonlinear autonomous oscillator with soft
exciting mode and give its statistical description on the basis of the method
proposed. All the main features of the method clearly come to light already in
this representative example. In Sect.4 with the help of our approach we
consider some problems from population dynamics relating to evolution of two
interaction populations living in a certain area..We show that in the case of
small populations statistical description leads to a number of differences
from the ordinary dynamical picture. On the other hand in the case of large
occupation numbers both descriptions are virtually identical. In conclusion we
discuss some generalizations of the method and its possible experimental
verification.
## II Description of the method
In this section we briefly remind the main points of the method proposed by
author earlier 1a which allows one to make the transition from known
dynamical equations of classical open system to the master equation for its
quantum analogue. The method based on the correspondence that can be set
between quantum master equation in the Lindblad form and the Liouville
equation for distribution function in phase space of the classical system of
interest. This correspondence allows one using classical equations of motion
to restore the form of all operators involved in the Lindblad equation.
Thereby we can apply the procedure of quantization at least in semiclassical
approximation in the case of a large class of nonhamiltonian dynamical
systems. In short (all details see in 1a ) the recipe of quantization
proposed consists of three consecutive steps.
Step1: The input classical dynamical equations should be presented in the form
allowed the quantization (FAQ). For the purposes of present paper the most
convenient form is complex representation of equations of motion:
$\frac{dz_{i}}{dt}=-i\cdot\frac{dH}{dz_{i}^{\ast}}+\mathop{\displaystyle\sum}\limits_{\alpha}\left(\overline{R_{\alpha}}\frac{dR_{\alpha}}{dz_{i}^{\ast}}-R_{\alpha}\frac{d\overline{R_{\alpha}}}{dz_{i}^{\ast}}\right),$
(2)
where $z_{i}=\frac{x_{i}+iy_{i}}{\sqrt{2}},$
$z_{i}^{\ast}=\frac{x_{i}-iy_{i}}{\sqrt{2}}$ are complex dynamical coordinates
of the system of interest, $H$, $R_{\alpha}$, $\overline{R_{\alpha}}$ are
functions of $z_{i}$, $z_{i}^{\ast}$ ( $H$ is real function, and $R_{\alpha}$,
$\overline{R_{\alpha}}$ are complex, $\overline{R_{\alpha}}$ means function
which conjugate to $R_{\alpha}$). It is necessary to emphasize that it is the
most delicate step of the method because it is difficult exactly to formalize
this point.
Step2. Having in hands representation Eq. (2) we can use classical function
$H$, $R_{\alpha}$, $\overline{R}_{\alpha}$ and with their help determine their
quantum analogues - operators $\widehat{H}$, $\widehat{R_{\alpha}}$,
$\widehat{R^{+}}$. For this purpose the variables $z_{i}$, $z_{i}^{\ast\text{
}}$must be replaced by the correspondence Bose operators $\widehat{a}_{i}$ and
$\widehat{a}_{i}^{+}$ with usual commutation rules:
$\left[\widehat{a}_{i},\widehat{a}_{j}^{+}\right]=\delta_{ij}$,
$\left[\widehat{a}_{i},\widehat{a}_{j}\right]=\left[\widehat{a}_{i}^{+},\widehat{a}_{j}^{+}\right]=0$.
Step3. The operators $\widehat{H}$, $\widehat{R_{\alpha}}$,
$\widehat{R_{\alpha}^{+}}$ found in this manner should be substituted into the
quantum Lindblad equation for the evolution of density matrix of the system:
$\frac{d\widehat{\rho}}{dt}=-i\left[\widehat{H},\widehat{\rho}\right]+\mathop{\displaystyle\sum}\limits_{\alpha}\left[\widehat{R_{\alpha}}\cdot\widehat{\rho},\widehat{R_{\alpha}^{+}}\right]+\left[\widehat{R_{\alpha}},\widehat{\rho}\cdot\widehat{R_{\alpha}^{+}}\right].$
(3)
The correspondence principle guarantees us that such approach will give
correct description of evolution of quantum open system at least with the
accuracy up to the first order in h .
## III Representative model. Autonomous nonlinear oscillator with self
exciting mode
It is convenient to demonstrate on concrete example all the features of the
approach proposed. We consider an oscillator with nonlinear damping in
situation when its equilibrium point loses stability and small fluctuations
switch the system to the new stationary state that corresponds to the closed
trajectory (limit cycle). The following system of equations gives the correct
mathematical description of the behavior of the oscillator in the vicinity of
bifurcation point (see 2a ):
$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle\omega y+\mu
x-x\left(x^{2}+y^{2}\right),$ (4) $\displaystyle\frac{dy}{dt}$
$\displaystyle=$ $\displaystyle-\omega x+\mu y-y\left(x^{2}+y^{2}\right),$
where $x$, $y$ are coordinates of oscillator in phase space and $\omega$ -its
frequency. The system of the equations of motion Eq. (4) can be written in the
complex form as one equation
$\frac{dz}{dt}=-i\omega z+\mu z-2z\left|z\right|^{2},$ (5)
where $z=\frac{x+iy}{\sqrt{2}}$. It is easy to show that equation Eq. (5) can
be represented in the FAC. For this purpose we introduce the functions
$H=\omega z^{\ast}z$, $R_{1}=\sqrt{\mu}z^{\ast}$ and $R_{2}=z^{2}$. One can
verify by direct checking that r.h.s. of Eq. (5) can be rewritten in the form
$\frac{dz}{dt}=-i\frac{\partial H}{\partial
z^{\ast}}+\left(\overline{R_{1}}\frac{\partial R_{1}}{\partial
z^{\ast}}-R_{1}\frac{\partial\overline{R_{1}}}{\partial
z^{\ast}}\right)+\left(\overline{R_{2}}\frac{\partial R_{2}}{\partial
z^{\ast}}-R_{2}\frac{\partial\overline{R_{2}}}{\partial z^{\ast}}\right).$ (6)
According to the recipe of quantization the Lindblad equation for quantum
analog of the system Eq. (4) can be written in the form
$\frac{d\rho}{dt}=-i\left[H,\rho\right]+\left[R_{1}\rho,R_{1}^{+}\right]+\left[R_{2}\rho,R_{2}^{+}\right]+h.c,$
(7)
where $\widehat{H}=\omega\widehat{a}^{+}\widehat{a}$,
$\widehat{R}_{1}=\sqrt{\mu}\widehat{a}^{+}$ and
$\widehat{R}_{2}=\widehat{a}^{2}$.
We are interesting in the stationary solution of Eq. 7 and from physical
considerations we will seek it in the form:
$\widehat{\rho}=\widehat{\rho}(\widehat{N})=\sum\left|n\right\rangle\rho_{n}\left\langle
n\right|,$ where $\left|n\right\rangle$ are the eigenfunctions of the operator
$\widehat{N}=\widehat{a}^{+}\widehat{a}$. It is convenient to introduce the
generating function $G(u)$ for the coefficients $\rho_{n}$. By definition
$G(u)=\sum\limits_{n=0}^{n=\infty}\rho_{n}\cdot u^{n}.$One can obtain from Eq.
7 in stationary case the following equation for the function $G(u)$:
$(1+u)\frac{d^{2}G}{du^{2}}-\mu u\frac{dG}{du}-\mu G(u)=0.$ (8)
The solution of Eq. 8 that satisfies all conditions of the problem may be
represented as
$G(u)=\frac{\Phi(1,\mu,\mu(1+u))}{\Phi(1,\mu,2\mu)},$ (9)
where $\Phi(a,c,x)$ is well known confluent hypergeometric function (see 3a )
- $\Phi(a,c,x)=1+\frac{ax}{c}+\frac{a(a+1)}{2!c(c+1)}x^{2}+...$. Note that
condition $G(u)=1$ corresponds to normalization of $\ \widehat{\rho_{st}}(n)-$
namely $\sum\limits_{n}\rho_{n}=1$. Having in hands the expression Eq. 9 for
the generation function one can find the average values for any physical
quantity of interest in the stationary state that is all moments of the
distribution $\rho(n)$. For example moments of first and second order are
determined by relations:
$\overline{n}=\sum\limits_{n}n\cdot\rho_{n}=\frac{dG}{du}$,
$\overline{n^{2}}-\overline{n}=\frac{d^{2}G}{du^{2}}$. (all derivatives are
taken at point $u=1$). Let us consider now the behavior of our system in two
limiting cases: $\mu\gg 1$ and $\ \mu$ $\ll 1$. When $\mu\gg 1$ using the
asymptotic formula for $\Phi(a,c,x)$ (see 3a )we obtain for $G\left(u\right)$
the next expression
$G(u)\approx e^{\mu(u-1)}\cdot\left(\frac{1+u}{2}\right)^{1-\mu},$ (10)
The average number of quanta in stationary state $\ \overline{n}$ and their
dispersion $\sigma=\overline{n^{2}}-\overline{n}^{2}$ in this case are:
$\overline{n}=\frac{\mu+1}{2}$ and $\ \sigma=\frac{3\mu+1}{4}$. The relative
fluctuation of quanta generated in stationary state is
$\frac{\sqrt{\sigma}}{\overline{n}}=\frac{\sqrt{3\mu+1}}{\mu+1}$ tends to zero
when $\mu\gg 1$. The obtained result testifies validity of deterministic
approach in this case since in classical case
$n_{cl}=\left|z\right|^{2}=\frac{\mu}{2}$ when the system is moving along the
limit cycle. Now let us consider the opposite case when $\mu\ll 1$. Using the
series expansion for $\ \Phi(a,c,x)$ 3a we obtained for $G(u)$ the next
expression (up to the first order in $\mu$)
$G(u)\backsimeq\frac{(2+u)(1+\mu)+\mu(1+u)^{2}}{3+7\mu}.$ (11)
The relation Eq. 11 implies that $G(u)$ tends to $G_{0}(u)=\frac{2+u}{3}$ when
$\mu$ tends to zero. The first two moments of the distribution $\rho(n)$ are:
$\overline{n}=\frac{1}{3}$ and $\sigma=\frac{2}{9}$. The relative fluctuation
of $n=\frac{\sqrt{\sigma}}{\overline{n}}$ is equal to $\sqrt{2\text{ }}$ in
this case. We see that in contrast to classical situation $\overline{n}\neq 0$
when $\mu$ tends to zero moreover fluctuations of occupation number turn out
to be large and very essential.
It is necessary to note that in classical system Eq. 4 in addition to variable
$\left|z\right|^{2}$ we have phase variable $\varphi$ that satisfies to
equation $\frac{d\varphi}{dt}=\omega$. However phase dynamics determines only
velocity along limit cycle but does not influence on stationary state itself.
In principle we can turn $\omega$ to zero (which implies that $H=0$) and all
foregoing results do not change. It is the point which explains why the
Lindblad equation can be successfully applied for statistical description of
classical open systems with integer variables. The reason is that in the case
when $H=0$ in Eq. 4, h may be eliminated from the Lindblad equation and wave
properties of quantum system turn out to be irrelevant. After this crucial
remark we can apply our approach to different classical systems with integer
variables in particular population dynamics models.
## IV Statistical description of population dynamics models.
We begin our consideration with well known Lotka-Volterra model (LVM) (see 4a
) describing ”the interaction” between two populations: prays and predators
(for example hares and lynxes) living in the same territory. Let $n_{1}(t)$
and $n_{2}(t)$ the current numbers of preys and predators correspondently.
Then input dynamical equations of the LVM can be written as
$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
n_{1}\left(a-2n_{2}\right),$ (12) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-n_{2}\left(b-2n_{1}\right),$
where coefficients $a$, $b$ are positive and have clear ecological meaning.
Note that for convenience we have chosen time unit thus to put coefficient of
the term $n_{1}n_{2}$ in Eq. 12 equal to 2. Now let us demonstrate that system
Eq. 12 can be represented in FAQ. For this purpose we introduce two auxiliary
complex variables $z_{1}$ and $z_{2}$ so that $\
n_{1}=\left|z_{1}\right|^{2}$and $n_{2}=\left|z_{2}\right|^{2}$. Let us assume
that evolution $z_{1}$, $z_{2}$ in time is governed by the following system of
equations:
$\displaystyle\ \frac{dz_{1}}{dt}$ $\displaystyle=$
$\displaystyle\lambda_{1}^{2}z_{1}-z_{1}\left|z_{2}\right|^{2},$ (13)
$\displaystyle\frac{dz_{2}}{dt}$ $\displaystyle=$
$\displaystyle-\lambda_{2}^{2}z_{2}+z_{2}\left|z_{1}\right|^{2}.$
It is easy to see that system Eq. 13 implies the following equations for
$n_{1}$, $n_{2}$
$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
2\lambda_{1}^{2}n_{1}-2n_{1}n_{2},$ (14) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-2\lambda_{2}^{2}n_{2}+2n_{1}n_{2},$
which is exactly coincides with Eq. 12 if one lets $a=2\lambda_{1}^{2}$,
$b=2\lambda_{2}^{2}$.
Now if we introduce three functions $R_{1}=\lambda_{1}z_{1}^{\ast}$,
$R_{2}=\lambda_{2}z_{2}$, and $R_{3}=z_{1}z_{2}^{\ast}$ it is easy to verify
that Eq. 13 can be represented in the form
$\displaystyle\frac{dz_{1}}{dt}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{3}\left(\overline{R_{i}}\frac{\partial
R_{i}}{\partial z_{1}^{\ast}}-R_{i}\frac{\partial\overline{R_{i}}}{\partial
z_{1}^{\ast}}\right),$ (15) $\displaystyle\frac{dz_{2}}{dt}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{3}\left(\overline{R_{i}}\ \frac{\partial
R_{i}}{\partial z_{2}^{\ast}}-R_{i}\frac{\partial\overline{R_{i}}}{\partial
z_{2}^{\ast}}\right).$
Using the foregoing recipe of quantization we maintain that description of LVM
which takes into account the discreteness of variables $n_{1}$ and $n_{2}$ and
their time fluctuations is given by the next master equation
$\frac{d\rho}{dt}=\sum\limits_{i=1}^{\
3}\left[\widehat{R}_{i}\rho,\widehat{R}_{i}^{+}\right]+h.c.,$ (16)
where $\widehat{R}_{1}=\lambda_{1}\widehat{a}_{1}^{+}$,
$\widehat{R}_{2}=\lambda_{2}\widehat{a}_{2}$ and
$\widehat{R}_{3}=\widehat{a}_{1}\widehat{a}_{2}^{+}$. Again we will
interesting in only the solutions of Eq. 16 which have the form
$\widehat{\rho}=\sum\limits_{n_{1,n_{2}}}\left|n_{1}n_{2}\right\rangle\rho_{n_{1,}n_{2}}\left\langle
n_{1}n_{2}\right|$. In this case for the coefficients of the expansion
$\rho_{n_{1}n_{2}}$ we obtain the next general equation
$\frac{d\rho_{n_{1}n_{2}}}{dt}=2\lambda_{1}^{2}\left[n_{1}\rho_{n_{1}-1,n_{2}}-\left(n_{1}+1\right)\rho_{n_{1}n_{2}}\right]+2\lambda_{2}^{2}\left[\left(n_{2}+1\right)\rho_{n_{1,}n_{2}+1}-n_{2}\rho_{n_{1}n_{2}}\right]+2\left[\left(n_{1}+1\right)n_{2}\rho_{n_{1}+1,n_{2}-1}-\left(n_{2}+1\right)n_{1}\rho_{n_{1}n_{2}}\right]$
(17)
. It is convenient to introduce the generating function
$G(u,v,t)=\sum\limits_{n_{1}n_{2}}\rho_{n_{1}n_{2}}\cdot u^{n_{1}}\cdot
v^{n_{2}}.$Then after the simple algebra we find that Eq. 17 implies the next
equation for $G(u,v,t)$:
$\frac{\partial G}{\partial t}=2\lambda_{1}^{2}(u-1)\frac{\partial}{\partial
u}(uG)+2\lambda_{2}^{2}(1-v)\frac{\partial G}{\partial
v}+2(v-u)\frac{\partial^{2}}{\partial u\partial v}(vG).$ (18)
The Eq. 17 and Eq. 18 in principle give us all the necessary information about
statistical behaviour of LVM. Now let us consider concrete results following
from these equations. Note that even in stationary case it is difficult to
find exact analytical solutions of Eq. 18 for arbitrary $\lambda_{1}$and
$\lambda_{2}$. But in the special case when
$\lambda_{2}^{2}=1+\lambda_{1}^{2}$ such solution easily can be found and have
the form $G(u,v)=\frac{\left(1-\varkappa\right)^{2}}{\left(1-\varkappa
u\right)\left(1-\varkappa v\right)}$ where
$\varkappa=\frac{\lambda_{1}^{2}}{1+\lambda_{1}^{2}}$. Since $G(u,v)=g(u)\cdot
g(v)$ it is clear that $n_{1}$and $n_{2}$ are independent variables in this
case and
$\overline{n_{1}}=\overline{n_{2}}=\frac{\varkappa}{1-\varkappa}=\lambda_{1}^{2}.$The
dispersion $\sigma_{1}^{2}$ in this case is equal to
$\lambda_{1}^{2}+\lambda_{1}^{4}$ and if we calculate relative fluctuation of
$n_{1}$ in stationary state we get the result: $\delta
n_{1}=\frac{\sqrt{\sigma_{1}}}{\overline{n_{1}}}=\sqrt{1+\frac{1}{\lambda_{1}^{2}}}$.
Thereby we see this quantity is not small and can be easily measured. More
detailed analysis of statistical behaviour of LVM following from Eq. 17 for
arbitrary $\lambda_{1\text{ }}$and $\lambda_{2}$ will be carried out
elsewhere. Here we show only that using Eq. 18 one can easy obtain a
collection of explicit relations connecting different moments of distribution
$\rho(n_{1},n_{2})$. For example if we differentiate stationary Eq. 18 with
respect to $u$ and after that put $u=v=1$ we obtain the simple relation
between moments of first and second order which reads as:
$\lambda_{1}^{2}\left(1+\overline{n_{1}}\right)-n_{1}-\overline{n_{1}n_{2}}=0.$
(19)
In a similar manner by differentiating of Eq. 18 with respect to v we obtain
the second relation:
$-\lambda_{2}^{2}\overline{n_{2}}\ +\overline{n_{1}}+\overline{n_{1}n_{2}}=0\
$ (20)
Relations Eq. 19 and Eq. 20 imply the helpful equation connecting
$\overline{n_{1}}$ and $\overline{n_{2}}$ namely
$\frac{\overline{n_{2}}}{1+\overline{n_{1}}}=\frac{\lambda_{1}^{2}}{\lambda_{2}^{2}}$.
It is worth to note that in classical LVM Eq. 14 similar relation exists:
$\frac{\overline{n_{2}}}{\overline{n_{1}}}=\frac{\lambda_{1}^{2}}{\lambda_{2}^{2}}$,
so we conclude that when $\overline{n_{1}}\gg 1$, $\overline{n_{2}}\gg 1$ the
results of statistical description completely coincide with pure dynamical
consideration. But in the case of small numbers $n_{1}$, $n_{2}$ the
difference between them may be essential. To demonstrate this distinction and
also to compare the approach proposed in the present paper with usual
Markovian description of such systems proposed in well known article of
Nicolis and Prigogine 5a and expanded in their later book 6a it is
appropriate to consider the truncated case of LVM, with
$\lambda_{1}=\lambda_{2}=0$. It is obvious that total number of individuals
N$=n_{1}+n_{2}$ in this model will be constant .This fact greatly simplifies
finding and analysis of solutions Eq. 18 which in this case takes the form:
$\frac{\partial G}{\partial t}=(v-u)\frac{\partial^{2}}{\partial u\partial
v}(vG)\ $ (21)
It is easy to see that Eq. 21 has solutions for any integer N in the form of
homogeneous polynomial in $u$ and $v$ of degree N , namely $G_{N\text{
}}(u,v,t)=\sum\limits_{k}A_{k}\left(t\right)u^{k}v^{N-k}$ where coefficients
$A_{k}$ satisfy the normalization condition $\sum\limits_{k}A_{k}=1$. Thereby
Eq. 21 is reduced to the linear system of equations of N+1order for
coefficients $A_{k}$ of the form$\ \frac{dA_{k}}{dt}=L_{km}A_{m}$, where
matrix elements $L_{km}$ can directly be found from Eq. 21 for any N. For
example in simplest case when N=2 the matrix $L_{km\text{ }}$has the
form$\begin{pmatrix}-2&0&0\\\ 2&-2&0\\\ 0&2&0\end{pmatrix}$.
Let us consider now this case more detail. Let
$G_{2}\left(u,v,t\right)=au^{2}+buv+cv^{2}$ is generating function of the
model. Then Eq. 21 implies the next system of equations for evolution of
coefficients $a,b,c$.
$\frac{da}{dt}=-2a,\frac{db}{dt}=2a-2b,\frac{dc}{dt}=2b.$ (22)
Together with normalization condition $a+b+c=1$ system Eq. 22 allows one to
give statistical description of the model at any time. In particular Eq. 22
implies that when t tends to infinity $\overline{n_{1}}$ tends to zero and
$\overline{n_{2\text{ }}}$ tends to 2.This completely agrees with solutions of
dynamical equations in this case. From the other hand let us consider now the
equation for generating function of this model obtained by Nicolis and
Prigogine (see eq .10.66 in their book 6a ) which in our notation has the form
$\frac{dG}{dt}=(v-u)v\frac{\partial^{2}G}{\partial u\partial v}.$ (23)
In the case $N=2$ Eq. 23 implies the system of equations for coefficients of
expansion $G_{2}=au^{2}+buv+cv^{2}$ which differs from 22 namely
$\frac{da}{dt}=0,\frac{db}{dt}=-b,\frac{dc}{dt}=b.$ (24)
Eq. (24) imply that when t tends to infinity both quantities
$\overline{n_{1}}$, $\overline{n_{2}}$ tend to nonzero values depending on
initial conditions what obviously does not agree with dynamical equations. But
when $N\gg 1$ it is easy to see that asymptotic behaviour solutions obtained
from equations Eq. (21) and Eq. (23) are virtually identical. Thus in this
example we see the benefits of approach proposed over the standard methods
which do not allow to consider explicitly the discreteness of variables of the
problem.
Finally in the last part of the paper using one concrete model of the
population dynamics we want to demonstrate that approach proposed let one to
give statistical description of the systems for which dynamical description in
the framework of mean field theory looks as oversimplified. For this purpose
we consider two ”competing” kins of cannibals eating each other so that
voracity of individuals in both kins is assumed to be distinctive. As one
knows cannibalism is widespread in living nature and plays important role in
evolution processes 7a . Besides many species possess special mechanisms which
let them to recognize relatives and to avoid of their eating 8a . Let us
assume that mutual eating is the major factor of changes of the number of
individuals in both kins. Then evolution of the number of individuals
$n_{1}$and $n_{2}$ in such model can be represented of simple system of
equation of the form
$\displaystyle\frac{dn_{1}}{dt}$ $\displaystyle=$ $\displaystyle
an_{1}n_{2}-bn_{1}n_{2},$ (25) $\displaystyle\frac{dn_{2}}{dt}$
$\displaystyle=$ $\displaystyle-an_{1}n_{2}+bn_{1}n_{2}.$
It is easy to see that total number of individuals in this model
N$=n_{1}+n_{2}$ conserves. But dynamical description of the system with the
help of Eq. (25) seems to be oversimplified. In particular it implies that for
any N when $a>b$ and t tends to infinity $n_{1}$tends to N , and $n_{2}$ tends
to zero. Now following the spirit of our method we will describe this system
by the help of two operators
$\widehat{R_{1}}=\lambda_{1}\widehat{a_{1}}\widehat{a_{2}^{+}}$ and
$\widehat{R_{2}}=\lambda_{2}\widehat{a_{1}^{+}}\widehat{a_{2}}$. Show that
such statistical version is completely consistent. Actually acting as in
previous examples we can write the master equation for the density matrix of
the system as
$\
\frac{d\widehat{\rho}}{dt}=\left[\widehat{R_{1}}\widehat{\rho,}\widehat{R_{1}^{+}}\right]+\left[\widehat{R_{2}}\widehat{\rho},\widehat{R_{2}^{+}}\right]+h.c..$
(26)
If again we are interesting in by solutions of the Eq. (26) of the form
$\widehat{\rho}=\sum\limits_{n_{1}n_{2}}\left|n_{1}n_{2}\right\rangle\rho_{n_{n}n_{2}}\left\langle
n_{1}n_{2}\right|$ then for the generating function of the problem
$G(u,v,t)=\sum\limits_{n_{1}n_{2}}\rho_{n_{1}n_{2}}u^{n_{1}}v^{n_{2}}$ we
obtain the equation
$\
\frac{dG}{dt}=a\left(u-v\right)\frac{d^{2}}{dudv}\left(uG\right)+b\left(v-u\right)\frac{d^{2}}{dudv}\left(vG\right),$
(27)
where $a=2\lambda_{2}^{2}$ and $b=2\lambda_{1}^{2}$. The equation Eq. (27)
implies that generating function of stationary state of this model in the case
when total number of individuals is equal to N can be represented in the form
$G_{st}\left(u,v\right)=C_{N}\left[\frac{\left(bv\right)^{N+1}-\left(au\right)^{N+1}}{bv-
au}\right]=C_{N}\left[\left(bv\right)^{N}+\left(bv\right)^{N-1}\left(au\right)+...\right]$
(28)
where $C_{N}$ is the normalization factor,
$C_{N}=\left(b^{N}+ab^{N-1}+...\right)^{-1}$ Having in hands expression Eq.
(28) for the generation function one can find all statistical characteristics
of the model for any N. In particular for N=2 for the average values of
individuals in both kins we obtain
$\overline{n_{1}}=\frac{2a^{2}+ab}{b^{2}+ab+a^{2}}$, and $\
\overline{n_{2}}=\frac{2b^{2}+ab}{b^{2}+ab+a^{2}}$. Now we want to show that
when N tends to infinity the properties of our statistical model will be
similar to ones of the dynamical model Eq. (25).Let us assume that $\ b>a$ and
let $\ \ \ \ \ \ \ \varkappa=\frac{a}{b},\left(\varkappa<1\right).$ Let us
calculate now the ratio $\frac{\overline{n_{1}}}{\overline{n_{2}}}$ in this
model. Using Eq. (28) we obtain
$\frac{\overline{n_{1}}}{\overleftarrow{n_{2}}}=\frac{Na^{N}+\left(N-1\right)a^{N-1}b+...}{Nb^{N}+\left(N-1\right)b^{N-1}a+...}=\frac{N\varkappa^{N}+\left(N-1\right)\varkappa^{N-1}+...\varkappa}{N\text{
}+\left(N-1\right)\varkappa+....\varkappa^{N-1}}$ (29)
. Expression Eq. (29) can be represented in the next convenient form
$\frac{\overline{n_{1}}}{\overline{n_{2}}}=\frac{\varkappa+\varkappa^{2}\frac{\partial
Lnf_{N}\left(\varkappa\right)}{\partial\varkappa}}{N-\varkappa\frac{\partial
Lnf_{N}\left(\varkappa\right)}{\partial\varkappa}},$ (30)
where $f_{N}\left(\varkappa\right)=\frac{1-\varkappa^{N}}{1-\varkappa}$. It is
easy to see that
$\frac{\partial\left(Lnf_{N}\right)}{\partial\varkappa}=\frac{\left(N-1\right)\varkappa^{N}-N\varkappa^{N-1}+1}{\left(1-\varkappa\right)\left(1-\varkappa^{N}\right)}$
. Obviously this expression tends to $\frac{1}{1-\varkappa}$ when
$\varkappa<1$ and N tends to infinity .Thus Eq. (30) implies that ratio
$\frac{\overline{n_{1}}}{\overline{n_{2}}}$ tends to zero but this case is
realized only for infinite population. From the other hand for any finite
population of cannibals all its statistical properties can be well described
by the approach proposed in our paper. In conclusion let us briefly summing up
our consideration. We propose the statistical method of describing of various
systems in physics,chemistry,ecology whose states can be represented by
integers. Although the method proposed is based on quantum Lindblad equation
nevertheless we have shown that it can be successfully used also for
description of classical open systems with integer variables (at least in the
cases when ”Hamiltonian” of the open system is equal to zero) .All examples
considered in present paper demonstrate that approach proposed results in
consistent conclusions and allows one in several cases to eliminate essential
inaccuracies of preceding considerations.
## V Acknowledgement
The author acknowledges I.V. Krive and S.N. Dolya for the discussions of the
results of the paper and valuable comments.
## References
* (1) E. D. Vol Phys. Rev. A 73, 062113 (2006).
* (2) V. I. Arnold Geometrical Methods in the Theory of Ordinary Differential Equations 2-nd ed.,Springer,New York,(1988).
* (3) Handbook of Mathematical Functions (ed. by M.Abramovitz and I.A. Stegun),NBS Applied Mathematical Series 55,Washington,DC(1964).
* (4) Maynard Smith J.,Models in Ecology, Cambridge University Press,London,(1974).
* (5) Nicolis G.,Prigogine I., Proc.Nat.Acad.Sci. USA, 68, 2102(1971).
* (6) NicolisG.,Prigogine I., Self-organization in Non Equilibrium Systems.-New York, Wiley,(1977).
* (7) Elgar M.A., Crespi B.J., (eds) Cannibalism, Oxford University Press,Oxford !(1992).
* (8) GD.W. Pfennig, Biosciences 47, 667,(1997).
|
arxiv-papers
| 2011-07-04T07:26:42 |
2024-09-04T02:49:20.278208
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. D. Vol",
"submitter": "Evgenii D. Vol",
"url": "https://arxiv.org/abs/1107.0548"
}
|
1107.0771
|
# Identifying Sneutrino Dark Matter: Interplay between the LHC and Direct
Search
Hye-Sung Lee hlee@bnl.gov Department of Physics, Brookhaven National
Laboratory, Upton, NY 11973, USA Yingchuan Li ycli@quark.phy.bnl.gov
Department of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA
(July 2011)
###### Abstract
Under $R$-parity, the lightest supersymmetric particle (LSP) is stable and may
serve as a good dark matter candidate. The $R$-parity can be naturally
introduced with a gauge origin at TeV scale. We go over why a TeV scale $B-L$
gauge extension of the minimal supersymmetric standard model (MSSM) is one of
the most natural, if not demanded, low energy supersymmetric models. In the
presence of a TeV scale Abelian gauge symmetry, the (predominantly) right-
handed sneutrino LSP can be a good dark matter candidate. Its identification
at the LHC is challenging because it does not carry any standard model charge.
We show how we can use the correlation between the LHC experiments (dilepton
resonance signals) and the direct dark matter search experiments (such as CDMS
and XENON) to identify the right-handed sneutrino LSP dark matter in the $B-L$
extended MSSM.
## I Introduction
There are strong evidences that about $22\%$ of the energy budget of the
Universe is in the form of dark matter (DM) Nakamura:2010zzi . The most
precise measurement comes from fitting the WMAP measured anisotropy of the
cosmic microwave background to the cosmological parameters Komatsu:2010fb .
One has to rely on the other methods including direct and indirect DM searches
as well as colliders to pinpoint the identity of the DM (see Ref. Feng:2010gw
for a review), which has far-reaching implications for particle physics. With
all standard model (SM) particles ruled out as viable DM candidates, DM is one
of the strongest empirical evidences for the beyond SM physics.
Large Hadron Collider (LHC) at CERN will explore the physics of the
electroweak (EW) symmetry breaking and beyond. The low energy supersymmetry
(SUSY), which is one of the most popular scenarios to stabilize the EW scale,
is expected to be largely explored at the LHC. In fact, the early search at
the LHC with total energy $\sqrt{s}=7~{}\text{TeV}$ and integrated luminosity
of $L=35~{}\text{pb}^{-1}$ has already started to put new constraints on SUSY
scenarios SUSYatLHC .
SUSY is one of the best-motivated new physics scenarios. It can address the
gauge hierarchy problem, help unification of three SM gauge coupling
constants, and may provide a natural DM candidate. Minimal supersymmetric
standard model (MSSM) consists of the SM fields, one more Higgs doublet and
their superpartners. Typically, the MSSM is accompanied by $R$-parity, which
can protect proton from decaying through renormalizable baryon number ($B$) or
lepton number ($L$) violating terms. Under the $R$-parity, the lightest
supersymmetric particle (LSP) is stable and may serve as a DM candidate. The
MSSM provides two natural LSP DM candidates: neutralino (superpartner of
neutral gauge bosons and Higgs bosons) and sneutrino (superpartner of
neutrinos) .
The neutralino LSP DM candidate has been extensively studied and proven to be
a good DM candidate Ellis:1983ew ; Jungman:1995df . Many studies have been
done also for the detection of the neutralino LSP signal at the collider
experiments. For example, the trilepton signals
($\chi^{\pm}_{1}+\chi^{0}_{2}\to 3\ell+\text{MET}$) can be used to look for
SUSY signal with the neutralino LSP final states, and the invariant mass
distribution of dilepton ($\chi^{0}_{2}\to\ell^{+}\ell^{-}+\chi^{0}_{1}$) can
be used to measure superparticle masses. (A brief summary of detecting the
neutralino LSP DM signals is included in a general SUSY review, Ref.
Martin:1997ns .)
On the other hand, the sneutrino (at earlier time, only the left-handed one)
LSP DM candidate has not been studied much, despite of the fact it is one of
only a few candidates in the SUSY scenario. It is basically because it was
excluded early as a viable DM candidate by a combination of cosmological (DM
relic density constraint) and terrestrial constraints (direct DM search by
nuclear recoil) Hagelin:1984wv ; Falk:1994es ; Ibanez:1983kw ; Arina:2007tm .
The major channel for the relic density and direct search is mediated by the
SM $Z$ boson, whose coupling to the left-handed sneutrino LSP is too large to
make it a good DM candidate.
It has been demonstrated, however, in Ref. Lee:2007mt that (predominantly)
right-handed (RH) sneutrino ($\tilde{\nu}_{R}$) can be a good cold DM
candidate, satisfying all the constraints for viable thermal DM candidate,
when there is a TeV scale neutral gauge boson $Z^{\prime}$ that couples to the
RH sneutrinos. (For an extensive review of heavy neutral gauge boson, see Ref.
Langacker:2008yv .) There are few studies in the RH sneutrino LSP search at
the collider experiments. Since the RH sneutrino LSP does not carry any SM
charge, we cannot use the methods developed for the neutralino LSP. In fact,
it would be very hard to see the signal related to
$Z^{\prime}\to\tilde{\nu}_{R}\tilde{\nu}_{R}^{*}$ at the LHC experiments.
In this paper, we aim to establish a correlation between the LHC experiments
and DM direct search experiments (such as CDMS and XENON) for a $U(1)$ gauge
symmetry and discuss how we can use it to confirm the RH sneutrino LSP DM. We
choose a TeV scale $U(1)_{B-L}$ gauge symmetry. As discussed in Section II,
this is a remarkably well-motivated (if not demanded) addition to the MSSM,
and further the economy of the model is also preserved in the sense that we do
not need the $R$-parity independently.
The rest of this paper is organized as follows. In Section II, we describe our
theoretical framework. In Section III, we discuss the correlation of the DM
direct search experiment and the LHC dilepton resonance search experiment. In
Section IV, we show various results of the numerical analysis. In Section V,
we summarize our results.
## II Theoretical framework
Here, we describe the theoretical framework in our study. The model we will
work on is a well-known extension of the MSSM: MSSM + three RH
neutrinos/sneutrinos + TeV scale $U(1)_{B-L}$ gauge symmetry.
The RH neutrinos are well-motivated to explain the observed neutrino
masses111Supersymmetric generation of the neutrino masses, which does not
require RH neutrinos, is possible only in the absence of the $R$-parity, which
is not within our context Hall:1983id ; Grossman:1998py .. They are also
necessary to introduce $B-L$ as an anomaly-free gauge symmetry.
The $U(1)_{B-L}$ is one of the most popular gauge extensions as we can see
from the plethora of the literature on the subject. (For very limited
instances, see Refs. Allahverdi:2007wt ; Khalil:2007dr ; Allahverdi:2008jm ;
Barger:2008wn ; Basso:2010pe ; Kajiyama:2010iq ; Khalil:2011tb ; Martin:1992mq
.) It has a strong motivation especially in the SUSY framework: (i) It is the
only possible flavor-independent Abelian gauge extension of the SM/MSSM
without introducing exotic fermions (except for the RH neutrinos which is well
motivated itself by neutrino masses). (ii) It can originate from Grand
Unification Theory (GUT) models such as $SO(10)$ and $E_{6}$. (iii) The
radiative $B-L$ symmetry breaking, similarly to the radiative EW symmetry
breaking, in the SUSY may be achievable Khalil:2007dr . (iv) It can contain
matter parity $(-1)^{3(B-L)}$, which is equivalent to $R$-parity
$(-1)^{3(B-L)+2S}$, as a residual discrete symmetry Martin:1992mq .
In particular, the MSSM already carries the $R$-parity in order to stabilize
the proton and the LSP DM candidate. When a discrete symmetry does not have a
gauge origin, it may be vulnerable from the Planck scale physics Krauss:1988zc
. Therefore it is more than natural to assume a $U(1)_{B-L}$ gauge symmetry,
which is a gauge origin of the $R$-parity.
Once an Abelian gauge symmetry is introduced in the SUSY models, its natural
scale is set to be the TeV scale. This is because the masses of sfermions
(such as stop) get an extra $D$-term contribution from a new $U(1)$ gauge
symmetry and we need to make sure the sfermion scale does not exceed the TeV
scale in order to keep the SUSY as a solution to the gauge hierarchy problem.
Since much lighter scale $U(1)$ with an ordinary size coupling should have
been discovered by the collider experiments, we can see that (roughly) TeV
scale is the right scale for the new $U(1)$ gauge symmetry in SUSY.
Therefore, replacing the $R$-parity with the TeV scale $U(1)_{B-L}$ gauge
symmetry is one of the most natural and economic extensions of the MSSM. One
of the direct consequences of this model is the existence of a TeV scale
$Z^{\prime}$ gauge boson, which couples to both quarks and leptons with
specific charges ($B$ for all quarks/squarks and $-L$ for all
leptons/sleptons). We assume one of the RH sneutrinos is the LSP. It does not
couple to any SM gauge boson, but it does couple to the $Z^{\prime}$ gauge
boson.
It would be appropriate to comment about more general cases at this point,
before we discuss our main findings. The aforementioned attractiveness does
not exclusively apply to the $B-L$. Some mixture with the hypercharge $Y$
(that is, $(B-L)+\alpha Y$ with some constant $\alpha$) or lepton flavor
dependent $U(1)$ gauge symmetry ($B-x_{i}L$) Lee:2010hf are also known to be
anomaly-free without introducing exotic fermions, and can have the matter
parity as a residual discrete symmetry. (For some references about discrete
symmetries from a gauge origin, see Refs. Ibanez:1991hv ; Ibanez:1991pr ;
Dreiner:2005rd ; Hur:2008sy ; Lee:2008zzl .) It would not be difficult to
distinguish them with the LHC experiments though. The forward-backward
asymmetry can tell about the $Z^{\prime}$ couplings Langacker:1984dc ;
Barger:1986hd . The $B-L$ is vectorial which can distinguish itself from the
axial coupling provided by the $Y$ in the forward-backward asymmetry
measurement. The lepton flavor dependence of couplings can be easily seen by
comparing the dilepton $Z^{\prime}$ resonance signals Lee:2010hf .
## III Correlation of two experiments
(a) (b)
Figure 1: (a) Sneutrino LSP dark matter direct search using nuclear recoil.
(b) Dilepton $Z^{\prime}$ resonance at the LHC.
In this section, we discuss the interplay between two experiments: the
dilepton $Z^{\prime}$ search at the LHC and the direct DM search experiments.
We will not consider the relic density constraints in our study. We are mainly
interested in establishing the correlation between the LHC and the direct DM
search with minimal assumptions. The relic density constraint in principle
depends on the cosmological assumptions (for example, whether the DM was
thermally in equilibrium in the early Universe or not). Furthermore, the
channels to reproduce the right DM relic density are not unique: it may
involve $Z^{\prime}$ as well as its superpartner $\tilde{Z}^{\prime}$. The
former suggests the RH sneutrino LSP DM mass is quite close to a half of
$Z^{\prime}$ mass, but the latter does not suggest it. (See Ref. Lee:2007mt
for details.) However, once the RH sneutrino is confirmed by our suggested
interplay of the LHC and the direct DM search, one can compare the measured DM
mass with those that can satisfy the relic density constraint to test
consistency with the standard cosmology.
The direct DM search experiments such as CDMS Ahmed:2009zw and XENON
Aprile:2011hi can detect the DM by observing the signal from the nuclear
recoil. For the RH sneutrino LSP DM, which is a SM singlet, it is mediated by
the $Z^{\prime}$. (See Fig. 1 (a).) Following the approach of Ref. Lee:2007mt
, we can see that the effective Lagrangian for the direct DM search in our
framework is given by
${\cal
L}=i\frac{g^{2}_{Z^{\prime}}}{M^{2}_{Z^{\prime}}}\left(-1\right)\left({\tilde{\nu}}^{*}_{R}\partial_{\mu}{\tilde{\nu}}_{R}-\partial_{\mu}{\tilde{\nu}}^{*}_{R}{\tilde{\nu}}_{R}\right)\sum_{i=u,d}\left(\frac{1}{3}\right){\bar{q}}_{i}\gamma_{\mu}q_{i}$
(1)
The spin-independent cross section per nucleon via a $Z^{\prime}$ gauge boson
exchange, in the non-relativistic limit, is given by
$\sigma^{\text{SI}}_{\text{nucleon}}=\frac{\left(Z\lambda_{p}+(A-Z)\lambda_{n}\right)^{2}}{\pi
A^{2}}\mu_{n}^{2}$ (2)
where the $\mu_{n}$ ($\simeq m_{\text{proton}}$ for $m_{\tilde{\nu}_{R}}\gg
m_{\text{proton}}$) is the effective mass of the nucleon and the DM. In
general, the $u$ and $d$ quarks would have different couplings to the
$Z^{\prime}$, and the cross section would depend on the detector type. Under
$B-L$, however, the $u$ and $d$ quarks carry the same charge, and the
$Z^{\prime}$ coupling to proton and neutron are the same
$\lambda_{p}=\lambda_{n}=-\frac{g_{Z^{\prime}}^{2}}{M_{Z^{\prime}}^{2}}$. Thus
Eq. (2) has a simple form of
$\sigma^{\text{SI}}_{\text{nucleon}}=\left(\frac{g_{Z^{\prime}}^{2}}{M_{Z^{\prime}}^{2}}\right)^{2}\frac{\mu_{n}^{2}}{\pi}$
(3)
which depends only on the $g_{Z^{\prime}}/M_{Z^{\prime}}$ regardless of the
detector type.
The process at LHC that is directly correlated with the direct search is the
di-sneutrino $Z^{\prime}$ resonance process ($q\bar{q}\to
Z^{\prime}\to\tilde{\nu}_{R}\tilde{\nu}_{R}^{*}$), whose observation would be
practically impossible since it does not leave anything but the missing
energy. Nevertheless, a typical dilepton $Z^{\prime}$ resonance ($q\bar{q}\to
Z^{\prime}\to\ell^{+}\ell^{-}$) can reveal the relevant information, because
all leptons and sleptons carry the same charge ($-L$), though the spin and
mass of the final particles are different. (See Fig. 1 (b).) If we neglect the
effect of the analysis cuts, the dilepton $Z^{\prime}$ resonance cross section
for the $B-L$ model is determined by 3 parameters: mass of $Z^{\prime}$
($M_{Z^{\prime}}$), width of $Z^{\prime}$ ($\Gamma_{Z^{\prime}}$), and gauge
coupling constant ($g_{Z^{\prime}}$).
The details of the dilepton $Z^{\prime}$ resonance at the hadron collider was
elegantly analyzed in Ref. Carena:2004xs although the focus was given for the
$p\bar{p}$ collider. In the narrow width approximation, one can write down the
dilepton $Z^{\prime}$ resonance cross section as
$\displaystyle\sigma_{\text{Dilepton}}$ $\displaystyle\equiv$
$\displaystyle\sigma(pp\to Z^{\prime}\to\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle\frac{\pi
g_{Z^{\prime}}^{2}}{48s}\left[2\cdot\left(\frac{1}{3}\right)^{2}w_{u}+2\cdot\left(\frac{1}{3}\right)^{2}w_{d}\right]\text{Br}(Z^{\prime}\to\ell^{+}\ell^{-})$
where the functions $w_{u}$ and $w_{d}$ includes the parton distribution
function information for the $u$ and $d$ quarks, respectively. (See Ref.
Carena:2004xs for details.) The branching ratio can be written as
$\text{Br}(Z^{\prime}\to\ell^{+}\ell^{-})=\frac{g_{Z^{\prime}}^{2}M_{Z^{\prime}}}{24\pi\Gamma_{Z^{\prime}}}\left[2\cdot(-1)^{2}\right].$
(5)
With $M_{Z^{\prime}}$ and $\Gamma_{Z^{\prime}}$ fixed, the $\sigma_{{\rm
Dilepton}}$ is proportional to $g^{4}_{Z^{\prime}}$, the same dependence as
the direct detection cross section $\sigma^{{\rm SI}}_{{\rm nucleon}}$. While
$\sigma^{{\rm SI}}_{{\rm nucleon}}$ is proportional to $M^{-4}_{Z^{\prime}}$,
the $\sigma_{{\rm Dilepton}}$ carries different and more complicated
dependence on the mass $M_{Z^{\prime}}$. The contribution to the $\sigma_{{\rm
Dilepton}}$ from the $Z^{\prime}$ propagator is
$[(M^{2}_{l^{+}l^{-}}-M^{2}_{Z^{\prime}})^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}]^{-1}\approx\pi\delta(M^{2}_{l^{+}l^{-}}-M^{2}_{Z^{\prime}})/M_{Z^{\prime}}\Gamma_{Z^{\prime}}$
in the narrow width approximation. The dependence of $\sigma_{{\rm Dilepton}}$
on parton distribution functions further makes the $M_{Z^{\prime}}$ dependence
more complicated. Moreover, the $\sigma_{{\rm Dilepton}}$ also depends on the
total width $\Gamma_{Z^{\prime}}$, which is an irrelevant parameter for
$\sigma^{{\rm SI}}_{{\rm nucleon}}$.
An appropriate quantity for the examination of the correlation is the ratio of
two cross sections $\sigma^{{\rm SI}}_{{\rm nucleon}}/\sigma_{{\rm
Dilepton}}$. The gauge coupling cancels and the ratio only depends on the mass
and width of $Z^{\prime}$. In practice, with signal events observed, the mass
and total width can be determined by fitting the resonance peak to the Breit-
Wigner form
$1/[(M^{2}_{l^{+}l^{-}}-M^{2}_{Z^{\prime}})^{2}+M^{2}_{Z^{\prime}}\Gamma^{2}_{Z^{\prime}}]$.
Thus, we can confirm the RH sneutrino LSP DM by checking if the experimental
results and theoretical predictions of the $\sigma^{{\rm SI}}_{{\rm
nucleon}}/\sigma_{{\rm Dilepton}}$ are consistent. (We will discuss it further
in the following section.) This method to identify the RH sneutrino LSP DM
using the interplay of the LHC and the direct DM search experiments is our
main finding in this paper.
Before the presentation of numerical analysis in the next section, we briefly
comment about the experimental bounds and the LHC discovery potential of the
model here. A dedicated study of this has been carried out in Ref.
Basso:2010pe , where the bounds on $g_{Z^{\prime}}$ and $M_{Z^{\prime}}$ from
LEP Cacciapaglia:2006pk and recent Tevatron search Abazov:2010ti ;
Aaltonen:2011gp have been discussed 222We notice the recent searches
Aad:2011xp ; Chatrchyan:2011wq at the LHC at 7 TeV with integrated luminosity
of 40 pb-1 put slightly stronger bounds than the Tevatron search., and the
reaches at LHC of 7, 10, and 14 TeV with various luminosity have been
explored. According to Ref. Basso:2010pe , the LHC will probe a large portion
of the region with $g_{Z^{\prime}}$ larger than 0.01 and $M_{Z^{\prime}}$
within a few TeV. The value of $\sigma^{{\rm SI}}_{{\rm nucleon}}$ in the
major portion of such parameter region is larger than $10^{-48}[{\rm
cm}^{2}]$. It would be explored by the upcoming direct detection experiments,
at SNOLAB and DUSEL for instance, if their precision can be improved by
another 2 to 3 orders of magnitude beyond the current most stringent bounds
from XENON100 Aprile:2011hi . We therefore conclude that there is a large
common region in the $g_{Z^{\prime}}-M_{Z^{\prime}}$ plane that will be probed
at both experiments. It is thus possible to test the model by the correlations
of these two phenomenological aspects.
## IV Numerical Analysis
$\sqrt{s}=14$TeV$\sqrt{s}=10$TeV$\sqrt{s}=7$TeV
Figure 2: The coupling normalized cross section of $pp\rightarrow
Z^{\prime}\rightarrow l^{+}l^{-}$ ($l=e,\mu$) with invariant mass cut
$|M_{l^{+}l^{-}}-M_{Z^{\prime}}|<3\Gamma_{Z^{\prime}}$ imposed in $U(1)_{B-L}$
model at the LHC with $\sqrt{s}=7$, 10, and 14 TeV. The $Z^{\prime}$ width
$\Gamma_{Z^{\prime}}$ is taken as 3$\%$ (red solid line) and 6$\%$ (blue
dashed line) of the mass $M_{Z^{\prime}}$.
In the following, we discuss the dilepton resonance production cross section
$\sigma_{{\rm Dilepton}}$ and the ratio $\sigma^{{\rm SI}}_{{\rm
nucleon}}/\sigma_{{\rm Dilepton}}$ as functions of the mass $M_{Z^{\prime}}$
for different values of width $\Gamma_{Z^{\prime}}$.
Taking into account the decay modes to SM particles only, we find the width of
$Z^{\prime}$ is roughly $\Gamma^{{\rm SM}}_{Z^{\prime}}\approx
0.2g^{2}_{Z^{\prime}}M_{Z^{\prime}}$. With all the possible decay channels
included, the total width $\Gamma_{Z^{\prime}}$ depends on the full mass
spectrum, with the $\Gamma^{{\rm SM}}_{Z^{\prime}}$ setting the minimum value.
For illustration purpose, we will take
$\Gamma_{Z^{\prime}}/M_{Z^{\prime}}=3\%$ and 6$\%$ in the analysis.
For the simulation of the dilepton resonance production process $pp\rightarrow
Z^{\prime}\rightarrow l^{+}l^{-}$ at the LHC, we use the CTEQ6.1L parton
distribution functions Pumplin:2002vw . We adopt the event selection criteria
with the basic cuts LHCcuts
$p_{T_{l}}>20~{}{\rm GeV},~{}~{}|\eta_{l}|<2.5,$ (6)
and we further impose cut on the invariant mass of lepton pair
$|M_{l^{+}l^{-}}-M_{Z^{\prime}}|<3\Gamma_{Z^{\prime}}.$ (7)
The cross sections $\sigma_{{\rm Dilepton}}$ normalized by gauge coupling for
the process $pp\rightarrow Z^{\prime}\rightarrow l^{+}l^{-}$ at the LHC of 7,
10, and 14 TeV, with cuts in Eq. (6), (7) imposed, are shown in Fig. 2.
$\sqrt{s}=7$TeV$\sqrt{s}=10$TeV$\sqrt{s}=14$TeV
Figure 3: The ratio of cross sections of the spin-independent sneutrino-
nucleus elastic scattering ${\tilde{\nu}}_{R}q\rightarrow{\tilde{\nu}}_{R}q$
(normalized to a single nucleon) at the DM direct detection experiments and
the process $pp\rightarrow Z^{\prime}\rightarrow
l^{+}l^{-}(|M_{l^{+}l^{-}}-M_{Z^{\prime}}|<3\Gamma_{Z^{\prime}})$ at the LHC
at 7, 10, and 14 TeV. The $Z^{\prime}$ width $\Gamma_{Z^{\prime}}$ is taken as
3$\%$ (red solid line) and 6$\%$ (blue dashed line) of the mass
$M_{Z^{\prime}}$.
In Fig. 3, we show the ratios $\sigma^{{\rm SI}}_{{\rm nucleon}}/\sigma_{{\rm
Dilepton}}$, for various center of mass energy 7, 10, and 14 TeV at the LHC,
as functions of $M_{Z^{\prime}}$ for $\Gamma_{Z^{\prime}}/M_{Z^{\prime}}=3\%$,
$6\%$. As the gauge coupling cancels, the ratio only depends on the mass
$M_{Z^{\prime}}$ and width $\Gamma_{Z^{\prime}}$ of $Z^{\prime}$.
The future direct detection experiments will reach the sensitivity beyond
$10^{-45}$ cm2 level. The future running of LHC at 7, 10, and 14 TeV will have
integrated luminosity ranging from a few fb-1 to a few 100 fb-1. Assuming the
background is negligible compared to the signal as is the case here, the
discovery at LHC at 3$\sigma$ and 5$\sigma$ significance requires 5 and 15
events, respectively. The LHC with integrated luminosity of 1 fb-1 (100 fb-1)
will be able to probe the cross section at 10 fb (0.1 fb) level. If positive
signals are observed at both experiments, and they obey the predicted ratio as
shown in Fig. 3, it should be taken as a rather strong hint for the sneutrino
LSP DM scenario. Otherwise, the model can be ruled out if positive signals are
observed in either or both experiments but not consistent with the predicted
ratio shown in Fig. 3.
The mass and width of $Z^{\prime}$ need to be determined from the LHC data for
the purpose of this examination of the ratio of cross sections. Since the
momentum resolution of $e^{\pm}$ is better than $\mu^{\pm}$ in the high
$P_{T}$ region, the $e^{+}e^{-}$ final state is more favorable than the
$\mu^{+}\mu^{-}$ final state for this purpose. There are errors in the
determination of width arising from momentum resolution as well as fitting to
the Breit-Wigner form with limited number of events. A quantitative study on
these errors is beyond the scope of this paper. However, these need to be
considered when a comparison of cross sections is carried out in the future
after positive signals are observed.
## V Summary
We study the sneutrino LSP DM scenario in the SUSY $U(1)_{B-L}$ model at the
LHC and direct detection experiments. The sneutrino only couples to the
$Z^{\prime}$, making it extremely hard to test this model at the LHC. However,
since charged leptons and sneutrinos carry the same $B-L$ charge, the charged
lepton $e^{\pm},\mu^{\pm}$ can serve as a good replacement of sneutrino for
diagnosing purpose.
Following this spirit, we propose to test this scenario at the LHC with the
process $pp\rightarrow Z^{\prime}\rightarrow l^{+}l^{-}(l=e,\mu)$. The cross
section of this process is tightly correlated with that of the sneutrino-
nucleus spin-independent elastic scattering in the direct detection
experiments. Since a large common region of the parameter space will be probed
by both experiments, the correlation can be used to confirm or rule out such
model. In particular, with the signal events of dilepton resonance production
observed at the LHC and with the $Z^{\prime}$ mass and width extracted from
the data, the ratio $\sigma^{{\rm SI}}_{{\rm nucleon}}/\sigma_{{\rm
Dilepton}}$ is fixed in this scenario and can be examined against the
experimental data.
###### Acknowledgements.
Acknowledgments: We thank T. Han and F. Paige for helpful discussions. We
further thank T. Han for providing the Fortran code HANLIB that is used in the
Monte Carlo simulations. This work is supported by the US DOE under Grant
Contract DEAC02-98CH10886.
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|
arxiv-papers
| 2011-07-05T02:54:29 |
2024-09-04T02:49:20.289623
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hye-Sung Lee, Yingchuan Li",
"submitter": "Yingchuan Li",
"url": "https://arxiv.org/abs/1107.0771"
}
|
1107.0826
|
A. Ivascenko
(aivascenko@astro.uni-wuerzburg.de)
# Semi-analytical model of cosmic ray electron transport
A. Ivascenko F. Spanier Universität Würzburg, ITPA, Lehrstuhl für
Astronomie, Emil-Fischer-Str. 31, 97074 Würzburg, Germany
(12 November 2010; 25 January 2011; 14 February 2011)
###### Abstract
We present a numerical extension to the analytical propagation model
introduced in Hein and Spanier (2008) to describe the leptonic population in
the galactic disc. The model is used to derive a possible identification of
the components that contribute to the leptonic cosmic ray spectrum, as
measured by PAMELA, Fermi and HESS, with an emphasis on secondary
$e^{+}-e^{-}$ production in collisions of cosmic ray particles with ambient
interstellar medium (ISM). We find that besides secondaries, an additional
source symmetric in $e^{+}$ and $e^{-}$ production is needed to explain both
the PAMELA anomaly and the Fermi bump, assuming a power-law primary electron
spectrum. Our model also allows us to derive constraints for some properties
of the ISM.
Recently the leptonic component of the cosmic ray spectrum has gained new
attention. New observations from ATIC, PAMELA and Fermi show a deviation from
a power-law in the form of an excess in both the electron and positron
spectra. Annihilating dark matter (Allahverdi et al., 2009) and nearby pulsars
(Büsching et al., 2008; Grimani, 2009; Blasi and Amato, 2011) (among other
hypotheses) have been proposed as possible sources of the excess leptons.
Regardless of the source, a propagation model is needed to connect the energy
spectrum measured on Earth with the injection spectra.
We present our numerical cosmic ray transport model in application to the high
energy electron transport in the ISM. Spatial and momentum diffusion, particle
escape, acceleration via Fermi I and continuous energy losses were taken into
account and their effects on the steady-state energy spectrum analyzed. In
solving the transport equation we employed quasi-linear transport theory, the
diffusion approximation and a separation of the spatial and momentum problem
to obtain the leaky-box-equation, which was then solved numerically. The
spatial problem was solved analytically in cylindrical and prolate spheroidal
coordinates.
The transport model was employed to calculate the spectrum of secondary
electrons in our galaxy. We assume the leptons from pioArXivn decay to be the
dominating component of the secondary spectrum. The lepton spectrum from pion
collisions of highly relativistic cosmic ray protons with thermal protons in
the ISM was calculated using the parametrization of pion production in p-p-
collisions from Kelner et al. (2006) and used as the injection spectrum for
the transport model. With realistic simulation parameters the resulting
positron flux in the vicinity of the solar system lies remarkably close to the
low-energy PAMELA data. Assuming a generic power-law primary $e^{-}$ spectrum
and an additional $e^{+}-e^{-}$ source allows us to fit the datasets from
PAMELA, Fermi and HESS very nicely and to put constraints on several transport
model parameters and the corresponding ISM properties.
## 1 The data
The leptonic cosmic ray spectrum gained new attention when the results of the
balloon experiment ATIC were published (Chang et al., 2008). The authors
claimed an anomalous excess in the spectrum just below 1 TeV. This started
wild speculations about the nature of the source for these leptons, which
should be in the relative vicinity of the solar system.
Those speculations were fueled further by new measurements from the PAMELA
satellite published in Adriani et al. (2009) claiming a rise in the positron
spectrum above 10 GeV. The lepton spectrum from the Fermi satellite (Abdo et
al., 2009) could not confirm the ATIC-peak quantitatively but showed a
deviation from a straight power-law in the same energy region.
The HESS atmospheric Cerenkov telescope extended the spectrum measurement
beyond 1 TeV showing a cut-off or a steeper power-law but could also not
confirm the ATIC excess (Aharonian et al., 2009).
Recently the Fermi lepton spectrum was extended to lower energies (7 GeV) in
Ackermann et al. (2010). Also the PAMELA results were updated with new
experimental data and different evaluation techniques in Adriani et al.
(2010).
To interpret these measurements, it is necessary to connect the injection
spectra at the (possible) sources with the observations in the solar system.
To do that we need a transport model that includes all the relevant processes
in the ISM (spatial and momentum diffusion, cooling, particle escape, etc.).
## 2 Transport model
The derivation of the transport equation follows closely Lerche and
Schlickeiser (1988) and Hein and Spanier (2008). Using Quasi Linear Theory and
the diffusion approximation the Vlasov equation is transformed into the
steady-state diffusion-convection equation for the phase space density
$f(\mathbf{x},p)$, which can be written in the form
$\mathcal{L}_{x}f+\mathcal{L}_{p}f+S(\textbf{x},p)=0$ (1)
with the spatial operator
$\mathcal{L}_{x}(\textbf{x},p)\equiv\nabla\left[\kappa(\textbf{x},p)\nabla\right]$
(2)
and the momentum operator
$\mathcal{L}_{p}(\textbf{x},p)\equiv\frac{1}{p^{2}}\frac{\partial}{\partial
p}\left(p^{2}D(\textbf{x},p)\frac{\partial}{\partial
p}-p^{2}\dot{p}(\textbf{x},p)\right)-\frac{1}{\tau(\textbf{x},p)}\textrm{.}$
(3)
This equation describes the diffusion (coefficients $\kappa$ and $D$ in space
and momentum, respectively), acceleration and cooling (rates $\dot{p}$) and
escape (timescale $\tau$) of an injected source distribution $S$.
Following Lerche and Schlickeiser (1988) the equation can be solved by
separating the operators and the source term in their spatial and momentum
dependencies. Then the resulting steady state particle distribution can be
written as an infinite sum:
$f(\textbf{x},r)=\sum_{i}A_{i}(\textbf{x})R_{i}(p)$ (4)
The spatial coefficients $A_{i}(\textbf{x})$ can be acquired by solving the
diffusion equation in the appropriate geometry
$\frac{\partial}{\partial
u}A_{i}\exp(-\omega_{i}^{2}u)=\nabla\left(\kappa\nabla
A_{i}\exp(-\omega_{i}^{2}u)\right)\textrm{,}$ (5)
where $\omega_{i}^{2}$ take the role of inverse escape times for the momentum
modes and $u$ is a convolution variable.
In this work we used the analytical solution for cylindrical geometries as
provided by Wang and Schlickeiser (1987) since this offers the best reflection
of the symmetries of a spiral galaxy. Analytical solutions in spherical and
prolate spheroidal coordinates are also available (Schlickeiser et al., 1987;
Hein and Spanier, 2008) and can be used to describe particle transport in
elliptical galaxies and galaxy clusters.
The momentum modes $R_{i}(p)$ are solutions of the ODE, commonly referred to
as the leaky-box equation:
$\frac{1}{p^{2}}\frac{\partial}{\partial
p}\left[\frac{a_{2}p^{4}}{\kappa(p)}\frac{\partial R_{i}}{\partial
p}-\frac{a_{1}p^{3}}{\kappa(p)}R_{i}+\dot{p}R_{i}\right]-\omega_{i}^{2}\kappa(p)R_{i}=-Q(p)$
(6)
The parameters $a_{1}$ and $a_{2}$ represent the absolute scales for Fermi I
and II processes and depend mainly on the Alfven and shock speed,
respectively. As Lerche and Schlickeiser (1988) have shown, an analytical
solution for the equation can be found, if the cooling rate $\dot{p}$ scales
linearly with the particle energy, which is a good assumption for high energy
protons and heavier nuclei. Electrons above a few GeV, however, are cooled
primarily by synchrotron radiation and IC scattering, both scaling
quadratically in particle energy. Therefore the leaky-box equation was solved
using numeric relaxation methods.
To summarize, the model describes a particle distribution that is injected by
a homogeneous population of time-independent sources in the galactic disc
(injection region), diffuses throughout the galactic disc and halo
(confinement region), cooling by bremsstrahlung, adiabatic deceleration,
synchrotron radiation and inverse Compton scattering, and leaves the
confinement region at some point. The matter, photon and magnetic field
distributions are assumed to be homogeneous over the whole confinement region.
A model with a uniform source distribution is not a good choice to simulate
high energy leptons from discrete sources (SNR, PWN…), since the short energy
loss timescales make the source distribution critical for the resulting
spectra, as discussed in Blasi and Amato (2011). Conversely, production of
secondaries from p-p collisions in the galactic halo can be approximated very
well as a homogeneous source, since there are no dramatic variations in the CR
and ISM densities.
A similar analysis by Moskalenko and Strong (1998) used an analogue model as
in this paper, but obviously did not include the new data above 1 GeV.
Stephens (2001) has basically the same drawbacks, in addition this work has a
more sophisticated secondary production model. Compared with the recent works
in this area like Blasi and Amato (2011), we concentrate more on the low
energy end of the current PAMELA data and the constraints it can put on
propagation parameters if interpreted as secondaries.
## 3 Secondary leptons
Fortunately there is a lepton production process that can be approximated with
a homogeneous spatial distribution: the secondary leptons from the decay of
products of proton collisions. To get a galaxy-averaged flux of secondaries we
consider collisions of high energy cosmic ray protons with ambient thermal
matter, assuming that both populations are homogeneously distributed with
average densities.
The pion production cross-section and the electron source function for a
single p-p collision used here are parametrizations of results from the SIBYLL
event generator from Kelner et al. (2006). Folding the source function
$F_{e}\left(\frac{E_{e}}{E_{p}},E_{e}\right)$ with the cosmic ray spectrum
$J_{p}(E_{p})$ and the inelastic collision cross-section
$\sigma_{\mathrm{inel}}(E_{p})$ yields the electron flux:
$\frac{dN_{e}}{dE}=4\pi
n_{H}\int\sigma_{\mathrm{inel}}(E_{p})J_{p}(E_{p})F_{e}\left(\frac{E_{e}}{E_{p}},E_{e}\right)\frac{dE_{p}}{E_{p}}$
(7)
Since the source function is sharply peaked, the CR spectrum can be cut off
exponentially at the first knee ($\approx 1000$ TeV) with negligible
deviations to the lepton spectrum below 10 TeV:
$J_{p}(E_{p})=0.252\left(\frac{E_{p}}{\mathrm{TeV}}\right)^{-2.677}e^{\frac{-E_{p}}{10^{3}\
\mathrm{TeV}}}\mathrm{m}^{-2}\mathrm{sr}^{-1}\mathrm{s}^{-1}\mathrm{TeV}^{-1}$
This parametrisation of the CR spectrum was obtained by fitting the CR data
between $10^{10}$ and $10^{14}$ eV in Cronin et al. (1997).
The average thermal matter density $n_{H}$ is treated as a free parameter. The
resulting electron flux with a slightly harder power-law (spectral index
$2.62$) is used as the injection function in the transport model. Since the
parametrizations in Kelner et al. (2006) only consider p-p interactions, the
actual flux of secondaries is underestimated by the contribution from heavier
nuclei.
## 4 Parameter constraints
Figure 1: Parameter studies for the secondary positron spectrum compared to
the PAMELA $e^{+}$ fraction multiplied by a fit of the Fermi $e^{+}+e^{-}$
data. Combined systematic errors of PAMELA and Fermi are shown by the grey
area. The thick black line with $n_{H}=1.5\cdot 10^{3}\,\mathrm{m}^{-3}$,
$B=3\,\mathrm{\mu G}$ and $t=t_{\mathrm{esc}}$ yields the best fit of the
positron spectrum, if combined with an extra high energy source as shown in
Fig. 2. The thin and dashed black lines show the slope change with changing
escape losses.
The principal shape of the resulting steady-state lepton spectrum is shown in
Fig. 1. The cooling, diffusion and escape processes introduce breaks in the
injected power-law, so that multiple energy intervals with different spectral
indices are formed, each of them defined by a different process. At the first
break the injected power-law is steepened by $1/3$ as the diffusion/escape
time scale becomes shorter than the time scale of bremsstrahlung and adiabatic
cooling (linear in energy). The second break appears when the time scale for
synchrotron and IC losses becomes shorter that the diffusion time scale. These
quadratic losses steepen the spectrum by 1. The absolute flux value is
determined mainly by the ISM density, since the injected secondary spectrum
has a linear $n_{H}$ dependence (Eq. (7)), and to a lesser extent by escape
and cooling losses, which transport particles out of the simulation region.
So the model has three major parameters: the linear energy loss time scale
that reflects the thermal gas density $n_{H}$, the quadratic loss time scale
that incorporates the energy densities of the EM field (composed of a constant
CMB density of $\approx 0.3$ eV/cm3 and a variable magnetic field $B$) and the
escape time scale $t_{\mathrm{esc}}$ that corresponds to the size of the
confinement region.
Measurements of the positron spectrum, in particular the newest results from
PAMELA, allow us to constraint those parameters. A rather conservative
statement can be made, if we assume that the secondaries have a negligible
contribution to the positron spectrum. As shown in Fig. 1, assuming an average
ISM density of $\approx 10^{3}$ m-3, the confinement region has to be less
than the assumed $30$ kpc or the diffusion process has to be more efficient,
so that the particles leave the galaxy 10 times faster (black dotted line). A
higher ISM density would require an even more efficient particle escape, so as
not to overshoot the measured positron spectrum. Since the spectral shape
wouldn’t play any role, the magnetic field can be almost arbitrary.
*
Figure 2: A fit of Fermi (Ackermann et al., 2010) and HESS (Aharonian et al.,
2009) lepton data (solid red line) and PAMELA (Adriani et al., 2010) positron
data (solid green line). Grey areas show the systematic errors. The components
that contribute to the spectra are secondary positrons (dashed green), a
generic power-law primary spectrum (dashed red) and an additional symmetric
$e^{+}-e^{-}$ source (solid and dashed black).
On the other hand, the PAMELA data can be fitted very well with a
superposition of two power-laws with a spectral index of about $3.6$ for the
low energy part, corresponding remarkably well with the secondary spectrum
steepened by synchrotron and IC losses. Assuming the low energy positrons to
be secondaries puts much stricter constraints on the parameters, since the
second break in the spectrum has to appear at $\approx 1$ GeV to match the
data. The break energy is determined by the equilibrium of the
diffusive/escape losses and the radiative losses. The thick black line in Fig.
1 is the best fit with $n_{H}=1.5\cdot 10^{3}$ m-3, $B=3\,\mu G$ and
$t=t_{\mathrm{esc}}$. The radiative losses in the galactic halo are dominated
by IC scattering on the CMB photons, so that lowering the magnetic field has
no effect on the break energy. A slight flattening at the low energy end of
the spectrum is needed for the best fit of the new PAMELA data. This
corresponds to a halo radius of $30$ kpc and a spatial diffusion coefficient
$\kappa(E=1\,\mathrm{GeV})=4.5\cdot 10^{24}$ m2 s-1, yielding an average
confinement time $t_{\mathrm{esc}}=2\cdot 10^{17}$ s. The absolute flux fixes
the third parameter, the ISM density, to $1.5\cdot 10^{3}$ m-3. Taking into
consideration that about $99\%$ of the confinement region represent the
galactic halo with an average gas density of about $10^{3}$ m-3 (McKee and
Ostriker, 1977), the parameters seem very realistic.
## 5 Lepton excess
In Fig. 2 the latest Fermi lepton measurement, in which the spectrum has been
extended down to $7$ GeV, is shown. A generic fit of it was used to calculate
a positron flux from the PAMELA positron fraction measurement. This data is
used because it was taken simultaneously in the same solar cycle making
interpretation easier. The HESS high-energy data was shifted down by $15\%$ to
better coincide with the Fermi data. This is still within the systematic error
margins of HESS, not to mention the systematic error of Fermi being of the
same order of magnitude.
Once the secondary background has been subtracted from the PAMELA positron
flux, the excess can be fitted very well with a simple power-law with a
spectral index of $2.3$ (Fig. 2, black dashed line). Remarkably, if we assume
the source of these “extra” positrons to be symmetric in $e^{+}-e^{-}$, we can
also fit the Fermi bump leaving a primary lepton component of the shape
$E^{-3.2}\cdot\exp(-E/1.46$ TeV) (red dashed line). The (shifted) high-energy
HESS data provides a means to fix the cut-off energy of the “extra” leptons to
about 920 GeV.
With the help of our semi-analytic transport model we were able to identify
the low-energy PAMELA data as secondary positrons from collisions of CR
protons with ISM gas using very realistic values for the simulation
parameters. An indirect implication from that is that lepton transport is
dominated by radiative losses (synchrotron and IC) above 1 GeV.
Assuming a generic primary lepton spectrum of the form
$E^{-3.2}\cdot\mathrm{exp}(-E/1.46$ TeV), the PAMELA anomaly, the Fermi bump
and the high energy HESS data can all be fitted by a single component, that is
symmetric in $e^{+}-e^{-}$ with a power-law spectral index of $2.3$ and a cut-
off energy of 920 GeV. This could either be a lepton population injected into
the ISM by nearby source with $E^{-2.3}$, so that the propagation time scale
is shorter than the radiative loss time scale, or a more distant source with a
much harder spectrum $E^{-1.3}$ which is then steepened by radiative losses. A
supernova remnant would be a good candidate in the first case, pulsars of
different ages would suite both cases.
###### Acknowledgements.
We acknowledge the support by the Deutsche Forschungsgesellschaft through the
Graduate School “Astroplasmaphysik” and through grant SP 1124-3.
Edited by: R. Vainio
Reviewed by: C. Grimani and another anonymous referee
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|
arxiv-papers
| 2011-07-05T09:20:03 |
2024-09-04T02:49:20.297279
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alex Ivascenko and Felix Spanier",
"submitter": "Alex Ivascenko",
"url": "https://arxiv.org/abs/1107.0826"
}
|
1107.0882
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
LHCb-PAPER-2011-005 CERN-PH-EP-2011-082
Measurement of $V^{0}$ production ratios in $pp$ collisions at $\sqrt{s}=$
0.9 and 7$\mathrm{\,Te\kern-2.07413ptV}$
The LHCb Collaboration 111Authors are listed on the following pages.
The $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ production
ratios are measured by the LHCb detector from $0.3\,\mathrm{nb}^{-1}$ of $pp$
collisions delivered by the LHC at $\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$
and $1.8\,\mathrm{nb}^{-1}$ at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. Both
ratios are presented as a function of transverse momentum, $p_{\mathrm{T}}$,
and rapidity, $y$, in the ranges
$0.15<p_{\mathrm{T}}<2.50{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y<4.5$. Results at the two energies are in good agreement as a function
of rapidity loss, $\Delta y=y_{\textrm{beam}}-y$, and are consistent with
previous measurements. The ratio $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$, measuring the transport
of baryon number from the collision into the detector, is smaller in data than
predicted in simulation, particularly at high rapidity. The ratio $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$, measuring the
baryon-to-meson suppression in strange quark hadronisation, is significantly
larger than expected.
The LHCb Collaboration
R. Aaij23, B. Adeva36, M. Adinolfi42, C. Adrover6, A. Affolder48, Z.
Ajaltouni5, J. Albrecht37, F. Alessio6,37, M. Alexander47, G. Alkhazov29, P.
Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J. Anderson39,
R.B. Appleby50, O. Aquines Gutierrez10, L. Arrabito53, A. Artamonov 34, M.
Artuso52,37, E. Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back44, D.S.
Bailey50, V. Balagura30,37, W. Baldini16, R.J. Barlow50, C. Barschel37, S.
Barsuk7, W. Barter43, A. Bates47, C. Bauer10, Th. Bauer23, A. Bay38, I.
Bediaga1, K. Belous34, I. Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson46, R. Bernet39, M.-O. Bettler17,37, M. van Beuzekom23,
A. Bien11, S. Bifani12, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake49, F.
Blanc38, C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A.
Bondar33, N. Bondar29, W. Bonivento15, S. Borghi47, A. Borgia52, T.J.V.
Bowcock48, C. Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D.
Brett50, S. Brisbane51, M. Britsch10, T. Britton52, N.H. Brook42, A. Büchler-
Germann39, I. Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, J.M.
Caicedo Carvajal37, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A.
Camboni35, P. Campana18,37, A. Carbone14, G. Carboni21,k, R. Cardinale19,i, A.
Cardini15, L. Carson36, K. Carvalho Akiba23, G. Casse48, M. Cattaneo37, M.
Charles51, Ph. Charpentier37, N. Chiapolini39, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, F. Constantin28, G. Conti38, A. Contu51, M.
Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, B. D’Almagne7, C.
D’Ambrosio37, P. David8, I. De Bonis4, S. De Capua21,k, M. De Cian39, F. De
Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi38,37, M. Deissenroth11, L. Del Buono8, C.
Deplano15, O. Deschamps5, F. Dettori15,d, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34,
C. Eames49, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D.G. d’Enterria35,o, D. Esperante Pereira36, L. Estève43, A.
Falabella16,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23, S.
Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32, C.
Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, C. Gaspar37, N. Gauvin38, M. Gersabeck37, T. Gershon44,
Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, S.
Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G. Haefeli38, C.
Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R. Harji49, N.
Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K. Hennessy48,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, W. Hofmann10, K.
Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12,
D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R.
Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F. Jansen23, P.
Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R. Jones43, B.
Jost37, S. Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, S.
Koblitz37, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M.
Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M.
Kucharczyk20,25, S. Kukulak25, R. Kumar14,37, T. Kvaratskheliya30,37, V.N. La
Thi38, D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert37,
E. Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, R. Le Gac6, J.
van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefran$c$cois7, O.
Leroy6, T. Lesiak25, L. Li3, Y.Y. Li43, L. Li Gioi5, M. Lieng9, R. Lindner37,
C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-
March38, J. Luisier38, F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O.
Maev29,37, J. Magnin1, S. Malde51, R.M.D. Mamunur37, G. Manca15,d, G.
Mancinelli6, N. Mangiafave43, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti22, A. Martens7, L. Martin51, A. Martín Sánchez7, D. Martinez
Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20, M. Matveev29, E.
Maurice6, B. Maynard52, A. Mazurov32,16,37, G. McGregor50, R. McNulty12, C.
Mclean14, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, S. Monteil5, D. Moran12, P.
Morawski25, J.V. Morris45, R. Mountain52, I. Mous23, F. Muheim46, K. Müller39,
R. Muresan28,38, B. Muryn26, M. Musy35, P. Naik42, T. Nakada38, R.
Nandakumar45, J. Nardulli45, I. Nasteva1, M. Nedos9, M. Needham46, N.
Neufeld37, C. Nguyen-Mau38,p, M. Nicol7, S. Nies9, V. Niess5, N. Nikitin31, A.
Oblakowska-Mucha26, V. Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41,
R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen49, B. Pal52,
J. Palacios39, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo13,b,
C. Parkes47,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49,
S.K. Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A.
Pazos Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, B. Pie Valls35, B. Pietrzyk4, T. Pilar44, D. Pinci22, R.
Plackett47, S. Playfer46, M. Plo Casasus36, G. Polok25, A. Poluektov44,33, E.
Polycarpo2, D. Popov10, B. Popovici28, C. Potterat35, A. Powell51, T. du
Pree23, J. Prisciandaro38, V. Pugatch41, A. Puig Navarro35, W. Qian52, J.H.
Rademacker42, B. Rakotomiaramanana38, I. Raniuk40, G. Raven24, S. Redford51,
M.M. Reid44, A.C. dos Reis1, S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues47, F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43,
V. Romanovsky34, J. Rouvinet38, T. Ruf37, H. Ruiz35, G. Sabatino21,k, J.J.
Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d, C. Salzmann39, M.
Sannino19,i, R. Santacesaria22, R. Santinelli37, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina30, P. Schaack49, M. Schiller11, S. Schleich9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, A.
Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N.
Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37,
P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40,
V. Shevchenko30, A. Shires49, R. Silva Coutinho54, H.P. Skottowe43, T.
Skwarnicki52, A.C. Smith37, N.A. Smith48, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro49, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, N. Styles46, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51,45, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, M.T. Tran38, A. Tsaregorodtsev6,
N. Tuning23, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14,
R. Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, K. Vervink37, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J.
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S. Wandernoth11, J. Wang52, D.R. Ward43, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Yang3, R. Young46, O.
Yushchenko34, M. Zavertyaev10,a, L. Zhang52, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin 37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Cracow, Poland
26Faculty of Physics & Applied Computer Science, Cracow, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oInstitució Catalana de Recerca i Estudis Avan$c$cats (ICREA), Barcelona,
Spain
pHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
While the underlying interactions of hadronic collisions and hadronisation are
understood within the Standard Model, exact computation of the processes
governed by QCD are difficult due to the highly non-linear nature of the
strong force. In the absence of full calculations, generators based on
phenomenological models have been devised and optimised, or “tuned”, to
accurately reproduce experimental observations. These generators predict how
Standard Model physics will behave at the LHC and constitute the reference for
discoveries of New Physics effects.
Strange quark production is a powerful probe for hadronisation processes at
$pp$ colliders since protons have no net strangeness. Recent experimental
results in the field have been published by STAR [1] from RHIC $pp$ collisions
at $\sqrt{s}=0.2\mathrm{\,Te\kern-1.00006ptV}$ and by ALICE [2], CMS [3] and
LHCb [4] from LHC $pp$ collisions at $\sqrt{s}=0.9$ and
7$\mathrm{\,Te\kern-1.00006ptV}$. LHCb can make an important contribution
thanks to a full instrumentation of the detector in the forward region that is
unique among the LHC experiments. Studies of data recorded at different
energies with the same apparatus help to control the experimental systematic
uncertainties.
In this paper we report on measurements of the efficiency corrected production
ratios of the strange particles $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$, $\mathchar 28931\relax$ and
$K^{0}_{\rm\scriptscriptstyle S}$ as observables related to the fundamental
processes behind parton fragmentation and hadronisation. The ratios
$\displaystyle\frac{\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}}{\mathchar 28931\relax}$
$\displaystyle=\frac{\sigma(pp\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}X)}{\sigma(pp\\!\rightarrow\mathchar 28931\relax
X)}$ (1) and $\displaystyle\frac{\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}}{K^{0}_{\rm\scriptscriptstyle S}}$
$\displaystyle=\frac{\sigma(pp\\!\rightarrow\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}X)}{\sigma(pp\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}X)}$ (2)
have predicted dependences on rapidity, $y$, and transverse momentum,
$p_{\mathrm{T}}$, which can vary strongly between different tunes of the
generators.
Measurements of the ratio $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ allow
the study of the transport of baryon number from $pp$ collisions to final
state hadrons and the ratio $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle
S}$ is a measure of baryon-to-meson suppression in strange quark
hadronisation.
## 2 The LHCb detector and data samples
The Large Hadron Collider beauty experiment (LHCb) at CERN is a single-arm
spectrometer covering the forward rapidity region. The analysis presented in
this paper relies exclusively on the tracking detectors. The high precision
tracking system begins with a silicon strip Vertex Locator (VELO), designed to
identify displaced secondary vertices up to about 65$\rm\,cm$ downstream of
the nominal interaction point. A large area silicon tracker follows upstream
of a dipole magnet and tracker stations, built with a mixture of straw tube
and silicon strip detectors, are located downstream. The LHCb coordinate
system is defined to be right-handed with its origin at the nominal
interaction point, the $z$ axis aligned along the beam line towards the magnet
and the $y$ axis pointing upwards. The bending plane is horizontal and the
magnet has a reversible field, with the positive $B_{y}$ polarity called “up”
and the negative “down”. Tracks reconstructed through the full spectrometer
experience an integrated magnetic field of around 4 Tm. The detector is
described in full elsewhere [5].
A loose minimum bias trigger is used for this analysis, requiring at least one
track segment in the downstream tracking stations. This trigger is more than
$99$ % efficient for offline selected events that contain at least two tracks
reconstructed through the full system.
Complementary data sets were recorded at two collision energies of
$\sqrt{s}=0.9$ and 7$\mathrm{\,Te\kern-1.00006ptV}$, with both polarities of
the dipole magnet. An integrated luminosity of $0.3\,\mathrm{nb}^{-1}$
(corresponding to 12.5 million triggers) was taken at the lower energy, of
which 48 % had the up magnetic field configuration. At the higher energy, 67 %
of a total $1.8\,\mathrm{nb}^{-1}$ (110.3 million triggers) was taken with
field up.
At injection energy ($\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$), the proton
beams are significantly broadened spatially compared to the accelerated beams
at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. To protect the detector, the two
halves of the VELO are retracted along the $x$ axis from their nominal
position of inner radius of 8$\rm\,mm$ to the beam, out to 18$\rm\,mm$, which
results in a reduction of the detector acceptance at small angles to the beam
axis by approximately 0.5 units of rapidity.
The beams collide with a crossing angle in the horizontal plane tuned to
compensate for LHCb’s magnetic field. The angle required varies as a function
of beam configuration and for the data taking period covered by this study was
set to 2.1$\rm\,mrad$ at $\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$ and 270
µrad at 7$\mathrm{\,Te\kern-1.00006ptV}$. Throughout this analysis $V^{0}$
momenta and any derived quantity such as rapidity are computed in the centre-
of-mass frame of the colliding protons.
Samples of Monte Carlo (MC) simulated events have been produced in close
approximation to the data-taking conditions described above for estimation of
efficiencies and systematic uncertainties. A total of 73 million simulated
minimum bias events were used for this analysis per magnet polarity at
$\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$ and 60 (69) million events at
7$\mathrm{\,Te\kern-1.00006ptV}$ for field up (down). LHCb MC simulations are
described in Ref. [6], with $pp$ collisions generated by Pythia 6 [7].
Emerging particles decay via EvtGen [8], with final state radiation handled
by Photos [9]. The resulting particles are transported through LHCb by Geant
4 [10], which models hits on the sensitive elements of the detector as well
as interactions between the particles and the detector material. Secondary
particles produced in these material interactions decay via Geant 4.
Additional samples of five million minimum bias events were generated for
studies of systematic uncertainties using Pythia 6 variants Perugia 0 (tuned
on experimental results from SPS, LEP and Tevatron) and Perugia NOCR (an
extreme model of baryon transport) [11]. Similarly sized samples of Pythia 8
[12] minimum bias diffractive events were also generated, including both hard
and soft diffraction 222Single- and double-diffractive process types are
considered: 92–94 in Pythia 6.421, with soft diffraction, and 103–105 in
Pythia 8.130, with soft and hard diffraction. [13].
## 3 Analysis procedure
$V^{0}$ hadrons are named after the “V”-shaped track signature of their
dominant decays: $\mathchar 28931\relax\\!\rightarrow p\pi^{-}$, $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}\\!\rightarrow\overline{}p\pi^{+}$ and
$K^{0}_{\rm\scriptscriptstyle S}\\!\rightarrow\pi^{+}\pi^{-}$, which are
reconstructed for this analysis. Only tracks with quality
$\chi^{2}/\textrm{ndf}<9$ are considered, with the $V^{0}$ required to decay
within the VELO and the daughter tracks to be reconstructed through the full
spectrometer. Any oppositely-charged pair is kept as a potential $V^{0}$
candidate if it forms a vertex with $\chi^{2}<9$ (with one degree of freedom
for a $V^{0}$ vertex). $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$, $\mathchar 28931\relax$ and
$K^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have invariant
masses within $\pm 50$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the PDG
values [14]. This mass window is large compared to the measured mass
resolutions of about 2${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$\mathchar 28931\relax$ ($\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$) and
5${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for $K^{0}_{\rm\scriptscriptstyle
S}$.
Combinatorial background is reduced with a Fisher discriminant based on the
impact parameters (IP) of the daughter tracks ($d^{\pm}$) and of the
reconstructed $V^{0}$ mother, where the impact parameter is defined as the
minimum distance of closest approach to the nearest reconstructed primary
interaction vertex measured in mm. The Fisher discriminant:
$\mathcal{F}_{\mathrm{IP}}=a\log_{10}(d^{+}_{\mathrm{IP}}/1\,\mathrm{mm})+b\log_{10}(d^{-}_{\mathrm{IP}}/1\,\mathrm{mm})+c\log_{10}(V^{0}\\!_{\mathrm{IP}}/1\,\mathrm{mm})$
(3)
is optimised for signal significance ($S/\sqrt{S+B}$) on simulated events
after the above quality criteria. The cut value,
$\mathcal{F}_{\mathrm{IP}}>1$, and coefficients, $a=b=-c=1$, were found to be
suitable for $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$, $\mathchar 28931\relax$ and
$K^{0}_{\rm\scriptscriptstyle S}$ at both collision energies (Fig. 1).
Figure 1: The Fisher discriminant $\mathcal{F}_{\mathrm{IP}}$ in 0.5 million
Monte Carlo simulated minimum bias events at
$\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$ for 1 $K^{0}_{\rm\scriptscriptstyle
S}$ and 1 $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$.
The $\mathchar 28931\relax$ ($\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$) signal significance is improved by a $\pm
4.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ veto around the PDG
$K^{0}_{\rm\scriptscriptstyle S}$ mass after re-calculation of each
candidate’s invariant mass with an alternative $\pi^{+}\pi^{-}$ daughter
hypothesis. A similar veto to remove $\mathchar 28931\relax$ ($\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$) with a $p\pi^{-}$ ($\overline{}p\pi^{+}$)
hypothesis from the $K^{0}_{\rm\scriptscriptstyle S}$ sample is not found to
improve significance so is not applied.
After the above selection, $V^{0}$ yields are estimated from data and
simulation by fits to the invariant mass distributions, examples of which are
shown in Fig. 2. These fits are carried out with the method of unbinned
extended maximum likelihood and are parametrised by a double Gaussian signal
peak (with a common mean) over a linear background. The mean values show a
small, but statistically significant, deviation from the known
$K^{0}_{\rm\scriptscriptstyle S}$ and $\mathchar 28931\relax$ ($\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$) masses [14], reflecting the status of the
momentum-scale calibration of the experiment. The width of the peak is
computed as the quadratic average of the two Gaussian widths, weighted by
their signal fractions. This width is found to be constant as a function of
$p_{\mathrm{T}}$ and increases linearly toward higher $y$, e.g. by 1.4
(0.8)${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ per unit rapidity for
$K^{0}_{\rm\scriptscriptstyle S}$ ($\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$) at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$.
The resulting signal yields are listed in Table 1.
Significant differences are observed between $V^{0}$ kinematic variables
reconstructed in data and in the simulation used for efficiency determination.
These differences can produce a bias for the measurement of $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ given the
different production kinematics of the baryon and meson. Simulated $V^{0}$
candidates are therefore weighted to match the two-dimensional
$p_{\mathrm{T}}$, $y$ distributions observed in data. These distributions are
shown projected along both axes in Fig. 3. The $V^{0}$ signal yield
$p_{\mathrm{T}}$, $y$ distributions are estimated from selected data and Monte
Carlo candidates using sideband subtraction. Two-dimensional fits, linear in
both $p_{\mathrm{T}}$ and $y$, are made to the ratios data/MC of these yields
independently for $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$, $\mathchar 28931\relax$ and
$K^{0}_{\rm\scriptscriptstyle S}$, for each magnet polarity and collision
energy. The resulting functions are used to weight generated and selected
$V^{0}$ candidates in the Monte Carlo simulation. These weights vary across
the measured $p_{\mathrm{T}}$, $y$ range between 0.4 and 2.1, with typical
values between 0.8 and 1.2.
Figure 2: Invariant mass peaks for 2 $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$ in the range
$0.25<p_{\mathrm{T}}<2.50{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ & $2.5<y<3.0$
and 2 $K^{0}_{\rm\scriptscriptstyle S}$ in the range
$0.65<p_{\mathrm{T}}<1.00{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ & $3.5<y<4.0$
at $\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$ with field up. Signal yields,
$N$, are found from fits (solid curves) with a double Gaussian peak with
common mean, $\mu$, over a linear background (dashed lines). The width,
$\sigma$, is computed as the quadratic average of the two Gaussian widths
weighted by their signal fractions. Table 1: Integrated signal yields
extracted by fits to the invariant mass distributions of selected $V^{0}$
candidates from data taken with magnetic field up and down at $\sqrt{s}=0.9$
and 7$\mathrm{\,Te\kern-0.90005ptV}$.
$\sqrt{s}$ | 0.9$\mathrm{\,Te\kern-0.90005ptV}$ | 7$\mathrm{\,Te\kern-0.90005ptV}$
---|---|---
Magnetic field | Up | Down | Up | Down
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$ | $3,440\pm 60$ | $4,100\pm 70$ | $258,930\pm 640$ | $132,550\pm 460$
$\mathchar 28931\relax$ | $4,880\pm 80$ | $5,420\pm 80$ | $294,010\pm 680$ | $141,860\pm 460$
$K^{0}_{\rm\scriptscriptstyle S}$ | $35,790\pm 200$ | $40,230\pm 220$ | $2,737,090\pm 1,940$ | $1,365,990\pm 1,370$
The measured ratios are presented in three complementary binning schemes:
projections over the full $p_{\mathrm{T}}$ range, the full $y$ range, and a
coarser two-dimensional binning. The rapidity range $2.0<y<4.0$ (4.5) is split
into 0.5-unit bins, while six bins in $p_{\mathrm{T}}$ are chosen to
approximately equalise signal $V^{0}$ statistics in data over the range 0.25
$(0.15)<p_{\mathrm{T}}<2.50$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ from
collisions at $\sqrt{s}=0.9$ (7)$\mathrm{\,Te\kern-1.00006ptV}$. The two-
dimensional binning combines pairs of $p_{\mathrm{T}}$ bins. The full analysis
procedure is carried out independently in each $p_{\mathrm{T}}$, $y$ bin.
The efficiency for selecting prompt $V^{0}$ decays is estimated from
simulation as
$\varepsilon=\frac{N(V^{0}\\!\rightarrow
d^{+}d^{-})_{\mathrm{Observed}}}{N(pp\rightarrow
V^{0}\\!X)_{\mathrm{Generated}}},$ (4)
where the denominator is the number of prompt $V^{0}$ hadrons generated in a
given $p_{\mathrm{T}}$, $y$ region after weighting and the numerator is the
number of those weighted candidates found from the selection and fitting
procedure described above. The efficiency therefore accounts for decays via
other channels and losses from interactions with the detector material. Prompt
$V^{0}$ hadrons are defined in Monte Carlo simulation by the cumulative
lifetimes of their ancestors
$\sum\limits_{i=1}^{n}c\tau_{i}<10^{-9}\rm\,m,$ (5)
where $\tau_{i}$ is the proper decay time of the $i^{\textrm{th}}$ ancestor.
This veto is defined such as to keep only $V^{0}$ hadrons created either
directly from the $pp$ collisions or from the strong or electromagnetic decays
of particles produced at those collisions, removing $V^{0}$ hadrons generated
from material interactions and weak decays. The Fisher discriminant
$\mathcal{F}_{\mathrm{IP}}$ strongly favours prompt $V^{0}$ hadrons, however
a small non-prompt contamination in data would lead to a systematic bias in
the ratios. The fractional contamination of selected events is determined from
simulation to be $2-6\,\%$ for $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$ and $\mathchar 28931\relax$, depending on the
measurement bin, and about 1 % for $K^{0}_{\rm\scriptscriptstyle S}$. This
effect is dominated by weak decays rather than material interactions. The
resulting absolute corrections to the ratios $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ are
approximately 0.01.
Figure 3: 3 Transverse momentum and 3 rapidity distributions for
$K^{0}_{\rm\scriptscriptstyle S}$ in data and Monte Carlo simulation at
$\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$. The difference between data and
Monte Carlo is reduced by weighting the simulated candidates.
## 4 Systematic uncertainties
The measured efficiency corrected ratios $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ are
subsequently corrected for non-prompt contamination as found from Monte Carlo
simulation and defined by Eq. 5. This procedure relies on simulation and the
corrections may be biased by the choice of the LHCb MC generator tune. To
estimate a systematic uncertainty on the correction for non-prompt $V^{0}$,
the contaminant fractions are also calculated using two alternative tunes of
Pythia 6: Perugia 0 and Perugia NOCR [11]. The maximum differences in non-
prompt fraction across the measurement range and at both energies are $<1\,\%$
for each $V^{0}$ species. The resulting absolute uncertainties on the ratios
are $<0.01$.
The efficiency of primary vertex reconstruction may introduce a bias on the
measured ratios if the detector occupancy is different for events containing
$K^{0}_{\rm\scriptscriptstyle S}$, $\mathchar 28931\relax$ or $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$. This efficiency is compared in data and
simulation using $V^{0}$ samples obtained with an alternative selection not
requiring a primary vertex. Instead, the $V^{0}$ flight vector is
extrapolated towards the beam axis to find the point of closest approach. The
$z$ coordinate of this point is used to define a pseudo-vertex, with $x=y=0$.
Candidates are kept if the impact parameters of their daughter tracks to this
pseudo-vertex are $>0.2$ mm. There is a large overlap of signal candidates
with the standard selection. The primary vertex finding efficiency is then
explored by taking the ratio of these selected events which do or do not have
a standard primary vertex. Calculated in bins of $p_{\mathrm{T}}$ and $y$,
this efficiency agrees between data and simulation to better than 2 % at both
$\sqrt{s}=0.9$ and 7$\mathrm{\,Te\kern-1.00006ptV}$. The resulting absolute
uncertainties on $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and
$\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ are $<0.02$ and
$<0.01$, respectively.
Figure 4: The double ratios 4 $(\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax)_{\mathrm{Data}}/(\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax)_{\mathrm{MC}}$ and 4
$(\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle
S})_{\mathrm{Data}}/(\kern 1.79997pt\overline{\kern-1.79997pt\mathchar
28931\relax\kern 0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle
S})_{\mathrm{MC}}$ are shown as a function of the material traversed, in units
of radiation length. Flat line fits, shown together with their respective
$\chi^{2}$ probabilities, give no evidence of a bias.
The primary vertex finding algorithm requires at least three reconstructed
tracks.333The minimum requirements for primary vertex reconstruction at LHCb
can be approximated in Monte Carlo simulation by a generator-level cut
requiring at least three charged particles from the collision with lifetime
$c\tau>10^{-9}\rm\,m$, momentum $p>0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and polar angle $15<\theta<460\rm\,mrad$. Therefore, the reconstruction highly
favours non-diffractive events due to the relatively low efficiency for
finding diffractive interaction vertices, which tend to produce fewer tracks.
In the LHCb MC simulation, the diffractive cross-section accounts for 28 (25)
% of the total minimum-bias cross-section of 65 (91) mb at 0.9
(7)$\mathrm{\,Te\kern-1.00006ptV}$ [6]. Due to the primary vertex
requirement, only about 3 % of the $V^{0}$ candidates selected in simulation
are produced in diffractive events. These fractions are determined using
Pythia 6 which models only soft diffraction. As a cross check, the fractions
are also calculated with Pythia 8 which includes both soft and hard
diffraction. The variation on the overall efficiency between models is about 2
% for both ratios at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ and close to 1
% at 0.9$\mathrm{\,Te\kern-1.00006ptV}$. Indeed, complete removal of
diffractive events only produces a change of $0.01-0.02$ in the ratios across
the measurement range.
The track reconstruction efficiency depends on particle momentum. In
particular, the tracking efficiency varies rapidly with momentum for tracks
below 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Any bias is expected to be
negligible for the ratio $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ but can
be larger for $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle
S}$ due to the different kinematics. Two complementary procedures are employed
to check this efficiency. First, track segments are reconstructed in the
tracking stations upstream of the magnet. These track segments are then paired
with the standard tracks reconstructed through the full detector and the pairs
are required to form a $K^{0}_{\rm\scriptscriptstyle S}$ to ensure only
genuine tracks are considered. This track matching gives a measure of the
tracking efficiency for the upstream tracking systems. The second procedure
uses the downstream stations to reconstruct track segments, which are
similarly paired with standard tracks to measure the efficiency of the
downstream tracking stations. The agreement between these efficiencies in data
and simulation is better than 5 %. To estimate the resulting uncertainty on
$\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$, both ratios
are re-calculated after weighting $V^{0}$ candidates by $95\,\%$ for each
daughter track with momentum below 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
The resulting systematic shifts in the ratios are $<0.01$.
Table 2: Absolute systematic errors are listed in descending order of
importance. Ranges indicate uncertainties that vary across the measurement
bins and/or by collision energy. Correlated sources of uncertainty between
field up and down are identified.
Sources of systematic uncertainty | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
_Correlated between field up and down_ : | |
Material interactions | $0.02$ | $0.02$
Diffractive event fraction | $0.01-0.02$ | $0.01-0.02$
Primary vertex finding | $<0.02$ | $<0.01$
Non-prompt fraction | $<0.01$ | $<0.01$
Track finding | negligible | $0.01$
_Uncorrelated_ : | |
Kinematic correction | $0.01-0.05$ | $<0.03$
Signal extraction from fit | $0.001$ | $0.001$
Total | $0.02-0.06$ | $0.02-0.03$
Particle interactions within the detector are simulated using the Geant 4
package, which implements interaction cross-sections for each particle
according to the LHEP physics list [10]. These simulated cross-sections have
been tested in the LHCb framework and are consistent with the LHEP values. The
small measured differences are propagated to $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ to estimate
absolute uncertainties on the ratios of about 0.02. $V^{0}$ absorption is
limited by the requirement that each $V^{0}$ decay occurs within the most
upstream tracker (the VELO). Secondary $V^{0}$ production in material is
suppressed by the Fisher discriminant, which rejects $V^{0}$ candidates with
large impact parameter. The potential bias on the ratios is explored by
measurement of both $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and
$\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ as a function
of material traversed (determined by the detector simulation), in units of
radiation length, $X_{0}$. Data and simulation are compared by their ratio,
shown in Fig. 4. These double ratios are consistent with a flat line as a
function of $X_{0}$, therefore any possible imperfections in the description
of the detector material in simulation do not have a large effect on the
$V^{0}$ ratios. Note that the double ratios are not expected to be unity
since simulations do not predict the same values for $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ as are observed
in data.
The potential bias from the Fisher discriminant, $\mathcal{F}_{\mathrm{IP}}$,
is investigated using a pre-selected sample, with only the track and vertex
quality cuts applied. The distributions of $\mathcal{F}_{\mathrm{IP}}$ for
$\mathchar 28931\relax$, $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$ and
$K^{0}_{\rm\scriptscriptstyle S}$ in data and Monte Carlo simulation are
estimated using sideband subtraction. The double ratios of data/MC
efficiencies are seen to be independent of the discriminant, implying that the
distribution is well modelled in the simulation. No systematic uncertainty is
assigned to this selection requirement.
A degradation is observed of the reconstructed impact parameter resolution in
data compared to simulation. The simulated $V^{0}$ impact parameters are
recalculated with smeared primary and secondary vertex positions to match the
resolution measured in data. There is a negligible effect on the $V^{0}$
ratio results.
A good estimate of the reconstructed yields and their uncertainties in both
data and simulation is provided by the fitting procedure but there may be a
residual systematic uncertainty from the choice of this method. Comparisons
are made using side-band subtraction and the resulting $V^{0}$ yields are in
agreement with the results of the fits at the 0.1 % level. The resulting
absolute uncertainties on the ratios are on the order of 0.001.
Simulated events are weighted to improve agreement between simulated $V^{0}$
kinematic distributions and data. As described in Section 3, these weights are
calculated from a two-dimensional fit, linear in both $p_{\mathrm{T}}$ and
$y$, to the distribution of the ratio between reconstructed data and simulated
Monte Carlo candidates. This choice of parametrisation could be a source of
systematic uncertainty, therefore alternative procedures are investigated
including a two-dimensional polynomial fit to $3^{\textrm{rd}}$ order in both
$p_{\mathrm{T}}$ and $y$ and a (non-parametric) bilinear interpolation. The
results from each method are compared across the measurement range to estimate
typical systematic uncertainties of $0.01-0.05$ for $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $<0.03$ for $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$.
The lifetime distributions of reconstructed and selected $V^{0}$ candidates
are consistent between data and simulation. The possible influence of
transverse $\mathchar 28931\relax$ ($\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$) polarisation was explored by simulations with
extreme values of polarisation and found to produce no significant effect on
the measured ratios. Potential acceptance effects were checked as a function
of azimuthal angle, with no evidence of systematic bias. The potential sources
of systematic uncertainty or bias are summarised in Table 2.
Figure 5: The ratios $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar
28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ from the full
analysis procedure at 5 & 5 $\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$ and 5
& 5 7$\mathrm{\,Te\kern-0.90005ptV}$ are shown as a function of rapidity,
compared across intervals of transverse momentum. Vertical lines show the
combined statistical and systematic uncertainties and the short horizontal
bars (where visible) show the statistical component.
## 5 Results
The $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ and $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ production
ratios are measured independently for each magnetic field polarity. These
measurements show good consistency after correction for detector acceptance.
Bin-by-bin comparisons in the two-dimensional binning scheme give $\chi^{2}$
probabilities for $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ ($\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$) of 3 (18) % at
$\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$ and 19 (97) % at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, with 12 (15) degrees of freedom.
The field up and down results are therefore combined to maximise statistical
significance. A weighted average is computed such that the result has minimal
variance while taking into account the correlations between sources of
systematic uncertainty identified in Table 2. These combined results are shown
as a function of $y$ in three intervals of $p_{\mathrm{T}}$ in Fig. 5 at
$\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$ and
7$\mathrm{\,Te\kern-1.00006ptV}$. The ratio $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ shows a strong
$p_{\mathrm{T}}$ dependence.
Figure 6: The ratios $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar
28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ at
$\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$ are compared with the predictions
of the LHCb MC, Perugia 0 and Perugia NOCR as a function of 6 & 6 rapidity and
6 & 6 transverse momentum. Vertical lines show the combined statistical and
systematic uncertainties and the short horizontal bars (where visible) show
the statistical component.
Figure 7: The ratios $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar
28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ at
$\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$ compared with the predictions of the
LHCb MC, Perugia 0 and Perugia NOCR as a function of 7 & 7 rapidity and 7 & 7
transverse momentum. Vertical lines show the combined statistical and
systematic uncertainties and the short horizontal bars (where visible) show
the statistical component.
Figure 8: The ratios 8 $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar
28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and 8
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ from LHCb are
compared at both $\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$ (triangles) and 7
TeV (circles) with the published results from STAR [1] (squares) as a function
of rapidity loss, $\Delta y=y_{\mathrm{beam}}-y$. Vertical lines show the
combined statistical and systematic uncertainties and the short horizontal
bars (where visible) show the statistical component.
Both measured ratios are compared to the predictions of the Pythia 6
generator tunes: LHCb MC, Perugia 0 and Perugia NOCR, as functions of
$p_{\mathrm{T}}$ and $y$ at $\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$ (Fig.
6) and at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ (Fig. 7). According to
Monte Carlo studies, as discussed in Section 4, the requirement for a
reconstructed primary vertex results in only a small contribution from
diffractive events to the selected $V^{0}$ sample, therefore non-diffractive
simulated events are used for these comparisons. The predictions of LHCb MC
and Perugia 0 are similar throughout. The ratio $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ is close to Perugia 0 at
low $y$ but becomes smaller with higher rapidity, approaching Perugia NOCR. In
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, this ratio is
consistent with Perugia 0 across the measured $p_{\mathrm{T}}$ range but is
closer to Perugia NOCR at $\sqrt{s}=0.9\mathrm{\,Te\kern-1.00006ptV}$. The
production ratio $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar
28931\relax\kern 0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle
S}$ is larger in data than predicted by Perugia 0 at both collision energies
and in all measurement bins, with the most significant differences observed at
high $p_{\mathrm{T}}$.
To compare results at both collision energies, and to probe scaling violation,
both production ratios are shown as a function of rapidity loss, $\Delta
y=y_{\textrm{beam}}-y$, in Fig. 8, where $y_{\textrm{beam}}$ is the rapidity
of the protons in the anti-clockwise LHC beam, which travels along the
positive $z$ direction through the detector. Excellent agreement is observed
between results at both $\sqrt{s}=0.9$ and 7$\mathrm{\,Te\kern-1.00006ptV}$ as
well as with results from STAR at $\sqrt{s}=0.2\mathrm{\,Te\kern-1.00006ptV}$.
The measured ratios are also consistent with results published by ALICE [2]
and CMS [3].
The combined field up and down results are also given in tables in Appendix A.
Results without applying the model dependent non-prompt correction, as
discussed in Section 3, are shown for comparison in Appendix B.
## 6 Conclusions
The ratio $\kern 1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$\mathchar 28931\relax$ is a measurement of the
transport of baryon number from $pp$ collisions to final state hadrons. There
is good agreement with Perugia 0 at low rapidity which is to be expected since
the past experimental results used to test this model have focused on that
rapidity region. At high rapidity however, the measurements favour the extreme
baryon transport model of Perugia NOCR. The measured ratio $\kern
1.99997pt\overline{\kern-1.99997pt\mathchar 28931\relax\kern
0.50003pt}\kern-0.50003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ is
significantly larger than predicted by Perugia 0, i.e. relatively more baryons
are produced in strange hadronisation in data than expected, particularly at
higher $p_{\mathrm{T}}$. Similar results are found at both $\sqrt{s}=0.9$ and
7$\mathrm{\,Te\kern-1.00006ptV}$.
When plotted as a function of rapidity loss, $\Delta y$, there is excellent
agreement between the measurements of both ratios at $\sqrt{s}=0.9$ and
7$\mathrm{\,Te\kern-1.00006ptV}$ as well as with STAR’s results published at
0.2$\mathrm{\,Te\kern-1.00006ptV}$. The broad coverage of the measurements in
$\Delta y$ provides a unique data set, which is complementary to previous
results. The $V^{0}$ production ratios presented in this paper will help the
development of hadronisation models to improve the predictions of Standard
Model physics at the LHC which will define the baseline for new discoveries.
## 7 Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XUNGAL and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the R$\mathrm{\acute{e}}$gion Auvergne.
## References
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Appendix
## Appendix A Tabulated results
Table 3: The production ratios $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$, measured at
$\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$, are quoted in percent with
statistical and systematic errors as a function of 3 & 3 rapidity, $y$, and 3
transverse momentum, $p_{\mathrm{T}}$
$[\\!{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}]$.
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$
---|---|---|---|---
$0.25<p_{\mathrm{T}}<2.50$ | 93.4$\pm$7.2$\pm$6.1 | 80.0$\pm$2.5$\pm$2.5 | 72.7$\pm$2.0$\pm$3.3 | 53.9$\pm$3.1$\pm$4.0
$0.25<p_{\mathrm{T}}<0.65$ | 162.2$\pm$48.2$\pm$6.6 | 90.4$\pm$6.6$\pm$3.0 | 61.0$\pm$4.2$\pm$3.5 | 42.0$\pm$12.4$\pm$5.3
$0.65<p_{\mathrm{T}}<1.00$ | 72.3$\pm$9.7$\pm$2.5 | 77.2$\pm$3.9$\pm$2.4 | 74.6$\pm$3.3$\pm$3.9 | 61.7$\pm$5.6$\pm$3.6
$1.00<p_{\mathrm{T}}<2.50$ | 90.4$\pm$11.3$\pm$2.8 | 74.5$\pm$4.6$\pm$2.4 | 75.7$\pm$3.4$\pm$3.1 | 48.5$\pm$3.8$\pm$2.2
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle S}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$
---|---|---|---|---
$0.25<p_{\mathrm{T}}<2.50$ | 28.5$\pm$1.8$\pm$2.6 | 26.3$\pm$0.7$\pm$2.1 | 25.8$\pm$0.6$\pm$2.1 | 25.2$\pm$1.1$\pm$2.0
$0.25<p_{\mathrm{T}}<0.65$ | 19.7$\pm$3.6$\pm$2.6 | 21.8$\pm$1.4$\pm$2.2 | 18.0$\pm$1.0$\pm$1.8 | 15.8$\pm$3.1$\pm$2.1
$0.65<p_{\mathrm{T}}<1.00$ | 31.6$\pm$2.9$\pm$2.5 | 30.6$\pm$1.3$\pm$2.3 | 30.0$\pm$1.2$\pm$2.2 | 29.9$\pm$2.1$\pm$2.2
$1.00<p_{\mathrm{T}}<2.50$ | 46.3$\pm$4.5$\pm$2.9 | 42.9$\pm$2.1$\pm$2.5 | 41.3$\pm$1.6$\pm$3.2 | 32.3$\pm$2.0$\pm$2.6
$2.0<y<4.0$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
$0.25<p_{\mathrm{T}}<0.50$ | 80.6$\pm$4.6$\pm$4.0 | 17.7$\pm$0.8$\pm$1.7
$0.50<p_{\mathrm{T}}<0.65$ | 73.1$\pm$3.6$\pm$3.2 | 21.8$\pm$0.9$\pm$1.8
$0.65<p_{\mathrm{T}}<0.80$ | 73.7$\pm$3.2$\pm$3.7 | 28.4$\pm$1.0$\pm$2.3
$0.80<p_{\mathrm{T}}<1.00$ | 77.5$\pm$3.2$\pm$3.7 | 32.3$\pm$1.2$\pm$2.4
$1.00<p_{\mathrm{T}}<1.20$ | 70.1$\pm$3.4$\pm$2.3 | 36.8$\pm$1.5$\pm$2.4
$1.20<p_{\mathrm{T}}<2.50$ | 74.5$\pm$3.0$\pm$2.5 | 44.2$\pm$1.5$\pm$2.8
Table 4: The production ratios $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$, measured at
$\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$, are quoted in percent with
statistical and systematic errors as a function of 4 & 4 rapidity, $y$, and 4
transverse momentum, $p_{\mathrm{T}}$
$[\\!{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}]$.
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
$0.15<p_{\mathrm{T}}<2.50$ | 97.8$\pm$2.8$\pm$3.8 | 95.2$\pm$1.2$\pm$3.2 | 93.1$\pm$0.8$\pm$3.1 | 88.9$\pm$1.1$\pm$3.1 | 81.0$\pm$2.2$\pm$3.5
$0.15<p_{\mathrm{T}}<0.65$ | 87.2$\pm$16.7$\pm$11.0 | 95.7$\pm$1.8$\pm$3.5 | 94.2$\pm$1.4$\pm$3.3 | 87.6$\pm$2.3$\pm$3.2 | 90.0$\pm$12.6$\pm$4.2
$0.65<p_{\mathrm{T}}<1.00$ | 97.4$\pm$5.3$\pm$3.9 | 96.8$\pm$2.2$\pm$3.5 | 92.4$\pm$1.3$\pm$3.3 | 89.6$\pm$1.8$\pm$3.2 | 86.2$\pm$4.2$\pm$3.2
$1.00<p_{\mathrm{T}}<2.50$ | 98.7$\pm$2.9$\pm$3.4 | 96.6$\pm$1.8$\pm$3.3 | 92.8$\pm$1.5$\pm$3.2 | 90.3$\pm$1.7$\pm$3.2 | 79.2$\pm$2.8$\pm$2.9
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle S}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
$0.15<p_{\mathrm{T}}<2.50$ | 29.4$\pm$0.6$\pm$2.9 | 27.9$\pm$0.3$\pm$2.8 | 27.4$\pm$0.2$\pm$2.7 | 27.6$\pm$0.3$\pm$2.6 | 28.6$\pm$0.6$\pm$2.9
$0.15<p_{\mathrm{T}}<0.65$ | 18.2$\pm$2.7$\pm$3.0 | 19.1$\pm$0.3$\pm$2.6 | 18.5$\pm$0.2$\pm$2.5 | 17.5$\pm$0.4$\pm$2.5 | 20.7$\pm$1.5$\pm$3.0
$0.65<p_{\mathrm{T}}<1.00$ | 32.0$\pm$1.3$\pm$3.0 | 32.8$\pm$0.6$\pm$3.0 | 31.5$\pm$0.4$\pm$2.8 | 29.9$\pm$0.5$\pm$2.8 | 32.1$\pm$1.2$\pm$2.9
$1.00<p_{\mathrm{T}}<2.50$ | 48.3$\pm$1.1$\pm$3.5 | 47.8$\pm$0.7$\pm$3.3 | 45.8$\pm$0.6$\pm$3.3 | 45.6$\pm$0.7$\pm$3.2 | 39.9$\pm$1.0$\pm$3.0
$2.0<y<4.5$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
$0.15<p_{\mathrm{T}}<0.50$ | 95.4$\pm$1.4$\pm$3.4 | 16.2$\pm$0.2$\pm$2.4
$0.50<p_{\mathrm{T}}<0.65$ | 93.0$\pm$1.4$\pm$3.3 | 23.1$\pm$0.3$\pm$2.5
$0.65<p_{\mathrm{T}}<0.80$ | 94.3$\pm$1.4$\pm$3.3 | 28.8$\pm$0.3$\pm$2.7
$0.80<p_{\mathrm{T}}<1.00$ | 92.3$\pm$1.3$\pm$3.2 | 35.1$\pm$0.4$\pm$2.8
$1.00<p_{\mathrm{T}}<1.20$ | 93.6$\pm$1.5$\pm$3.2 | 41.2$\pm$0.6$\pm$3.0
$1.20<p_{\mathrm{T}}<2.50$ | 91.9$\pm$1.1$\pm$3.1 | 49.2$\pm$0.5$\pm$3.4
## Appendix B Tabulated results before non-prompt correction
Table 5: The production ratios $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ without non-
prompt corrections at $\sqrt{s}=0.9\mathrm{\,Te\kern-0.90005ptV}$ are quoted
in percent with statistical and systematic errors as a function of 5 & 5
rapidity, $y$, and 5 transverse momentum, $p_{\mathrm{T}}$
$[\\!{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}]$.
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$
---|---|---|---|---
$0.25<p_{\mathrm{T}}<2.50$ | 93.1$\pm$7.2$\pm$6.0 | 79.3$\pm$2.5$\pm$2.4 | 73.2$\pm$2.0$\pm$3.2 | 54.1$\pm$3.1$\pm$3.9
$0.25<p_{\mathrm{T}}<0.65$ | 163.7$\pm$48.2$\pm$6.5 | 89.2$\pm$6.6$\pm$2.8 | 61.5$\pm$4.2$\pm$3.4 | 41.4$\pm$12.4$\pm$5.3
$0.65<p_{\mathrm{T}}<1.00$ | 71.8$\pm$9.7$\pm$2.4 | 76.5$\pm$3.9$\pm$2.2 | 75.2$\pm$3.3$\pm$3.8 | 62.0$\pm$5.6$\pm$3.5
$1.00<p_{\mathrm{T}}<2.50$ | 89.9$\pm$11.3$\pm$2.7 | 74.2$\pm$4.6$\pm$2.3 | 75.7$\pm$3.4$\pm$3.0 | 48.5$\pm$3.8$\pm$2.1
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle S}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$
---|---|---|---|---
$0.25<p_{\mathrm{T}}<2.50$ | 28.9$\pm$1.8$\pm$2.4 | 27.2$\pm$0.7$\pm$1.9 | 26.6$\pm$0.6$\pm$1.9 | 25.6$\pm$1.1$\pm$1.8
$0.25<p_{\mathrm{T}}<0.65$ | 20.7$\pm$3.6$\pm$2.4 | 23.0$\pm$1.4$\pm$2.0 | 18.9$\pm$1.0$\pm$1.6 | 16.3$\pm$3.1$\pm$1.9
$0.65<p_{\mathrm{T}}<1.00$ | 31.9$\pm$2.9$\pm$2.3 | 31.5$\pm$1.3$\pm$2.1 | 31.0$\pm$1.2$\pm$2.0 | 30.6$\pm$2.1$\pm$2.0
$1.00<p_{\mathrm{T}}<2.50$ | 46.7$\pm$4.5$\pm$2.8 | 43.1$\pm$2.1$\pm$2.4 | 41.9$\pm$1.6$\pm$3.0 | 32.5$\pm$2.0$\pm$2.4
$2.0<y<4.0$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
$0.25<p_{\mathrm{T}}<0.50$ | 80.1$\pm$4.6$\pm$3.9 | 18.8$\pm$0.8$\pm$1.5
$0.50<p_{\mathrm{T}}<0.65$ | 72.9$\pm$3.6$\pm$3.1 | 22.9$\pm$0.9$\pm$1.6
$0.65<p_{\mathrm{T}}<0.80$ | 73.9$\pm$3.2$\pm$3.6 | 29.5$\pm$1.0$\pm$2.1
$0.80<p_{\mathrm{T}}<1.00$ | 77.5$\pm$3.2$\pm$3.5 | 33.1$\pm$1.2$\pm$2.3
$1.00<p_{\mathrm{T}}<1.20$ | 70.1$\pm$3.4$\pm$2.1 | 37.2$\pm$1.5$\pm$2.2
$1.20<p_{\mathrm{T}}<2.50$ | 74.4$\pm$3.0$\pm$2.3 | 44.5$\pm$1.5$\pm$2.6
Table 6: The production ratios $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ and $\kern
1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern
0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$ without non-
prompt corrections at $\sqrt{s}=7\mathrm{\,Te\kern-0.90005ptV}$ are quoted in
percent with statistical and systematic errors as a function of 6 & 6
rapidity, $y$, and 6 transverse momentum, $p_{\mathrm{T}}$
$[\\!{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}]$.
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/\mathchar 28931\relax$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
$0.15<p_{\mathrm{T}}<2.50$ | 97.3$\pm$2.8$\pm$3.6 | 95.1$\pm$1.2$\pm$3.1 | 92.7$\pm$0.8$\pm$3.0 | 88.6$\pm$1.1$\pm$2.9 | 80.9$\pm$2.2$\pm$3.4
$0.15<p_{\mathrm{T}}<0.65$ | 85.6$\pm$16.7$\pm$11.0 | 95.4$\pm$1.8$\pm$3.4 | 93.9$\pm$1.4$\pm$3.2 | 87.3$\pm$2.3$\pm$3.1 | 90.1$\pm$12.6$\pm$4.1
$0.65<p_{\mathrm{T}}<1.00$ | 97.5$\pm$5.3$\pm$3.8 | 96.5$\pm$2.2$\pm$3.4 | 91.8$\pm$1.3$\pm$3.1 | 89.5$\pm$1.8$\pm$3.1 | 86.2$\pm$4.2$\pm$3.0
$1.00<p_{\mathrm{T}}<2.50$ | 98.2$\pm$2.9$\pm$3.3 | 96.6$\pm$1.8$\pm$3.2 | 92.5$\pm$1.5$\pm$3.1 | 90.0$\pm$1.7$\pm$3.1 | 79.0$\pm$2.8$\pm$2.8
$\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}/K^{0}_{\rm\scriptscriptstyle S}$ | $2.0<y<2.5$ | $2.5<y<3.0$ | $3.0<y<3.5$ | $3.5<y<4.0$ | $4.0<y<4.5$
---|---|---|---|---|---
$0.15<p_{\mathrm{T}}<2.50$ | 29.4$\pm$0.6$\pm$2.8 | 28.4$\pm$0.3$\pm$2.6 | 28.0$\pm$0.2$\pm$2.5 | 27.9$\pm$0.3$\pm$2.5 | 28.7$\pm$0.6$\pm$2.7
$0.15<p_{\mathrm{T}}<0.65$ | 18.5$\pm$2.7$\pm$2.9 | 20.0$\pm$0.3$\pm$2.5 | 19.2$\pm$0.2$\pm$2.3 | 17.9$\pm$0.4$\pm$2.3 | 21.1$\pm$1.5$\pm$2.9
$0.65<p_{\mathrm{T}}<1.00$ | 32.3$\pm$1.3$\pm$2.9 | 33.3$\pm$0.6$\pm$2.8 | 32.2$\pm$0.4$\pm$2.7 | 30.2$\pm$0.5$\pm$2.6 | 32.2$\pm$1.2$\pm$2.7
$1.00<p_{\mathrm{T}}<2.50$ | 47.9$\pm$1.1$\pm$3.3 | 47.5$\pm$0.7$\pm$3.2 | 45.7$\pm$0.6$\pm$3.2 | 45.6$\pm$0.7$\pm$3.1 | 39.5$\pm$1.0$\pm$2.8
$2.0<y<4.5$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$\mathchar 28931\relax$ | $\kern 1.79997pt\overline{\kern-1.79997pt\mathchar 28931\relax\kern 0.45003pt}\kern-0.45003pt{}$/$K^{0}_{\rm\scriptscriptstyle S}$
---|---|---
$0.15<p_{\mathrm{T}}<0.50$ | 95.0$\pm$1.4$\pm$3.2 | 16.9$\pm$0.2$\pm$2.3
$0.50<p_{\mathrm{T}}<0.65$ | 92.9$\pm$1.4$\pm$3.2 | 23.8$\pm$0.3$\pm$2.4
$0.65<p_{\mathrm{T}}<0.80$ | 94.0$\pm$1.4$\pm$3.2 | 29.4$\pm$0.3$\pm$2.5
$0.80<p_{\mathrm{T}}<1.00$ | 91.9$\pm$1.3$\pm$3.1 | 35.5$\pm$0.4$\pm$2.7
$1.00<p_{\mathrm{T}}<1.20$ | 93.1$\pm$1.5$\pm$3.1 | 41.3$\pm$0.6$\pm$2.9
$1.20<p_{\mathrm{T}}<2.50$ | 91.8$\pm$1.1$\pm$3.0 | 48.9$\pm$0.5$\pm$3.2
|
arxiv-papers
| 2011-07-05T14:27:37 |
2024-09-04T02:49:20.305213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, G. Alkhazov,\n P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J. Amoraal, J.\n Anderson, R.B. Appleby, O. Aquines Gutierrez, L. Arrabito, A. Artamonov, M.\n Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S. Bailey, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, A.\n Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, K. Belous, I. Belyaev, E.\n Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, A. Bizzeti, P.M. Bj{\\o}rnstad,\n T. Blake, F. Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A.\n Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, S. Brisbane, M. Britsch,\n T. Britton, N.H. Brook, A. B\\\"uchler-Germann, A. Bursche, J. Buytaert, S.\n Cadeddu, J.M. Caicedo Carvajal, O. Callot, M. Calvi, M. Calvo Gomez, A.\n Camboni, P. Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L.\n Carson, K. Carvalho Akiba, G. Casse, M. Cattaneo, M. Charles, Ph.\n Charpentier, N. Chiapolini, X. Cid Vidal, P.E.L. Clarke, M. Clemencic, H.V.\n Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, P. Collins, F. Constantin, G.\n Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G.A. Cowan, R. Currie, B.\n D'Almagne, C. D'Ambrosio, P. David, I. De Bonis, S. De Capua, M. De Cian, F.\n De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del Buono, C. Deplano, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, D. Dossett,\n A. Dovbnya, F. Dupertuis, R. Dzhelyadin, C. Eames, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, Ch. Elsasser, D.G. d'Enterria, D. Esperante Pereira, L. Est\\`eve, A.\n Falabella, E. Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas,\n A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier,\n J. Garofoli, L. Garrido, C. Gaspar, N. Gauvin, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, S. Gregson, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, G. Haefeli, S.C. Haines, T. Hampson, S.\n Hansmann-Menzemer, R. Harji, N. Harnew, J. Harrison, P.F. Harrison, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, W.\n Hofmann, K. Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S.\n Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F.\n Jing, M. John, D. Johnson, C.R. Jones, B. Jost, S. Kandybei, T.M. Karbach, J.\n Keaveney, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y.M. Kim, M. Knecht, S.\n Koblitz, P. Koppenburg, A. Kozlinskiy, L. Kravchuk, K. Kreplin, G. Krocker,\n P. Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, S. Kukulak, R. Kumar, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, R. Le\n Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois,\n O. Leroy, T. Lesiak, L. Li, Y.Y. Li, L. Li Gioi, M. Lieng, R. Lindner, C.\n Linn, B. Liu, G. Liu, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, J.\n Luisier, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, A.\n Maier, S. Malde, R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U.\n Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R.\n McNulty, C. Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S.\n Miglioranzi, D.A. Milanes, M.-N. Minard, S. Monteil, D. Moran, P. Morawski,\n J.V. Morris, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, M. Musy, P. Naik, T. Nakada, R. Nandakumar, J. Nardulli, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, S. Nies, V. Niess, N. Nikitin,\n A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, B. Pal, J. Palacios, M.\n Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson,\n G. Passaleva, G.D. Patel, M. Patel, S.K. Paterson, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, D.L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrella, A.\n Petrolini, B. Pie Valls, B. Pietrzyk, T. Pilar, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, T. du Pree, V. Pugatch, A. Puig Navarro, W.\n Qian, J.H. Rademacker, B. Rakotomiaramanana, I. Raniuk, G. Raven, S. Redford,\n M.M. Reid, A.C. dos Reis, S. Ricciardi, K. Rinnert, D.A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, C. Rodriguez Cobo, P. Rodriguez Perez,\n G.J. Rogers, V. Romanovsky, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J.J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M. Sannino, R.\n Santacesaria, R. Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, S.\n Schleich, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, N. Serra,\n J. Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, H.P. Skottowe, T. Skwarnicki, A.C. Smith, N.A. Smith, K.\n Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, U. Straumann, N. Styles, S. Swientek, M. Szczekowski, P.\n Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E.\n Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, M.T.\n Tran, A. Tsaregorodtsev, N. Tuning, A. Ukleja, P. Urquijo, U. Uwer, V.\n Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J.\n Velthuis, M. Veltri, K. Vervink, B. Viaud, I. Videau, X. Vilasis-Cardona, J.\n Visniakov, A. Vollhardt, D. Voong, A. Vorobyev, H. Voss, K. Wacker, S.\n Wandernoth, J. Wang, D.R. Ward, A.D. Webber, D. Websdale, M. Whitehead, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y. Xie, F. Xing,\n Z. Yang, R. Young, O. Yushchenko, M. Zavertyaev, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Christopher Blanks",
"url": "https://arxiv.org/abs/1107.0882"
}
|
1107.1457
|
LMU-ASC 28/11
CERN-PH-TH/2011-166
Reflections of NS5 branes
Adrian Mertens111adrian.mertens@physik.uni-muenchen.de
Arnold Sommerfeld Center for Theoretical Physics
LMU, Theresienstr. 37, D-80333 Munich, Germany
Department of Physics, CERN Theory Divison
CH-1211 Geneva 23 Switzerland
Abstract
We study complex structure monodromies of certain Calabi-Yau fibrations and
find evidence that they are mirror to Calabi-Yau manifolds with NS5 brane on a
divisor. This gives a simple way to construct mirrors to any Calabi-Yau
hypersurface with NS5 branes wrapped on divisors and a complementary
interpretation of some recent calculations in open string mirror symmetry.
## 1 Introduction
The equivalence of NS5 branes and certain Ricci flat geometries under
T-duality was first shown in [1] by a study of the conformal field theory for
ALE spaces with $A_{N-1}$ singularities. These geometries are $S^{1}$
fibrations with $N$ vanishing fibers. A T-duality along the fiber turns it
into $N$ parallel NS5 branes in flat space. The T-duality acts in a normal
direction to the resulting NS5 brane, the localization of the brane in this
direction is due to instanton effects [2]. The following geometric explanation
based on [3] was already given in [1]: The effect of an NS5 brane localized on
a point in a Torus $Z^{*}$ and a point $\mathbb{C}$ is a monodromy of the
$B$-field, $B\to B+2\pi$ around the brane in $\mathbb{C}$. This gives a
monodromy $\rho\to\rho+1$ of the complexified Kähler class
$\rho=\frac{B}{2\pi}+i\sqrt{G}$. Mirror symmetry, or T-duality in one $S^{1}$
of the torus, exchanges the Kähler class with the complex structure. After
T-duality one thus expects a monodromy $\tau\to\tau+1$ for the dual torus $Z$.
To get such a monodromy the dual geometry has to be a fibration of $Z$ over
$\mathbb{C}$. Instead of an NS5 brane there is a singular fiber with a
shrinking $S^{1}$.
Mirror symmetry should also geometrize NS5 branes on divisors in higher
dimensional Calabi-Yau (CY) spaces [4]. In the Strominger Yau Zaslow picture
[5] one of the T-dualities in the Lagrangian torus fiber acts in a normal
direction of a generic divisor. While T-dualities in an internal direction map
an NS5 brane to another NS5 brane, the single T-duality in a normal direction
should turn it into a locus of a vanishing $S^{1}$. The resulting fibration
has to be a consistent background preserving the same amount of supersymmetry,
so it should be a non compact CY space. Based on this idea we propose a global
description of a dual geometry for NS5 branes localized on a point in
$\mathbb{C}$ and a divisor in a n-dimensional CY hypersurface $Z^{*}$ in a
toric ambient space. The dual geometry $X$ is a fibration of the mirror CY $Z$
over $\mathbb{C}$ and is itself a non-compact n+1 dimensional CY hypersurface.
It can be constructed by toric methods. We study the complex structure
monodromy of the fibers and find perfect agreement with the dual Kähler
monodromies created by NS5 branes.
More concretely we propose that specific non-compact CY manifolds that already
appeared in [6, 7, 8] can be interpreted in this way. These papers discuss
superpotentials for branes wrapped on curves in CY 3-folds $Z^{*}$. The curves
are first immersed into a divisor. Then the unobstructed deformation space of
the divisor inside the CY is used to calculate volumes of chains ending on
curves within the divisor. These relative period integrals were seen to be
equivalent to period integrals of a non compact CY 4-fold $X^{*}$. The mirror
$X$ to this non compact 4-fold is the geometry we will mainly study in this
note. The matching of relative period integrals for divisors in $Z^{*}$ with
quantum corrected volumes of cycles in $X$ can be interpreted as first
evidence for the present proposal. In [9] this match was explained by a
different chain of dualities starting from 7-branes on the divisor and
involving heterotic/F-theory duality. It would be interesting to close both
proposals to a cycle of dualities. The present proposal appeared implicitly
already in [4], where a similar construction involving NS5 branes is used to
calculate superpotentials.
The paper is organized as follows: In chapter 2 we explain the construction of
the non compact manifold $X$ for a given divisor on a CY hypersurface $Z^{*}$.
In chapter 3 we consider the example of an NS5 brane on a torus. We calculate
the monodromy of the complex structure of the fiber in the proposed dual
geometry and show that it matches with the shift of the $B$-field. In chapter
4 we take $X$ to be a K3 fibration and study the central fiber in detail.
Chapter 5 contains the main observation. The complex structure monodromies for
the fibers of $X$ always map to the expected shift of a $B$-field. We use
toric methods and the relation between monomials and divisors of mirror
manifolds. Chapter 6 contains further examples and chapter 7 some conclusions.
We also comment on a ”mirror” mapping between the divisor and the degeneration
locus. This could be interesting as generically the degeneration locus is not
CY.
## 2 The dual geometry
We start by repeating the construction of the non compact CY fibration $X$,
[10, 6, 7]. We use standard notation for polytopes in mirror symmetry, see
e.g. [11]. Some details are summarized in 5.
A CY hypersurface $Z^{*}$ in a toric variety is given as the vanishing locus
of an equation
$\tilde{P}(Z^{*})=\sum_{i}a_{i}\tilde{x}^{\nu_{i}}\,.$
The monomials $\tilde{x}^{\nu_{i}}=\prod_{j}\tilde{x}_{j}^{\nu_{i,j}}$
appearing in this equation are labeled by integral points $\nu_{i}$ of some
reflexive lattice polytope $\Delta^{*}$. There are relations
$\sum_{i}l_{i}^{a}\nu_{i}=0$ between these points and therefore between the
monomials, $\prod_{i}(\tilde{x}^{\nu_{i}})^{l_{i}^{a}}=1$. These relations can
be used to derive a Picard-Fuchs system for the periods of $Z^{*}$ and to
define the gauged linear sigma model (GLSM) [12] of the mirror CY $Z$.
In $Z^{*}$ we study the most general divisor $\mathcal{D}$ of a given divisor
class without any rigid component. If the degree of the defining equation
$\tilde{Q}=0$ for this divisor is not higher than the degree of $\tilde{P}$,
it can be expressed as111If the degree is higher a straightforward analogous
construction is still possible. In this case further new points can be added
to the extended polytope. We will not consider this case to avoid cluttering
the notation.
$\tilde{Q}=(b_{1}\tilde{x}^{\nu_{a}}+b_{2}\tilde{x}^{\nu_{b}}+...+b_{n}\tilde{x}^{\nu_{*}})/\operatorname{g\\!\;\\!c\\!\;\\!f}\,,$
(1)
where $\operatorname{g\\!\;\\!c\\!\;\\!f}$ is the greatest common factor of
the monomials in $\tilde{Q}$. There are new relations between the monomials of
$\tilde{Q}$ and the monomials of $\tilde{P}$. They lead to a Picard-Fuchs
system governing the volume of chains ending on the divisor $\mathcal{D}$,
[13, 6, 7]. For bookkeeping we express them as relations
$\sum_{i}\hat{l}_{i}^{a}\hat{\nu}_{i}=0$ between points $\hat{\nu}_{i}$ of an
enlarged lattice polytope $\hat{\Delta}^{*}$. To construct it we embed
$\Delta^{*}$ in a lattice with one additional dimension and add one point for
every monomial in $\tilde{Q}$,
$\hat{\Delta}^{*}=\\{(\Delta^{*},0),(\nu_{a},1),(\nu_{b},1),...,(\nu_{*},1)\\}$.
In the following we use the notation $\hat{l}$ only for the new relations that
involve some of the additional points $(*,1)$. Relations involving only the
points $(\Delta^{*},0)$ are called $l$.
The GLSM defined by a basis for these relations gives a non compact CY $X$.
This is the geometry we will mainly study in the following. It is always an
$Z$ fibration over $\mathbb{C}$ with a single singular fiber. In the singular
fiber an $S^{1}$ shrinks over a codimension two locus. As we will show in the
following, the complex structure monodromy of $Z$ around this singular fiber
matches the $B$-field monodromy of $Z^{*}$ for an NS5 brane wrapped on
$\mathcal{D}$. Moreover the relative periods of the pair $(Z^{*},\mathcal{D})$
are mirror to cycles of $X$, including quantum corrections. In particular the
moduli of the divisor $\mathcal{D}$ are mapped to Kähler moduli controlling
the location of the shrinking $S^{1}$ in the singular fiber.
We thus conjecture that the geometry $X$ is dual to $Z^{*}\times\mathbb{C}$
with an NS5 brane wrapped on the divisor $\mathcal{D}$ and localized at a
point in $\mathbb{C}$. This statement appeared implicitly already in
[4].222There an NS5 brane on a divisor is geometrized by a T-duality and the
resulting geometry is used to calculate superpotentials. Instead of
$Z^{*}\times\mathbb{C}$ [4] starts with a CY 3-fold $Z^{*}$ times
$S^{1}\times\mathbb{R}$ and performs a T-duality on the $S^{1}$ to get a
4-fold $Y$ without branes. It was noted that 3 dimensional mirror symmetry of
$Z^{*}$ should also geometrize the NS5 brane and that the resulting geometry
could be the (4 dimensional) mirror of $Y$. On the level of period integrals
the mirror symmetry between $Y$ and $X$ was checked. $Y$ is however not
identical with the mirror $X^{*}$ of the non compact CY $X$ as it appeared in
[6, 7]. The pair $(X,X^{*})$ can be compactified to a mirror pair of compact
CY hypersurfaces. The internal directions of the NS5 brane fill the divisor in
$Z^{*}$ and the remaining unspecified directions, the dimension of $Z^{*}$
does not matter. By mirror symmetry, instanton effects for $X$ are naturally
captured by the classical geometry $X^{*}$ or the pair $(Z^{*},\mathcal{D})$.
From a supergravity point of view they cause the localization of the NS5 brane
in the transverse circle [2, 14], for a recent discussion within ”doubled
geometry” see [15].
An NS5 brane localized on a divisor and a point in $\mathbb{C}$, and the dual
geometry $X$.
## 3 NS5 brane on a Torus
Consider a torus $Z^{*}$, defined by a hypersurface equation
$\tilde{P}=a_{1}\tilde{x}_{1}^{3}+a_{2}\tilde{x}_{2}^{3}+a_{3}\tilde{x}_{3}^{3}+a_{0}\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}=0$
in $\mathbb{P}^{2}/\mathbb{Z}_{3}$. To this geometry we add an NS5 brane at
$\tilde{Q}=b_{1}\tilde{x}_{1}^{2}+b_{0}\tilde{x}_{2}\tilde{x}_{3}=0$,
localized at the origin of $\mathbb{C}$ and wrapping $\mathbb{R}^{6}$. If we
forget about the NS5 brane and apply mirror symmetry to the torus we get a
dual Torus $Z$. From [1] we learn that we get a fibration of the dual torus
$Z$ over $\mathbb{C}$ if we take the NS5 brane into account. In the case at
hand we wrap an NS5 brane around a divisor of class $2[pt]$.333This is the
simplest example, we will explain later how to construct the dual geometry as
well for a Torus with NS5 brane on a minimal divisor. As the class of a point,
$[pt]$, is dual to the Kähler class, this gives a monodromy $\rho\to\rho+2$.
We conjecture that the dual non-compact CY 2-fold is given by the GLSM
(2) | $P$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $y_{0}$ | $y_{1}$
---|---|---|---|---|---|---
$l$ | -3 | 1 | 1 | 1 | 0 | 0
$\hat{l}$ | -1 | 1 | 0 | 0 | 1 | -1
.
The most general hypersurface equation for these charges is
$P=x_{1}p^{2}(x_{1}y_{1},\,x_{2},\,x_{3})+y_{0}\,q^{3}(x_{1}y_{1},\,x_{2},\,x_{3})+\mathcal{O}(y_{0}y_{1})\,,$
where $p^{2}$ and $q^{3}$ are arbitrary degree two and three polynomials in
$x_{1}y_{1}$, $x_{2}$ and $x_{3}$. This geometry is a $Z$ torus fibration over
$\mathbb{C}$, the coordinate on $\mathbb{C}$ is $y_{0}y_{1}$. We can see this
as follows.
$\\{(x_{1},x_{2},x_{3})\in\operatorname{\mathbb{P}}^{2}|P(x_{1},x_{2},x_{3},y_{0},y_{1})=0\\}$
is a torus whose complex structure depends on $y_{0}$ and $y_{1}$. By the
D-term constraint for the charge vector $\hat{l}$,
$|x_{1}|^{2}+|y_{0}|^{2}-|y_{1}|^{2}=\hat{t}$, the two coordinates $y_{0}$ and
$y_{1}$ are not independent. We can use this constraint together with the
corresponding $U(1)_{\hat{l}}$ action to fix $y_{0}$ and $y_{1}$ once the
product $y_{0}y_{1}\in\mathbb{C}$ is given. We have thus a torus over each
generic point $y_{0}y_{1}$ of the base. The only non generic point is
$y_{0}y_{1}=0$, where a $S^{1}$ shrinks in the central fiber. For
$|x_{1}|^{2}=\hat{t}$ both $y_{0}$ and $y_{1}$ vanish and $U(1)_{\hat{l}}$
acts only on the phase of $x_{1}$. By construction this action is compatible
with the hypersurface constraint for the torus at $y_{0}=y_{1}=0$,
$P=x_{1}p^{2}(x_{2},x_{3})$. So we can use a cylinder $a<|x_{1}|<b$ with
$a<\sqrt{\hat{t}}<b$ as coordinate patch for the torus and the
$U(1)_{\hat{l}}$ action cuts the cylinder into a union of two cones. As there
are two solutions to $P=x_{1}p^{2}(x_{2},x_{3})=0$ with $|x_{1}|^{2}=\hat{t}$,
this happens twice. The two loci are mirror to the two points
$\tilde{Q}=b_{1}\tilde{x}_{1}^{2}+b_{0}\tilde{x}_{2}\tilde{x}_{3}=0$. The
Kähler modulus $\hat{t}$ that determines the position of the degenerating
$S^{1}$ in the torus $T$ is mirror to the modulus
$\hat{z}=\frac{a_{1}b_{0}}{a_{0}b_{1}}$ that determines the position of the
NS5 branes in $Z^{*}$. For details on the mirror map in slightly more
complicated examples see [6, 7].
To calculate the monodromy around the origin we consider $y_{0},\,y_{1}$ as
(redundant) parameters that determine the complex structure of the fiber. The
period integrals can be brought into the standard form by a rescaling
$x_{1}\to x_{1}\frac{1}{y_{1}^{2/3}},\,x_{2}\to x_{2}y_{1}^{1/3},\,x_{3}\to
x_{3}y_{1}^{1/3}$,
$\int\frac{\Xi}{x_{1}p^{2}(x_{1}y_{1},\,x_{2},\,x_{3})+y_{0}\,q^{3}(x_{1}y_{1},\,x_{2},\,x_{3})}=\int\frac{\Xi}{x_{1}p^{2}(x_{1},x_{2},x_{3})+y_{0}y_{1}\,p^{3}(x_{1},x_{2},x_{3})}\,,$
where $\Xi$ is the holomorphic 2-form of $\operatorname{\mathbb{P}}^{2}$.
After this rescaling $P$ depends on $y_{0}$ and $y_{1}$ only in the
combination $y_{0}y_{1}$, so we can treat it as hypersurface equation of the
fiber depending on the position of the base,
$P(x_{1},x_{2},x_{3};y_{0}y_{1})$. The geometry (2) is a blow-up of the
fibration
$\\{(x_{1},x_{2},x_{3})\in\operatorname{\mathbb{P}}^{2}|P(x_{1},x_{2},x_{3};y_{0}y_{1})=0\\}\to\mathbb{C}$.
We discuss this in more detail in the next example. Close to $y_{0}y_{1}=0$
all monomials containing only $x_{2}$ and $x_{3}$ are suppressed. After some
coordinate redefinitions these are only the two monomials $x_{2}^{3}$ and
$x_{3}^{3}$. Moving in a sufficiently small circle around $y_{0}y_{1}=0$ these
are the only monomials whose prefactors in the hypersurface equation of the
fiber vary and we can use standard methods [16] to determine the complex
structure. We find $\tau=2\ln(y_{0}y_{1})+\mathcal{O}(y_{0}y_{1})$ near
$y_{0}y_{1}=0$. The factor of 2 comes about as both the monomials $x_{2}^{3}$
and $x_{3}^{3}$ are suppressed by $y_{0}y_{1}$. Alternatively, after an
additional rescaling, only one of the monomials e.g. $x_{2}^{3}$ is suppressed
by $(y_{0}y_{1})^{2}$. This gives the expected monodromy $\tau\to\tau+2$. The
logarithmic singularity at $y_{0}y_{1}=0$ is in accordance with the expected
backreaction of an NS5 brane, for a recent study of this situation in the
heterotic string see [17, 18].
## 4 NS5 brane on a K3
We now want to extend this construction to more complicated geometries. The
Strominger Yau Zaslow picture of mirror symmetry [5] seems to indicate that
this is possible. It explains mirror symmetry as simultaneous T-dualities in
all directions of a Lagrangian torus fibration. One of this directions is
normal, the others transversal to a holomorphic divisor. A T-duality performed
in an internal direction maps an NS5 brane to another NS5 brane, a T-duality
in a normal direction should turn it into a locus where the T-dual $S^{1}$
shrinks. Mirror symmetry should thus geometrize the NS5 brane. As it exchanges
Kähler and complex structure moduli, die shift of the $B$ field that signals
the presence of an NS5 brane should be mapped to a monodromy of the complex
structure.
We will study generalizations of the geometry (2) and show that the complex
structure of the fiber $Z$ has a monodromy around the origin of the base. This
monodromy is in agreement with the interpretation as mirrors of a CY $Z^{*}$
with NS5 brane. To make contact with the easier case of the torus we consider
an elliptically fibered K3 $Z$ that is fibered over $\mathbb{C}$,
(3) | $P$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $y_{0}$ | $y_{1}$
---|---|---|---|---|---|---|---|---
$l^{1}$ | -3 | 1 | 1 | 1 | 0 | 0 | 0 | 0
$l^{2}$ | 0 | 0 | 0 | -2 | 1 | 1 | 0 | 0
$\hat{l}$ | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1
.
The coordinate on $\mathbb{C}$ is $y_{0}y_{1}$. We call the whole fibration
again $X$. It has a singular fiber over the origin of $\mathbb{C}$, in this
singular K3 the elliptic fiber degenerates. Now we concentrate on a
neighborhood of the vanishing $S^{1}$ in the degenerate elliptic fiber.
Locally the geometry is a cone ($uv=0$) over
$\operatorname{\mathbb{P}}^{1}\times\mathbb{C}$. This should turn into an NS5
brane, if we can consistently implement a duality transformation that involves
one T-duality in the elliptic fiber of the K3 $Z$. Mirror symmetry in the $Z$
fiber over each point in $\mathbb{C}$ is such a duality, its maps the $Z$
fibration to a $Z^{*}$ fibration over $\mathbb{C}$. As the Kähler structure of
$Z$ fiber does not vary in (3), the complex structure of $Z^{*}$ is constant
in the dual fibration.
The complex structure of the fiber $Z$ however does vary. The hypersurface
equation
$P=x_{1}p^{2}(x_{1}y_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})+y_{0}q^{3}(x_{1}y_{1},x_{2},..)+\mathcal{O}(y_{0}y_{1})\,,$
(4)
depends on parameters $y_{0}$ and $y_{1}$. With $x_{i}$ we denote coordinates
for the smooth blow-up of $\operatorname{\mathbb{P}}_{1122}$. In the following
this blow-up is understood whenever we write
$\operatorname{\mathbb{P}}_{1122}$ or $\operatorname{\mathbb{P}}_{112}$.
In the period integrals we can rescale $x_{1}\to x_{1}/y_{1}^{2/3},\,x_{2}\to
x_{2}y_{1}^{1/3},\,x_{3}\to x_{3}y_{1}^{1/3}$,
$\int\frac{\Xi}{P}=\int\frac{\Xi}{x_{1}p^{2}(x_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})+y_{0}y_{1}q^{3}(x_{1},x_{2},..)}\,,$
so the complex structure only depends on the product $y_{0}y_{1}$, as it
should. Here we claimed that the complex structure of the $Z$ fiber is the
same as the complex structure of the hypersurface
$P^{\prime}=x_{1}p^{2}(x_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})+zq^{3}(x_{1},x_{2},..)+\mathcal{O}(z^{2})$
in $\operatorname{\mathbb{P}}_{1122}$. Let us look at the two geometries more
carefully. Both geometries fall apart into two components at $y_{0}y_{1}=0$
and $z=0$ respectively. For the fiber $Z|_{y_{0}y_{1}=0}$ we have the
components
$\displaystyle|x_{1}|^{2}\leq\hat{t}\,,$
$\displaystyle\;y_{1}=0\;\;\text{and}$
$\displaystyle|x_{1}|^{2}\geq\hat{t}\,,$ $\displaystyle\;y_{0}=0\,,$
where $\hat{t}$ is the Kähler modulus for the charge vector $\hat{l}$,
$|x_{1}|^{2}+|y_{0}|^{2}-|y_{1}|^{2}=\hat{t}$. For the hypersurface
$P^{\prime}=0$ we have
$\displaystyle\\{x_{1}=0\\}\in\operatorname{\mathbb{P}}_{1122}\simeq\operatorname{\mathbb{P}}_{112}\;\;\text{and}$
$\displaystyle\\{p^{2}(x_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})=0\\}\in\operatorname{\mathbb{P}}_{1122}\,.$
In the first component of the fiber $Z|_{y_{0}y_{1}=0}$, $y_{1}=0$, we have
the equation
$x_{1}p^{2}(0,x_{2},x_{3}x_{4}^{2},..)+y_{0}q^{3}(0,x_{2},x_{3}x_{4}^{2},..)=0$.
This can unambiguously be solved for $\frac{x_{1}}{y_{0}}$ for any
${(x_{2}:x_{3}:x_{4}:x_{5})}\in\operatorname{\mathbb{P}}_{112}$ away from
$p^{2}(x_{2},x_{3}x_{4}^{2},..)=q^{3}(x_{2},x_{3}x_{4}^{2},..)=0$. Once
$\frac{x_{1}}{y_{0}}$ is fixed, $x_{1}$ and $y_{0}$ are determined by the
Kähler parameter $\hat{t}$. However, at
$p^{2}(x_{2},x_{3}x_{4}^{2},..)=q^{3}(x_{2},x_{3}x_{4}^{2},..)=0$, the ratio
$\frac{x_{1}}{y_{0}}$ is free and $(x_{1}:y_{0})$ parameterize a
$\operatorname{\mathbb{P}}^{1}$. So the first component is a
$\operatorname{\mathbb{P}}_{112}$, with the locus
$p^{2}(x_{2},x_{3}x_{4}^{2},..)=q^{3}(x_{2},x_{3}x_{4}^{2},..)=0$ blown up by
a $\operatorname{\mathbb{P}}^{1}$. The size of this
$\operatorname{\mathbb{P}}^{1}$ is the Kähler modulus $\hat{t}$. In the second
component of the fiber, $y_{0}=0$, we have the equation
$x_{1}p^{2}(x_{1}y_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})=0$.
As $x_{1}\not=0$ in this component we have
$\\{p^{2}(x_{1}y_{1},x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})=0\\}\in\operatorname{\mathbb{P}}_{1122}$
as for the second component of the hypersurface $P^{\prime}=0$. The
coordinates on $\operatorname{\mathbb{P}}_{1122}$ in this case are
$(y_{1}:x_{2}:x_{3}:x_{4}:x_{5})$, so $x_{1}$ is exchanged for $y_{1}$. Away
from the singular point we have an isomorphism between the K3 fiber and the
hypersurface $P^{\prime}$ by the rescaling given above.444We see that the
singular fiber is a union of two Fano varieties.
$Y_{1}=\\{p^{2}(x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})=0\\}\in\operatorname{\mathbb{P}}_{1122}$
and $Y_{2}$ is a blow-up of $\operatorname{\mathbb{P}}_{112}$. They intersect
over a Torus
$D=\\{p^{2}(x_{2},x_{3}x_{4}^{2},x_{3}x_{5}^{2},x_{3}x_{4}x_{5})=0\\}\in\operatorname{\mathbb{P}}_{112}$,
$K_{D}=0$ so $D\in|-K_{Y_{i}}|$. The singular fiber is a normal crossing of
the type described in [19], while the whole non-compact 3-fold defined by (3)
is its smoothing. This is a generic property, one can see the toric
constructions introduced in [6, 7] as a prescription how to cut a CY
hypersurface into a normal crossing of Fano varieties.
The difference between the fibration (3) and the fibration of $P^{\prime}$
over the $z$-plane is the additional Kähler modulus $\hat{t}$. For $\hat{t}=0$
the additional $\operatorname{\mathbb{P}}^{1}$ shrinks and the two geometries
agree, $y_{0}y_{1}=0$ implies $y_{0}=0$ in this case. Especially the first
component of the singular fiber $Z|_{y_{0}y_{1}=0}$ is $x_{1}=0$ and the
coordinates for the second one are $(x_{1}:x_{2}:x_{3}:x_{4})$ in both cases.
The complex structure of the fibration is singular at $y_{0}y_{1}=0$/$z=0$ and
has a monodromy if we move around this point. The dual Kähler monodromy of
$Z^{*}$ is a shift in the $B$-field. This signals the presence of an NS5 brane
on a divisor dual to the class of the corresponding $B$-field. In the next
chapter we show that this is indeed the divisor (1) whose relative periods
obey a GKZ system with charges (3). The modulus of this divisor is mirror to
the additional Kähler modulus $\hat{t}$.
We did not discuss possible $\mathcal{O}(y_{0}y_{1})$ terms in the equation
(4). Such terms signal the additional freedom in the variation of the complex
structure of the fiber over the base. Depending on the choice of these terms,
the dual geometry is the trivial fibration $Z^{*}\times\mathbb{C}$ or an
honest fibration with a varying Kähler structure.
The same construction is possible for any realization of a K3 surface or for 3
or 4 dimensional CY hypersurfaces. Above we started with an elliptic K3 to
make contact with the torus. But note that locally, at the vanishing locus of
the $S^{1}$, the singular fiber always looks like the product of the
degeneration locus times a cone. Mirror Symmetry in the SYZ picture always
involves one T-duality in the transverse geometry, so applying Mirror Symmetry
fiberwise should give rise to a dual geometry involving NS5 branes. In the
following we use toric methods to show that the complex structure monodromy
around the central fiber always maps to the monodromy in the $B$-field caused
by an NS5 brane.
## 5 Divisors and Monomials
First we fix the notation and repeat some facts about reflexive polytopes and
associated CY hypersurfaces that we will need in the following. For more
information see [20, 11]. $\nu_{i}\in\Delta^{*}$ are integral points of the
lattice polytope $\Delta^{*}$ of the CY $Z^{*}$, $\mu_{j}\in\Delta$ are
integral points in the dual lattice polytope $\Delta$ of $Z$. $\nu_{0}$ and
$\mu_{0}$ are the unique interior points and
${\langle\nu_{i},\mu_{j}\rangle}={\langle\mu_{j},\nu_{i}\rangle}\in\mathbb{Z}$
is the natural pairing. We take the whole polytope to lie in an affine plane
of distance $1$ to the origin, such that ${\langle\nu_{0},\mu_{j}\rangle}=1$
for all $\mu_{j}$ and ${\langle\mu_{0},\nu_{i}\rangle}=1$ for all $\nu_{i}$.
Taking the vectors $\nu_{i}-\nu_{0}$ as generators of one dimensional cones,
we can construct the fan of the ambient space of $Z$ from $\Delta^{*}$ and
likewise the fan of the ambient space of $Z^{*}$ from $\Delta$. One
dimensional cones correspond to divisors $x_{i}=0$ of the ambient space and by
restriction onto the hypersurface to toric divisors of $Z$. So there is a
correspondence $\nu_{i}\leftrightarrow x_{i}=0$ and
$\mu_{j}\leftrightarrow\tilde{x}_{j}=0$, $i\,,j\not=0$, between integral
points and divisors and we choose to label the coordinates $x$ and $\tilde{x}$
with the same indices as $\nu$ and $\mu$.
Moreover all integral points $\mu_{j}$ of the polytope $\Delta$ correspond to
a monomial $x^{\mu_{j}}$ in the hypersurface equation $P=0$ of $Z$ and
likewise $\nu_{i}$ to monomials $\tilde{x}^{\nu_{i}}$ in $\tilde{P}=0$. Here
we use the notation
$x^{\mu_{j}}:=\prod_{i}x_{i}^{{\langle\mu_{j},\nu_{i}\rangle}}$ and
$\tilde{x}^{\nu_{i}}:=\prod_{j}\tilde{x}_{j}^{{\langle\nu_{i},\mu_{j}\rangle}}$.
The integral points $\mu_{i}$, $i\not=0$ correspond thus both to a monomial in
the defining equation $P=0$ of $Z$ and to a toric divisor of $Z^{*}$. Mirror
symmetry exchanges this data. Close to the large volume point in the Kähler
moduli space of $Z^{*}$ and to the point of maximal unipotent monodromy in the
complex structure moduli space of $Z$ this identification gives rise to the
”monomial divisor mirror map” [21]. A change of the Kähler volume of a two
cycle dual to a given toric divisor is mapped to a change of the prefactor of
the corresponding monomial in the hypersurface equation $P=0$ and thus to a
change of complex structure. In particular, at the point of maximal unipotent
monodromy this prefactor vanishes and moving around this point we get a
monodromy $\tau\to\tau+1$ in the complex structure moduli space of $Z$ and
$t\to t+1$ in the Kähler moduli space of $Z^{*}$.
Kähler classes of the ambient space555Most of these restrict to Kähler classes
of the CY. of $Z$ are in one to one correspondence with a certain base for the
set of linear relations between points of the polytope $\Delta^{*}$,
$\sum_{i}l_{i}^{m}\nu_{i}=0$. For this base, the entries of the charge vectors
$l^{m}$ are the intersection numbers between a curve dual to the corresponding
Kähler class and the divisors $x_{i}=0$. Divisors with the same entries for
all $l^{m}$ and thus the same intersection numbers are equivalent and dual to
the same Kähler class. The relation $\sum_{i}l_{i}^{m}\nu_{i}=0$ translates to
the condition that all monomials $x^{\mu_{j}}$ of the hypersurface equation
$P=0$ are in the divisor class of the anticanonical bundle.
With the construction of chapter 2 we can choose any divisor $\mathcal{D}$ in
$Z^{*}$ given by
$\tilde{Q}=(\tilde{x}^{\nu_{1}}+\tilde{x}^{\nu_{2}}+..+\tilde{x}^{\nu_{n}})/\operatorname{g\\!\;\\!c\\!\;\\!f}$,
where $\operatorname{g\\!\;\\!c\\!\;\\!f}$ is the greatest common factor of
the appearing monomials $x^{\nu_{i}}$. In the following we explain how to
identify the divisor class in terms of one dimensional cones generated by
$\mu_{a}-\mu_{0}$ and thus in terms of points $\mu_{a}$ of the dual polytope.
Next we study the proposed mirror geometry and determine which monomials of
$P$ depend on the base coordinate $y_{1}...y_{n}$ of the $CY$ fibration. We
will see that exactly the monomials $x^{\mu_{a}}$ get suppressed in the
central fiber over $y_{1}...y_{n}=0$, where $\mu_{a}$ are the points that
correspond to the divisor class of $\mathcal{D}$. The monomial divisor mirror
map [21] then assures a monodromy of the complex structure in agreement with
the proposed picture of a geometrization of NS5 branes by mirror symmetry.
In the simplest case we have only two monomials that determine the divisor,
$\tilde{Q}=(b_{1}\tilde{x}^{\nu_{a}}+b_{2}\tilde{x}^{\nu_{b}})/\operatorname{g\\!\;\\!c\\!\;\\!f}$.
The divisor class can be read of either the nominator or the denominator of
$\frac{\tilde{x}^{\nu_{a}}}{\tilde{x}^{\nu_{b}}}=\tilde{x}^{\nu_{a}-\nu_{b}}$.
Choosing the nominator we find the divisor
$\tilde{x}_{1}^{k_{1}}\tilde{x}_{2}^{k_{2}}..=0$ with the multiplicities
$\displaystyle\begin{array}[]{cc}k_{j}={\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}&\text{if}\;{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}>0\,,\\\
k_{j}=0&\text{if}\;{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}\leq 0\,.\\\
\end{array}$ (7)
The proposed mirror geometry is a CY fibration over $\mathbb{C}$. Enlarging
the polytope $\Delta^{*}$ to a polytope $\hat{\Delta}^{*}$ with points
$(\Delta^{*},0)$ and $(\nu_{a},1)$, $(\nu_{b},1)$ we find a new relation
$\hat{l}$ between the points of $\hat{\Delta}^{*}$ and thus a condition on the
possible monomials in the coordinates $x_{i}$ and $y_{1}$, $y_{2}$.666If one
of the lattice points $\nu_{a/b}$ is the interior point $\nu_{0}$, the
coordinate corresponding to $(\nu_{a/b},0)$ is $P$ and not $x_{a/b}$. This
does not change the following discussion however.
(8) | $(\nu_{a},0)$ | $(\nu_{b},0)$ | … | $(\nu_{a},1)$ | $(\nu_{b},1)$
---|---|---|---|---|---
| $x_{a}$ | $x_{b}$ | … | $y_{a}$ | $y_{b}$
$\hat{l}$ | -1 | 1 | … | 1 | -1
,
In addition we have all the relations $l^{m}$ between the points $\nu_{i}$.
Imposing them gives the set of $x^{\mu_{j}}$ as possible monomials of the
hypersurface equation $P=\sum_{j}a_{j}x^{\mu_{j}}=0$ of a general fiber. The
additional condition forces us to multiply some of these monomials with
$y_{a}$ or $y_{b}$. We get a hypersurface equation
$P=\sum_{j}a_{j}(y_{a},y_{b})\;x^{\mu_{j}}$. We are interested in the behavior
of the coefficients
$a_{j}(y_{a},y_{b})=y_{a}^{k_{j}}y_{b}^{l_{j}}(a_{j}^{0}+\mathcal{O}(y_{a}y_{b}))$
close to $y_{a}y_{b}=0$ so we neglect the subleading contributions
$\mathcal{O}(y_{a}y_{b})$. The monomials have to be neutral under the charges
of the vector $\hat{l}$. The power of $x_{a}$ in $x^{\mu_{j}}$ is
${\langle\nu_{a},\mu_{j}\rangle}$ and similarly for $x_{b}$, so we get
monomials $x^{\mu_{j}}y_{a}^{k_{j}}y_{b}^{l_{j}}$, where
$\begin{array}[]{ccc}k_{j}={\langle\nu_{a}-\nu_{b},\mu_{j}\rangle},&l_{j}=0\,,&\text{if}\;{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}>0\,,\\\
k_{j}=0\,,&l_{j}=0\,,&\text{if}\;{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}=0\,,\\\
k_{j}=0\,,&l_{j}=-{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}\,,&\text{if}\;{\langle\nu_{a}-\nu_{b},\mu_{j}\rangle}<0\,.\end{array}$
(9)
By a rescaling of the $x_{i}$ that leaves the holomorphic $(n,0)$ form and
thus the period integrals invariant it is always possible to combine $y_{a}$
and $y_{b}$ to the product $y_{a}y_{b}$. In the monomials this replaces e.g.
$y_{a}\to y_{a}y_{b}$ and $y_{b}\to 1$ and we are left with
$P=\sum_{j}(y_{a}y_{b})^{k_{j}}x^{\mu_{j}}(a_{j}^{0}+\mathcal{O}(y_{a}y_{b}))$.
Comparing conditions (7) and (9) we see that monomials corresponding to a
point $\mu_{j}$ are suppressed with a power $k_{j}$, if the divisor
$\mathcal{D}$ contains the divisor $\tilde{x}_{j}=0$ $k_{j}$ times. The
monomial divisor mirror map thus assures fitting monodromies.
By a different rescaling of the $x_{i}$ we could as well replace $y_{a}\to 1$
and $y_{b}\to y_{a}y_{b}$. This would suppress monomials that started out with
positive powers of $y_{b}$ by $(y_{a}y_{b})^{l_{j}}$ . The corresponding
points $\mu_{j}$ correspond to the divisors in the denominator of
$\tilde{x}^{\nu_{a}-\nu_{b}}$. This reflects the equivalence of the divisor
classes.
This can be generalized to divisors
$\tilde{Q}=(b_{1}\tilde{x}^{\nu_{a}}+b_{2}\tilde{x}^{\nu_{b}}...+b_{n}\tilde{x}^{\nu_{*}})/\operatorname{g\\!\;\\!c\\!\;\\!f}$
with more than two monomials. For each additional monomial we get a new
independent relation $\hat{l}_{m}$. Imposing one relation we multiply all
monomials $x^{\mu_{i}}$ with new coordinates $y_{*}$ that correspond to
divisors $\tilde{x}_{j}=0$ that differ between two of the monomials of
$\tilde{Q}$. After imposing all relations, all monomials $x^{\mu_{i}}$ are
multiplied by some power of some new coordinates $y_{*}$ up to monomials that
correspond to divisors in the $\operatorname{g\\!\;\\!c\\!\;\\!f}$ of
$\tilde{Q}$. A rescaling of $x_{i}$ again collects all $y_{*}$ into the
coordinate on the base of the fibration. In the following we give some
explicit examples of divisors with several moduli.
## 6 Further examples
### 6.1 Torus, charge 3
As an example with more than two monomials in the divisor equation we again
consider a torus $Z^{*}$, defined by
$\tilde{P}=a_{1}\tilde{x}_{1}^{3}+a_{2}\tilde{x}_{2}^{3}+a_{3}\tilde{x}_{3}^{3}+a_{0}\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}=0$
in $\mathbb{P}^{2}/\mathbb{Z}_{3}$. This time we add an NS5 brane on the
divisor
$\tilde{Q}=b_{1}\tilde{x}_{1}^{3}+b_{2}\tilde{x}_{2}^{3}+b_{3}\tilde{x}_{3}^{3}+b_{0}\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}=0$,
again localized at the origin of $\mathbb{C}$ and wrapping $\mathbb{R}^{6}$.
The dual non compact 2-fold is given by the GLSM
(10) | $P$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $y_{0}$ | $y_{1}$ | $y_{2}$ | $y_{3}$
---|---|---|---|---|---|---|---|---
$l$ | -3 | 1 | 1 | 1 | 0 | 0 | 0 | 0
$\hat{l}^{1}$ | -1 | 1 | 0 | 0 | 1 | -1 | 0 | 0
$\hat{l}^{2}$ | -1 | 0 | 1 | 0 | 1 | 0 | -1 | 0
$\hat{l}^{3}$ | -1 | 0 | 0 | 1 | 1 | 0 | 0 | -1
,
with hypersurface
$P=x_{1}x_{2}x_{3}+y_{0}\,q^{3}(x_{1}y_{1},\,x_{2}y_{2},\,x_{3}y_{3})+\mathcal{O}(y_{0}y_{1}y_{2}y_{3})\,.$
(11)
This is a fibration of the dual torus $Z$ over $\mathbb{C}$, the coordinate on
$\mathbb{C}$ is $y_{0}y_{1}y_{2}y_{3}$. The period integrals can be brought
into the standard form by a rescaling $x_{1}\to
x_{1}\frac{(y_{2}y_{3})^{1/3}}{y_{1}^{2/3}},\,x_{2}\to
x_{2}\frac{(y_{1}y_{3})^{1/3}}{y_{2}^{2/3}},\,x_{3}\to
x_{3}\frac{(y_{1}y_{2})^{1/3}}{y_{3}^{2/3}}$,
$\int\frac{\Xi}{x_{1}x_{2}x_{3}+y_{0}\,q^{3}(x_{1}y_{1},\,x_{2}y_{2},\,x_{3}y_{3})}=\int\frac{\Xi}{x_{1}x_{2}x_{3}+y_{0}y_{1}y_{2}y_{3}\,q^{3}(x_{1},x_{2},x_{3})}\,.$
The complex structure of the fiber behaves like
$\tau=3\ln(y_{0}y_{1}y_{2}y_{3})+\mathcal{O}(y_{0}y_{1}y_{2}y_{3})$ near
$y_{0}y_{1}y_{2}y_{3}=0$. We get the factor 3 as all monomials $x_{i}^{3}$ are
suppressed by $y_{0}y_{1}y_{2}y_{3}$. Alternatively, after a rescaling a
single monomial e.g. $x_{3}^{3}$ is suppressed by
$(y_{0}y_{1}y_{2}y_{3})^{3}$. The monomials $x_{i}^{3}$ are related to the
divisors $\tilde{x}_{i}=0$ by the monomial-divisor mirror map. The class of a
point is dual to the Kähler class, so we would indeed expect a monodromy
$\rho\to\rho+3$ for an NS5 brane wrapped on the divisor
$\tilde{Q}=b_{1}\tilde{x}_{1}^{3}+b_{2}\tilde{x}_{2}^{3}+b_{3}\tilde{x}_{3}^{3}+b_{0}\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}=0$.
### 6.2 Torus, charge 1
We already described NS5 branes on divisors of class $2[pt]$ and $3[pt]$ in a
torus, but not the elementary situation of a single brane localized on one
point. The toric realization of the dual geometry is a little bit more
complicated but straightforward after the general discussion of section 5.
This time we realize the torus $T^{*}$ as a degree 3 hypersurface
$\tilde{P}=0$ in $\operatorname{\mathbb{P}}^{3}$. There are ten possible
monomials $\tilde{x}^{\nu_{i}}$ out of which we can choose two to define
$\tilde{Q}$. Usually one restricts the number of monomials by
$PGL(3,\mathbb{C})$ coordinate changes and only keeps $\tilde{x}_{1}^{3}$,
$\tilde{x}_{2}^{3}$, $\tilde{x}_{3}^{3}$ and
$\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}$. In the polytope the other monomials
correspond to interior points of a codimension one face. On the mirror side
these points correspond to divisors in the ambient space that are not hit by
the generic CY hypersurface. However, if we want to express
$\tilde{Q}=\tilde{x}_{1}+\tilde{x}_{2}$ in terms of monomials
$\tilde{x}^{\nu_{i}}$ we have to use at least one of these additional points,
e.g.
$\tilde{Q}=(b_{1}\tilde{x}_{1}^{3}+b_{2}\tilde{x}_{1}^{2}\tilde{x}_{2})/x_{1}^{2}$.
The GLSM for of the dual geometry is given by
| $P$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $y_{1}$ | $y_{2}$
---|---|---|---|---|---|---|---
$l^{1}$ | -3 | 1 | 1 | 1 | 0 | 0 | 0
$l^{2}$ | 0 | 2 | 1 | 0 | -3 | 0 | 0
| | | | $\vdots$ | | |
$\hat{l}$ | 0 | -1 | 0 | 0 | 1 | 1 | -1
,
where $x_{4}$ is the coordinate for one of the blow-ups of the singularities
of $\operatorname{\mathbb{P}}^{2}/\mathbb{Z}^{3}$ and for ease of notation we
omitted further blow-up coordinates and relations for them. These relations
however have to be included to determine the allowed monomials for $P$.777In
constructing the dual polytope one does so automatically, we focus here on the
relations as we have to include the additional constraint by $\hat{l}$. We
find
$P=x_{1}^{3}x_{4}^{2}y_{1}+x_{2}^{3}x_{4}y_{2}+x_{3}^{3}+x_{1}x_{2}x_{3}x_{4}+\mathcal{O}(y_{1}y_{2})\,.$
After a rescaling of $x_{i}$ either the monomial with $x_{1}^{3}$ or
$x_{2}^{3}$ is suppressed close to $y_{1}y_{2}=0$ and we get the expected
monodromy $\tau\to\tau+1$.
### 6.3 Quintic
For CY 3-folds, geometries of the type discussed in chapter 2 were already
used in [6, 7] to calculate superpotentials. The simplest example is the
mirror quintic
$\tilde{P}=\tilde{x}_{1}^{5}+\tilde{x}_{2}^{5}+\tilde{x}_{3}^{5}+\tilde{x}_{4}^{5}+\tilde{x}_{5}^{5}+\tilde{x}_{1}\tilde{x}_{2}\tilde{x}_{3}\tilde{x}_{4}\tilde{x}_{5}$
with NS5 brane on
$\tilde{Q}=\tilde{x}_{1}^{4}+\tilde{x}_{2}\tilde{x}_{3}\tilde{x}_{4}\tilde{x}_{5}$.
Here all additional coordinates needed to describe the blow-ups are scaled to
one for ease of notation. The intersection of $\tilde{P}$ and $\tilde{Q}$ is a
covering of a K3 surface, for more details see [7, 8]. The dual quintic
fibration is
| $P$ | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $y_{0}$ | $y_{1}$
---|---|---|---|---|---|---|---|---
$l$ | -5 | 1 | 1 | 1 | 1 | 1 | 0 | 0
$\hat{l}$ | -1 | 0 | 0 | 0 | 0 | 0 | 1 | -1
,
with hypersurface
$P=x_{1}p^{4}(x_{1}y_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5})+y_{0}\,q^{5}(x_{1}y_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5})+\mathcal{O}(y_{0}y_{1})\,.$
After a rescaling of $x_{i}$ all monomials without $x_{1}$ are suppressed by
$y_{0}y_{1}$. As shown in section 5 this are the monomials that correspond to
the divisor class of $\mathcal{D}$.
The singular locus in the central fiber is $p^{4}(x_{2},x_{3},x_{4},x_{5})=0$
and $x_{1}=\sqrt{\hat{t}}$ in $\operatorname{\mathbb{P}}^{4}$, where $\hat{t}$
is again the Kähler parameter associated with $\hat{l}$. This is a K3 surface
and it is the mirror of the K3 surface whose covering is wrapped by the NS5
brane in the mirror quintic.
## 7 Conclusions
We presented evidence that CY fibrations of the type discussed in chapter 2
can be interpreted as mirrors of CY hypersurfaces with NS5 brane on a divisor.
This gives a new interpretation of recent calculations in open string mirror
symmetry.
A generalization to complete intersection CY manifolds should be
straightforward and the idea should also carry over to other CY that were
studied in open string mirror symmetry [22]. The construction allows to study
mirror symmetry for a pair of a CY and divisor without specifying an A-type
brane on the mirror. Nevertheless the geometry should encode information of an
A-type brane as discussed in [6, 7]. The role of the A-type brane is played by
the degeneration locus in the singular fiber. It would be interesting to
investigate such a correspondence, e.g. by a lift to M-theory.
We would like to note some observations. We saw in the Quintic example 6.3
that the degeneration locus is the mirror of the K3 surface that determines
the subset of open periods. This is true also for all examples in [8], where
these K3 ”subsystems” in $Z^{*}$ were used to calculate numbers of disks
ending on Lagrangian submanifolds in $Z$. It might be rewarding to study this
”mirror symmetry” between degeneration locus and divisor in the light of the
Strominger Yau Zaslow conjecture. The Lagrangian torus fibration has always
one leg in a normal direction to the divisor. If the remaining directions
restrict to a Lagrangian torus fibre of the divisor, one would expect the
degeneration locus to be the mirror. Note that the construction is possible
for any non rigid divisor.
For $d-1$ dimensional divisors with more then one modulus the degeneration
locus in the dual geometry falls apart into different components that only
meet in complex codimension one. As ${dim}(H^{(d-1,0)})>1$ for such divisors
this is what one would expect for the mirror geometry, there should be more
then one class of points. Such a structure appeared e.g. in [23].
## Acknowledgements
I thank Murad Alim, Michael Hecht, Hans Jockers, Peter Mayr and Masoud Soroush
for collaboration on closely related topics and Andres Collinucci, Stefan
Groot Nibbelink, Michael Kay, Christian Roemmelsberger and Ahmad Zein Assi for
discussions or comments. The work was supported by the Studienstiftung des
deutschen Volkes.
## References
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* [2] R. Gregory, J. A. Harvey, and G. W. Moore, Unwinding strings and T-duality of Kaluza-Klein and H- monopoles, Adv. Theor. Math. Phys. 1 (1997) 283–297, [hep-th/9708086].
* [3] B. R. Greene, A. D. Shapere, C. Vafa, and S.-T. Yau, Stringy Cosmic Strings and Noncompact Calabi-Yau Manifolds, Nucl.Phys. B337 (1990) 1.
* [4] M. Aganagic and C. Beem, The Geometry of D-Brane Superpotentials, arXiv:0909.2245.
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* [7] M. Alim et. al., Hints for Off-Shell Mirror Symmetry in type II/F-theory Compactifications, Nucl. Phys. B841 (2010) 303–338, [arXiv:0909.1842].
* [8] M. Alim, M. Hecht, H. Jockers, P. Mayr, A. Mertens, and M. Soroush, Type II/F-theory Superpotentials with Several Deformations and N=1 Mirror Symmetry, arXiv:1010.0977.
* [9] H. Jockers, P. Mayr, and J. Walcher, On N=1 4d Effective Couplings for F-theory and Heterotic Vacua, arXiv:0912.3265.
* [10] P. Mayr, N = 1 mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys. 5 (2002) 213–242, [hep-th/0108229].
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* [13] S. Li, B. H. Lian, and S.-T. Yau, Picard-Fuchs Equations for Relative Periods and Abel- Jacobi Map for Calabi-Yau Hypersurfaces, arXiv:0910.4215.
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* [15] S. Jensen, The KK-Monopole/NS5-Brane in Doubled Geometry, arXiv:1106.1174.
* [16] S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301–350, [hep-th/9308122].
* [17] M. Blaszczyk, S. G. Nibbelink, and F. Ruehle, Green-Schwarz Mechanism in Heterotic (2,0) Gauged Linear Sigma Models: Torsion and NS5 Branes, arXiv:1107.0320.
* [18] C. Quigley and S. Sethi, Linear Sigma Models with Torsion, arXiv:1107.0714.
* [19] N.-H. Lee, Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds, JHEP 1004 (2010) 088, [arXiv:1004.1858].
* [20] V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493–545.
* [21] P. S. Aspinwall, B. R. Greene, and D. R. Morrison, The Monomial-Divisor Mirror Map, alg-geom/9309007.
* [22] M. Shimizu and H. Suzuki, Open mirror symmetry for Pfaffian Calabi-Yau 3-folds, JHEP 1103 (2011) 083, [arXiv:1011.2350].
* [23] A. Kapustin, L. Katzarkov, D. Orlov, and M. Yotov, Homological Mirror Symmetry for manifolds of general type, ArXiv e-prints (Apr., 2010) [arXiv:1004.0129].
|
arxiv-papers
| 2011-07-07T17:17:32 |
2024-09-04T02:49:20.332296
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Adrian Mertens",
"submitter": "Adrian Mertens",
"url": "https://arxiv.org/abs/1107.1457"
}
|
1107.1542
|
# Outflow activities in the young high-mass stellar object G23.44-0.18
Jeremy Zhiyuan Ren1, Tie Liu1, Yuefang Wu1, and Lixin Li1,2
1Department of Astronomy, Peking University, 100871, Beijing China, E-mail:
rzy,ywu@pku.edu.cn
2The Kavli Institute for Astronomy and Astrophysics, Peking University, Yi He
Yuan Lu 5, Hai Dian Qu, Beijing 100871, P. R. China
###### Abstract
We present an observational study towards the young high-mass star forming
region G23.44-0.18 using the Submillimeter Array. Two massive, radio-quiet
dusty cores MM1 and MM2 are observed in 1.3 mm continuum emission and dense
molecular gas tracers including thermal CH3OH, CH3CN, HNCO, SO, and OCS lines.
The 12CO (2–1) line reveals a strong bipolar outflow originated from MM2. The
outflow consists of a low-velocity component with wide-angle quasi-parabolic
shape and a more compact and collimated high-velocity component. The overall
geometry resembles the outflow system observed in the low-mass protostar which
has a jet-driven fast flow and entrained gas shell. The outflow has a
dynamical age of $6\times 10^{3}$ years and a mass ejection rate $\sim
10^{-3}~{}M_{\odot}$ year-1. A prominent shock emission in the outflow is
observed in SO and OCS, and also detected in CH3OH and HNCO. We investigated
the chemistry of MM1, MM2 and the shocked region. The dense core MM2 have
molecular abundances of 3 to 4 times higher than those in MM1. The abundance
excess, we suggest, can be a net effect of the stellar evolution and embedded
shocks in MM2 that calls for further inspection.
###### keywords:
stars: pre-main sequence — ISM: molecules — ISM: abundances — ISM: kinematics
and dynamics — ISM: individual (G23.44-0.18) — stars: formation
††pagerange: Outflow activities in the young high-mass stellar object
G23.44-0.18–LABEL:lastpage††pubyear: 2011
## 1 Introduction
The outflows take place in high-mass star forming regions at very early
evolutionary stages (e.g. Birkmann et al., 2006; Beuther & Sridharan, 2007;
Longmore et al., 2011). They inject large amount of hot gas and kinetic energy
into the molecular cloud, causing intense shock waves that dramatically alter
the chemistry of the surrounding environment (van Dishoeck & Blake, 1998).
However, the morphological and dynamical properties of the outflows on smaller
scales, as well as their chemical effect to the young high-mass stellar cores
are still to be further characterized.
The high-mass star forming region G23.44-0.18 (G23.44 hereafter) was
previously detected as a group of strong CH3OH masers (Walsh et al., 1998)
which indicate the presence of young massive stars. It has a trigonometric-
parallax distance of 5.88 kpc (Brunthaler et al., 2009). In this letter, we
report an observation of the massive dusty cores MM1 and MM2 associated with
the CH3OH masers, as well as a high-velocity, intense bipolar outflow revealed
in CO (2–1). The outflow is causing a shock emission and may be affecting the
chemistry in MM2. Section 3 presents the observational results. A further
discussion on the outflow is given in Section 4.1 and 4.2. The properties of
the dusty cores are discussed in Section 4.3. A summary is given in Section 5.
## 2 Observations and Data Reduction
The observational data is taken from the released SMA data
archive111http://www.cfa.harvard.edu/sma/. The observation toward G23.44 was
carried out in September 2008. The phase tracking center is
RA.(J2000)=18h34m39.25s,
Dec.(J2000)=$-8^{\circ}31^{\prime}36.2^{\prime\prime}$. The pointing accuracy
of the SMA is $\sim 3^{\prime\prime}$. The observation employed 8 antennas in
their compact configuration. The correlator has a total bandwidth of 4 GHz,
centered at 220 GHz (LSB) and 230 GHz (USB). The on-source integration time is
80 min. The SMA primary beam size (field of view) at this waveband (1.3 mm) is
$55^{\prime\prime}$. The correlator has a channel width of 0.812 MHz (1.1 km
s-1). The system temperature is between 120 to 150 K. QSO 3c454.3 and Neptune
were taken as bandpass and flux calibrators, respectively. QSO 1733-130 and
1911-201 were observed in every 30 and 50 minutes respectively during the
observation sequence to track the antenna gains. The interferometer array has
a shortest baseline of 17 m, corresponding to an u-v coverage for structures
smaller than $20^{\prime\prime}$.
The calibration and imaging were performed in Miriad1. The absolute flux level
has an uncertainty of $\sim 10\%$. The continuum map was extracted from the
line-free channels in LSB. As the visibility data are INVERTed to the image
domain, the synthesized beam size is $4.1^{\prime\prime}\times
3.7^{\prime\prime}$.
Images of the G23.44 region at other wavelengths were also extracted from
several public archives. The Spitzer/IRAC images and the point-source
catalogue are taken the database of the GLIMPSE sky survey in the NASA/IPAC
Infrared Science Archive222http://irsa.ipac.caltech.edu/. We looked for the
near-infrared counterparts in the 2MASS point-source catalogue2. The continuum
image at JCMT 450 $\micron$ was also obtained to measure the flux densities.
They are taken from the Canadian Astronomy Data Center (CADC) repository of
the SCUBA Legacy Fundamental Object Catalogue333http://www4.cadc-ccda.hia-
iha.nrc-cnrc.gc.ca.
Figure 1: The 1.3 mm continuum emission observed with the SMA overlayed on the
IRAC composite image (blue=3.6, green=4.5, and red=8.0 $\micron$) image. The
image center is RA.(J2000)=18h34m39.25s and
Dec.(J2000)=$-8^{\circ}31^{\prime}36.2^{\prime\prime}$. The contour levels are
-5, 5, 15, 25, 35, 45, and 55 $\sigma$ (0.013 Jy beam-1). The synthesized beam
($4.1^{\prime\prime}\times 3.7^{\prime\prime}$) with $PA=7^{\circ}$ from the
north to NW. The squares denote the two strongest CH3OH masers. The crosses
are the centers of the IRAC sources. Table 1: Physical parameters of the
cores.
parameter | MM1 | MM2 | Unit
---|---|---|---
$m_{J}~{}^{a}$ | $12.315\pm 0.049$ | $>15.8$ | mag
$m_{H}~{}^{a}$ | $11.702\pm 0.028$ | $>15.1$ | …
$m_{K_{s}}~{}^{a}$ | $11.435\pm 0.023$ | $>14.3$ | …
$F(3.6\micron)^{b}$ | $1.42\pm 0.28$ | $0.63\pm 0.08$ | mJy
$F(4.5\micron)^{b}$ | $25.65\pm 2.23$ | $14.48\pm 1.96$ | …
$F(5.8\micron)^{b}$ | $47.84\pm 2.22$ | $53.16\pm 2.11$ | …
$F(8.0\micron)^{b}$ | $32.92\pm 2.04$ | $40.64\pm 2.86$ | …
$F(450\micron)^{c}$ | $27.3\pm 1.5$ | $38.6\pm 1.5$ | Jy
F(1.3 mm) | $0.65\pm 0.03$ | $1.42\pm 0.03$ | …
$T_{\rm rot}$(CH3CN) | $65\pm 10$ | $110\pm 10$ | K
$T_{\rm rot}$(CH3OH) | $66\pm 5$ | $60\pm 6$ | …
Mass | $120\pm 30$ | $140\pm 10$ | $M_{\odot}$
$N({\rm H_{2}})$ | $1.9\pm 0.2$ | $2.5\pm 0.2$ | $10^{23}$ cm-2
Note. The flux densities at IR and sub-millimeter wavebands are taken from
a2MASS, bSpitzer/IRAC, cJCMT/SCUBA. The remaining data all come from the SMA
observation.
## 3 Results
Figure 1 shows the 1.3 mm continuum emission (white contours) overlayed on the
RGB image of the three IRAC bands. It reveals two dust cores aligned from the
north to south, denoted as MM1 (northern core) and MM2 (southern core).
Deconvoluted with the beam size, MM1 has an mean diameter of
$7.3^{\prime\prime}$ (0.2 pc) while MM2 is more elliptical, with an extent of
$9.7^{\prime\prime}\times 6.7^{\prime\prime}$ ($0.27\times 0.19$ pc) and
position angle $PA=45^{\circ}$ northeast. The strongest 6.7 GHz CH3OH masers
(Walsh et al., 1998) coincide with the centers of MM1 and MM2. We referred to
the MAGPIS 6 cm survey444http://third.ucllnl.org/gps/index.html for the
potential radio continuum emission from the ionized gas. It turns out that
neither MM1 nor MM2 has detectable emission above the sensitivity level of 2.9
mJy, indicating the both cores being prior to forming an Ultra-Compact Hii(UC
Hii) region. The fluxes (or magnitudes) of MM1 and MM2 at near- to far-
Infrared are drawn from the 2MASS and IRAC point-source catalogues. The 450
$\micron$ fluxes are measured directly from the JCMT images. The obtained
values are shown in Table 1.
Figure 2: (a) The velocity-integrated images of the molecular lines (contours)
overlayed on the 1.3 mm continuum (gray). The integration is from 95 to 110 km
s-1. For C18O (2–1), the contours are -4, 4, 8, 12… 40 $\sigma$ ($1.35$ Jy
beam-1 km s-1). For CH3CN $(12_{2}-11_{2})$, the contours are -4, 4, 8… 24
$\sigma$ ($0.6$ Jy beam-1 km s-1). For HNCO ($10_{0,10}-9_{0,9}$), the
contours are -4, 4, 6, 8… 20 $\sigma$ ($0.34$ Jy beam-1 km s-1). For CH3OH
$(8_{0}-7_{1})$, the contours are -4, 4, 8… 48 $\sigma$ ($0.5$ Jy beam-1 km
s-1). For SO ($6_{5}-5_{4}$), the contours are -4, 4, 8, 12… 32 $\sigma$
($0.6$ Jy beam-1 km s-1). For OCS (19–18), the contours are -4, 4, 5… 11
$\sigma$ ($0.3$ Jy beam-1 km s-1). The cross denotes the position of the
shocked clump at
$(\Delta\alpha=-11^{\prime\prime},\Delta\delta=10^{\prime\prime})$. (b) The
components of the 12CO (2–1) outflow. The low velocity component (LVC) is
integrated in (83,93) and (113,123) km s-1 for its blue- and redshifted lobes,
respectively, while the high velocity component (HVC) is integrated in (65,75)
and (140,170) km s-1 for the two lobes. For each lobe, the contours are -10,
10, 20, …, 90% of the peak value. The dashed line denotes the axis of the
outflow as indicated by the HVC. (c) The beam-averaged SO and OCS spectra at
the center of MM1 and MM2, and the shocked clump. The vertical lines denote
the systemic velocity of $v_{\rm lsr}=101$ km s-1.
The $J=2-1$ transition of 12CO, 13CO and C18O and a number of molecular
species tracing dense gas are detected in our sidebands, including HNCO
($10_{0,10}-9_{0,9}$), SO ($6_{5}-5_{4}$), OCS (19–18), CH3CN $12_{K}-11_{K}$
($K=0$ to 6), and four thermally excited CH3OH lines. In the online material,
their spectra and physical parameters are provided in Figure S1 and Table S1.
The 2MASS $K_{s}$ and JCMT 450 $\micron$ images are presented in Figure S2 and
S3.
Figure 2a shows the integrated images of the C18O, CH3CN, HNCO, CH3OH
$(8_{0}-7_{1})$, SO, and OCS lines. The C18O has a broad distribution, with
the emission peaks reasonably coincident with the dust cores. The other
species show more compact emissions closely associated with the dust cores.
The C18O emission has comparable intensities for MM1 and MM2, while the
emissions of other dense-gas tracers are much brighter in MM2. Since the dust
continuum and C18O emission all closely follow the H2 distributions, the
intensity contrast suggests that MM2 have higher molecular abundances than
MM1, as discussed in Section 4.3.
A strong bipolar outflow is observed in 12CO (2–1). Figure 2b presents the CO
(2–1) image integrated in the four velocity intervals. The outflow can be
divided into a pair of low-velocity component (LVC) and high-velocity
component (HVC) based on their distinct morphologies. The HVC roughly has
$|v|\geq 20$ km s-1 from the systemic velocity (101 km s-1). The bulk of the
outflow emission is in the southeast-northwest direction, with a position
angle of $PA=40^{\circ}$ northwest. Figure 2c shows the beam-averaged SO and
OCS spectra at MM1 and MM2 center, and the shocked region
$(\Delta\alpha=-11^{\prime\prime},\Delta\delta=10^{\prime\prime})$. The 13CO
emission shows a similar feature with 12CO despite being weak in the HVC. We
present the channel maps of 12CO and 13CO in Figure S4 and S5, respectively.
## 4 discussion
### 4.1 The physical properties of the outflow
The presence of intense bipolar outflow in G23.44 suggests that the central
stars are likely being formed via a disk-mediated accretion (Zhang et al.,
2007). The HVC is well collimated along the outflow axis, while the LVC,
especially its red wing, has a more opened parabolic shape with a decreasing
brightness from the edge to the center. The overall geometry of the outflow is
reminiscent of the outflow system in the low-mass protostar HH211 (Gueth &
Guilloteau, 1999). HH211 consists of a high-velocity jet-driven flow and a
entrained low-velocity gas shell. The G23.44 outflow may represent a scaled-up
version of this system, despite its collimation and shell structure being much
less perfect than HH211. However, the size of the outflow in HH211 is only
$\sim 10^{4}$ AU, while the G23.44 outflow ($10^{5}$ AU) has a 10-time larger
spatial extent. On such a large scale, the outflow structure might begin to
deteriorate due to increased turbulence in its environment.
Table 2: Abundance of the detected molecules.
species | $N_{\rm x}$ | $f_{\rm x}$ | $f_{x,2/1}^{a}$
---|---|---|---
(X) | (cm-2) | |
MM1 $(-1^{\prime\prime},10^{\prime\prime})$
C18O | $3.16\pm 0.05(16)$ | $1.7\pm 0.1(-7)$ | $-$
CH3OH | $2.5\pm 1.2(15)$ | $1.3\pm 0.6(-8)$ | $-$
CH3CN | $5.2\pm 0.6(14)$ | $2.7\pm 0.5(-9)$ | $-$
HNCO | $3.8\pm 0.2(14)$ | $2.0\pm 0.4(-9)$ | $-$
SO | $2.5\pm 0.2(14)$ | $1.3\pm 0.3(-9)$ | $-$
OCS | $3.7\pm 0.2(14)$ | $1.9\pm 0.3(-9)$ | $-$
MM2 $(1^{\prime\prime},-3^{\prime\prime})$
C18O | $7.30\pm 0.07(16)$ | $2.9\pm 0.1(-7)$ | $1.7\pm 0.2$
CH3OH | $9.8\pm 0.8(15)$ | $3.9\pm 0.9(-8)$ | $3.0\pm 1.5$
CH3CN | $1.6\pm 0.2(15)$ | $6.3\pm 0.7(-9)$ | $2.3\pm 0.8$
HNCO | $2.20\pm 0.05(15)$ | $8.8\pm 1.5(-9)$ | $4.4\pm 1.4$
SO | $1.45\pm 0.06(15)$ | $5.8\pm 0.9(-9)$ | $4.5\pm 1.4$
OCS | $1.20\pm 0.05(15)$ | $4.8\pm 0.8(-9)$ | $2.5\pm 0.7$
outflow-shock $(-11^{\prime\prime},10^{\prime\prime})$
C18O | $2.4\pm 0.07(16)$ | $=2.0(-7)^{b}$ | $-$
CH3OH | $2.8\pm 0.5(15)$ | $2.3\pm 0.5(-8)$ | $-$
HNCO | $4.1\pm 0.2(14)$ | $8.3\pm 0.4(-10)$ | $-$
SO | $3.5\pm 0.3(14)$ | $3.1\pm 0.3(-9)$ | $-$
OCS | $8.0\pm 0.3(14)$ | $6.6\pm 0.3(-9)$ | $-$
$a.$ The abundance ratio of MM2 to MM1.
$b.$ Assumed as the typical value in ISM. Abundances of the other molecules
are then calculated by comparing to $N({\rm C^{18}O})$.
The CO outflow, especially the LVC also exhibits a weak emission feature in
opposite direction to the major one, i.e. a blueshifted emission in the NW and
a redshifted one in the SE. This can be a second flow by chance aligned in the
same position angle with the major one. Alternatively, once the outflow
direction is close to the plane of the sky and its inclination angle becomes
smaller than the opening angle $(\sim 30^{\circ})$, its front and back sides
will exhibit opposite Doppler shift. This scenario is well demonstrated by Yen
et al. (2010, Figure 10 therein). We speculate the second case to be more
plausible.
Assuming a local thermal equilibrium (LTE) and low optical depth for the 12CO
line-wing emission, the CO column density and the outflow mass can be inferred
following the approach of Garden et al. (1991). The excitation temperature of
the outflow is speculated to be slightly lower than in the dense core MM2
($\sim 70$ K, Section 4.3). A value of $T_{\rm ex}=50$ K is adopted here. The
outflow mass is estimated in each 1 km s-1 interval, then the total mass,
momentum, and kinetic energy are calculated from $M=\Sigma_{v}m(v)$,
$P=\Sigma_{v}m(v)v$, and $E_{k}=\Sigma_{v}m(v)v^{2}/2$. In calculation, we
assume an inclination angle of $\theta=20^{\circ}$ to correct the velocity
projection. The outflow turns out to have $M\simeq 5.5~{}M_{\odot}$, $P\simeq
360~{}M_{\odot}$ km s-1, $E_{k}\simeq 9\times 10^{46}$ erg.
Each of the blue and red wings in the HVC is resolved into two major clumps
(Figure 2b, lower panel) denoted B1, B2 and R1, R2. These clumps in the
outflow may indicate an episodic mass ejection (e.g. Qiu et al., 2009). The
average central distance is $24^{\prime\prime}$ (0.68 pc) for the pair of
B1-R1, and $5.0^{\prime\prime}$ (0.14 pc) for B2-R2. The two pairs both show a
reasonable geometrical symmetry. However, the clumps are not exactly aligned
across the MM2 center, instead with an offset to its west. This may indicate
that the driven agent, or the stellar disk system is also displaced from the
MM2 center. The dynamical age of the outflow $t_{\rm dyn}$ is estimated by
dividing its spatial extent with the typical velocity ($v=40/\sin\theta$ km
s-1). That yields $t_{\rm dyn}=6.1\times 10^{3}$ years for the B1-R1 pair and
$t_{\rm dyn}=1.3\times 10^{3}$ years for B2-R2. Adopting the first value, we
calculate the mass ejection rate to be $\dot{M}=M/t_{\rm dyn}\simeq 1.0\times
10^{-3}M_{\odot}$ yr-1. One can see that the outflow is young and performing
an intense mass ejection.
### 4.2 Chemical enhancement in the outflow and shock
In both low- and high-mass young stellar objects the outflows have led to a
rich chemistry in CH3OH, HNCO, and sulphides (Blake et al., 1987; van der Tak
et al., 2003; Jørgensen et al., 2004; Rodríguez-Fernández et al., 2010, etc.).
In G23.44, the SO and OCS emission shows a noticeable emission clump at
offset=$(-11^{\prime\prime},10^{\prime\prime})$ (Figure 2a). The clump is
almost perfectly aligned with the HVC axis of the CO (2–1), and reasonably
coincides with the peak R2, despite being closer to MM2 center. This strongly
suggests that the clump traces a shock within the jet/outflow. The clump is
also weakly detected in HNCO and CH3OH (spectra shown in Figure S1). Their
column densities are estimated using Equation (1) and (2) in Section 4.3. In
calculation a temperature of $T_{\rm ex}=50$ K [same with that of CO (2–1)] is
adopted, and the C18O abundance is taken to be a constant of $2\times 10^{-7}$
as the typical ISM abundance which is close to $f_{\rm C^{18}O}$ measured in
MM1 and MM2. The abundances of other molecules are then inferred from their
intensity ratio relative to C18O (2–1). The abundances of the shocked clump
are listed in Table 2.
In the low-mass protostars, the outflows often lead to a rich chemistry. In a
few cases, the column density of HNCO and S-bearing molecules can been
enhanced for orders of magnitude (e.g. L1157 and L1448, Rodríguez-Fernández et
al., 2010; Taffala et al., 2010). In comparison, the shocked region at
$(-11^{\prime\prime},10^{\prime\prime})$ shows moderate abundances which are
comparable or even slightly lower than in MM2. The enhancement level can be
limited by the temperature and density in the shocked region, as well as the
efficiency of the previous grain-surface chemistry. In addition, since the
outflow has a short $t_{\rm dyn}$, the shock chemistry may still be undergoing
and have not yet attained its maximum level.
The blue lobe of CO $(2-1)$ outflow are devoid of the SO and OCS emissions. As
a possible explanation, the S-bearing species (e.g. H2S and CS as their
progenitors) would be rather tightly bound to the dust grains (Blake et al.,
1987), and and a threshold shock velocity may be required for their
desorption. Higher resolution and primary shock products like SiO and H2S may
better reveal the shock chemistry.
Figure 3: (a) The solid line is the CH3CN ($12_{K}-11_{K}$) towards the center
of MM1 (top panel) and MM2 (bottom panel). The dashed line is the spectrum of
the best fit from the LTE gas model. (b) The rotation diagram of the CH3OH
lines. The filled circles are for the observed transitions. The vertical bars
indicate the $3\sigma$ errors. The linear least-squares fit is shown as the
solid line.
### 4.3 The dense core properties
The dust cores MM1 and MM2 coincide with dark patches on the extended 8
$\micron$ emission (Figure 1). The two cores are slightly connected with each
other, suggesting that they might be the fragmentation products from the same
natal cloud.
The CH3CN ($12_{K}-11_{K}$) lines provide a temperature diagnosis for the
cores (Figure 3a). We take an LTE radiative transfer model with a uniform
rotation temperature and gas density (Chen et al., 2006; Zhang et al., 2007)
to reproduce the observed CH3CN spectra at the dust-core center. We found the
best fit at $T_{\rm rot}=65$ K for MM1 and $T_{\rm rot}=110$ K for MM2. The
modeled spectra of $K=0$ and 1 towards MM2 is much lower than the observed
line profile while the $K>1$ lines are all well fitted. The excess emission in
$K=0$ and 1 lines may largely arise from the cold outer-envelope gas.
The rotation temperature is also evaluated from the thermally excited CH3OH
lines at the core center using the rotation diagram method (Goldsmith &
Langer, 1999; Liu et al., 2001; Bisschop et al., 2007). The column density of
the upper level $N_{\rm u}$ is related to the line intensity in the form
$\qquad\qquad\frac{N_{\rm u}}{g_{\rm u}}=\frac{3k}{8\pi^{3}\nu S\mu^{2}}\int
T_{\rm b}\,{\rm d}v$ (1)
where $g_{\rm u}$ is the upper-level degeneracy. $S$ is the line strength,
$\mu$ is the dipole moment. $N_{\rm u}$ depends on the total column density
$N_{\rm T}$ in the form
$\qquad\qquad\ln(\frac{N_{\rm u}}{g_{\rm u}})=\ln(\frac{N_{\rm T}}{Q_{\rm
rot}})-\frac{E_{\rm u}}{T_{\rm rot}}$ (2)
where the partition function $Q_{\rm rot}$ can be adopted from Blake et al.
(1987). $T_{\rm rot}$ is then estimated from a least-square fit (Figure 3b).
The derived values are listed in Table 1.
The $T_{\rm rot}$ derived from CH3OH and CH3CN are remarkably similar for MM1.
For MM2, however, the $T_{\rm rot}$ of CH3OH is lower than that of CH3CN. As a
probable reason, the CH3OH lines generally have lower $E_{\rm u}$ than CH3CN
($12_{K}-11_{K}$), then the cooler outer-part gas would contribute more
significantly to the CH3OH lines than to the CH3CN (in particular $K=2$ to 6).
In addition the CH3OH can be easily released from the dust grains by the
shocks while the CH3CN production requires subsequent gas-phase reactions and
stellar heating (van Dishoeck & Blake, 1998). CH3CN may therefore be
distributed closer to the MM2 center (as seen in Figure 2a), and showing a
higher $T_{\rm rot}$.
For the other species, an independent temperature estimation is currently
unavailable. We adopt $T_{\rm ex}=60$ K for MM1 and $T_{\rm ex}=70$ K for MM2
assuming an LTE excitation, the column density of the molecules are calculated
from their spectral lines towards the dust-core center, using equation (1) and
(2).
Assuming optically thin 1.3 mm dust continuum emission, the total dust-and-gas
mass can be estimated using $S_{\nu}=M_{\rm core}\kappa_{\nu}B_{\nu}(T_{\rm
d})/gD^{2}$. In the formula $S_{\nu}$ is the total flux density at 1.3 mm.
$g=100$ is the dust-gas mass ratio. $B_{\nu}(T_{\rm
d})=(2h\nu^{3}/c^{2})/[\exp(T_{0}/T_{\rm d})-1]$ is the Planck function
($T_{0}=h\nu/k$). The dust opacity is set to be $\kappa_{\nu}=\kappa_{\rm
230GHz}=0.9$ cm2 g-1 of MRN dust grains with thin ice mantles at $n_{\rm
H_{2}}=10^{6}$ cm-3 (Ossenkopf & Henning, 1994). In the expression of
$S_{\nu}$, using the beam-averaged flux density towards the core center, and
dividing the obtained mass with $m_{\rm H_{2}}$, one can also calculate the
peak column density N(H2) at MM1 and MM2. The derived values are presented in
Table 1. MM2 has a comparable but slightly higher N(H2) than MM1.
The molecular abundances are estimated using $f_{\rm x}=N(X)/N({\rm H_{2}})$.
The $f_{\rm x}$ and its ratio between MM2 and MM1 are listed in Table 2. For
the high-density tracers, the abundances in MM2 are generally 3 to 4 times
higher than those in MM1. The SO shows the highest abundance ratio of 4.5. We
note that the temperature adopted for MM2 (70 K) is conservative. A higher
$T_{\rm ex}$ would yield a lower N(H2) and higher $N(X)$ for most species,
hence leading to even higher abundances. For instance, adopting the value of
$T_{\rm rot}({\rm CH_{3}CN})=110$ K as the $T_{\rm ex}$ for MM2, we will have
$N({\rm HNCO})=6.7\times 10^{15}$ cm2 that is 3 times larger than the current
value.
The molecular distribution appears to be influenced by the outflow. Besides
the coherent elongation of SO and OCS with the HVC of 12CO emission (Figure
2b), the CH3OH and HNCO emissions are also largely distributed to the west of
MM2. In addition, all the molecular lines have broader line widths in MM2 than
in MM1, e.g. The SO line has $\Delta v_{\rm FWHM}=11.3$ in MM2 and 3.4 km s-1
in MM1. The line widths of MM2 can be broadened by a kinetic energy input from
the outflow. Considering its influence to the molecular line profiles, the
outflow may also contribute to the higher abundances in MM2. Specifically,
there would be underlying shocks which are induced as the outflow is
interacting with the envelope and/or infalling gas (Arce & Sargent, 2004; Wu
et al., 2009).
At the near- to mid-IR and sub-millimeter wavebands, the two cores have
similar flux densities (Table 1). At IRAC bands MM2 is brighter than MM1 at
5.8 and 8 $\micron$ but weaker at 3.6 and 4.5 $\micron$, while at 2MASS $J$,
$H$, $K_{s}$ bands, only MM1 is significantly detected (Figure S2). Having a
spectral energy distribution (SED) obviously skewed to the shorter wavelengths
($\lambda<4.5~{}\micron$), MM1 may have a more evolved stage for its stellar
disk system (Robitaille et al., 2006). However, since the outflow in MM2 is
close to the plane of the sky, it is also possible that a disk/toroid
structure lies edge-on in MM2 thus would largely absorb the short-wave
emissions. A higher resolution will be helpful in revealing the morphology of
the stellar system and underlying shocks in MM2.
## 5 summary
We present a multi-wavelength study towards the young high-mass star forming
region G23.44-0.18. Two dusty cores, MM1 and MM2 are observed in 1.3 mm
continuum emission and the molecular lines of CH3OH, CH3CN, HNCO, SO, and OCS.
The both cores have a pre-UC Hii evolutionary stage. A strong bipolar outflow
is arising from MM2 as revealed by CO (2–1). The outflow consists of a
collimated, bi-polar high-velocity component and a more extended, parabolic
flow at lower velocities. A clump of shocked gas is observed in SO
$(6_{5}-5_{4})$ and OCS $(19-18)$ lines and also detected in HNCO and CH3OH.
The outflow shows a high momentum and mass-loss rate, and a short timescale of
$t_{\rm dyn}\sim 6\times 10^{3}$ years.
In MM2 the high-density tracers CH3OH, CH3CN, HNCO, SO, and OCS show
abundances of 3 to 4 times higher than those in MM1. To explain this
difference, we suggest a combined effect of the stellar heating and underlying
shock interactions in MM2. A further study with higher excited lines and
improved resolution can be performed to probe the chemistry, possible disk-jet
system and mass accretion close to the MM2 center.
## Acknowledgment
We are grateful to the SMA observers and the SMA data archive. We would thank
the anonymous referee for the useful comments that helped to improve the
presentation and interpretation. This work is supported by grants of
No.10733033, 10873019, 10973003, 2009CB24901, and the Doctoral Candidate
Innovation Research Support Program (kjdb201001-1) from Science & Technology
Review.
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Table S1: Parameters of the molecular lines. Frequency | Species | | $V_{\rm lsr}$ | $T_{\rm b}^{\rm peak}$ | $\Delta V_{\rm FWHM}$
---|---|---|---|---|---
(GHz) | (X) | Transition | (km s-1) | (K) | (km s-1)
MM1 $(-1^{\prime\prime},10^{\prime\prime})$
219.5603… | C18O | $2-1$ | 102.2 | $11\pm 0.4$ | $3.2\pm 0.5$
219.7982… | HNCO | $10_{0,10}-9_{0,9}$ | 101.4 | $1.8\pm 0.1$ | $3.6\pm 0.4$
219.9494… | SO | $6_{5}-5_{4}$ | 101.3 | $4.2\pm 0.1$ | $3.4\pm 0.4$
220.0784… | CH3OH | $8_{0,8}-7_{1,6}$ | 101.5 | $1.0\pm 0.3$ | $4.2\pm 0.6$
220.7302… | CH3CN | $12_{2}-11_{2}$ | 101.6 | $1.3\pm 0.2$ | $5.5\pm 0.8$
229.7588… | CH3OH | $8_{-1}-7_{0}$ | 101.8 | $3.9\pm 0.1$ | $3.5\pm 0.5$
230.0270… | CH3OH | $3_{-2,2}-4_{-1,4}$ | 102.1 | $1.3\pm 0.3$ | $2.9\pm 0.6$
231.0609… | OCS | $19-18$ | 102.1 | $1.9\pm 0.2$ | $4.5\pm 0.4$
231.2810… | CH3OH | $10_{2,9}-9_{3,6}$ | 101.3 | $0.6\pm 0.2$ | $2.3\pm 0.4$
MM2 $(1^{\prime\prime},-3^{\prime\prime})$
219.5603… | C18O | $2-1$ | 99.8 | $6.4\pm 0.5$ | $10.5\pm 1.5$
219.7982… | HNCO | $10_{0,10}-9_{0,9}$ | 100.6 | $3.5\pm 0.1$ | $8.8\pm 0.6$
219.9494… | SO | $6_{5}-5_{4}$ | 101.5 | $6.0\pm 0.1$ | $11.3\pm 0.7$
220.0784… | CH3OH | $8_{0,8}-7_{1,6}$ | 100.5 | $3.5\pm 0.3$ | $6.7\pm 0.8$
220.7302… | CH3CN | $12_{3}-11_{3}$ | 101.7 | $2.8\pm 0.1$ | $6.6\pm 0.7$
229.7588… | CH3OH | $8_{-1}-7_{0}$ | 102.1 | $15.0\pm 0.1$ | $8.5\pm 0.8$
230.0270… | CH3OH | $3_{-2,2}-4_{-1,4}$ | 102.1 | $2.0\pm 0.2$ | $4.3\pm 0.5$
231.0609… | OCS | $19-18$ | 102.1 | $3.2\pm 0.1$ | $6.7\pm 0.4$
231.2810… | CH3OH | $10_{2,9}-9_{3,6}$ | 102.1 | $1.4\pm 0.3$ | $4.3\pm 0.3$
Shock $(-11^{\prime\prime},10^{\prime\prime})$
219.5603… | C18O | $2-1$ | 102.1 | $8.0\pm 0.3$ | $2.7\pm 0.4$
219.7982… | HNCO | $10_{0,10}-9_{0,9}$ | 96.3 | $0.8\pm 0.2$ | $1.3\pm 0.4$
219.9494… | SO | $6_{5}-5_{4}$ | 102.3 | $4.9\pm 0.3$ | $5.6\pm 0.5$
220.0784… | CH3OH | $8_{0,8}-7_{1,6}$ | 102.1 | $0.9\pm 0.2$ | $3.7\pm 0.6$
220.7302… | CH3CN | $12_{2}-11_{2}$ | $-$ | $-$ | $-$
229.7588… | CH3OH | $8_{-1}-7_{0}$ | 103.2 | $8.0\pm 0.2$ | $4.2\pm 0.4$
230.0270… | CH3OH | $3_{-2,2}-4_{-1,4}$ | $-$ | $-$ | $-$
231.0609… | OCS | $19-18$ | 102.1 | $2.0\pm 0.2$ | $3.3\pm 0.5$
231.2810… | CH3OH | $10_{2,9}-9_{3,6}$ | $-$ | $-$ | $-$
Figure S1: The molecular lines detected in the observational sidebands at the
three positions.
Figure S2: The near to mid-IR RGB-coded image of G23.44. The blue is the 2MASS
$K_{s}$, the green and red colors are IRAC 3.6 and 4.5 micron, respectively.
The image center is RA.(J2000)=18h34m39.25s and
Dec.(J2000)=$-8^{\circ}31^{\prime}36.2^{\prime\prime}$. The contour levels are
-5, 5, 15, 25, 35, 45, and 55 $\sigma$ (0.013 Jy beam-1). The synthesized beam
($4.1^{\prime\prime}\times 3.7^{\prime\prime}$) with $PA=7^{\circ}$ to the
west. The squares are the two strongest CH3OH masers. The crosses are the
centers of the IRAC sources.
Figure S3: The SCUBA 450 $\micron$ image of G23.44. Contours are 10 to 90 per
cent of the peak brightness (34.5 Jy beam-1). The beam size is
$7.5^{\prime\prime}$. The cross denotes the center position
RA.(J2000)=18h34m39.25s and
Dec.(J2000)=$-8^{\circ}31^{\prime}36.2^{\prime\prime}$.
Figure S4: The 12CO (2–1) channel images. The contours are -10, 20 to 90
percent of the peak intensity, which is 45 Jy beam-1 for the velocity range of
[63,90] km s-1 and 18.9 Jy beam-1 for v=[110,152] km s-1. Gray is the SMA 1.3
mm continuum emission.
Figure S5: The 13CO (2–1) channel images. The contours are -10, 20 to 90
percent of the peak intensity which is 12.2 Jy beam-1 for the velocity range
of [84,93] km s-1 and 8.8 Jy beam-1 for v=[108,117] km s-1.
|
arxiv-papers
| 2011-07-08T01:33:57 |
2024-09-04T02:49:20.342133
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jeremy Zhiyuan Ren, Tie Liu, Yuefang Wu, and Lixin Li",
"submitter": "Zhiyuan Ren",
"url": "https://arxiv.org/abs/1107.1542"
}
|
1107.1798
|
# Comparison of seismic signatures of flares obtained by SOHO/MDI and GONG
instruments
S. Zharkov11affiliation: Mullard Space Science Laboratory, University College
London, Holmbury St. Mary, Dorking, RH5 6NT, UK , V.V.Zharkova22affiliation:
Horton D building, Department of Mathematics, University of Bradford,
Bradford, BD7 1DP, UK , S.A. Matthews11affiliation: Mullard Space Science
Laboratory, University College London, Holmbury St. Mary, Dorking, RH5 6NT, UK
###### Abstract
The first observations of seismic responses to solar flares were carried out
using time-distance (TD) and holography techniques applied to SOHO/MDI
Dopplergrams obtained from space and un-affected by terrestrial atmospheric
disturbances. However, the ground-based network GONG is potentially a very
valuable source of sunquake observations, especially in cases where space
observations are unavailable. In this paper we present updated technique for
pre-processing of GONG observations for application of subjacent vantage
holography. Using this method and TD diagrams we investigate several sunquakes
observed in association with M and X-class solar flares and compare the
outcomes with those reported earlier using MDI data. In both GONG and MDI
datasets, for the first time, we also detect the TD ridge associated with the
September 9, 2001 flare. Our results show reassuringly positive identification
of sunquakes from GONG data that can provide further information about the
physics of seismic processes associated with solar flares.
Sun: photosphere; Sun:helioseismology; Sun:flares; Sun: oscillations; Sun:data
analysis
## 1 Introduction
Discovery of sunquakes associated with solar flares (Kosovichev & Zharkova
1998; Donea et al. 1999) has opened a new era in the investigation of energy
and momentum transport mechanisms from the upper atmosphere to the photosphere
and beneath, uncovering structure of these spectacular events. Sunquakes, seen
as circular or elliptical waves - ripples, propagating outward from impulsive
hard X-ray (HXR) solar flares along the solar surface, appear 20-60 minutes
after the flare onsets. The surface ripples are also associated with strong
downward shocks preceding these ripples with close (1-4 minutes) temporal
correlation with the start of HXR flares, indicating that high energy
particles play some role in initiation of sunquakes (Zharkova & Zharkov 2007;
Martínez-Oliveros et al. 2008).
Even though every flare is expected to inject particle beams of one kind or
another into a flaring atmosphere, inducing either shocks or magnetic
impulses, not many of them have recorded measurable signatures of seismic
activity. Initially only X-class flares were considered as candidates for
producing sunquakes. The first flare detection used time-distance diagrams and
reported well distinguished ripples emanating from the center of the location
of a hard X-ray source in the X1.1 flare 9 July 1996 (Kosovichev & Zharkova
1998; Donea et al. 1999). It was followed by detection of quakes associated
with two extremely powerful solar flares of class X17 and X10 which erupted in
NOAA Active region 10486 on October 28 and 29, 2003. These two flares, known
as the Halloween 2003 flares, were extensively investigated in Donea & Lindsey
(2005) by applying subjacent vantage acoustic holography to SOHO/MDI
Dopplergram data. The acoustic signatures were also shown to co-align with the
hard X-ray signatures and GONG intensity observations revealed significant
radiative emission with a sudden onset in the compact region encompassing the
acoustic signature.
Further analysis of the 29 October 2003 quake was presented in Lindsey & Donea
(2008), where the authors proposed a new method for correcting intensity data
recorded by the Global Oscillation Network Group (GONG), that allowed
comparison of acoustic kernels with white-light traces of the flare. Later
Zharkova & Zharkov (2006); Kosovichev (2006); Zharkova & Zharkov (2007)
investigated the Halloween flare of 28 October 2003 using the time-distance
diagram method applied to SOHO/MDI data and detected three distinct seismic
sources that coincided with the holographic sources from Donea & Lindsey
(2005). For the 29 October 2003 flare there were no time-distance ridges found
by the authors of Zharkova & Zharkov (2007) when they analyzed MDI
dopplergrams, however one was later reported in Kosovichev (2006). The first
M-class flare in which seismic signatures were detected by means of acoustic
holography using Doppler velocity data from SOHO/MDI instrument was the flare
that occurred in NOAA Active Region 9608 on September 9, 2001 (Donea et al.
2006a).
Later the list of acoustically active flares was significantly extended.
Beşliu-Ionescu et al. (2005) reported another six such flares, adding another
four in Donea, A.-C. et al. (2006b). In fact, during the period of
observations with SOHO/MDI instrument there have been 17 flares showing signs
of acoustic activity possibly related to sunquakes111see
http://users.monash.edu.au/$∼$dionescu/sunquakes/sunquakes.html. This
expanding list motivated researchers to look further and to explore the data
from the ground-based GONG observatories, which offers better coverage of
helioseismic data compared to SOHO/MDI.
In the unusually quiet solar minimum between cycles 23 and 24 more attention
was paid to each new flare occurring on the Sun, one of which was the flare of
14 December 2006. The latter was observed only by the GONG instruments and
revealed some noticeable seismic signatures in both time-distance ridges and
egression powers (Matthews et al. 2011). In order to validate these findings,
a comparison is required of the signatures of sunquakes derived for both the
time-distance and holographic techniques from GONG data with those from
SOHO/MDI for a number flares with distinct seismic signatures. Such a
comparison will allow us to understand the differences in appearances and to
derive recommendations for the reliable detection of sunquakes from GONG data.
In this study we consider three acoustically active flares that have the
luxury of helioseismic observations available from both the GONG and the
SOHO/MDI instrument. The available Dopplergrams (GONG and MDI) are used to
analyze the acoustic signatures of the flares by using both the time-distance
diagram technique (TD method; Kosovichev & Zharkova 1998) and acoustic
holography (e.g. Braun & Lindsey 1999; Lindsey & Braun 2000). The description
of data and additional corrections for the technique applied to GONG data are
presented in section 2, the results of the comparison are described in section
3 and conclusions with recommendations are drawn in section 4.
## 2 Description of data and techniques
In this study we use three acoustically active flares with strong seismic
signals detected by SOHO/MDI. The first flare is an M-class flare that
occurred in NOAA Active Region 9608 on September 9, 2001. The solar quake
associated with the flare was the first one detected for M-class flares and
investigated by means of acoustic holography in Donea et al. (2006a) using
data from SOHO MDI. The other two are extremely powerful X-class flares that
erupted in NOAA Active region 10486 on October 28 and 29, 2003, with
associated sunquakes first detected using acoustic holography by Donea &
Lindsey (2005). The Halloween flares have also been investigated by applying
time-distance method to SOHO/MDI data in Zharkova & Zharkov (2007), where
three distinct sources were detected for October 28 flare. However, after
extensively analyzing MDI velocity data for the 29 October 2003 flare, the
authors of Zharkova & Zharkov (2007) did not find a time-distance ridge in any
of time-distance diagrams computed around the flare location.
### 2.1 Helioseismic techniques for quake detection
In order to detect and analyze the solar quake associated with a flare we use
both time-distance analysis (Kosovichev & Zharkova 1998) and acoustic
holography (Donea et al. 1999). Time-distance analysis is applied to detect
the circular ripples generated by the quake. This consists of rewriting the
observed surface signal in polar coordinates relative to the source, i.e.
$v(r,\theta,t)$, and using azimuthal transformation
$V_{m}(r,t)=\int_{0}^{2\pi}v(r,\theta,t)e^{-im\theta}d\theta,$ (1)
to study the $m=0$ component for evidence of the propagating wave. Then if
seen, the quake manifests itself as a time-distance ridge, thus providing
estimates of the surface propagation speed and the time of excitation. In this
work the GONG high-cadence velocity data were used in the time-distance
analysis.
Acoustic holography is applied to calculate the egression power maps from
observations. The holography method (Braun & Lindsey 1999; Donea et al. 1999;
Braun & Lindsey 2000; Lindsey & Braun 2000) works by essentially
“backtracking” the observed surface signal, $\psi({\bf r},t)$, by using
Green’s function, $G_{+}(|{\bf r}-{\bf r}^{\prime}|,t-t^{\prime})$, which
prescribes the acoustic wave propagation from a point source. This allows us
to reconstruct egression images showing the subsurface acoustic sources and
sinks. Following Donea & Lindsey (2005), in temporal Fourier domain we have
$\hat{H}_{+}({{\bf r},\nu})=\int_{a<|{\bf r}-{\bf r}^{\prime}|<b}d^{2}{\bf
r}^{\prime}\hat{G}_{+}(|{\bf r}-{\bf r}^{\prime}|,\nu)\hat{\psi}({\bf
r}^{\prime},\nu),$ (2)
where $a,b$ define the holographic pupil and $\hat{H}_{+}({{\bf r},\nu})$ is
the temporal Fourier transform of $H_{+}({{\bf r},t}).$ Then
$H_{+}({{\bf r},t})=\int_{\Delta\nu}d\nu\,e^{2\pi i\nu t}\,\hat{H}_{+}({{\bf
r},\nu}),$ (3)
whence the square amplitude of egression is called the egression power
$P({\bf r},t)=|H_{+}({{\bf r},t})|^{2}dt.$ (4)
Green’s functions built using a geometrical-optics approach are used in this
study. As flare acoustic signatures can be submerged by ambient noise for the
relatively long periods over which the egression power maps are integrated,
again we follow Donea & Lindsey (2005) using egression-power ’snapshots’ to
discriminate flare emission from the noise with pass-band integration in
equation (3) performed over positive frequencies only in order to reduce
noise. Such a snapshot is simply a sample of the egression power within a time
$\Delta t=\frac{1}{2{\rm\ mHz}}=500{\rm s}$. The snapshots used in this work
are taken from the egression power, $P({\bf r},t)$, at selected times, $t$.
### 2.2 Observations and data reduction
An M9.5-class flare occurred in NOAA Active Region 9608 around 20:40 UT on
September 9, 2001 at around $104^{\circ}$ Carrington longitude and
$26^{\circ}$ latitude south. GOES soft X-ray flux reached a peak at 20:46 UT,
with the background emission remaining above the ”C” level for most of the
flare period. The flare of October 28, 2003, occurred in NOAA Active Region
10486 at $287^{\circ}$ Carrington longitude and $8^{\circ}$ south latitude. It
is classified as X17.2, one of the most powerful amongst the quake producing
flares recorded. The GOES satellite detected increased X-ray flux starting at
09:51 UT, reaching a maximum at 11:10 UT. On the following day in the same
Active Region an X10 class flare occurred at around $270^{\circ}$ Carrington
longitude, $10^{\circ}$ latitude in the southern hemisphere. The X-ray flux
observed by GOES began to increase at 20:41 UT, reaching maximum at 20:49 UT
and ending at 21:01 UT.
The helioseismic observations analyzed in this study were obtained by the GONG
(Harvey et al. 1988, 1996) ground-based observatories, and by the MDI
instrument (Scherrer et al. 1995) onboard the SOHO spacecraft. Both GONG and
MDI observe using the photospheric Ni I 6768 Å line. GONG observations are
made with one minute cadence and normally include full-disk Dopplergrams,
line-of-sight magnetograms and intensity images. In this work we use MDI full-
disk Doppler images obtained with a one minute cadence. Velocity measurements
in both cases are made from Doppler shifts of the Ni spectral line. MDI
estimates the velocities from line instensities (filtergrams) scanned in
several locations across the line, while GONG relies on a fast Fourier
tachometric scans across the line (Harvey et al. 1988) to derive the surface-
velocity images based on a standard response of the line profile to Doppler
motion caused by propagation of acoustic waves.
For the October 28, 2003 flare we use two-hour-long full-disk velocity
observations with one minute cadence from the SOHO/MDI instrument and GONG
starting at 10:46 UT. The October 29, 2003 and September 9, 2001 series
commence at 20.00 UT on the corresponding dates. In addition, for the
Halloween flares we use one-minute cadence intensity observations available
from the GONG for the same period. Unfortunately, there are no such intensity
observations available for the duration of the September 9 flare. Following
the standard approach in local helioseismology, we extract datacubes centered
on the region of interest from each full-disk series to remap the data onto
the heliographic grid using Postel-projection and to remove a differential
rotation at the Snodgrass rate. For the velocity data the series average full-
disk velocity image is subtracted from each observations before the procedure,
in order to remove the rotation gradient. Due to different resolution of the
instruments, SOHO MDI data is remapped at 0.125 degrees per pixel resolution,
while GONG datacubes are at 0.15 degrees per pixel.
For acoustic holography we use Green’s functions centered at 6mHz, so the
datacubes are filtered in the frequency domain using a bandpass filter
allowing the full signal in the 5-7 mHz band, with steep Gaussian roll-offs on
each side. The pupil dimensions for each dataset for the selected flares are
presented in Table 1.
### 2.3 Additional corrections for the GONG data
As GONG is a ground-based network, its data are affected by visibility
conditions at the time of observation. Effects such as atmospheric smearing
and local stochastic translation introduce spurious temporal variations in
magnetized regions that can easily dominate over a genuine seismic signal.
Another concern can be related to GONG usage of tachometric scanning of the Ni
line, which, due to variations in atmospheric conditions between the start and
end of the scan, is likely to be more affected (see, for example, Grigor’ev &
Kobanov 1988).
The spurious Doppler shifts cited by Grigor’ev & Kobanov (1988) are applicable
to radiation incidence away from normal incidence (above 2∘) passing through a
Fabrey-Perot etalon, which has effective path differences of $2.2\times
10^{4}$ wavelengths implemented for GONG (Harvey & The GONG Instrument Team
1995). The GONG optics ingeniously avoid this problem by directing the long
optical path through glass and the shorter through air, the geometrical paths
being the same to within about a micron (Title & Ramsey 1980; Harvey & The
GONG Instrument Team 1995).
In the presence of a strong magnetic field (e.g. sunspot umbrae) Zeeman
splitting of a magnetic line introduces spurious phase shifts in the
measurements (Rajaguru et al. 2007). One possible reason for such effect is
the reduced line intensities within a sunspot (Toner & Labonte 1993; Bruls
1993; Norton et al. 2006) which cause noise such as due to variable
atmospheric smearing to introduce spurious intensity observations from
surrounding region into the desired pixel (Toner & Labonte 1993; Braun 1997).
This, for instance, leads to significant differences in the computed acoustic
power maps between the sunspot data from MDI and GONG, with the space-based
data generally showing suppression of the acoustic power over a sunspot region
(e.g. Gizon et al. 2009, and references therein), while the ground-based GONG
data demonstrating a large power increase at the same location (see top row of
Figure 1 for example) that is clearly noise-related due to the reasoning
above.
To correct the atmospheric contribution in GONG observations in the first
instance we use the method developed in Lindsey & Donea (2008) where the
intensity data are available, e.g. Halloween flares. The method works by
measuring atmospheric seeing effects such as translation and smearing of GONG
intensity observations in relation to a reference image and then removing
their contribution from the intensity data. As both intensity and velocity
data come from the same instrument, we apply the parameters extracted from the
intensity series to correct the line of sight velocity data.
In addition, since the atmospheric noise affects mostly the measurements taken
over magnetized regions, we seek to minimize the contribution of such data to
the quake-specific computation of egression power. First, we note that in the
magnetized regions the atmospheric noise manifests itself as spurious velocity
fluctuations leading to a substantial increase of the observed acoustic power.
Such an increase in GONG data is assumed to be induced by the atmospheric
seeing effects. Second, it is known that flares and associated with them solar
quake sources are usually located over or near a sunspot. On the other hand,
quake signatures such as ripples and time-distance ridges are normally seen in
the surrounding non-magnetised region. This is, at least in part, due to the
complex and less understood propagation of magneto-acoustic waves generated by
the quake in the sunspot itself. Also, for our estimates of the egression
power we use Green’s function based on a non-magnetic model of the solar
interior as it is intrinsic to the acoustic holography. In the view of
equations (3-4) it is then reasonable to minimize the contribution of the
velocity data taken over magnetic regions (MRs) such as sunspot.
For smaller sunspots this can be achieved by the choice of pupil, ensuring
that the smallest radius is always selected outside of MR. When a sunspot is
large, other methods will need to be considered such as weighting of the
sunspot data in the velocity measurements. One possible option is to fully
neglect such data, i.e. using zero as weights for sunspot velocities. This, of
course, introduces artificial inhomogeneities in the computed egression power,
but the qualitative strength of the quake source can still be evaluated by
comparison with the egression power of surrounding plasma. In our experience,
however, the best results are achieved by weighting all measurements by the
inverse averaged acoustic power computed for the filtered velocity series.
This is equivalent to normalizing the acoustic power of the filtered datacube,
similar to the approach of Rajaguru et al. (2006) developed for time-distance
helioseismology and phase-speed filtering. All of the GONG data used in the
following sections are processed as described unless otherwise stated.
Flare | MDI pupil size | GONG pupil size
---|---|---
September 9, 2001 | 15-60 Mm | 25-70 Mm
October 28, 2003 | 15-45 Mm | 25-95 Mm
October 29, 2003 | 15-45 Mm | 20-55 Mm
Table 1: Holography pupil sizes for each dataset for the selected flares.
## 3 Comparison of the GONG and MDI results
### 3.1 Comparison of acoustic holography results
For illustration purposes, the results of calculations of the total egression
power in 5-7 mHz range estimated from the MDI and GONG datacubes corresponding
to observations of the September 9, 2001 flare, are presented in the bottom
row of Figure 1. One can see that even after the corrections, the acoustic
source suppression over the sunspot region is considerably weaker in the map
computed from GONG data. This is generally the case for other observations we
have considered and is believed to occur because of the lower resolution and
atmospheric noise contamination in the ground-based network’s data. For these
reasons, in order to reduce such contamination we consider larger pupil sizes
when working with GONG data as shown in Table 1. By choosing the larger lower
limit on pupil dimensions we ensure the minimised contribution of the
measurements taken over MR for egression power computation at the points near
and around sunspots.
#### 3.1.1 Holography: September 9, 2001
Egression power snapshots computed for the September 9 flare are presented in
Figure 2, with the MDI data plotted in the left column and the GONG data on
the right. Velocity images averaged over the series duration for both
instruments (located at the top of the Figure 2) demonstrate clearly the
differences in the original datasets, which are due to a further loss of
resolution due to the atmospheric effects as described by Lindsey & Donea
(2008). Our MDI egression measurements for this flare essentially duplicate
the results of Donea et al. (2006a). Comparing these with the obtained GONG
snapshots (Figure 2), it is clear that even after the corrections, though many
similarities are present, there is a significant variation between the two
sets of images.
The most obvious difference is the apparent absence of the region with low
acoustic emission around the sunspot in the GONG produced data. Such an
absence is clearly related to a much weaker signature of this phenomena in the
GONG egression power seen in Figure 1. This can be explained by the spurious
atmospheric noise affecting GONG measurements over the regions with strong
magnetic field. Nonetheless, the quake’s signature is clearly present in the
GONG measurements, with the locations of acoustic kernels agreeing very well
for the two instruments. We note, however, the difference in acoustic kernel
shape. This is, most likely, due to the reduced spatial resolution of GONG
data suppressing higher-$l$ contribution to the egression. The possibility of
atmospheric noise contamination is discussed later in Section 4.
#### 3.1.2 Holography: October 28, 2003
Egression power snapshots computed for the October 28 flare are presented in
Figure 3 with the reference GONG intensity image located in the top left
corner, followed to the right by the MDI egression power snapshot taken at
11:07 UT with the arrows pointing to the detected acoustic kernels. This image
essentially duplicates the panel c) of Figure 3 in Donea & Lindsey (2005). For
a reference, the GONG snapshot for the same time is presented in the bottom
row with and without arrows. Here, one can see the acoustic signatures at the
same locations as in the MDI data. Again, we note the region with weaker lower
acoustic emission as seen by GONG, which affects the visible contrast of the
quake kernels relative to their surroundings. It is also clear that while the
locations are the same, the shape and strength of each of the four kernels
varies from one instrument to another. For example, source 1 (see Figure 3)
appears to be more prominent and extended in GONG measurements compared with
MDI. Given the reservations about ground-based data outlined above with the
fact that our correction procedure rather artificially modifies the
oscillation amplitudes in GONG data, it is clear that MDI measurements are
closer to the true picture of the event. Nonetheless, we reiterate that quake
signatures are clearly visible in GONG measurements in the same locations as
those detected by MDI.
#### 3.1.3 Holography: October 29, 2003
Egression power snapshots computed for the October 29 flare are presented in
Figure 4 with the reference GONG intensity image located in the top left
corner, followed to the right by the MDI egression power snapshot at 20:43 UT
with the arrows pointing to the detected acoustic kernels. This image is
essentially equivalent to the middle panel of Figure 6 in Donea & Lindsey
(2005). The corresponding GONG egression snapshot with the quake signature is
plotted in the lower row on the left clearly present at approximately the same
location as in the MDI plot. As an example, the egression power computed from
the GONG velocity observation with the masked sunspot area is plotted in the
bottom row to the right.
### 3.2 Time-Distance diagrams
#### 3.2.1 9 September 2001 flare
The time-distance diagrams extracted from the MDI and GONG data are presented
in Figure 5. The ridge representing the quake in the MDI image is relatively
weak but can be clearly seen. As far as we know, this is the first time-
distance ridge for a solar quake associated with an M-class flare has been
found. By comparing the MDI image with GONG time-distance diagram one can also
detect in GONG a very similar disturbance located at the same part of the
image. Although weaker and less defined than in MDI, nevertheless, the ridge
is definitely present. Once again the relative weakness of the ridge can be
explained by the fact that it is obscured by a significant noise contribution.
As additional re-assurance, there is a near perfect coincidence between the
MDI and GONG time-distance source locations. Also, as can be seen from Figure
2 where the location of time distance source is marked as plus sign on
selected GONG plots, there is a good agreement between the egression acoustic
kernels and time-distance source locations.
#### 3.2.2 Halloween flares
We were not able to find any distinguishable time-distance ridges for the
October 29 flare in either MDI or GONG data, similar to the previous attempts
(e.g. Zharkova & Zharkov 2007). The other GONG dataset for the October 28
flare, has a gap of about ten minutes between 11:30 and 11:40 UT. However, we
have attempted to build the time-distance diagram and the time-distance plot
obtained from the interrupted GONG data. This is presented in Figure 6. It
shows (at least a part of) a ridge, with the location corresponding to Source
1 in Zharkova & Zharkov (2007), reproduced in the top row of Figure 6. Once
again, the location of the time distance ridge coincides with that of MDI and
agrees with the egression measurements presented in section 3.1.2.
## 4 General discussion and conclusions
In this study we have compared egression power maps and time-distance diagrams
derived from GONG and MDI data. SOHO MDI and GONG velocity datasets were used
for three flares: M-class September 9, 2001 (Figures 1-2, 5), X-class October
28, 2003 (Figures 3 and 6), X-class October 29, 2003 (Figure 4).
Reassuringly, the egression power map snapshots show seismic signatures common
to both instruments for all flares. These signatures display an excellent
agreement between the two instruments in terms of their time and location. We
note, however, that even after the pre-processing, as outlined in Section 2.3,
GONG egression measurements remain relatively noisy. This leads to important
differences from the MDI produced egression maps, which are only partially
compensated by increasing the pupil sizes when working with GONG data. One
such difference is the apparent variance in shape and strength of the detected
acoustic kernels as seen by these two instruments. Another is the relative
weakness of the suppression of acoustic sources below the sunspot. Since solar
quakes are often located in the sunspot, this means that identifying such
seismic signatures in GONG egression measurements is a harder task due to its
lower signal to noise ratio over the sunspot region.
One such method of verification is the computation of the time-distance
diagram. As we have demonstrated for September 9, 2001 and October 28, 2003
flares, such diagrams computed from GONG velocity data can present the
additional evidence of the quake. Figures 5 and 6 clearly show that, in spite
of being less sensitive, the GONG data can respond to the time-distance
analysis producing noticeable ridges similar to those observed from the
higher-quality MDI data. Results of the comparison with MDI time-distance
measurements have again revealed an excellent spatial agreement between the
two instruments in terms of the time-distance source location. Additionally,
the fact that the locations of the sources observed with the GONG time-
distance diagrams coincide with the acoustic kernels deduced from the GONG
egression snapshots confirms that with these different techniques one observes
the same events - seismic signatures induced by solar flares. Therefore, we
conclude that the GONG data can respond to time-distance analysis. Obviously,
due to the characteristic noise, not every quake can be expected to produce
the time-distance ridge in GONG diagrams, but if a ridge is seen in the GONG
ground-based data, one can expect that it will also be observed by using the
higher-quality satellite MDI or HMI data.
We believe, the results of this study show that quake detection based on
helioseismic reduction of GONG observations is possible. However, as the data
are subject to atmospheric smearing and other related instrumental effects,
GONG observations clearly have less intrinsic sensitivity than the space-borne
observations. A useful prospective object for further study might be the
quantitative comparison of intrinsic sensitivities of ground- and space-based
helioseismic observations under various seeing conditions. Nonetheless these
results should allow us to add to the list of known sun-quakes by
investigating the known flares in the Solar Cycle 23 using GONG data when MDI
observations were not available. This will provide further information about
the physics of seismic processes associated with solar flares.
## 5 Acknowledgments
The authors would like to thank Dr Charlie Lindsey for many useful discussions
providing the stimulus to complete this work.
Figure 1: September 9, 2001 M-class flare: comparison of unprocessed GONG and
MDI integrated high frequency (5-7 mHz range) acoustic power maps (top row)
and 5-7 mHz range egression power maps (bottom row) computed using MDI (pupil
15 - 60 Mm) and GONG (pupil 25 - 70 Mm) with correction for variable
atmospheric smearing applied to GONG data.
Figure 2: September 9, 2001 M-class flare: MDI data is in the left column,
GONG is on the right. averaged MDI/GONG velocity image (top), followed by
egression power snapshots. MDI egression plots reproduce the results in Donea
et al. (2006a). Quake location is indicated by an arrow. The plus sign in two
upper right frames indicates the source position assumed for the diagnostics
specified by equation (1) presented in Figure 5.
Figure 3: October 28, 2003 X-class flare: Averaged GONG intensity image is
top left, followed by MDI egression snapshot to the right with arrows
indicating sources in Donea & Lindsey (2005) with #1, #3 and #4 corresponding
respectively to time-distance sources #2, #1 and #3 in Zharkova & Zharkov
(2007). At the bottom row: GONG egression power snapshot at 11:07 (left), and
on the right is the same image with arrows indicating sources in Donea &
Lindsey (2005) as above.
Figure 4: October 29, 2003 X-class flare: (a) is GONG intensity image; (b) MDI
egression power snapshot at 20:43; (c-d) GONG egression power snapshots around
the quake time; (e) egression power computed from GONG velocity data using
sunspot mask averaged over one hour. Location of the quake is indicated by an
arrow; (f) egression power snapshot computed from masked GONG data used in (e)
taken around the quake time.
Figure 5: September 9, 2001 flare: time-distance diagram computed from MDI
velocity data (top row), and GONG dopplergram observations (bottom).
Figure 6: October 28, 2003 flare: (Top row):time-distance diagram computed
from MDI data. The plots are reproduced from Zharkova & Zharkov (2007).
(Bottom row): time-distance diagram computed from GONG data. 0 along the
$y$-axis corresponds to 11:00 UT.
## References
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* Harvey et al. (1988) Harvey, J. W., Abdel-Gawad, K., Ball, W., et al. 1988, in ESA Special Publication, Vol. 286, Seismology of the Sun and Sun-Like Stars, ed. E. J. Rolfe, 203–208
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|
arxiv-papers
| 2011-07-09T16:24:05 |
2024-09-04T02:49:20.356002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Zharkov, V.V. Zharkova, S.A. Matthews",
"submitter": "Sergei Zharkov Dr",
"url": "https://arxiv.org/abs/1107.1798"
}
|
1107.1804
|
# Effect of Dust on Lyman-alpha Photon Transfer in Optically Thick Halo
Yang Yang11affiliation: Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA , Ishani Roy22affiliation: Computing Laboratory,
University of Oxford, Oxford, OX1 3QD, United Kingdom , Chi-Wang
Shu11affiliation: Division of Applied Mathematics, Brown University,
Providence, RI 02912, USA and Li-Zhi Fang33affiliation: Department of Physics,
University of Arizona, Tucson, AZ 85721, USA
###### Abstract
We investigate the effects of dust on Ly$\alpha$ photons emergent from an
optically thick medium by solving the integro-differential equation of the
radiative transfer of resonant photons. To solve the differential equations
numerically we use the Weighted Essentially Non-oscillatory method (WENO).
Although the effects of dust on radiative transfer is well known, the resonant
scattering of Ly$\alpha$ photons makes the problem non-trivial. For instance,
if the medium has the optical depth of dust absorption and scattering to be
$\tau_{a}\gg 1$, $\tau\gg 1$, and $\tau\gg\tau_{a}$, the effective absorption
optical depth in a random walk scenario would be equal to
$\sqrt{\tau_{a}(\tau_{a}+\tau)}$. We show, however, that for a resonant
scattering at frequency $\nu_{0}$, the effective absorption optical depth
would be even larger than $\tau(\nu_{0})$. If the cross section of dust
scattering and absorption is frequency-independent, the double-peaked
structure of the frequency profile given by the resonant scattering is
basically dust-independent. That is, dust causes neither narrowing nor
widening of the width of the double peaked profile. One more result is that
the time scales of the Ly$\alpha$ photon transfer in the optically thick halo
are also basically independent of the dust scattering, even when the
scattering is anisotropic. This is because those time scales are mainly
determined by the transfer in the frequency space, while dust scattering,
either isotropic or anisotropic, does not affect the behavior of the transfer
in the frequency space when the cross section of scattering is wavelength-
independent. This result does not support the speculation that dust will lead
to the smoothing of the brightness distribution of Ly$\alpha$ photon source
with optical thick halo.
cosmology: theory - intergalactic medium - radiation transfer - scattering
## 1 Introduction
Ly$\alpha$ photons have been widely applied to study the physics of luminous
objects at various epochs of the universe, such as Ly$\alpha$ emitters,
Ly$\alpha$ blob, damped Ly$\alpha$ system, Ly$\alpha$ forest, fluorescent
Ly$\alpha$ emission, star-forming galaxies, quasars at high redshifts as well
as optical afterglow of gamma ray bursts (Haiman et al. 2000; Fardal et al.
2001; Dijkstra & Loeb. 2009; Latif et al. 2011). The resonant scattering of
Ly$\alpha$ photons with neutral hydrogen atoms has a profound effect on the
time, space and frequency dependencies of Ly$\alpha$ photons transfer in an
optically thick medium. Ly$\alpha$ photons emergent from an optically thick
medium would carry rich information of photon sources and halo surrounding the
source of the Ly$\alpha$ photon. The profiles of the emission and absorption
of the Ly$\alpha$ radiation are powerful tools to constrain the mass density,
velocity, temperature and the fraction of neutral hydrogen of the optically
thick medium. Radiation transfer of Ly$\alpha$ photons in an optically thick
medium is fundamentally important.
The radiative transfer of Ly$\alpha$ photons in a medium consisting of neutral
hydrogen atoms has been extensively studied either analytically or
numerically. Yet, there have been relatively few results which are directly
based on the solutions of the integro-differential equation of the resonant
radiative transfer. Besides the Field solution (Field 1959, Rybicki &
Dell’Antonio 1994), analytical solutions with and without dust mostly are
based on the Fokker-Planck (P-F) approximation (Harrington 1973, Neufeld 1990,
Dijkstra et al. 2006). The P-F equation might miss the detailed balance
relationship of resonant scattering (Rybicki 2006), and therefore, the
analytical solutions cannot describe the formation and evolution of the
Wouthuysen-Field (W-F) local thermalization of the Ly$\alpha$ photon frequency
distribution (Wouthuysen 1952, Field 1958), which is important for the
emission and absorption of the hydrogen 21 cm line (e.g. Fang 2009). The
features of the Ly$\alpha$ photon transfer related to the W-F local
thermalization are also missed. An early effort (Adams et al. 1971) trying to
directly solve the integro-differential equation of the resonant radiative
transfer with numerical method. It still is, however, of a time-independent
approximation.
Recently, a state-of-the-art numerical method has been introduced to solve the
integro-differential equation of the radiative transfer with resonant
scattering (Qiu et al. 2006, 2007, 2008, Roy et al. 2009a). The solver is
based on the weighted essentially non-oscillatory (WENO) scheme (Jiang & Shu
1996). With the WENO solver, many physical features of the transfer of
Ly$\alpha$ photons in an optically thick medium (Roy et al. 2009b, 2009c,
2010), which are missed in the Fokker-Planck equation approximations, have
been revealed. For instance, the WENO solution shows that the time scale of
the formation of the W-F local thermal equilibrium actually is only about a
few hundred times of the resonant scattering. It also shows that the double
peaked frequency profile of the Ly$\alpha$ photon emergent from an optically
thick medium does not follow the time-independent solutions of the P-F
equation. These results directly indicate the needs of re-visiting problems
which have been studied only via the F-P time-independent approximation.
We will investigate, in this paper, the effects of the dust on the Ly$\alpha$
photons transfer in an optically thick medium. Dust can be produced at epochs
of low and moderate redshifts, and even at redshift as high as 6 (Stratta et
al. 2007). Absorption and scattering of dust have been used to explain the
observations on Ly$\alpha$ emission and absorption (Hummer & Kunasz 1980),
such as the escaping fraction of Ly$\alpha$ photons (Hayes et al. 2010, 2011,
Blanc et al. 2010); the redshift-dependence of the ratio between Ly$\alpha$
emitters and Lyman Break galaxies (Verhamme et al. 2008); and the “evolution”
of the double-peaked profile (Laursen et al. 2009).
Nevertheless, it is still unclear whether the time scale of photon escaping
from optically thick halo will be increasing (or decreasing) when the halo is
dusty. It is also unclear whether the effects of dust absorption can be
estimated by the random walk picture (Hansen & Oh 2006). As for the dust
effect on the double-peaked profile, the current results given by different
studies seem to be contradictory: some claims that the dust absorption leads
to the narrowing of the double-peaked profile (Lauresen et al 2009), while
others result that the width between the two peaks apparently should be
increasing due to the dust absorption (Verhamme et al. 2006). We will focus on
these basic problems, and examine them with the solution of the integro-
differential equation of radiative transfer.
This paper is organized in the following way: section 2 presents the theory of
the Ly$\alpha$ photon transfer in an optically thick medium with dust. The
equations of the intensity and flux of resonant photons in a dusty medium are
given. We will study three models of the interaction between dust and photons:
(1) dust causes only scattering with photons; (2) dust causes both scattering
and absorption; and (3) dust causes only absorption of photons. Section 3
gives the solutions of Ly$\alpha$ photons escaping from an optically thick
spherical halos with dust. The dusty effect on the double-peaked profile will
be studied in Section 4. The discussion and conclusion are given in Section 5.
Some mathematical derivations of the equations and numerical implementation
details are given in the Appendix.
## 2 Basic theory
### 2.1 Radiative transfer equation of dusty halo
We study the transfer of Ly$\alpha$ photons in a spherical halo with radius
$R$ around an optical source. The halo is assumed to consist of uniformly
distributed HI gas and dust. The optical depth of HI scattering over a light
path $dl$ is $d\tau=\sigma(\nu)n_{\rm HI}dl$, where $n_{\rm HI}$ is the number
density of HI, and $\sigma(\nu)$ is the cross section of the resonant
scattering of Ly$\alpha$ photons by neutral hydrogen, which is given by
$\sigma(x)=\sigma_{0}\phi(x,a)=\sigma_{0}\frac{a}{\pi^{3/2}}\int^{\infty}_{-\infty}dy\frac{e^{-y^{2}}}{(x-y)^{2}+a^{2}}$
(1)
where $\phi(x,a)$ is the normalized Voigt profile (Hummer 1965). As usual, the
photon frequency $\nu$ in eq.(1) is described by the dimensionless frequency
$x\equiv(\nu-\nu_{0})/\Delta\nu_{D}$, with $\nu_{0}=2.46\times 10^{15}$ s-1
being the resonant frequency, $\Delta\nu_{D}=\nu_{0}(v_{T}/c)=1.06\times
10^{11}(T/10^{4})^{1/2}$ Hz the Doppler broadening, $v_{T}=\sqrt{2k_{B}T/m}$
the thermal velocity, and $T$ the gas temperature of the halo.
$\sigma_{0}/\pi^{1/2}$ is the cross section of scattering at the the resonant
frequency $\nu_{0}$. The parameter $a$ in eq.(1) is the ratio of the natural
to the Doppler broadening. For the Ly$\alpha$ line, $a=4.7\times
10^{-4}(T/10^{4})^{-1/2}$. The optical depth of Ly$\alpha$ photons with
respect to HI resonant scattering is $\tau_{s}(x)=n_{\rm
HI}R\sigma(x)=\tau_{0}\phi(x,a)$, where $\tau_{0}=n_{\rm HI}\sigma_{0}R$.
If the absorption and scattering of dust are described by effective cross-
section per hydrogen atom $\sigma_{d}(x)$, the total optical depth is given by
$\tau(x)=\tau_{0}\phi(x,a)+\tau_{d}(x)$ (2)
where the dust optical depth $\tau_{d}(x)=n_{\rm HI}\sigma_{d}(x)R$. This is
equal to assume that dust is uniformly distributed in IGM. The effects of
inhomogeneous density distributions of dust (Neufeld 1991; Haiman & Spaans
1999) will not be studied in this paper.
The radiative transfer equation of Ly$\alpha$ photons in a spherical halo with
dust is given by
$\displaystyle{\partial I\over\partial\eta}+\mu\frac{\partial I}{\partial
r}+\frac{(1-\mu^{2})}{r}\frac{\partial I}{\partial\mu}-\gamma\frac{\partial
I}{\partial x}=$
$\displaystyle-\phi(x;a)I+\int\mathcal{R}(x,x^{\prime};a)I(\eta,r,x^{\prime},\mu^{\prime})dx^{\prime}d\mu^{\prime}/2$
$\displaystyle-\kappa(x)I+A\kappa(x)\int\mathcal{R}^{d}(x,x^{\prime};\mu,\mu^{\prime};a)I(\eta,r,x^{\prime},\mu^{\prime})dx^{\prime}d\mu^{\prime}+S$
where $I(t,r_{p},x,\mu)$ is the specific intensity, which is a function of
time $t$, radial coordinate $r_{p}$, frequency $x$ and the direction angle,
$\mu=\cos\theta$, with respect to the radial vector ${\bf r}$.
In eq.(3), we use the dimensionless time $\eta$ defined as $\eta=cn_{\rm
HI}\sigma_{0}t$ and the dimensionless radial coordinate $r$ defined as
$r=n_{\rm HI}\sigma_{0}r_{p}$. That is, $\eta$ and $r$ are, respectively, in
the units of mean free flight-time and mean free path of photon $\nu_{0}$ with
respect to the resonant scattering without dust scattering and absorption.
Without resonant scattering, a signal propagates in the radial direction with
the speed of light, the orbit of the signal is then $r=\eta+{\rm const}$. With
dimensionless variable, the size of the halo $R$ is equal to $\tau_{0}$.
The re-distribution function $\mathcal{R}(x,x^{\prime};a)$ gives the
probability of a photon absorbed at the frequency $x^{\prime}$, and re-emitted
at the frequency $x$. It depends on the details of the scattering (Henyey &
Greestein 1941; Hummer 1962; Hummer 1969). If we consider coherent scattering
without recoil, the re-distribution function with the Voigt profile can be
written as,
$\displaystyle\mathcal{R}(x,x^{\prime};a)=$
$\displaystyle\frac{1}{\pi^{3/2}}\int^{\infty}_{|x-x^{\prime}|/2}e^{-u^{2}}\left[\tan^{-1}\left(\frac{x_{\min}+u}{a}\right)-\tan^{-1}\left(\frac{x_{\max}-u}{a}\right)\right]du$
where $x_{\min}=\min(x,x^{\prime})$ and $x_{\max}=\max(x,x^{\prime})$. In the
case of $a=0$, i.e. considering only the Doppler broadening, the re-
distribution function is
$\mathcal{R}(x,x^{\prime})=\frac{1}{2}{\rm erfc}[{\rm
max}(|x|,|x^{\prime}|)].$ (5)
The re-distribution function of equation (5) is normalized as
$\int_{-\infty}^{\infty}\mathcal{R}(x,x^{\prime})dx^{\prime}=\phi(x,0)=\pi^{-1/2}e^{-x^{2}}$.
With this normalization, the total number of photons is conserved in the
evolution described by equation (3). That is, the destruction processes of
Ly$\alpha$ photons, such as the two-photon process (Spitzer & Greenstein 1951;
Osterbrock 1962), are ignored in equation (3). The recoil of atoms is also not
considered in equation (4) or (5). The effect of recoil actually is under
control (Roy et al. 2009c, 2010). We will address it in next section.
The absorption and scattering of dust are described by the term $\kappa(x)I$
of eq.(3), where $\kappa(x)=\sigma_{d}/\sigma_{0}$, which is of the order of
$10^{-8}(T/10^{4})^{1/2}$ (Draine & Lee 1984; Draine 2003). The term with $A$
of eq.(3) describes albedo, i.e. $A\equiv\sigma_{s}/\sigma_{d}$, where
$\sigma_{s}$ is the cross section of dust scattering. Generally, $A$ lies
approximately between 0.3 and 0.4 (Pei 1992; Weingartner & Draine 2001).
Since dust generally is much heavier than a single atoms, the recoil of dust
particles can be neglected when colliding with a photon. Under this “heavy
dust” approximation, photons do not change their frequency during the
collision with dust. The redistribution function of dust $\mathcal{R}^{d}$ is
independent of $x$ and $x^{\prime}$, and is simply given by a phase function
as
$\mathcal{R}^{d}(\mu,\mu^{\prime})=\frac{1}{4\pi}\int^{2\pi}_{0}d\phi^{\prime}\frac{1-g^{2}}{(1+g^{2}-2g\bar{\mu})^{3/2}}=\sum_{l=0}^{\infty}\frac{(2l+1)}{2}g^{l}P_{l}(\mu)P_{l}(\mu^{\prime}),$
(6)
where $\bar{\mu}=\mu\mu^{\prime}+\sqrt{(1-\mu^{2})(1-\mu^{\prime
2})}\rm{cos}\phi^{\prime}$ and $P_{l}$ is the Legendre function. The factor
$g$ in eq.(6) is the asymmetry parameter. For isotropic scattering, $g=0$. The
cases of $g=+1$ and -1 correspond to complete forward and backward scattering,
respectively. Generally, the factor $g$ is a function of the wavelength. For
the Ly$\alpha$ photon, we will take $g=0.73$ for realistic dust scattering (Li
& Draine 2001). The integral of eq.(6) is performed in Appendix A.
In eq. (3), the term with the parameter $\gamma$ is due to the expansion of
the universe. If $n_{\rm H}$ is equal to the mean of the number density of
cosmic hydrogen, we have $\gamma=\tau_{GP}^{-1}$, and $\tau_{GP}$ is the Gunn-
Peterson optical depth. Since the Gunn-Peterson optical depth is of the order
of $10^{6}$ at high redshift (e.g. Roy et al. 2009c), the parameter $\gamma$
is of the order of $10^{-5}-10^{-6}$. Therefore, if the optical depth of halos
is equal to or less than 106, the term with $\gamma$ of eq.(3) can be ignored.
In eq.(3) we neglect the effect of collision transition from $H(2p)$ state to
$H(2s)$ state, which can significantly affect on the escape of Ly$\alpha$
photons when HI column density is higher than $10^{21}$ cm-2 and dust
absorption is very small (Neufeld, 1990). This generally is out of the
parameter range used below. We are also not considering the effects of bulk
motion of the medium of halos (e.g. Spaans & Silk 2006, Xu & Wu, 2010).
### 2.2 Eddington approximation
Eq.(6) indicates that the transfer equation (3) can be solved with the
Legendre expansion $I(\eta,r,x,\mu)=\sum_{l}I_{l}(\eta,r,x)P_{l}(\mu)$. If we
take only the first two terms, $l=0$ and 1, it is the Eddington approximation
as
$I(\eta,r,x,\mu)\simeq J(\eta,r,x)+3\mu F(\eta,r,x)$ (7)
where
$J(\eta,r,x)=\frac{1}{2}\int_{-1}^{+1}I(\eta,r,x,\mu)d\mu,\hskip
14.22636ptF(\eta,r,x)=\frac{1}{2}\int_{-1}^{+1}\mu I(\eta,r,x,\mu)d\mu.$ (8)
They are, respectively, the angularly averaged specific intensity and flux.
Defining $j=r^{2}J$ and $f=r^{2}F$, Eq.(3) yields the equations of $j$ and $f$
as
$\displaystyle{\partial j\over\partial\eta}+\frac{\partial f}{\partial r}$
$\displaystyle=$ $\displaystyle-(1-A)\kappa
j-\phi(x;a)j+\int\mathcal{R}(x,x^{\prime};a)jdx^{\prime}+\gamma\frac{\partial
j}{\partial x}+r^{2}S,$ (9) $\displaystyle\frac{\partial
f}{\partial\eta}+\frac{1}{3}\frac{\partial j}{\partial
r}-\frac{2}{3}\frac{j}{r}$ $\displaystyle=$ $\displaystyle-(1-Ag)\kappa
f+\gamma\frac{\partial f}{\partial x}-\phi(x;a)f.$ (10)
The mean intensity $j(\eta,r,x)$ describes the $x$ photons trapped in the
position $r$ at time $\eta$ by the resonant scattering, while the flux
$f(\eta,r,x)$ describes the photons in transit.
The source term $S$ in the equations (3) and (9) can be described by a
boundary condition of $j$ and $f$ at $r=r_{0}$. We can take $r_{0}=0$. Thus,
the boundary condition is
$j(\eta,0,x)=0,\hskip 28.45274ptf(\eta,0,x)=S_{0}\phi_{s}(x),$ (11)
where $S_{0}$, and $\phi_{s}(x)$ are, respectively, the intensity and
normalized frequency profile of the sources. Since equation (3) is linear, the
solutions of $j(x)$ and $f(x)$ for given $S_{0}=S$ are equal to $Sj_{1}(x)$
and $Sf_{1}(x)$, where $j_{1}(x)$ and $f_{1}(x)$ are the solutions of
$S_{0}=1$. On the other hand, the equation (3) is not linear with respect to
the function $\phi_{s}(x)$. The solution $f(x)$ for a given $\phi_{s}(x)$ is
not equal to $\phi_{s}(x)f_{1}(x)$, where $f_{1}(x)$ is the solution of
$\phi_{s}(x)=1$.
In the range outside the halo, $r>R$, no photons propagate in the direction
$\mu<0$. The boundary condition at $r=R$ given by $\int_{0}^{-1}\mu
J(\eta,R,x,\mu)d\mu=0$ is then (Unno 1955)
$j(\eta,R,x)=2f(\eta,R,x).$ (12)
There is no photon in the field before $t=0$. Therefore, the initial condition
is
$j(0,r,x)=f(0,r,x)=0.$ (13)
We will solve equations (9) and (10) with boundary and initial conditions
eqs.(11) - (13) by using the WENO solver (Roy et al. 2009a, b, c, 2010). Some
details of this method is given in Appendix B.
### 2.3 Dust models
We consider three models of the dust as follows:
I. pure scattering, $A=1,\ g=0.73$: dust causes only anisotropic scattering,
but no absorption;
II. scattering and absorption. $A=0.32,\ g=0.73$: dust causes both absorption
and anisotropic scattering.
III. pure absorption. $A=0$: dust causes only absorption, but no scattering;
Models I and III do not occur in reality. They are, however, helpful to reveal
the effects of pure scattering and absorption on the radiative transfer.
Since $\kappa(x)$ is on the order of $10^{-8}$, its effect will be significant
only for halos with halos with optical depth $\tau_{0}\geq 10^{6}$, and
ignorable for $\tau_{0}\leq 10^{5}$. To illustrate the dust effect, we use
halos of $R=\tau_{0}\leq 10^{4}$, and take larger $\kappa$ to be $\simeq
10^{-4}-10^{-2}$. We also assume that $\kappa$ is frequency-independent. We
consider below only the case of grey dust, i.e. $\kappa$ is independent of
frequency $x$. This certainly is not realistic dust. Yet, the frequency range
given in solution below mostly are in the range $|x|<4$. Therefore, the
approximation of grey dust would be proper if cross section of dust is not
significantly frequency dependent in the range $|x|<4$.
### 2.4 Numerical example: Wouthuysen-Field thermalization
As the first example of numerical solutions, we show the Wouthuysen-Field
(W-F) effect, which requires that the distribution of Ly$\alpha$ photons in
the frequency space should be thermalized near the resonant frequency
$\nu_{0}$. The W-F effect illustrates the difference between the analytical
solutions of the Fokker-Planck approximation and that of eqs. (9) and (10).
The former can not show the local thermalization (Neufeld 1990), while the
latter can (Roy et al. 2009b). All problems related to the W-F local thermal
equilibrium should be studied with the integro-differential equation (3).
Figure 1: The mean intensity $j(\eta,r,x)$ at $\eta=500$ and $r=100$ for dust
models I (left panel), II (middle panel) and III (right panel). The source is
$S_{0}=1$ and $\phi_{s}(x)=(1/\sqrt{\pi})e^{-x^{2}}$. The parameter
$a=10^{-3}$. In each panel, $\kappa$ is taken to be 0, 10-4, 10-3 and 10-2.
Figure 1 presents a solution of mean intensity $j(\eta,r,x)$ at time radial
$\eta=500$ coordinate $r=10^{2}$ for halo with size $R\gg r=10^{2}$. The three
panels correspond to dust models I (left panel), II (middle panel) and III
(right panel). The source is taken to have a Gaussian profile
$\phi_{s}(x)=(1/\sqrt{\pi})e^{-x^{2}}$ and unit intensity $S_{0}=1$. The
solutions of Figure 1 actually are independent of $R$, if $R\gg 10^{2}$. The
intensity of $j$ is decreasing from left to right in Figure 1, because the
absorption is increasing with the models from I to III.
A remarkable feature shown in Figure 1 is that all $j(\eta,r,x)$ have a flat
plateau in the range $|x|\leq 2$. This gives the frequency range of the W-F
local thermalization (Roy et al, 2009b, c). The range of the flat plateau
$|x|\leq 2$ is almost dust-independent, either for model I or for models II
and III. This is expected, as neither the absorption nor scattering given by
the $\kappa$ term of eq.(3) changes the frequency distribution of photons. The
redistribution function (6) also does not change the frequency distribution of
photons. This point can also be seen from eqs.(9) and (10), in which the
$\kappa$ terms are frequency-independent. The evolution of the frequency
distribution of photons is due only to the resonant scattering.
Since thermalization will erase all frequency features within the range
$|x|\leq 2$, the double-peaked structure does not retain information of the
photon frequency distribution within $|x|<2$ at the source. That is, the
results in Figure 1 will hold for any source $S_{0}\phi_{s}(x)$ with arbitrary
$\phi_{s}(x)$ which is non-zero within $|x|<2$ (Roy et al. 2009b, c). This
property can also be used as a test of the simulation code. It requires that
simulation results of flat plateau should be hold, regardless of the source to
be monochromatic or with finite width around $\nu_{0}$.
## 3 Dust effects on photon escape
### 3.1 Model I: scattering of dust
To study the effects of dust scattering on the Ly$\alpha$ photon escape, we
show in Figure 2 the flux $f(\eta,r,x)$ of Ly$\alpha$ photons emergent from
halos at the boundary $r=R=10^{2}$ for Model I. The three panels of Figure 2
correspond to $\kappa=10^{-4}$, 10-3, and 10-2 from left to right,
respectively. The source starts to emit photons at $\eta=0$ with a stable
luminosity $S_{0}=1$, and with a Gaussian profile
$\phi_{s}(x)=(1/\sqrt{\pi})e^{-x^{2}}$.
Figure 2: Flux $f(\eta,r,x)$ of Ly$\alpha$ photons emergent from halos at the
boundary $R=10^{2}$, and for the dust model I $A=1,\ g=0.73$. The parameter
$\kappa$ is taken to be 10-4 (left), 10-3(middle) and 10-2 (right). The source
is $S_{0}=1$ and $\phi_{s}(x)=(1/\sqrt{\pi})e^{-x^{2}}$. The parameter
$a=10^{-3}$.
Figure 2 clearly shows that the time-evolution of $f(\eta,r,x)$ is
$\kappa$-independent. Although the cross section of dust scattering increases
about 100 times from $\kappa=10^{-4}$ to $\kappa=10^{-2}$, the curves of the
left and right panels in Figure 2 actually are almost identical.
According to the scenario of “single longest excursion”, photon escape is not
a process of Brownian random walk in the spatial space, but a transfer in the
frequency space (Osterbrock 1962; Avery & House 1968; Adams, 1972, 1975;
Harrington 1973; Bonilha et al. 1979). Photon will escape, once its frequency
is transferred from $|x|<2$ to $|x|>2$, on which the medium is transparent. On
the other hand, dust scattering given by the redistribution function eq.(6)
does not change photon frequency. Dust scattering has no effect on the
transfer in the frequency space.
Moreover, photons with frequency $|x|<2$ are quickly thermalized after a few
hundred resonant scattering. In the local thermal equilibrium state, the
angular distribution of photons is isotropic. Thus, even if the dust
scattering is anisotropic $g\neq 0$ with respect to the direction of the
incident particle, the angular distribution will keep isotropic undergoing a
$g\neq 0$ scattering. Hence, dust scattering also has no effect on the angular
distribution.
### 3.2 Model III: absorption of dust
Similar to Figure 2, we present in Figure 3 the flux of Model III, i.e. dust
causes only absorption without scattering. All other parameters of Figure 3
are the same as in Figure 2. In the left panel of Figure 3, the curves at the
time $\eta=2000$ and $3000$ are the same. It means the flux $f(\eta,R,x)$ at
the boundary $R$ is already stable, or saturated at the time $\eta\geq 2000$.
The small difference between the curves of $\eta=1000$ and $\eta\geq 2000$ of
the left panel indicates that the flux is still not yet completely saturated
at the time $\eta=1000$. However, comparing the middle and right panels of
Figure 3, we see that for $\kappa=10^{-3}$, the flux has already saturated at
$\eta=1600$, while it has saturated at $\eta=800$ for $\kappa=10^{-2}$. That
is, the stronger the dust absorption, the shorter the saturation time scale.
The time scales of escape or saturation do not increase by dust absorption,
and even decrease with respect to the medium without dust. Stronger absorption
leads to shorter time scale of saturation.
Figure 3: Flux $f(\eta,r,x)$ of Ly$\alpha$ photons emergent from halos at the
boundary $r=R=10^{2}$. The parameters of the dust are $A=0$ and $\kappa=$ 10-4
(left), 10-3 (middle) and 10-2 (right). Other parameters are the same as in
Figure 2.
Obviously, dust absorption does not help in producing photons for the “single
longest excursion”. Therefore, dust absorption can not make the time scale of
producing photons for “single longest excursion” to be smaller. However, dust
absorptions are effective in reducing the number of photons trapped in the
state of local thermalized equilibrium $|x|<2$ (see also §4.2). This leads to
the fact that the higher the value of $\kappa$, shorter the time scale of
saturation.
### 3.3 Effective absorption optical depth
Since Ly$\alpha$ photons underwent a large number of resonant scattering
before escaping from halo with optical depth $\tau_{0}\gg 1$, it is generally
believed that a small absorption of dust will lead to a significant decrease
of the flux. However, it is still unclear what the exact relationship between
the dust absorption and the resonant scattering is. This problem should be
measured by the effective optical depth of dust absorption of Ly$\alpha$
photons in $R=\tau_{0}\gg 1$ halos.
To calculate the effective optical depth, we first give the total flux of
Ly$\alpha$ photons emergent from halo of radius $R$, which is defined as
$F(\eta)=\int f(\eta,R,x)dx$. Figure 4 plots $F(\eta)$ as a function of time
$\eta$ for halo with sizes $R=\tau_{0}=10^{2}$ and $10^{4}$. The curves
typically are the time-evolution of growing and then saturating. The three
panels correspond to the dust models I, II and III from left to right. The
upper panels are of $R=10^{2}$, and lower panels for $R=10^{4}$. In each panel
of $R=10^{2}$, we have three curves corresponding to $\kappa=$ 10-4, 10-3 and
10-2, respectively. In cases of $R=10^{4}$, we take $\kappa=$ 10-4 and 10-3.
Figure 4: The time evolution of the total flux $F(\eta)$ at the boundary of
halos with $R=\tau_{0}=10^{2}$ (upper panels), and $R=\tau_{0}=10^{4}$ (lower
Panels). The source of $S_{0}=1$ and $\phi_{s}(x)=(1/\sqrt{\pi})e^{-x^{2}}$
starts to emit photons at time $\eta=0$. The parameters of dust are
$(A=1,g=0.73)$ (left); $(A=0.32,\ g=0.73)$ (middle) and $A=0$ (right). In each
panel of $R=10^{2}$, $\kappa$ is taken to be 10-4, 10-3 and 10-2. In the cases
of $R=10^{4}$, $\kappa$ is taken to be 10-4, 10-3.
The left panel of Figure 4 shows that the three curves of $\kappa=10^{-4}$,
10-3 and 10-2 are almost the same. This is consistent with Figure 2 that for
Model I, the time-evolution of $f$ are $\kappa$-independent for the pure
scattering dust. For the pure absorption dust (the right panel of Figure 4),
the saturated flux is smaller for larger $\kappa$. We can also see from Figure
4 that the time scale of approaching saturation is smaller for larger
$\kappa$. The result of model II is in between that for models I and III.
With the saturated flux of Figure 4, one can define the effective absorption
optical depth by $\tau_{\rm effect}\equiv-(1/\kappa)\ln f_{S}$. The results
are shown in Table 1, in which $\tau_{a}$ is the dust absorption depth. It is
interested to see that the effective absorption optical depth is always equal
to about a few times of the optical depth of resonant scattering $\tau_{0}$,
regardless whether $\tau_{a}$ is less than 1. Namely, the effective absorption
depth $\tau_{\rm effect}$ of dust is roughly proportional to $\tau_{0}$.
Table 1. Effective absorption optical depth $\tau_{\rm effect}$
| Model II | Model III
---|---|---
$R=\tau_{0}$ | $\kappa$ | $\tau_{a}$ | $f_{S}$ | $\tau_{\rm effect}$ | $\tau_{a}$ | $f_{S}$ | $\tau_{\rm effect}$
$10^{2}$ | $10^{-4}$ | 0.0068 | 0.978 | $2.2\times 10^{2}$ | 0.01 | 0.963 | $3.8\times 10^{2}$
$10^{2}$ | $10^{-3}$ | 0.068 | 0.760 | $2.7\times 10^{2}$ | 0.10 | 0.670 | $4.0\times 10^{2}$
$10^{2}$ | $10^{-2}$ | 0.68 | 0.116 | $2.2\times 10^{2}$ | 1.00 | 0.057 | $2.9\times 10^{2}$
$10^{4}$ | $10^{-4}$ | 0.68 | $6.28\times 10^{-2}$ | $2.8\times 10^{4}$ | 1.00 | $3.02\times 10^{-2}$ | $3.5\times 10^{4}$
$10^{4}$ | $10^{-3}$ | 6.8 | $4.07\times 10^{-7}$ | $1.5\times 10^{4}$ | 10.0 | $2.87\times 10^{-9}$ | $1.97\times 10^{4}$
According to the random walk scenario, if a medium has optical depths of
absorption $\tau_{a}$ and scattering $\tau_{s}$, the effective absorption
optical depth should be equal to $\tau_{\rm
effect}=\sqrt{\tau_{a}(\tau_{a}+\tau_{s})}$ (Rybicki & Lightman 1979).
However, the results of the last line of Table 1 show that the random walk
scenario does not work for the dust effect on resonant photon transfer. This
result is consistent with Figures 2 and 3. When optical depth of dust is lower
than the optical depth of resonant scattering $\tau_{0}$, the time scale of
photon escaping basically is not affected by the dust, but is proportional to
$\tau_{0}$, and therefore, the absorption is also proportional to $\tau_{0}$.
### 3.4 Escape coefficient
With the total flux, we can define the escaping coefficient of Ly$\alpha$
photon as $f_{\rm esc}(\eta,\tau_{0})\equiv F(\eta)/F_{0}$, where $F_{0}$ is
the flux of the center source. Figure 5 shows $f_{\rm esc}(\eta,\tau_{0})$ at
three times $\eta=5\times 10^{3}$, 104 and 3.2$\times 10^{4}$ for Model II and
$\kappa=10^{-3}$. At $\eta=5\times 10^{3}$, the flux of halos with
$\tau_{0}\leq 10^{3}$ is saturated. At $\eta=10^{4}$, halos with $\tau_{0}\leq
3\times 10^{3}$ are saturated, and all halos of $\tau_{0}\leq 10^{4}$ are
saturated at $\eta=3.2\times 10^{4}$.
Figure 5: Escaping coefficient $f_{\rm esc}(\eta)$ as a function of the
optical depth $\tau_{0}$ of halo at time $\eta=5\times 10^{3}$, 104, and
3.2$\times 10^{4}$ from bottom to up. Dust is modeled by II, $A=0.32,\
g=0.73$, and $\kappa=10^{-3}$.
## 4 Dust effects on double-peaked profile
### 4.1 Dust and the frequency of double peaks
A remarkable feature of Ly$\alpha$ photon emergent from optically thick medium
is the double-peaked profile. Figures 1, 2 and 3 have shown that the double
peak frequencies $x_{+}=|x_{-}|$ are almost independent of either the
scattering or the absorption of dust. In this section, we consider halos with
size $R$ or $\tau_{0}$ larger than $10^{2}$. Figure 6 presents the double peak
frequency $|x_{\pm}|$ as a function of $a\tau_{0}$, where the parameter $a$ is
taken to be 10-2 (left) and $5\times 10^{-3}$ (right). Comparing the curves
with dust and without dust in Figure 6 we can say that the dust effect on
$|x_{\pm}|$ is very small till $a\tau_{0}=aR=10^{2}$.
Figure 6: The two-peak frequencies $x_{+}=|x_{-}|$ as a function of
$a\tau_{0}$. The parameter $a$ is taken to be 10-2 (left) and $5\times
10^{-3}$ (right). Dust model III (pure absorption) is used, and $\kappa$ is
taken to be $10^{-3}$. The dashed straight line gives $\log x_{\pm}$-$\log
a\tau$ with slope 1/3, which is to show the $(a\tau)^{1/3}$-law of $x_{\pm}$.
In the range $a\tau_{0}<20$, the $|x_{\pm}|$-$\tau_{0}$ relation is almost
flat with $|x_{\pm}|\simeq 2$. It is because the double-peaked profile is
given by the frequency range of the locally thermal equilibrium. The positions
of the two peaks, $x_{+}$ and $x_{-}$, basically are at the maximum and
minimum frequencies of the local thermalization. The frequency range of the
local thermal equilibrium state is mainly determined by the Doppler
broadening, and weakly dependent on $\tau_{0}$. Thus, we always have
$x_{\pm}\simeq\pm 2$. When the optical depth is larger, $a\tau_{0}\sim
10^{2}$, more and more photons of the flux are attributed to the resonant
scattering by the Lorentzian wing of the Voigt profile. In this phase,
$|x_{\pm}|$ will increase with $\tau_{0}$.
Figure 6 shows also a line $x_{\pm}=\pm(a\tau_{0})^{1/3}$, which is given by
the analytical solution of the Fokker-Planck approximation, in which the
Doppler broadening core in the Voigt profile is ignored (Harrington 1973,
Neufeld 1990, Dijkstra 2006). The numerical solutions of eqs (3) or (9) and
(10) deviate from the $(a\tau_{0})^{1/3}$-law at all parameter range of Figure
6. The deviation at $a\tau_{0}<20$ is due to that the Doppler broadening core
in the Voigt profile is ignored in the Fokker-Planck approximation, and then,
no locally thermal equilibrium can be reached. Therefore, in the range
$a\tau_{0}<20$, $|x_{\pm}|$ of the WENO solution is larger than the
$(a\tau_{0})^{1/3}$-law. In the range of $a\tau_{0}>20$, the Fokker-Planck
approximation yields a faster diffusion of photons in the frequency space.
This point can be seen in the comparison between a Fokker-Planck solution with
Field’s analytical solution (Figure 1 in Rybicki & Dell’Antonio 1994). In this
range, the numerical results of $|x_{\pm}|$ is less than the
$(a\tau_{0})^{1/3}$-law.
### 4.2 No narrowing and no widening
The dust effect has been used to explain the narrowing of the width between
the two peaks (Laursen et al. 2009). Oppositely, it is also used to explain
the widening of the width between the two peaks (Verhamme et al. 2006).
However, Figures 1, 2, 3 and 6 already show that the width between the two
peaks of the profile is very weakly dependent on dust scattering and
absorption. This result supports, at least in the parameter range considered
in Figures 1, 2, 3, neither the narrowing nor the widening of the two peaks.
Figure 7: $\ln[f(\eta,r,x,\kappa=0)/f(\eta,r,x,\kappa)]$ as function of $x$
for model II (up), and III (bottom), and $\kappa=10^{-3}$ (left) and 10-2
(right). Other parameters are the same as in Figure 2.
If dust absorption can cause narrowing, the absorption should be weaker at
$|x|\sim 0$, and stronger at $|x|\geq 2$. Similarly, if dust absorption can
cause widening, the absorption should be weaker at $|x|\sim 2$, and stronger
at $|x|\sim 0$. To test these assumptions, Figure 7 plots
$\ln[f(\eta,r,x,\kappa=0)/f(\eta,r,x,\kappa)]$ as a function of $x$. It
measures the $x$(frequency)-dependence of the flux ratio with and without dust
absorption. We take large $\eta$, and then the fluxes in Figure 7 are
saturated. Figure 7 shows that the absorption in the range $|x|<2$ is much
stronger than that of $|x|>2$, and therefore, the assumption of the narrowing
is ruled out. Figure 7 shows also that the curves of
$\ln[f(\eta,r,x,\kappa=0)/f(\eta,r,x,\kappa=10^{-3})]$ are almost flat in the
range $|x|<2$. Therefore, the assumption of widening of the two peaks can also
be ruled out.
Since the cross sections of dust absorption and scattering are assumed to be
frequency-independent. Eqs. (9) and (10) do not contain any frequency scales
other than that from resonant scattering. However, either narrowing or
widening would require to have frequency scales different from that of
resonant scattering. This is occurence is not possible if the dust is gray.
### 4.3 Profile of absorption spectrum
If the radiation from the sources has a continuum spectrum, the effect of
neutral hydrogen halos is to produce an absorption line at $\nu=\nu_{0}$. The
profile of the absorption line can also be found by solving equations (9) and
(10), but replacing the boundary equation (11) by
$j(\eta,0,x)=0,\hskip 28.45274ptf(\eta,0,x)=S_{0}.$ (14)
That is, we assume that the original spectrum is flat in the frequency space.
The spectrum of the flux emergent from halo of $R=10^{2}$ and $10^{4}$ with
central source of eq.(14) for dust models I, II and III are shown in Figure 8.
All curves are for large $\eta$, i.e. they are saturated.
Figure 8: The spectrum of the flux emergent from halo of $R=10^{2}$ (upper
panels) and 104 (lower panels) with central source of eq.(14) for the dust
model I (left), II (middle) and III (right). Other parameters are the same as
in Figure 2.
The optical depths at the frequency $|x|>4$ are small, and therefore, the
Eddington approximation might no longer be proper. However, those photons do
not strongly involve the resonant scattering, and hence they do not
significantly affect the solution around $x=0$. The solutions of Figure 8 is
still useful to study the profiles of $f$ around $x=0$.
The flux profile of Figure 8 typically are absorption lines with width given
by the double peaks similar to the double peaked structure of the emission
line. The flux at the double peaks is even higher than the flat wing. It is
because more photons are stored in the frequency range $|x|<2$. According to
the redistribution function eq.(4), the probability of transferring a
$x^{\prime}$ photon to a $|x|<|x^{\prime}|$ photon is larger than that from
$|x^{\prime}|$ to $|x|>|x^{\prime}|$. Therefore, if the original spectrum is
flat, the net effect of resonant scattering is to bring photons with frequency
$|x|>2$ to $|x|<2$. Photons stored $|x|<2$ are thermalized, and therefore, in
the range $|x|<2$, the profile will be the same as the emission line, and the
double peaks can be higher than the wing. It makes the shoulder at $|x|\sim
2$.
As expected, for model I (left panels of Figure 8), the double profile is
completely $\kappa$-independent. Dusty scattering does not change the flux and
its profile. For models II and III, the higher the $\kappa$, the lower the
flux of the wing, because the dust absorption is assumed to be frequency-
independent. The positions of the double peaks, $x$, in the absorption
spectrum are also $\kappa$-independent. This once again shows that dust
absorption and scattering causes neither narrowing nor widening of the double-
peaked profile. However, for higher $\kappa$ the flux of the peaks is lower.
When the absorption is very strong, the double-peaked structure might
disappear, but will never be narrowed or widened.
## 5 Discussions and conclusions
The study of dust effects on radiative transfer has had a long history related
to extinction. However, dust effects on radiative transfer of resonant photons
actually have not been carefully investigated. Existing works are mostly based
on the solutions of the Fokker-Planck approximation, or Monte Carlo
simulation. These results are important. We revisited these problems with the
WENO solver of the integro-differential equation of the resonant radiative
transfer, and have found some features which have not been addressed in
previous works. These features are summarized as follows.
First, the random walk picture in the physical space will no longer be
available for estimating the effective optical depth of dust absorption. For a
medium with the optical depth of absorption and resonant scattering to be
$\tau_{a}\gg 1$, $\tau(\nu_{0})\gg 1$ and $\tau_{s}(\nu_{0})\gg\tau_{a}$, the
effective absorption optical depth is found to be almost independent of
$\tau_{a}$, and to be equal to about a few times of $\tau_{s}(\nu_{0})$.
Second, dust absorption will, of course, yield the decrease of the flux of
Ly$\alpha$ photons emergent from optical thick medium. However, if the
absorption cross-section of dust is frequency independent, the double-peaked
structure of the frequency profile is basically dust-independent. The double-
peaked structure does not get narrowed or widened by the absorption and
scattering of dust.
Third, the time scales of Ly$\alpha$ photon transfer basically are independent
of dust scattering and absorption. It is because those time scales are mainly
determined by the kinetics in the frequency space, while dust does not affect
the behavior of the transfer in the frequency space if the cross section of
the dust is wavelength-independent. The local thermal equilibrium makes the
anisotropic scattering to be ineffective on the angular distribution of
photons. Dust absorption and scattering do not lead to the increase or
decrease of the time of storing Ly$\alpha$ photons in the halos.
The differences between the time-independent solutions of the Fokker-Planck
approximation, or Monte Carlo simulation and the WENO solution of eq.(3) is
mainly related to the W-F effect. Therefore, all above-mentioned features can
already be clearly seen with halos of $\tau_{0}\sim 10^{2}$, in which the W-F
local thermal equilibrium has been well established.
In this context, most calculation in this paper is on holes with
$\tau_{0}<10^{5}$. This range of $\tau_{0}$ certainly is unable to describe
halos with column number density of HI larger than 1017 cm-2 (e.g. Roy et al.
2010). Nevertheless, the result of $\tau_{0}<10^{5}$ would already be useful
for studying the 21 cm region around high-redshift sources, of which the
optical depth typically is (Liu et al 2007; Roy, et al. 2009c).
$\tau_{0}=3.9\times 10^{5}f_{\rm HI}\left(\frac{T}{10^{4}{\rm
K}}\right)^{-1/2}\left(\frac{1+z}{10}\right)^{3}\left(\frac{\Omega_{b}h^{2}}{0.022}\right)\left(\frac{R_{\rm
ph}}{10{\rm kpc}}\right),$ (15)
where $f_{\rm HI}$ is the fraction of HI. All other parameters in eq. (15) is
taken from the concordance $\Lambda$CDM mode. For these objects the relation
between dimensionless $\eta$ and physical time $t$ is given by
$t=5.4\times 10^{-2}f^{-1}_{\rm HI}\left(\frac{T}{10^{4}{\rm
K}}\right)^{1/2}\left(\frac{1+z}{10}\right)^{-3}\left(\frac{\Omega_{b}h^{2}}{0.022}\right)^{-1}\eta,\
\ {\rm yr}.$ (16)
The 21 cm emission rely on the W-F effect. On the other hand, the time-scale
of the evolution of the 21 region is short. The effect of dust on the time-
scales of Ly$\alpha$ evolution should be considered.
We have not considered the Ly$\alpha$ photons produced by the recombination in
the ionized halo. If the halo is optical thick, photons from the recombination
will also be thermalized. The information of where the photon comes from will
be forgotten during the thermalization. Therefore, photons from recombination
should not show any difference from those emitted from central sources. Only
the photons formed at the place very close to the boundary of the halo will
not be thermalized, and may yield different behavior.
This research is partially supported by ARO grants W911NF-08-1-0520 and
W911NF-11-1-0091.
## Appendix A Integral of the phase function [eq.(6)]
Eq.(6) can be rewritten as
$\mathcal{R}^{d}(\mu,\mu^{\prime})=\frac{1}{4\pi}\int^{2\pi}_{0}d\phi^{\prime}\frac{1-g^{2}}{|{\bf
I}-g{\bf I^{\prime}}|^{\frac{3}{2}}}$ (A1)
where ${\bf I}$ and ${\bf I^{\prime}}$ are unit vector on the direction of
polar angle $\theta$ and $\theta^{\prime}$, and azimuth angle $\phi$ and
$\phi^{\prime}$, respectively. That is ${\bf I}\cdot{\bf I}={\bf
I^{\prime}}\cdot{\bf I^{\prime}}=1$ and ${\bf I}\cdot{\bf
I^{\prime}}=\cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime})$,
and $\mu=\cos\theta$, $\mu^{\prime}=\cos\theta$. We have
$\displaystyle\frac{d}{dg}\frac{1}{|{\bf I}-g{\bf
I^{\prime}}|^{1/2}}=\frac{1-g^{2}}{2g|{\bf I}-g{\bf
I^{\prime}}|^{3/2}}-\frac{1}{2g|{\bf I}-g{\bf I^{\prime}}|^{1/2}},$ (A2)
and therefore,
$\frac{1-g^{2}}{|{\bf I}-g{\bf
I^{\prime}}|^{3/2}}=2g\frac{d}{dg}\frac{1}{|{\bf I}-g{\bf
I^{\prime}}|^{1/2}}+\frac{1}{|{\bf I}-g{\bf I^{\prime}}|^{1/2}}.$ (A3)
The expansion with Legendre functions $P_{l}(\cos\gamma)$ gives
$\frac{1}{|{\bf I}-g{\bf
I^{\prime}}|^{1/2}}=\sum_{l=0}^{\infty}g^{l}P_{l}(\cos\gamma),$ (A4)
and then
$\frac{1-g^{2}}{|{\bf I}-g{\bf
I^{\prime}}|^{3/2}}=\sum_{l=1}^{\infty}2lg^{l}P_{l}(\cos\gamma)+\sum_{l=0}^{\infty}g^{l}P_{l}(\cos\gamma).$
(A5)
Since
$\cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime})$,
we have the following identity for the Legendre function $P_{l}(\cos\gamma)$
as
$P_{l}(\cos\gamma)=P_{l}(\cos\theta)P_{l}(\cos\theta^{\prime})+2\sum_{m=1}^{m=l}\frac{(l-m)!}{(l+m)!}P^{m}_{l}(\cos\theta)P^{m}_{l}(\cos\theta^{\prime})\cos[m(\phi-\phi^{\prime})].$
(A6)
The integral of $\phi^{\prime}$ in eq.(A1) kills the second term of eq.(A6),
we have then
$\displaystyle\mathcal{R}^{d}(\mu,\mu^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}2\pi\left[\sum_{l=1}^{\infty}2lg^{l}P_{l}(\cos\theta)P_{l}(\cos\theta^{\prime})+\sum_{l=0}^{\infty}g^{l}P_{l}(\cos\theta)P_{l}(\cos\theta^{\prime})\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\sum_{l=1}^{\infty}2lg^{l}P_{l}(\mu)P_{l}(\mu^{\prime})+\sum_{l=0}^{\infty}g^{l}P_{l}(\mu)P_{l}(\mu^{\prime})\right].$
Using the orthogonal relation
$\int_{-1}^{1}P_{l}(\mu)P_{l^{\prime}}(\mu)d\mu=\frac{2}{2l+1}\delta_{l,l^{\prime}}$,
we have
$R_{0}(g)=\frac{1}{2}\int_{-1}^{1}d\mu\int_{-1}^{1}d\mu^{\prime}R^{d}(\mu,\mu^{\prime})=1,$
(A8)
for which only the term $l=0$ in eq.(A7) has contribution. Similarly,
$R_{1}(g)=\frac{1}{2}\int_{-1}^{1}d\mu\int_{-1}^{1}d\mu^{\prime}\mu
R^{d}(\mu,\mu^{\prime})=\frac{1}{2}\int_{-1}^{1}d\mu\int_{-1}^{1}d\mu^{\prime}\mu^{\prime}R^{d}(\mu,\mu^{\prime})=0,$
(A9)
$R_{2}(g)=\frac{1}{2}\int_{-1}^{1}d\mu\int_{-1}^{1}d\mu^{\prime}\mu\mu^{\prime}R^{d}(\mu,\mu^{\prime})=\frac{g}{3}.$
(A10)
These results are used in deriving eqs.(9) and (10).
## Appendix B Numerical algorithm
To solve Equations $(9)$ and $(10)$ as a system, our computational domain is
$(r,x)\in[0,r_{\textrm{max}}]\times[x_{\textrm{left}},x_{\textrm{right}}],$
where $r_{\textrm{max}},x_{\textrm{left}}$ and $x_{\textrm{right}}$ are chosen
such that the solution vanishes to zero outside the boundaries. We choose mesh
sizes with grid refinement tests to ensure proper numerical resolution. In the
following, we describe numerical techniques involved in our algorithm,
including approximations to spatial derivatives, numerical boundary condition,
and time evolution.
### B.1 The WENO Algorithm: Approximations to the Spacial Derivatives
The spacial derivative terms in Equation $(9)$ and $(10)$ are approximated by
a fifth-order finite difference WENO scheme.
We first give the WENO reconstruction procedure in approximating
$\frac{\partial j}{\partial x},$
$\frac{\partial j(\eta^{n},r_{i},x_{j})}{\partial x}\approx\frac{1}{\Delta
x}(\hat{h}_{j+1/2}-\hat{h}_{j-1/2})$
with fixed $\eta=\eta^{n}$ and $r=r_{i}.$ The numerical flux $\hat{h}_{j+1/2}$
is obtained by the fifth-order WENO approximation in an upwind fashion,
because the wind direction is fixed. Denote
$h_{j}=j(\eta^{n},r_{i},x_{j}),\quad\quad j=-2,-1,\cdots,N+3$
with fixed $n$ and $i$. The numerical flux from the WENO procedure is obtained
by
$\hat{h}_{j+1/2}=\omega_{1}\hat{h}^{(1)}_{j+1/2}+\omega_{2}\hat{h}_{j+1/2}^{(2)}+\omega_{3}\hat{h}_{j+1/2}^{(3)}$
where $\hat{h}_{j+1/2}^{(m)}$ are the three third-order fluxes on three
different stencils given by
$\displaystyle\hat{h}_{j+1/2}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}h_{j-1}+\frac{5}{6}h_{j}+\frac{1}{3}h_{j+1},$
$\displaystyle\hat{h}_{j+1/2}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}h_{j}+\frac{5}{6}h_{j+1}-\frac{1}{6}h_{j+2},$
$\displaystyle\hat{h}_{j+1/2}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{11}{6}h_{j+1}-\frac{7}{6}h_{j+2}+\frac{1}{3}h_{j+3}.$
and the nonlinear weights $\omega_{m}$ are given by
$\omega_{m}=\frac{\breve{\omega}_{m}}{\sum_{l=1}^{3}\breve{\omega}_{l}},\quad\breve{\omega}_{l}=\frac{\gamma_{l}}{(\epsilon+\beta_{l})^{2}}$
where $\epsilon$ is a parameter to avoid the denominator to become zero and is
taken as $\epsilon=10^{-8}$. The linear weights $\gamma_{l}$ are given by
$\gamma_{1}=\frac{3}{10},\quad\gamma_{2}=\frac{3}{5},\quad\gamma_{3}=\frac{1}{10},$
and the smoothness indicators $\beta_{l}$ are given by
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{13}{12}(h_{j-1}-2h_{j}+h_{j+1})^{2}+\frac{1}{4}(h_{j-1}-4h_{j}+3h_{j+1})^{2},$
$\displaystyle\beta_{2}$ $\displaystyle=$
$\displaystyle\frac{13}{12}(h_{j}-2h_{j+1}+h_{j+2})^{2}+\frac{1}{4}(h_{j}-h_{j+2})^{2},$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\frac{13}{12}(h_{j+1}-2h_{j+2}+h_{j+3})^{2}+\frac{1}{4}(3h_{j+1}-4h_{j+2}+h_{j+3})^{2}.$
To approximate the $r$-derivatives in the system of Equations $(9)$ and
$(10)$, we need to perform the WENO procedure based on a characteristic
decomposition. We write the left-hand side of Equations $(9)$ and $(10)$ as
${\bf u}_{t}+A{\bf u}_{r},$
where ${\bf u}=(j,f)^{T}$ and
$A=\left(\begin{array}[]{cc}0&1\\\ \frac{1}{3}&0\end{array}\right)$
is a constant matrix. To perform the characteristic decomposition, we first
compute the eigenvalues, the right eigenvectors and the left eigenvectors of A
and denote them by $\Lambda$, $R$ and $R^{-1}$. We then project u to the local
characteristic fields v with ${\bf v}=R^{-1}{\bf u}.$ Now ${\bf u}_{t}+A{\bf
u}_{r}$ of the original system is decoupled as two independent equations as
${\bf v}_{t}+\Lambda{\bf v}_{r}.$ We approximate the derivative ${\bf v}_{r}$
component by component, each with the correct upwind direction, with the WENO
reconstruction procedure similar to the procedure described above for
$\frac{\partial j}{\partial x}$. In the end, we transform ${\bf v}_{r}$ back
to the physical space by ${\bf u}_{r}=R{\bf v}_{r}$. We refer the readers to
Cockburn et al. 1998 for more implementation details.
### B.2 Numerical Boundary Condition
To implement the boundary condition $(12)$, we also need to perform a
characteristic decomposition as discussed above. Using the same notation as
before, we project u to the local characteristic fields v with ${\bf
v}=R^{-1}{\bf u}$. Denote ${\bf v}=(v_{1},v_{2})^{T}$, now ${\bf u}_{t}+A{\bf
u}_{r}$ of the original system is decoupled to two independent scalar
operators given by
$\frac{\partial v_{1}}{\partial t}+\lambda_{1}\frac{\partial v_{1}}{\partial
r};\qquad\frac{\partial v_{2}}{\partial t}+\lambda_{2}\frac{\partial
v_{2}}{\partial r}$
where $\lambda_{1}=\frac{\sqrt{3}}{3}$ and $\lambda_{2}=-\frac{\sqrt{3}}{3}$.
The characteristic line starting from the boundary $r=r_{\textrm{max}}$ for
the first equation is pointing outside the computational domain while the one
for the second equation is pointing inside. For well-posedness of our system,
we need to impose the boundary condition there as
$v_{2}=\alpha v_{1}+\beta$
with constants $\alpha$ and $\beta$. We can calculate the values of $\alpha$
and $\beta$ based on equation $(12)$ and the left and right eigenvectors of
$A$. For example, if we take
$R=\left(\begin{array}[]{cc}\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}\\\
\frac{1}{2}&-\frac{1}{2}\end{array}\right),$
we can compute that $\alpha=7-4\sqrt{3}$ and $\beta=0$. We use extrapolation
to obtain the value of $v_{1}$ and then compute the value $v_{2}$. In the end,
we transfer ${\bf v}$ back to the physical space by ${\bf u}=R{\bf v}$.
### B.3 Time Evolution
To evolve in time, we use the third-order TVD Runge-Kutta time discretization
(Shu & Osher 1988). For system of ODEs $u_{t}=L(u)$, the third order Runge-
Kutta method is
$\displaystyle u^{(1)}$ $\displaystyle=$ $\displaystyle u^{n}+\Delta\tau
L(u^{n},\tau^{n}),$ $\displaystyle u^{(2)}$ $\displaystyle=$
$\displaystyle\frac{3}{4}u^{n}+\frac{1}{4}(u^{(1)}+\Delta\tau
L(u^{(1)},\tau^{n}+\Delta\tau)),$ $\displaystyle u^{n+1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}u^{n}+\frac{2}{3}(u^{(2)}+\Delta\tau
L(u^{(2)},\tau^{n}+\frac{1}{2}\Delta\tau)).$
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|
arxiv-papers
| 2011-07-09T17:54:38 |
2024-09-04T02:49:20.363758
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yang Yang, Ishani Roy, Chi-Wang Shu and Li-Zhi Fang",
"submitter": "Yang Yang",
"url": "https://arxiv.org/abs/1107.1804"
}
|
1107.1851
|
# Task swapping networks in distributed systems
Dohan Kim dohankim1@gmail.com
###### Abstract
In this paper we propose task swapping networks for task reassignments by
using task swappings in distributed systems. Some classes of task
reassignments are achieved by using iterative local task swappings between
software agents in distributed systems. We use group-theoretic methods to find
a minimum-length sequence of adjacent task swappings needed from a source task
assignment to a target task assignment in a task swapping network of several
well-known topologies.
###### keywords:
Task swapping network, Task swapping graph, Task assignment, Permutation
group, Distributed system
12pt
## 1 Introduction
A distributed system is defined as a collection of independent computers or
processors that are connected by an arbitrary interconnection network [1, 2].
The objective of a task assignment [3, 1] in a distributed system is to find
an optimal or suboptimal assignment of tasks to processors (or agents), while
satisfying temporal and spatial constraints imposed on the system. A task
assignment is either preemptive or non-preemptive [4]. If a task assignment is
preemptive, a task reassignment is allowed in such a way that tasks are
transferred between processors (or agents) during their execution for
improving the system performance [4, 5]. Recent advances in software agent
technology [6, 7, 8] in distributed systems allow software entities to observe
their environment, and cooperate with other entities if necessary to
accomplish their goals. Task migration 111In this paper we use task migration
and process migration interchangeably. between software agents intends to
improve the system throughput in a distributed system in which the loads
incurred by tasks vary over time [8]. A task reassignment can be achieved by
iterative local task swappings, where a task swapping involves task migrations
between a pair of agents as a method of local task reassignment [9]. A
subclass of task assignment (or reassignment) problems involves an equal
number of tasks and agents, finding a bijective task assignment between tasks
and agents in such a way that the total task assignment (or reassignment)
benefit is maximized [10]. Those bijective task assignment problems and their
variants appear in a wide variety of areas in computer science and mathematics
[11, 12, 10, 13]. Given $n$ tasks and $n$ agents whose connections are
described by a (connected) network topology, task swapping cost between two
agents often relies on a distance in the network topology; the larger the
distance between two agents in the network topology, the larger communication
delays of migrating tasks caused by the network. Therefore, we need to
consider how task swappings are performed on a given network topology for task
reassignments. In this type of problems a group theory can be used to
represent task reassignments including iterative task swappings. A group-
theoretic approach to representing task assignments or reassignments have
already been researched [14, 15]. However, little work has done about task
swappings for task reassignments between tasks and agents by using group
theory. We propose a group-theoretic model of global task reassignments
involving $n$ tasks and $n$ agents by using local task swappings in a
distributed network of several well-known topologies.
This paper is organized as follows. Section 2 presents the problem formulation
and its assumptions. We give the necessary definitions for the problem
formulation in this section. Section 3 gives an introduction to permutation
groups and Cayley graphs. We also discuss transposition graphs of several
network topologies and their relationship to permutation sortings in this
section. We present task swapping graphs and their examples in distributed
systems in Section 4. Group-theoretic properties of task swapping graphs and
their examples are discussed in this section. Finally, we conclude in Section
5.
## 2 Problem formulation and assumptions
An _agent graph_ $G_{a}=(V_{a},E_{a})$ [16] is an undirected graph, where
$V_{a}$ denotes a set of agents in a distributed system and $E_{a}$ denotes
communication links between agents. Let $T=\\{t_{1},t_{2},\ldots,t_{n}\\}$ be
a set of $n$ tasks and $A=\\{a_{1},a_{2},\ldots,a_{n}\\}$ be a set of $n$
agents represented by an agent graph $G_{a}=(V_{a},E_{a})$. Now, each task is
assigned to each agent in $G_{a}=(V_{a},E_{a})$ bijectively as an initial task
assignment by using a task assignment algorithm. Suppose the load on each
agent varies over time, implying that an initial task assignment may not
remain optimal or suboptimal. The _load rebalancing_ intends to find a task
reassignment in order to decrease makespan [17], or to achieve a target load
balance level along with minimum task migration cost [18]. Task migration
between agents incurs a task migration cost [19] to transfer the state of the
running task (e.g., execution state, I/O state, etc) of one agent to the other
agent. This cost is not trivial, following that only nearby agents (or
processors) in the network topology are often involved to perform a task
migration [18, 20]. Moreover, if we restrict each task reassignment to be
bijective between tasks and agents, then iterative local task swappings can be
used for a global task reassignment. A _swapping distance_ is defined as the
distance between agents in an agent graph $G_{a}$ for a task swapping. For
example, a task swapping of swapping distance 1 is a task swapping between
adjacent agents in $G_{a}$, and a task swapping of swapping distance 2 is a
task swapping between agents whose distance is 2 in $G_{a}$. Now, the overall
cost difference before task swapping and after task swapping for
computationally intensive tasks (rather than communicationally intensive
tasks) is roughly the cost of a target task assignment after task swapping
subtracted by the cost of a source task assignment before task swapping, and
added by the task swapping cost itself. In the remainder of this paper we
assume the followings:
1. 1.
Each task and agent are not necessarily homogeneous, and the load on each
agent may vary over time.
2. 2.
An agent network described by $G_{a}=(V_{a},E_{a})$ is static without taking
mobility (e.g., dynamic join or leave by agents [21]) into account.
3. 3.
A task swapping cost is uniform between adjacent agents in the agent network.
4. 4.
A task swapping cost is assumed to be proportional to a task swapping
distance.
5. 5.
Each task is long-lived and computationally intensive.
6. 6.
A global task reassignment is obtained by using iterative and sequential local
task swappings.
7. 7.
A task startup cost and the message passing overhead for task migration
protocol are ignored.
8. 8.
Every agent in $G_{a}=(V_{a},E_{a})$ is cooperative [22], pursuing the same
goal to improve their collective performance for task assignments.
It follows that the expected total cost of task migrations hinges on how many
local task swappings are needed from a source task assignment to a target task
assignment. A decision making by the coordinator agent(s) whether or not a
task reassignment is performed is based on the information regarding the total
cost of task migrations along with the cost difference involving a source and
target task assignment. Let $g_{1}$ be a bijective source task assignment
between a set of $n$ tasks $T=\\{t_{1},t_{2},\ldots,t_{n}\\}$ and a set of $n$
agents $A=\\{a_{1},a_{2},\ldots,a_{n}\\}$ before task migrations. Let $g_{2}$
be a feasible target task assignment between $T$ and $A$ after task
migrations, which implies that $g_{2}$ can be obtained by using iterative
local task swappings. Let $h_{1}$ be the cost of task assignment $g_{1}$ and
let $h_{2}$ be the cost of task assignment $g_{2}$. Let
$f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k})$ be the total cost of task
migrations, where $s_{i}$ for $1\leq i\leq k<n$ is the number of local task
swappings of swapping distance $i$ involved in converting $g_{1}$ to $g_{2}$.
Then, a task reassignment benefit $b(h_{1},h_{2},f)$ is defined as
$-(h_{2}-h_{1}+f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k}))$, i.e.,
$h_{1}-h_{2}-f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k})$. The higer value of
$b(h_{1},h_{2},f)$ implies that a task reassignment is more desirable, while
the negative value of $b(h_{1},h_{2},f)$ implies that a task reassignment from
$g_{1}$ to $g_{2}$ is not desirable at all. If we restrict a local task
swapping to be a task swapping between adjacent agents in
$G_{a}=(V_{a},E_{a})$, $f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k})$ is simply
$cs_{1}$, where $c$ is constant. If we restrict a local task swapping to be a
task swapping between agents in $G_{a}=(V_{a},E_{a})$ within distance $m$,
$f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k})$ is $cs_{1}+2cs_{2}+\cdots+mcs_{m}$,
where $c$ is constant and $s_{r}$ $(1\leq r\leq m)$ is the number of task
swappings of swapping distance $r$ involved in converting $g_{1}$ to $g_{2}$.
The problem is formulated as follows:
Given a source task assignment $g_{1}$ and a feasible target task assignment
$g_{2}$ on the agent network described by an agent graph
$G_{a}=(V_{a},E_{a})$, find the minimum total cost of task migrations
$f(g_{1},g_{2},s_{1},s_{2},\ldots,s_{k})$ to reach from $g_{1}$ to $g_{2}$.
In this paper we only concern local task swappings of swapping distance 1,
which are adjacent task swappings between agents in $G_{a}=(V_{a},E_{a})$.
Since we assumed that a task swapping cost is uniform between adjacent agents
in $G_{a}=(V_{a},E_{a})$, the problem is reduced to find the minimum number of
adjacent task swappings needed from a source task assignment to reach a target
task assignment in $G_{a}=(V_{a},E_{a})$.
## 3 Transposition and Cayley graphs
We first give a brief introduction to finite groups found in [23, 24, 25, 15].
A _group_ $(G,\,\cdot\,)$ is a nonempty set _G_ , closed under a binary
operation $\cdot:G\times G\rightarrow G$, such that the following axioms are
satisfied: (i) $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ for all $a,b,c\in G$; (ii)
there is an element _e_ $\in$ _G_ such that for all $x\in G,~{}e\cdot x=x\cdot
e=x$; (iii) for each element $a\in G$, there is an element $a^{-1}\in G$ such
that $a\cdot a^{-1}=a^{-1}\cdot a=e$. If $H$ is a nonempty subset of $G$ and
is also a group under the binary operation $\cdot$ in $G$, then $H$ is called
a _subgroup_ of $G$.
The group of all bijections $I_{n}\rightarrow I_{n}$, where
$I_{n}=\\{1,2,\ldots,n\\}$, is called the _symmetric group on n letters_ and
denoted $\mathfrak{S}_{n}$. Since $\mathfrak{S}_{n}$ is the group of all
permutations of $I_{n}=\\{1,2,\ldots,n\\}$, it has order $n!$. A _permutation
group_ is a subgroup of some $\mathfrak{S}_{n}$.
Let $i_{1},i_{2},\ldots,i_{n}$ be distinct elements of
$I_{n}=\\{1,2,\ldots,n\\}$. Then,
$\begin{pmatrix}1&2&\cdots&n\\\ i_{1}&i_{2}&\cdots&i_{n}\end{pmatrix}$
$\stackrel{{\scriptstyle\rm{def}}}{{=}}i_{1}\,i_{2}\,\cdots\,i_{n}$$\in\mathfrak{S}_{n}$
denotes the permutation that maps $1\mapsto i_{1},2\mapsto
i_{2},\ldots,n\mapsto i_{n}$.
Let $i_{1},i_{2},\ldots,i_{r}\,(r\leq n)$ be distinct elements of
$I_{n}=\\{1,2,\ldots,n\\}$. Then $(i_{1}\,i_{2}\cdots i_{r})$ is defined as
the permutation that maps $i_{1}\mapsto i_{2}$, $i_{2}\mapsto i_{3}$,…,
$i_{r-1}$$\mapsto i_{r}$ and $i_{r}\mapsto i_{1}$, and every other element of
$I_{n}$ maps onto itself. Then, $(i_{1}\,i_{2}\cdots i_{r})$ is called a cycle
of length $r$ or an $r$-cycle; a 2-cycle is called a _transposition_.
Let $G$ be a group and let $s_{i}\in G$ for $i\in I$. The subgroup generated
by $S=\\{s_{i}:i\in I\\}$ is the smallest subgroup of $G$ containing the set
$S$. If this subgroup is all of $G$, then $S$ is called a _generating set_ of
$G$.
An (right) action of a group $G$ on a set $X$ is a function $X\times
G\rightarrow X$ (usually denoted by $(x,\,g)\mapsto xg$) such that for all
$x\in X$ and $g_{1},g_{2}\in G$: (i) $xe=x$; (ii)
$x(g_{1}g_{2})=(xg_{1})g_{2}$. When such an action is given, we say that $G$
acts (right) on the set $X$. The set $X$ is called a (right) $G$-set. If $X$
is $G$ as a set, we say that $G$ acts on itself.
###### Theorem 3.1 ([24]).
Every non-identity permutation in $\mathfrak{S}_{n}$ can be expressed as a
product of disjoint cycles of length at least 2. Further, $\mathfrak{S}_{n}$
can be expressed as a product of (not necessarily disjoint) transpositions. ∎
For example, $p=2\,1\,6\,3\,4\,5\in\mathfrak{S}_{6}$ is written as
$(1\,2)(3\,6\,5\,4)$ as a product of disjoint cycles or
$(1\,2)(3\,4)(3\,5)(3\,6)$ as a product of transpositions.
Let $G$ be a finite group and $S$ be a generating set of $G$. Then,
$\text{Cay}(G,S)$ denotes the _Cayley graph_ [26, 25, 27] of $G$ with the
generating set $S$, where the set of vertices $V$ of $\text{Cay}(G,S)$
corresponds to the elements of $G$, and the set of edges $E$ of
$\text{Cay}(G,S)$ corresponds to the action of generators in such a way that
$E=\\{<x,y>_{g}:x,y\in G\text{ and }g\in S\text{ such that }y=x\cdot g\\}$. In
this paper we assume that $G$ is a finite permutation group and the elements
of $S$ are transpositions in $\text{Cay}(G,S)$. It follows that $S$ is closed
under inverses, and $E$ is undirected. In other words, an edge $<x,y>$ of the
resulting $\text{Cay}(G,S)$ is viewed as both $<x,y>_{g}$ and $<y,x>_{g}$. If
$G$ is a finite group and $S$ is a set of transpositions that generates $G$,
then a _transposition graph_ $T=(<n>,S)$ [27] is an undirected graph in which
$<n>$ denotes a vertex set of cardinality $n$ and each edge $<i,\,j>$ denotes
transposition $(i\,j)$. If a transposition graph is a tree, we call the
resulting transposition graph as a _transposition tree_. We next provide
examples of the generating sets for permutation groups including their
transposition trees.
Let
$S_{1}=\\{(i\,i+1):1\leq i<n\\},\\\ $ (1) $S_{2}=\\{(2i-1\,2i):1\leq i\leq
n\\},\\\ $ (2) $S_{3}=\\{(1\,i):2\leq i\leq n\\},\\\ $ (3)
$S_{4}=\\{(i\,j):1\leq i<j\leq n\\},\\\ $ (4) $S_{5}=\\{(i\,j):1\leq i\leq
k<j\leq n\\},\\\ $ (5) $S_{6}=S_{1}\cup\\{(1\,n)\\}.$ (6)
Fig. 1: Examples of transposition graphs $T=(<n>,S)$ [27].
Fig. 1 shows different kinds of transposition graphs corresponding to the
generating sets $S_{1},\,S_{2},\ldots,\,S_{6}$ for some $n$ and $k$. In this
paper a transposition graph having $n$ vertices are labeled from 1 to $n$
without any duplication. The Cayley graph generated by $S_{1}$ is called the
_bubble-sort_ graph $\text{BS}_{n}$ [26, 27], by $S_{2}$ is called the
_hypercube_ graph $\text{HC}_{n}$ [25], by $S_{3}$ is called the _star_ graph
$\text{ST}_{n}$ [26, 27, 25], by $S_{4}$ is called the _complete
transposition_ graph $\text{CT}_{n}$ [27], by $S_{5}$ is called the
_generalized star_ graph $\text{GST}_{n,k}$ [25], and by $S_{6}$ is called the
_modified bubble sort_ graph $\text{MBS}_{n}$ [27].
Fig. 2: The bubble sort graph $\text{BS}_{4}$ [25].
A path $p$ from vertex $v_{1}$ to vertex $v_{2}$ in $\text{Cay}(G,S)$ can be
represented by a sequence of generators $g_{1},g_{2},\ldots,g_{k}$, where
$g_{i}\in S$ for $1\leq i\leq k$. By abuse of notation, we let
$p=g_{1}g_{2}\ldots g_{k}$. Note that $p$ is also a path from vertex
$v_{2}^{-1}v_{1}$ to vertex $I$ (i.e., $v_{1}p=v_{2}$ if and only if
$v_{2}^{-1}v_{1}p=I$ for the identity permutation $I$ in $G$). Thus, to find a
path from vertex $v_{1}$ to vertex $v_{2}$ is reduced to find a path from
vertex $v_{2}^{-1}v_{1}$ to vertex $I$, which in turn is reduced to the
problem of sorting $v_{2}^{-1}v_{1}$ to $I$ using the generating set $S$ [26].
Fig. 2 shows the bubble sort graph $\text{BS}_{4}$ of the transposition tree
$T=(<4>,S_{1})$. Since adjacent transpositions $(1\,2),(2\,3)$, and $(3\,4)$
generate $\mathfrak{S}_{4}$, $\text{BS}_{4}$ has $|\mathfrak{S}_{4}|=24$
vertices, each of which has degree 3. To find a path from permutation $p_{1}$
to permutation $p_{2}$ in $\text{BS}_{4}$ is obtained by using the bubble sort
algorithm [28]. For example, let vertex $v_{1}$ be $2\,1\,3\,4$ and vertex
$v_{2}$ be $3\,1\,4\,2$ in $\text{BS}_{4}$. Then, to find path $p$ from
$v_{1}=2\,1\,3\,4$ to $v_{2}=3\,1\,4\,2$ is equivalent to find path $p$ from
$v_{2}^{-1}v_{1}=4\,2\,1\,3$ to permutation $I$:
$4\,2\,1\,3\xrightarrow{<1,2>}$$2\,4\,1\,3\xrightarrow{<2,3>}$$2\,1\,4\,3\xrightarrow{<3,4>}$$2\,1\,3\,4\xrightarrow{<1,2>}$$1\,2\,3\,4$,
where the label of each arrow denotes an edge in $\text{BS}_{4}$. Thus, we
have $p=(1\,2)(2\,3)(3\,4)(1\,2)$.
Fig. 3: The star graph $\text{ ST}_{4}$ [25].
Fig. 3 shows the star graph $\text{ ST}_{4}$ of the transposition tree
$T=(<4>,S_{3})$. Similarly to the bubble sort graph $\text{ BS}_{4}$, star
transpositions $(1\,2),(1\,3)$, and $(1\,4)$ generate $\mathfrak{S}_{4}$.
Therefore, $\text{ST}_{4}$ has $|\mathfrak{S}_{4}|=24$ vertices, each of which
has degree 3.
Note that transpositions denoted by a transposition tree of order $n$ labeled
from 1 to $n$ generate $\mathfrak{S}_{n}$ [26]. However, not every
transposition graph having $n\,(n\geq 2)$ vertices yields a Cayley graph of
$\mathfrak{S}_{n}$. For example, Fig. 1(b) shows a transposition graph having
6 vertices, but it yields a Cayley graph of a group isomorphic to $C_{2}\times
C_{2}\times C_{2}$ instead of $\mathfrak{S}_{6}$, where $C_{2}$ is the cyclic
group of order 2.
The _minimum generator sequence_ [29] for permutation $p$ of a permutation
group $G$ using the generating set $S$ is a minimum-length sequence consisting
of generators in $S$ whose composition is $p$. For example, permutation
$\pi=4\,2\,3\,1\in\mathfrak{S}_{4}$ can be expressed as
$\pi=(1\,2)(2\,3)(3\,4)(2\,3)(1\,2)$ of length 5 or $\pi=(1\,4)$ of length 1
using the generating set $S_{6}^{\prime}=\\{(i\,i+1):1\leq
i<4\\}\cup\\{(1\,4)\\}$ in which $\pi=(1\,4)$ is the minimum generator
sequence of length 1. Further, if an element $s$ in the generating set $S$ can
be expressed as the product of elements in $S$ other than $s$, then the
generating set is called _redundant_ [27]. For example, $S_{6}$ is redundant,
since $(1\,n)$ can be expressed as
$(1\,2)(2\,3)\cdots(n-1\,n)\cdots(2\,3)(1\,2)$.
Recall that the _diameter_ [30] of a connected graph is the length of the
”longest shortest path” between two vertices of the graph. Thus, the diameter
of $\text{Cay}(G,S)$ is an upper bound of distance $d(\sigma,I)$ from an
arbitrary vertex $\sigma$ to vertex $I$ in $\text{Cay}(G,S)$, where the
computation of $d(\sigma,I)$ is obtained by sorting permutation $\sigma$ to
the identity permutation $I$ in $G$ by means of the minimum generator sequence
using the generating set $S$ [25]. Thus, the diameter of $\text{Cay}(G,S)$ is
an upper bound of the lengths of minimum generator sequences for permutations
in $G$ using the generating set $S$ [26].
Table 1: Properties of some known Cayley graphs [27]. Types | Number of Vertices | Degree | Diameter
---|---|---|---
$\text{BS}_{n}$ | $n!$ | $n-1$ | $n(n-1)/2$
$\text{ST}_{n}$ | $n!$ | $n-1$ | $\lfloor 3(n-1)/2\rfloor$
$\text{CT}_{n}$ | $n!$ | $n(n-1)/2$ | $n-1$
$\text{GST}_{n,k}$ | $n!$ | $k(n-k)$ | $n-1+\max{(\lfloor k/2\rfloor,\lfloor(n-k)/2\rfloor)}$
$\text{MBS}_{n}$ | $n!$ | $n$ | Unknown
$\text{HC}_{n}$ | $2^{n}$ | $n$ | $n$
| | |
| | |
Now, consider a _permutation puzzle_ [26] on a transposition tree described as
follows: Given a transposition tree $T$ having $n$ vertices labeled from 1 to
$n$, place $n$ markers, each of which is labeled from 1 to $n$, at the
vertices of $T$ arbitrarily in such a way that each vertex of $T$ is paired
with an exactly one marker. Let $P$ be such an arbitrary initial position of
markers. Each legal move of the puzzle is to interchange the markers placed at
the ends of an edge in $T$. The terminal position of markers, denoted $Q$, is
the position in which each marker is paired with each vertex of $T$ having the
same label. The puzzle is to find a sequence of legal moves from a given
initial position $P$ to the final position $Q$ with the minimum number of
legal moves.
###### Theorem 3.2 ([26]).
Let $T$ be a transposition tree having $n$ vertices. Given an initial position
$P$ for a permutation puzzle on $T$, the final position $Q$ for the puzzle can
be reached by legal moves in the following number of steps
$c(p)-n+\displaystyle\sum_{i=1}^{n}{d(i,p(i))}$,
where $p$ is the permutation for $P$ as an assignment of markers to the
vertices of $T$, $c(p)$ is the number of cycles in $p$, and $d(i,j)$ is the
distance between vertex $i$ and vertex $j$ in $T$. ∎
###### Corollary 3.1 ([25]).
Let Diam(Cay(G, S)) be the diameter of Cay(G, S) of a transposition tree $T$
having $n$ vertices. Let $P$ be an initial position and $Q$ be the final
position for a permutation puzzle on $T$. Then,
$\text{Diam(Cay(G, S))}\leq\displaystyle\max_{p\in
G}\\{c(p)-n+\sum_{i=1}^{n}{d(i,p(i))}\\}$,
where $p$ is the permutation for $P$ as an assignment of markers to the
vertices of $T$, $c(p)$ is the number of cycles in $p$, and $d(i,j)$ is the
distance between vertex $i$ and vertex $j$ in $T$. ∎
The Cay(G, S) of a transposition tree $T$ can be viewed as the state diagram
of a permutation puzzle on $T$, where the vertices of Cay(G, S) is the
possible positions of markers on $T$ and each edge of Cay(G, S) corresponds to
a legal move of the permutation puzzle on $T$. An upper bound of the minimum
number of legal moves to reach the final position $Q$ for a given initial
position $P$ is indeed the diameter of Cay(G, S) [26] of a transposition tree
$T$. Moreover, the inequality in Corollary 3.1 can be replaced by the equality
for the cases of bubble sort and star graphs [25].
We close this section by describing the main ideas behind Theorem 3.2 (an
interested reader may refer to [26] for further details). Consider a
permutation puzzle on a transposition tree $T$ having $n$ vertices. Let $p$ be
a permutation for position $P$ as an assignment of $n$ markers to the $n$
vertices of $T$. The marker $i$ is said to be _homed_ if $i=p(i)$. If there
exists any unhomed marker in the assignment, either one of two cases occurs.
The first case is that there exists an edge of $T$ involving two unhomed
markers $x$ and $y$ such that they need to move toward each other for the
final position $Q$. The second case is that there exists an edge of $T$
involving one unhomed marker $x$ and one homed marker $y$ such that $x$ needs
to move toward $y$ for the final position $Q$. In both cases, interchanging
markers $x$ and $y$ reduces the number of steps in Theorem 3.2. Finally, if
there is no such case in the assignment (i.e., all markers are homed), then
the number is reduced to 0.
## 4 Task swapping networks
In Section 3 we discussed a permutation puzzle on a transposition tree $T$
having $n$ vertices and its legal moves. We now consider a transposition tree
as a network topology in which each vertex of $T$ corresponds to an agent and
each marker to a task. Further, consider a position of $n$ markers placed at
$n$ vertices of $T$ as a task assignment involving $n$ tasks and $n$ agents.
Each legal move of a permutation puzzle then corresponds to an adjacent task
swapping in the network topology. We apply the idea of the permutation puzzle
on a transposition tree to task assignments in a task swapping network. We
first define a _task swapping graph_.
A _task swapping graph_ $\Gamma$ having $n$ vertices is a connected graph in
which each vertex represents an agent and each edge represents a direct
communication between agents. The label next to each vertex denotes an agent
ID in $\Gamma$. Now, $n$ tasks are assigned to $n$ vertices of $\Gamma$ as a
task assignment in which the label of each vertex in $\Gamma$ denotes a task
ID. We assume that each task and agent ID are distinct numbers from the set
$\\{1,2,\ldots,n\\}$, where $n$ is the number of vertices in $\Gamma$.
Therefore, each task assignment in $\Gamma$ is represented by permutation
$p=\begin{pmatrix}1&2&\cdots&n\\\ t_{1}&t_{2}&\cdots&t_{n}\end{pmatrix}$
$\in\mathfrak{S}_{n}$, where each element in the first row of $p$ denotes an
agent ID and each element in the second row of $p$ denotes a task ID. If it is
clear from the context, we simply denote $p$ as a one-line notation
$t_{1}\,t_{2}\,\cdots\,t_{n}\in\mathfrak{S}_{n}$. For example, task
assignments in the task swapping graphs in Fig. 4(a) and Fig. 5(a) are
represented by $2\,5\,6\,3\,1\,4\,8\,7$ and $5\,4\,2\,1\,6\,9\,7\,8\,3$,
respectively. Let $p\in\mathfrak{S}_{n}$ be a permutation representing a task
assignment in $\Gamma$. Then, we denote this labeled task swapping graph
$\Gamma$ as $\Gamma_{p}$. Now, we define a task swapping of swapping distance
$k$ in $\Gamma_{p}$. A task swapping of swapping distance $k$ is the swapping
of tasks between agents whose distance is $k$ in $\Gamma_{p}$. Recall that a
right multiplication of permutation $p$ by transposition $(a\,b)$ exchanges
the values in position $a$ and position $b$ of $p$. It follows that a task
swapping of swapping distance $k$ in $\Gamma_{p}$ is represented by a right
multiplication of $p$ by transposition
$t=(i(v_{1})\,i(v_{2}))\in\mathfrak{S}_{n}$, where the distance between vertex
$v_{1}$ and vertex $v_{2}$ is $k$ and $i(v)$ denotes the agent ID for vertex
$v$ in $\Gamma_{p}$. Therefore, it converts $\Gamma_{p}$ into $\Gamma_{pt}$,
A _task swapping network_ is simply a distributed network represented by a
task swapping graph $\Gamma$. In this paper a task swapping network is
referred to as a task swapping graph $\Gamma$ unless otherwise stated. Based
on the assumptions and the problem formulation in Section 2, the problem is
now reduced and rephrased by using a task swapping graph $\Gamma$:
Given a source task assignment $t_{1}$ and a feasible target task assignment
$t_{2}$ in a task swapping graph $\Gamma$, find a minimum-length sequence of
adjacent task swappings (i.e., task swappings of swapping distance 1) needed
from $\Gamma_{t_{1}}$ to reach $\Gamma_{t_{2}}$. If $t_{1}=t_{2}$, then
nothing needs to be done. Similarly, a task swapping graph having one vertex
involves no task swapping, which returns the empty sequence. A task swapping
graph of two vertices involves only a single task swapping, which gives an
immediate solution. In the remainder of this paper we assume that $t_{1}\neq
t_{2}$ and a task swapping graph has at least three vertices unless otherwise
stated.
Fig. 4: Task swapping graphs of line topology.
We find the solution of the problem for task swapping graphs of several key
topologies. We first find the solution of the problem for a task swapping
graph of line topology.
Line topology [21, 31, 32, 10] is one of the simplest interconnection network
topologies, where each agent is connected to exactly two neighboring agents
other than the two end agents that are connected to only one neighboring agent
(see Fig. 4). We assume that each agent in a task swapping graph of line
topology is labeled in ascending order from left to right as shown in Fig. 4.
A task swapping graph of line topology having $n$ agents with their task
assignment represented by permutation $p\in\mathfrak{S}_{n}$ is denoted as
$\Gamma_{p}^{L(n)}$. For example, a task swapping graph of Fig. 4(a) is
denoted as $\Gamma_{p_{1}}^{L(8)}$ and a task swapping graph of Fig. 4(b) is
denoted as $\Gamma_{p_{2}}^{L(8)}$, respectively, where
$p_{1}=2\,5\,6\,3\,1\,4\,8\,7\in\mathfrak{S}_{8}$ and
$p_{2}=1\,6\,2\,3\,4\,5\,7\,8\in\mathfrak{S}_{8}$. Recall that a right
multiplication of permutation $p$ by transposition $(i\,i+1)$ exchanges the
values in position $i$ and position $i+1$ of $p$. Since only adjacent task
swappings are allowed, we use the generating set $S_{1}=\\{(i\,i+1):1\leq
i<n\\}$ to find a minimum-length sequence of adjacent task swappings needed
from $\Gamma_{\pi_{1}}^{L(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{L(n)}$ for $\pi_{2}\in\mathfrak{S}_{n}$. For example, the
number of the minimum adjacent task swappings needed from
$\Gamma_{p_{1}}^{L(8)}$ in Fig. 4(a) to reach $\Gamma_{p_{2}}^{L(8)}$ in Fig.
4(b) is the minimum length of a permutation factorization of
${p_{1}}^{-1}p_{2}$ using the generating set $S_{1}^{\prime}=\\{(i\,i+1):1\leq
i<8\\}$. Observe that ${p_{1}}^{-1}p_{2}$ takes $p_{1}$ to $p_{2}$, i.e.,
$p_{1}({p_{1}}^{-1}p_{2})=(p_{1}{p_{1}}^{-1})p_{2}=p_{2}$. Equivalently, it is
viewed as taking $p_{2}^{-1}p_{1}$ to the identity permutation $I$, i.e.,
$p_{2}^{-1}p_{1}({p_{1}}^{-1}p_{2})=I$. Thus, a minimum-length permutation
factorization of ${p_{1}}^{-1}p_{2}$ using the generating set $S_{1}^{\prime}$
corresponds to a shortest path from vertex $p_{2}^{-1}p_{1}$ to vertex $I$ in
the bubble sort Cayley graph $\text{BS}_{8}$. Thus, it is reduced to find a
shortest path from $p_{2}^{-1}p_{1}=3\,6\,2\,4\,1\,5\,8\,7$ to permutation $I$
in the bubble sort Cayley graph $\text{BS}_{8}$:
$3\,6\,2\,4\,1\,5\,8\,7\xrightarrow{<2,3>}3\,2\,6\,4\,1\,5\,8\,7$$\xrightarrow{<3,4>}3\,2\,4\,6\,1\,5\,8\,7\xrightarrow{<4,5>}3\,2\,4\,1\,6\,5\,8\,7$$\xrightarrow{<5,6>}3\,2\,4\,1\,5\,6\,8\,7\xrightarrow{<7,8>}3\,2\,4\,1\,5\,6\,7\,8$$\xrightarrow{<1,2>}2\,3\,4\,1\,5\,6\,7\,8\xrightarrow{<3,4>}2\,3\,1\,4\,5\,6\,7\,8$$\xrightarrow{<2,3>}2\,1\,3\,4\,5\,6\,7\,8\xrightarrow{<1,2>}1\,2\,3\,4\,5\,6\,7\,8$,
where the label of each arrow denotes an edge in $\text{BS}_{8}$. Therefore,
${p_{1}}^{-1}p_{2}={(p_{2}^{-1}p_{1})}^{-1}=(2\,3)(3\,4)(4\,5)(5\,6)(7\,8)(1\,2)(3\,4)(2\,3)(1\,2)$.
It follows that the resulting minimum-length sequence of adjacent task
swappings needed from $\Gamma_{p_{1}}^{L(8)}$ to reach $\Gamma_{p_{2}}^{L(8)}$
is $((2\,3)$, $(3\,4)$, $(4\,5)$, $(5\,6)$, $(7\,8)$, $(1\,2)$, $(3\,4)$,
$(2\,3)$, $(1\,2))$. It is interpreted as a sequence of task swappings so that
a task swapping between agents in the first term (agent 2 and agent 3) is
followed by a task swapping between agents in the second term (agent 3 and
agent 4), and so on, until arriving at a task swapping between agents in the
last term (agent 1 and agent 2) of the sequence. We see that the length of the
sequence is 9, implying that at least 9 adjacent task swappings are needed
from $\Gamma_{p_{1}}^{L(8)}$ to reach $\Gamma_{p_{2}}^{L(8)}$. Algorithm 1
describes the procedure of converting $\Gamma_{\pi_{1}}^{L(n)}$ into
$\Gamma_{\pi_{2}}^{L(n)}$ by using a minimum-length sequence of adjacent task
swappings. It is known from group theory that the minimum length of
permutation $p\in\mathfrak{S}_{n}$ using the generating set $S_{1}$ is the
_inversion number_ [33, 28] of $p$, where the inversion number of $p$ is
defined as $|\\{(i,j):i<j,p(i)>p(j)\\}|$. The maximum inversion number of
permutations of $n$ elements is $n(n-1)/2$, which corresponds to permutation
$n\,n-1\,\cdots\,2\,1$ [33]. Now, the minimum number of adjacent task
swappings needed from $\Gamma_{\pi_{1}}^{L(n)}$ for
$\pi_{1}\in\mathfrak{S}_{n}$ to reach $\Gamma_{\pi_{2}}^{L(n)}$ for
$\pi_{2}\in\mathfrak{S}_{n}$ is the inversion number of
${\pi_{1}}^{-1}\pi_{2}$ (or ${\pi_{2}}^{-1}\pi_{1})$. It follows that the
least upper bound of the minimum number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{L(n)}$ to reach $\Gamma_{\pi_{2}}^{L(n)}$ is $n(n-1)/2$,
which is the maximum inversion number of permutations of $n$ elements. We also
see that it coincides the diameter of bubble sort Cayley graph
$\text{BS}_{n}$.
Input: A source and a target task assignment in a task swapping graph of line
topology $\Gamma_{\pi_{1}}^{L(n)}$ and $\Gamma_{\pi_{2}}^{L(n)}$,
respectively.
Output: A minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{L(n)}$ to reach $\Gamma_{\pi_{2}}^{L(n)}$.
begin
Find a minimum-length permutation factorization of $\pi_{1}^{-1}\pi_{2}$ using
a shortest path from vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ in the bubble-
sort Cayley graph $\text{BS}_{n}$;
Obtain a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{L(n)}$ to reach $\Gamma_{\pi_{2}}^{L(n)}$ by using the
above permutation factorization of $\pi_{1}^{-1}\pi_{2}$;
end
Algorithm 1 A task reassignment by using adjacent task swappings in a task
swapping graph of line topology
In Algorithm 1 222An alternative method to find a minimum-length permutation
factorization of permutation $\pi\in\mathfrak{S}_{n}$ using the generating set
$S_{1}=\\{(i\,i+1):1\leq i<n\\}$ is to apply _numbers game_ [34, 35] of finite
Coxeter group of type $A_{n-1}$ [36, 33]. An interested reader may refer to
[34, 35, 36, 33] for further details., we do not need to generate every
bubble-sort Cayley graph $\text{BS}_{n}$ to find a shortest path from vertex
$\pi_{2}^{-1}\pi_{1}$ to vertex $I$. If a right multiplication of
$\pi_{2}^{-1}\pi_{1}$ by an adjacent transposition reduces an inversion number
by 1, the vertex of the resulting permutation comes closer to vertex $I$ in
terms of a distance in $\text{BS}_{n}$ in which permutation $I$ has the 0
inversion number [33, 26]. By swapping adjacent out-of-order elements in the
permutation using the bubble-sort algorithm, it reduces an inversion number by
1. Therefore, we may apply a bubble-sort algorithm of $O(n^{2})$ complexity
[28] to $\pi_{2}^{-1}\pi_{1}$ in order to keep track of a shortest path from
vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ as shown by the example in this
section. Note that a minimum-length sequence of adjacent task swappings needed
from $\Gamma_{\pi_{1}}^{L(n)}$ to reach $\Gamma_{\pi_{2}}^{L(n)}$ is not
necessarily unique. However, the length of a minimum-length sequence is
unique, which is the inversion number of permutation $\pi_{2}^{-1}\pi_{1}$. It
is also the distance from vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ in
$\text{BS}_{n}$.
Fig. 5: Task swapping graphs of star topology.
###### Proposition 4.1.
An upper bound for the minimum number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{L(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{L(n)}$ for $\pi_{2}\in\mathfrak{S}_{n}$ is $n(n-1)/2$.
###### Proof 1.
Reduction from the diameter of Cayley graph $\text{BS}_{n}$ [27, 25] and from
Algorithm 1. ∎
We next discuss a task swapping graph of star topology. A star topology
consists of a supervisor agent and worker agents, where a supervisor agent
communicates directly to worker agents and each worker agent communicates
indirectly to other worker agent(s) [37, 21, 38]. We assume that a task
swapping graph of star topology having $n\,(n\geq 3)$ agents is labeled in
such a way that a supervisor agent is labeled 1 and worker agents are labeled
in ascending order (clockwise) starting from 2 to $n$. A task swapping graph
of star topology having $n$ agents with their task assignment represented by
permutation $p\in\mathfrak{S}_{n}$ is denoted as $\Gamma_{p}^{S(n)}$. For
example, a task swapping graph of Fig. 5(a) is denoted as
$\Gamma_{p_{1}}^{S(9)}$ and a task swapping graph of Fig. 5(b) is denoted as
$\Gamma_{p_{2}}^{S(9)}$, respectively, where
$p_{1}=5\,4\,2\,1\,6\,9\,7\,8\,3\in\mathfrak{S}_{9}$ and
$p_{2}=7\,8\,9\,2\,4\,5\,1\,6\,3\in\mathfrak{S}_{9}$. Let
$p\in\mathfrak{S}_{n}$ be a permutation representing a task assignment in
$\Gamma_{p}^{S(n)}$. We see that a right multiplication of permutation $p$ by
a _star transposition_ [39, 40] $(1\,i)$ for $2\leq i\leq n$ exchanges the
values in position $1$ (a supervisor agent’s position) of $p$ and position $i$
(a worker agent’s position) of $p$. Therefore, to find a minimum-length
sequence of adjacent task swappings needed from $\Gamma_{\pi_{1}}^{S(n)}$ for
$\pi_{1}\in\mathfrak{S}_{n}$ to reach $\Gamma_{\pi_{2}}^{S(n)}$ for
$\pi_{2}\in\mathfrak{S}_{n}$ is equivalent to finding a minimum-length
permutation factorization of $\pi_{1}^{-1}\pi_{2}$ using the generating set
$S_{3}=\\{(1\,i):2\leq i\leq n\\}$. Observe that every non-identity
permutation is denoted as a product of disjoint cycles, one of which includes
element 1. Therefore, a non-identity permutation $\pi\in\mathfrak{S}_{n}$ is
expressed as $\pi=(1\,q_{2}\cdots q_{s})(p_{1}^{1}\cdots
p_{l_{1}}^{1})\cdots(p_{1}^{m}\cdots p_{l_{m}}^{m})\in\mathfrak{S}_{n}$ if
$s\geq 2$ [40]. If a cycle including element 1 is a cycle of length 1, then
permutation $\pi\in\mathfrak{S}_{n}$ is expressed as $\pi=(p_{1}^{1}\cdots
p_{l_{1}}^{1})\cdots(p_{1}^{m}\cdots p_{l_{m}}^{m})\in\mathfrak{S}_{n}$. It is
easily verified that $(1\,q_{2}\cdots q_{s})$ in $\pi$ for $s\geq 2$ is
factorized into $(1\,q_{s})(1\,q_{s-1})\cdots(1\,q_{2})$ using the generating
set $S_{3}$, whose length is $s-1$. Similarly, if $l_{k}\geq 2$, then
$(p_{1}^{k}\cdots p_{l_{k}}^{k})$ in $\pi$ is factorized into
$(1\,p_{1}^{k})$$(1\,p_{l_{k}}^{k})(1\,p_{l_{k}-1}^{k})\cdots(1\,p_{1}^{k})$
of length $l_{k}+1$ using the generating set $S_{3}$ [40]. It turns out these
factorizations are minimal in the sense that no other way of factorizations
can have less length when using the generating set $S_{3}$ [40]. Therefore,
the minimum length of the above $\pi\in\mathfrak{S}_{n}$ using the generating
set $S_{3}$ is $n+m-k-1$, where $k$ is the number of cycle(s) of length 1. For
example, if $\pi=(1\,2\,3\,4)(5\,6\,7)\in\mathfrak{S}_{8}$, then $\pi$ is
factorized into $\pi=(1\,4)(1\,3)(1\,2)(1\,5)(1\,7)(1\,6)(1\,5)$ using the
generating set $S_{3}^{\prime}=\\{(1\,i):2\leq i\leq 8\\}$, whose length is
$8+1-1-1=7$.
Input: A source and a target task assignment in a task swapping graph of star
topology $\Gamma_{\pi_{1}}^{S(n)}$ and $\Gamma_{\pi_{2}}^{S(n)}$,
respectively.
Output: A minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{S(n)}$ to reach $\Gamma_{\pi_{2}}^{S(n)}$.
begin
Compute $\pi_{1}^{-1}\pi_{2}$ and set it as $\pi$;
Denote $\pi$ as a product of disjoint cycles such that $(1\,q_{2}\cdots
q_{s})(p_{1}^{1}\cdots p_{l_{1}}^{1})\cdots(p_{1}^{m}\cdots
p_{l_{m}}^{m})\in\mathfrak{S}_{n}$ if $s\geq 2$. Denote $\pi$ as
$(p_{1}^{1}\cdots p_{l_{1}}^{1})\cdots(p_{1}^{m}\cdots
p_{l_{m}}^{m})\in\mathfrak{S}_{n}$ otherwise;
if _$s\geq 3$_ then
$(1\,q_{2}\cdots q_{s})$ in $\pi$ is factorized into
$(1\,q_{s})(1\,q_{s-1})\cdots(1\,q_{2})$;
end if
for _$k\leftarrow 1$ to $m$_ do
if _$l_{k}\geq 2$_ then
$(p_{1}^{k}\cdots p_{l_{k}}^{k})$ in $\pi$ is factorized into
$(1\,p_{1}^{k})$$(1\,p_{l_{k}}^{k})(1\,p_{l_{k}-1}^{k})\cdots(1\,p_{1}^{k})$;
end if
end for
Obtain a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{S(n)}$ to reach $\Gamma_{\pi_{2}}^{S(n)}$ by using the
above permutation factorization of $\pi_{1}^{-1}\pi_{2}$;
end
Algorithm 2 A task reassignment by using adjacent task swappings in a task
swapping graph of star topology
Algorithm 2 describes the procedure of converting $\Gamma_{\pi_{1}}^{S(n)}$
into $\Gamma_{\pi_{2}}^{S(n)}$ by using a minimum-length sequence of adjacent
task swappings. For example, we find a minimum-length sequence of adjacent
task swappings needed from $\Gamma_{p_{1}}^{S(9)}$ to reach
$\Gamma_{p_{2}}^{S(9)}$ in Fig. 5, where
$p_{1}=5\,4\,2\,1\,6\,9\,7\,8\,3\in\mathfrak{S}_{9}$ and
$p_{2}=7\,8\,9\,2\,4\,5\,1\,6\,3\in\mathfrak{S}_{9}$. A simple computation
shows that
$p_{1}^{-1}p_{2}=7\,8\,6\,3\,2\,1\,4\,5\,9=(1\,7\,4\,3\,6)(2\,8\,5)\in\mathfrak{S}_{9}$.
We factorize $p_{1}^{-1}p_{2}$ into a product of star transpositions, i.e.,
$p_{1}^{-1}p_{2}=(1\,6)(1\,3)(1\,4)(1\,7)(1\,2)(1\,5)(1\,8)(1\,2)$ by applying
Algorithm 2. Now, a minimum-length sequence of adjacent task swappings needed
from $\Gamma_{p_{1}}^{S(9)}$ to reach $\Gamma_{p_{2}}^{S(9)}$ is $((1\,6)$,
$(1\,3)$, $(1\,4)$, $(1\,7)$, $(1\,2)$, $(1\,5)$, $(1\,8)$, $(1\,2))$ of
length 8.
Observe that a sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{S(n)}$ to reach $\Gamma_{\pi_{2}}^{S(n)}$ in Algorithm 2
corresponds to a path from vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ in a
star graph $\text{ST}_{n}$. Therefore, an upper bound of the minimum number of
adjacent task swappings needed from $\Gamma_{\pi_{1}}^{S(n)}$ to reach
$\Gamma_{\pi_{2}}^{S(n)}$ in Algorithm 2 is the diameter of a star graph
$\text{ST}_{n}$, which is $\lfloor 3(n-1)/2\rfloor$ [26, 27].
###### Proposition 4.2.
An upper bound for the minimum number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{S(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{S(n)}$ for $\pi_{2}\in\mathfrak{S}_{n}$ is $\lfloor
3(n-1)/2\rfloor$.
###### Proof 2.
Reduction from the diameter of Cayley graph $\text{ST}_{n}$ [27, 26] and from
Algorithm 2. ∎
Fig. 6: Task swapping graphs of complete topology.
A task swapping graph of complete topology is a fully-connected task swapping
graph in which each agent has direct links with all other agents in the
topology. Although the complete topology provides redundancy in terms of
communication links between pairs of agents, the cost is often too high when
setting up communication links between agents in the topology (i.e.,
$n(n-1)/2$ total communication links are required for $n$ agents in the
complete topology) [37, 21]. A task swapping graph of complete topology having
$n$ agents with their task assignment represented by permutation
$p\in\mathfrak{S}_{n}$ is denoted as $\Gamma_{p}^{C(n)}$. For example, a task
swapping graph of Fig. 6(a) is denoted as $\Gamma_{p_{1}}^{C(6)}$ and a task
swapping graph of Fig. 6(b) is denoted as $\Gamma_{p_{2}}^{C(6)}$,
respectively, where $p_{1}=1\,3\,4\,2\,6\,5\in\mathfrak{S}_{6}$ and
$p_{2}=1\,5\,6\,3\,2\,4\in\mathfrak{S}_{6}$. Since each agent has direct links
with all other agents in the complete topology, an adjacent task swapping may
occur between any pair of agents in the topology. Now, to find a minimum-
length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{C(n)}$ to reach $\Gamma_{\pi_{2}}^{C(n)}$ is reduced to
find a minimum-length permutation factorization of $\pi_{1}^{-1}\pi_{2}$ using
the generating set $S_{4}=\\{(i\,j):1\leq i<j\leq n\\}$. Verify that each
cycle $(q_{1}\,q_{2}\cdots q_{l})$ of length $l>2$ can be factorized into a
product of $l-1$ transpositions
$(q_{1}\,q_{l})(q_{1}\,q_{l-1})\cdots(q_{1}\,q_{2})$. It is known from group
theory that a cycle of length $l>2$ cannot be written as a product of fewer
than $l-1$ transpositions in the generating set $S_{4}$ [41, 42]. Thus, a
minimum-length permutation factorization of $\pi_{1}^{-1}\pi_{2}$ is obtained
by first denoting it as a product of disjoint cycles, then factorizing all the
cycle(s) of length greater than 2 using the generating set $S_{4}$ as
described.
Input: A source and a target task assignment in a task swapping graph of
complete topology $\Gamma_{\pi_{1}}^{C(n)}$ and $\Gamma_{\pi_{2}}^{C(n)}$,
respectively.
Output: A minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{C(n)}$ to reach $\Gamma_{\pi_{2}}^{C(n)}$.
begin
Compute $\pi_{1}^{-1}\pi_{2}$ and set it as $\pi$;
Denote $\pi$ as a product of disjoint cycles such that $(q_{1}^{1}\,\cdots
q_{l_{1}}^{1})(q_{1}^{2}\cdots q_{l_{2}}^{2})\cdots(q_{1}^{m}\cdots
q_{l_{m}}^{m})\in\mathfrak{S}_{n}$;
for _$k\leftarrow 1$ to $m$_ do
if _$l_{k}\geq 3$_ then
$(q_{1}^{k}\,q_{2}^{k}\cdots q_{l}^{k})$ in $\pi$ is factorized into
$(q_{1}^{k}\,q_{l}^{k})\cdots(q_{1}^{k}\,q_{2}^{k})$;
end if
end for
Obtain a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{C(n)}$ to reach $\Gamma_{\pi_{2}}^{C(n)}$ by using the
above permutation factorization of $\pi_{1}^{-1}\pi_{2}$;
end
Algorithm 3 A task reassignment by using adjacent task swappings in a task
swapping graph of complete topology
Algorithm 3 describes the procedure of converting $\Gamma_{\pi_{1}}^{C(n)}$
into $\Gamma_{\pi_{2}}^{C(n)}$ by using a minimum-length sequence of adjacent
task swappings. We see that a minimum length of permutation
$\pi\in\mathfrak{S}_{n}$ using the generating set $S_{4}$ is $n-r$, where
$\pi$ consists of $r$ disjoint cycles.
Now, we find a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{p_{1}}^{C(6)}$ to reach $\Gamma_{p_{2}}^{C(6)}$ in Fig. 6, where
$p_{1}=1\,3\,4\,2\,6\,5\in\mathfrak{S}_{6}$ and
$p_{2}=1\,5\,6\,3\,2\,4\in\mathfrak{S}_{6}$. A direct computation shows that
$p_{1}^{-1}p_{2}=1\,6\,5\,2\,4\,3=(2\,6\,3\,5\,4)\in\mathfrak{S}_{6}$. Then,
we factorize $p_{1}^{-1}p_{2}$ as a product of transpositions, i.e.,
$p_{1}^{-1}p_{2}=(2\,4)$$(2\,5)$$(2\,3)$$(2\,6)$ by applying Algorithm 3.
Therefore, a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{p_{1}}^{C(6)}$ to reach $\Gamma_{p_{2}}^{C(6)}$ is $((2\,4)$,
$(2\,5)$, $(2\,3)$, $(2\,6))$ of length 4.
In Section 3 we discussed that a Cayley graph of $\mathfrak{S}_{n}$ generated
by $S_{4}=\\{(i\,j):1\leq i<j\leq n\\}$ is the complete transposition graph
$\text{CT}_{n}$. It follows that a sequence of adjacent task swappings needed
from $\Gamma_{\pi_{1}}^{C(n)}$ to reach $\Gamma_{\pi_{2}}^{C(n)}$ in Algorithm
3 corresponds to a path from vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ in the
complete transposition graph $\text{CT}_{n}$. To find a shortest path from
vertex $\pi_{2}^{-1}\pi_{1}$ to vertex $I$ in $\text{CT}_{n}$ [27], one may
apply a greedy algorithm to transpose and locate each element in the
permutation to its homed position iteratively from left to right until
arriving at $I$. For example, a shortest path from vertex
$p_{2}^{-1}p_{1}={(p_{1}^{-1}p_{2})}^{-1}=1\,4\,6\,5\,3\,2\in\mathfrak{S}_{6}$
to vertex $I$ in $\text{CT}_{6}$ is as follows:
$1\,4\,6\,5\,3\,2\xrightarrow{<2,4>}1\,5\,6\,4\,3\,2\xrightarrow{<2,5>}1\,3\,6\,4\,5\,2$$\xrightarrow{<2,3>}1\,6\,3\,4\,5\,2\xrightarrow{<2,6>}1\,2\,3\,4\,5\,6$,
where the label of each arrow denotes an edge in $\text{CT}_{6}$. Therefore,
${p_{1}}^{-1}p_{2}={(p_{2}^{-1}p_{1})}^{-1}=(2\,4)$$(2\,5)$$(2\,3)$$(2\,6)$,
which coincides the above permutation factorization of ${p_{1}}^{-1}p_{2}$
using the generating set $S_{4}$.
It follows that an upper bound for the minimum number of adjacent task
swappings needed from $\Gamma_{\pi_{1}}^{C(n)}$ to reach
$\Gamma_{\pi_{2}}^{C(n)}$ in Algorithm 3 is the diameter of a complete
transposition graph $\text{CT}_{n}$, which is $n-1$ [27].
###### Proposition 4.3.
An upper bound for the minimum number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{C(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{C(n)}$ for $\pi_{2}\in\mathfrak{S}_{n}$ is $n-1$.
###### Proof 3.
Reduction from the diameter of Cayley graph $\text{CT}_{n}$ [27, 25] and from
Algorithm 3. ∎
Fig. 7: Task swapping graphs of complete bipartite topology.
Recall that a graph is _bipartite_ if its vertex set admits a partition into
two classes in such a way that every edge has its ends in two different
classes [43, 44]. A _complete bipartite graph_ [44] is a bipartite graph in
which every two vertices from two different classes are adjacent. A task
swapping graph of _complete bipartite topology_ [37, 45] is a complete
bipartite task swapping graph, where agents are grouped in two layers or
classes (the upper and lower) in such a way that agents between different
layers are fully-connected, while agents in the same layer are not directly
connected. The agents at the upper layer have a centralized control over the
agents at the lower layer in a distributed manner in this topology. We assume
that each agent in a task swapping graph of complete bipartite topology is
labeled in ascending order from the upper left to bottom right as shown in
Fig. 7. A task swapping graph of complete bipartite topology having $n$ agents
with their task assignment represented by permutation $p\in\mathfrak{S}_{n}$
is denoted as $\Gamma_{p}^{B(n,k)}$, where $k$ is the number of agents at the
upper layer in $\Gamma_{p}^{B(n,k)}$. We call $k$ as _bipartite index_. As
with other task swapping graphs, we assume $n\geq 3$ for
$\Gamma_{p}^{B(n,k)}$. We also assume $1\leq k<n$ for $\Gamma_{p}^{B(n,k)}$.
Note that $\Gamma_{p}^{B(n,k)}$ is simply $\Gamma_{p}^{S(n)}$ when bipartite
index $k$ is 1. For example, the task swapping graph of Fig. 7(a) is denoted
as $\Gamma_{p_{1}}^{B(8,3)}$ and the task swapping graph of Fig. 7(b) is
denoted as $\Gamma_{p_{2}}^{B(8,3)}$, respectively, where
$p_{1}=3\,7\,1\,6\,5\,4\,8\,2\in\mathfrak{S}_{8}$ and
$p_{2}=2\,4\,7\,8\,6\,1\,3\,5\in\mathfrak{S}_{8}$. We see that a right
multiplication of permutation $p_{1}\in\mathfrak{S}_{8}$ by transposition
$(1\,4)$ represents an adjacent task swapping between agent 1 and agent 4 in
$\Gamma_{p_{1}}^{B(8,3)}$, while a right multiplication of permutation
$p_{1}\in\mathfrak{S}_{8}$ by transposition $(1\,2)$ does not represent an
adjacent task swapping in $\Gamma_{p_{1}}^{B(8,3)}$. Therefore, the generating
set $S_{5}^{\prime}=\\{(i\,j):1\leq i\leq 3<j\leq 8\\}$ is required for
finding a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{p_{1}}^{B(8,3)}$ to reach $\Gamma_{p_{2}}^{B(8,3)}$.
We next find a minimum-length permutation factorization of
$\pi_{1}^{-1}\pi_{2}$ using the generating set $S_{5}=\\{(i\,j):1\leq i\leq
k<j\leq n\\}$ by which we obtain a minimum-length sequence of adjacent task
swappings needed from $\Gamma_{\pi_{1}}^{B(n,k)}$ for
$\pi_{1}\in\mathfrak{S}_{n}$ to reach $\Gamma_{\pi_{2}}^{B(n,k)}$ for
$\pi_{2}\in\mathfrak{S}_{n}$. We consider several types of cycles to factorize
$\pi_{1}^{-1}\pi_{2}$ using the generating set $S_{5}=\\{(i\,j):1\leq i\leq
k<j\leq n\\}$. Set $\pi=\pi_{1}^{-1}\pi_{2}$ and represent it as a product of
(commutative) disjoint cycles $C_{s}$ for $1\leq s\leq t$ such that
$\pi=C_{1}\cdots C_{i}C_{i+1}\cdots C_{e}C_{e+1}\cdots C_{t}$, where each
element in each cycle of $C_{1}\cdots C_{i}$ is less than or equal to
(bipartite index) $k$, each element in each cycle of $C_{i+1}\cdots C_{e}$ is
greater than $k$, and each cycle of $C_{e+1}\cdots C_{t}$ has both element(s)
less than or equal to $k$ and element(s) greater than $k$. We first consider
the first type of a cycle, referred to as an _internal cycle_ [45], which is a
cycle in $C_{1}\cdots C_{i}$ of $\pi$. Let $C_{x}$ be an internal cycle such
that $C_{x}=(c_{1}\,c_{2}\,\cdots\,c_{v})$, where $1\leq c_{u}\leq k$ for
$1\leq u\leq v$. Then, $C_{x}$ is factorized into
$(c_{1}\,t)(c_{v}\,t)\cdots(c_{2}\,t)(c_{1}\,t)$ using the generating set
$S_{5}=\\{(i\,j):1\leq i\leq k<j\leq n\\}$, where $t$ is an arbitrary number
satisfying $k<t\leq n$. We next consider the second type of a cycle, referred
to as an _external cycle_ [45], which is a cycle in $C_{i+1}\cdots C_{e}$ of
$\pi$. Let $C_{y}$ be an external cycle, i.e.,
$C_{y}=(c_{1}\,\cdots\,c_{q-1}\,c_{q})$, where $k<c_{p}\leq n$ for $1\leq
p\leq q$. Then, $C_{y}=(c_{1}\,\cdots\,c_{q-1}\,c_{q})$ is factorized into
$(u\,c_{q})(u\,c_{q-1})\cdots(u\,c_{1})(u\,c_{q})$ using the generating set
$S_{5}$, where $u$ is an arbitrary number satisfying $1\leq u\leq k$. Now, we
consider the final type of a cycle, referred to as a _mixed cycle_ [45], which
is a cycle in $C_{e+1}\cdots C_{t}$ of $\pi$. Each mixed cycle $C_{m}$ for
$e+1\leq m\leq t$ is written as $(E_{m_{1}}E_{m_{2}}\cdots E_{m_{s}})$, where
each $E_{m_{i}}$ for $1\leq i\leq s$ can be denoted as a concatenation of two
blocks of numbers [45]. Each number of the first block of $E_{m_{i}}$ for
$1\leq i\leq s$ is less than or equal to $k$, while each number of the second
block of $E_{m_{i}}$ for $1\leq i\leq s$ is greater than $k$. We call
$E_{m_{i}}$ as a _simple cycle_. For example, a mixed cycle
$C_{m}^{\prime}=(1\,3\,4\,5\,7\,2\,6\,8)\in\mathfrak{S}_{8}$ for $k=3$ is
written as $(E_{m_{1}}^{\prime}E_{m_{2}}^{\prime})$, where
$E_{m_{1}}^{\prime}=1\,3\,4\,5\,7$ and $E_{m_{2}}^{\prime}=2\,6\,8$. It
follows that the first block of $E_{m_{1}}^{\prime}$ is $1\,3$, while the
second block of $E_{m_{1}}^{\prime}$ is $4\,5\,7$.
Let $E_{m_{i}}=i_{1}\,i_{2}\cdots i_{a}\,j_{1}\,j_{2}\cdots j_{b}$, where
$1\leq i_{u}\leq k$ for $1\leq u\leq a$ and $k<j_{v}\leq n$ for $1\leq v\leq
b$. Observe that $(E_{m_{i}})$ is factorized into
$(i_{1}\,j_{b})\cdots(i_{1}\,j_{2})(i_{a}\,j_{1})\cdots(i_{2}\,j_{1})(i_{1}\,j_{1})$
using the generating set $S_{5}$. For example,
$(1\,3\,4\,5\,7)\in\mathfrak{S}_{8}$ is factorized into
$(1\,7)(1\,5)(3\,4)(1\,4)$ using the generating set
$S_{5}^{\prime}=\\{(i\,j):1\leq i\leq 3<j\leq 8\\}$. Further, observe that a
mixed cycle $C_{m}^{\prime}=(E_{m_{1}}^{\prime}E_{m_{2}}^{\prime})$ is written
as $(1\,2)(E_{m_{1}}^{\prime})(E_{m_{2}}^{\prime})$, where 1 in $(1\,2)$ is
the first element in $E_{m_{1}}^{\prime}$ and 2 in $(1\,2)$ is the first
element in $E_{m_{2}}^{\prime}$. Thus,
$C_{m}^{\prime}=(E_{m_{1}}^{\prime}E_{m_{2}}^{\prime})=(1\,2)(E_{m_{1}}^{\prime})(E_{m_{2}}^{\prime})=(1\,2)(1\,7)(1\,5)(3\,4)(1\,4)(2\,8)(2\,6)$.
However, $(1\,2)$ in $C_{m}^{\prime}$ is not a transposition in
$S_{5}^{\prime}$. Therefore, we use a transposition $(1\,7)$ next to $(1\,2)$
and convert $(1\,2)(1\,7)$ into $(1\,7)(2\,7)$ in which transposition $(1\,7)$
and transposition $(2\,7)$ are transpositions in $S_{5}^{\prime}$.
In general $C_{m}=(E_{m_{1}}E_{m_{2}}\cdots E_{m_{s}})$ is recursively
factorized into $C_{m}=(v_{1}\,v_{2})(E_{m_{1}})(E_{m_{2}}\cdots E_{m_{s}})$,
where $v_{1}$ is the first element of $E_{m_{1}}$ and $v_{2}$ is the first
element of $E_{m_{2}}$ [45]. Let $(w_{1}\,w_{2})$ be the first transposition
of a factorization of $E_{m_{1}}$ using the generating set $S_{5}$. Then, we
have $v_{1}=w_{1}$. Now, we see that $E_{m_{1}}$ is factorized using
transpositions in the generating set $S_{5}$ such that
$(v_{1}\,v_{2})(w_{1}\,w_{2})$ with $v_{1}=w_{1}$ is rearranged into
$(v_{1}\,w_{2})(v_{2}\,w_{2})$. Therefore, $C_{m}$ is recursively factorized
using transpositions in the generating set $S_{5}$. Furthermore, it turns out
that the above way of factorizing an arbitrary permutation
$p\in\mathfrak{S}_{n}$ using the generating set $S_{5}$ for a given $1\leq
k<n$ yields the length of $p$ not greater than $n-1+\max{(\lfloor
k/2\rfloor,\lfloor(n-k)/2\rfloor)}$ [45]. Algorithm 4 describes the procedure
of converting $\Gamma_{\pi_{1}}^{B(n,k)}$ into $\Gamma_{\pi_{2}}^{B(n,k)}$ by
using a minimum-length sequence of adjacent task swappings. Now, we obtain a
minimum-length sequence of adjacent task swappings needed from a task swapping
graph $\Gamma_{p_{1}}^{B(8,3)}$ in Fig. 7(a) to reach a task swapping graph
$\Gamma_{p_{2}}^{B(8,3)}$ in Fig. 7(b), where
$p_{1}=3\,7\,1\,6\,5\,4\,8\,2\in\mathfrak{S}_{8}$ and
$p_{2}=2\,4\,7\,8\,6\,1\,3\,5\in\mathfrak{S}_{8}$. We first compute
$p_{1}^{-1}p_{2}$, which is
$8\,6\,2\,7\,4\,3\,1\,5=(1\,8\,5\,4\,7)(3\,2\,6)\in\mathfrak{S}_{8}$. Then, we
factorize $p_{1}^{-1}p_{2}$ using the generating set $S_{5}^{\prime}$. Observe
that $p_{1}^{-1}p_{2}$ is the product of simple cycles $(E_{1})(E_{2})$, where
$E_{1}=1\,8\,5\,4\,7$ and $E_{2}=3\,2\,6$. Then,
$(E_{1})=(1\,7)(1\,4)(1\,5)(1\,8)$ and $(E_{2})=(2\,6)(3\,6)$. Therefore, a
minimum-length sequence of adjacent task swapping needed from
$\Gamma_{p_{1}}^{B(8,3)}$ in Fig. 7(a) to reach $\Gamma_{p_{2}}^{B(8,3)}$ in
Fig. 7(b) is $((1\,7),(1\,4),(1\,5),(1\,8),(2\,6),(3\,6))$ of length 6.
Input: A source and a target task assignment in a task swapping graph of
complete bipartite topology $\Gamma_{\pi_{1}}^{B(n,k)}$ and
$\Gamma_{\pi_{2}}^{B(n,k)}$, respectively.
Output: A minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{B(n,k)}$ to reach $\Gamma_{\pi_{2}}^{B(n,k)}$.
begin
Compute $\pi_{1}^{-1}\pi_{2}$ and set it as $\pi$;
Write $\pi$ as a product of disjoint cycles $C_{m}$ for $1\leq m\leq t$ such
that $\pi=C_{1}\cdots C_{i}C_{i+1}\cdots C_{e}C_{e+1}\cdots C_{t}$, where each
cycle of $C_{1}\cdots C_{i}$ is an internal cycle, each cycle of
$C_{i+1}\cdots C_{e}$ is an external cycle, and each cycle of $C_{e+1}\cdots
C_{t}$ is a mixed cycle;
for _$m\leftarrow 1$ to $t$_ do
if _$C_{m}$ is an $r$-cycle for $r\geq 3$_ then
// see Alogorithm 5 BipartiteCycleFactorization ($C_{m}$, $k$);
end if
end for
Obtain a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{B(n,k)}$ to reach $\Gamma_{\pi_{2}}^{B(n,k)}$ by using the
above permutation factorization of $\pi_{1}^{-1}\pi_{2}$;
end
Algorithm 4 A task reassignment by using adjacent task swappings in a task
swapping graph of complete bipartite topology
Input: An $r$-cycle $C_{m}$ ($3\leq r\leq n$) and bipartite index $k$ ($1\leq
k<n$).
Output: A factorization of $C_{m}$ into a product of transpositions in
$S_{5}$.
begin
if _$C_{m}$ is an internal cycle_ then
Let $C_{m}=(c_{1}\,c_{2}\,\cdots\,c_{v})$. Then, $C_{m}$ is factorized into
$(c_{1}\,t)(c_{v}\,t)\cdots(c_{2}\,t)(c_{1}\,t)$, where $t$ is any number in
$k<t\leq n$;
end if
else if _$C_{m}$ is an external cycle_ then
Let $C_{m}=(c_{1}\,\cdots\,c_{q-1}\,c_{q})$. Then, $C_{m}$ is factorized into
$(u\,c_{q})(u\,c_{q-1})\cdots(u\,c_{1})(u\,c_{q})$, where $u$ is any number in
$1\leq u\leq k$.
end if
else // $C_{m}$ is a mixed cycle
if _$C_{m}$ is a simple cycle_ then
Let $C_{m}=(i_{1}\,i_{2}\cdots i_{a}\,j_{1}\,j_{2}\cdots j_{b})$, where $1\leq
i_{u}\leq k$ for $1\leq u\leq a$ and $k<j_{v}\leq n$ for $1\leq v\leq b$.
Then, $C_{m}$ is factorized into
$(i_{1}\,j_{b})\cdots(i_{1}\,j_{2})(i_{a}\,j_{1})\cdots(i_{2}\,j_{1})(i_{1}\,j_{1})$;
end if
else
Let $C_{m}=(E_{m_{1}}E_{m_{2}}\cdots E_{m_{t}})$, where each $(E_{m_{i}})$ for
$1\leq i\leq t$ is a simple cycle. Then,
$C_{m}=(v_{1}\,v_{2})(E_{m_{1}})(E_{m_{2}}\cdots E_{m_{t}})$, where $v_{1}$
and $v_{2}$ are the first elements of $E_{m_{1}}$ and $E_{m_{2}}$,
respectively. Factorize a simple cycle $(E_{m_{1}})$ as indicated above. Let
$(w_{1}\,w_{2})$ be the first transposition of a factorization of $E_{m_{1}}$.
If $v_{1}=w_{1}$, rearrange $(v_{1}\,v_{2})(w_{1}\,w_{2})$ into
$(w_{1}\,w_{2})(v_{2}\,w_{2})$. Otherwise, rearrange
$(v_{1}\,v_{2})(w_{1}\,w_{2})$ into
$(w_{1}\,w_{2})(v_{2}\,w_{2})(v_{1}\,w_{2})(v_{2}\,w_{2})$. Then, $C_{m}$ is
written as a product of transpositions in $S_{5}$ followed by
$(E_{m_{2}}\cdots E_{m_{t}})$. We repeat this process recursively to
$(E_{m_{2}}\cdots E_{m_{t}})$ until we completely factorize $C_{m}$ using
$S_{5}$.
end if
end if
end
Algorithm 5 A bipartite cycle factorization using the generating set
$S_{5}=\\{(i\,j):1\leq i\leq k<j\leq n\\}$ [45]
As discussed in Section 3, a Cayley graph of $\mathfrak{S}_{n}$ generated by
$S_{5}=\\{(i\,j):1\leq i\leq k<j\leq n\\}$ is the generalized star graph
$\text{GST}_{n,k}$. Therefore, an upper bound of the minimum number of
adjacent task swappings needed from $\Gamma_{\pi_{1}}^{B(n,k)}$ for
$\pi_{1}\in\mathfrak{S}_{n}$ to reach $\Gamma_{\pi_{2}}^{B(n,k)}$ for
$\pi_{2}\in\mathfrak{S}_{n}$ in Algorithm 4 is the diameter of a generalized
star graph $\text{GST}_{n,k}$, which is $n-1+\max{(\lfloor
k/2\rfloor,\lfloor(n-k)/2\rfloor)}$ [45].
###### Proposition 4.4.
An upper bound for the minimum number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{B(n,k)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{B(n,k)}$ for $\pi_{2}\in\mathfrak{S}_{n}$ is
$n-1+\max{(\lfloor k/2\rfloor,\lfloor(n-k)/2\rfloor)}$.
###### Proof 4.
Reduction from the diameter of Cayley graph $\text{GST}_{n,k}$ [25, 45], a
bipartite cycle factorization algorithm given in [45], and Algorithm 4. ∎
Fig. 8: Task swapping graphs of ring topology.
A task swapping graph of ring topology is a circular task swapping graph in
which each agent has direct links with exactly two other agents in the
topology. In case any direct link of two agents is removed, a ring topology is
changed into a line topology [21]. We assume that a task swapping graph of
ring topology having $n$ agents for $n\geq 3$ is labeled in such a way that
$n$ agents are labeled clockwise in ascending order starting from 1 to $n$
(see Fig. 8). Now, we denote a task swapping graph of ring topology having $n$
agents with their task assignment represented by permutation
$p\in\mathfrak{S}_{n}$ as $\Gamma_{p}^{R(n)}$. We call the corresponding
permutation $p\in\mathfrak{S}_{n}$ for $\Gamma_{p}^{R(n)}$ as a _circular
permutation_ [46] in which position $i$ of a circular permutation
$p\in\mathfrak{S}_{n}$ is referred to as verex (agent) $i$ for $1\leq i\leq n$
in $\Gamma_{p}^{R(n)}$. For example, a task swapping graph of Fig. 8(a) is
denoted as $\Gamma_{p_{1}}^{R(8)}$ and a task swapping graph of Fig. 8(b) is
denoted as $\Gamma_{p_{2}}^{R(8)}$, respectively, where
$p_{1}=5\,7\,3\,4\,8\,2\,6\,1\in\mathfrak{S}_{8}$ and
$p_{2}=3\,2\,8\,4\,7\,1\,5\,6\in\mathfrak{S}_{8}$.
Now, observe that a minimum-length sequence of adjacent task swappings needed
from $\Gamma_{\pi_{1}}^{R(n)}$ to reach $\Gamma_{\pi_{2}}^{R(n)}$ is obtained
by finding a minimum-length permutation factorization of $\pi_{1}^{-1}\pi_{2}$
using the generating set $S_{6}=\\{(i\,i+1):1\leq i<n\\}\cup\\{(1\,n)\\}$.
Note that adjacent task swappings in $\Gamma_{p}^{R(n)}$ are adjacent task
swappings in $\Gamma_{p}^{L(n)}$ along with an adjacent task swapping between
agent 1 and agent $n$.
A _displacement vector_ [29, 46] $d=(d_{1},d_{2},d_{3},\ldots,d_{n})$ of a
circular permutation $p\in\mathfrak{S}_{n}$ is introduced to sort a circular
permutation $p$ into $I$ using the generating set $S_{6}=\\{(i\,i+1):1\leq
i<n\\}\cup\\{(1\,n)\\}$ by which we obtain a permutation factorization of $p$
using the same generating set. Each component $d_{i}$ in $d$ is defined as
$d_{i}=j-i$, where $p(j)=i$ for $1\leq i,j\leq n$. For any displacement vector
$d=(d_{1},d_{2},d_{3},\ldots,d_{n})$, we have $\sum_{i=1}^{n}{d_{i}}=0$, and
each $d_{i}=0$ if $d$ is a displacement vector of the identity permutation of
$\mathfrak{S}_{n}$. For example, a displacement vector of a circular
permutation $5\,7\,3\,4\,8\,2\,6\,1\in\mathfrak{S}_{8}$ in Fig. 8(a) is
$(7,4,0,0,-4,1,-5,-3)$. Intuitively, each $|d_{i}|$ in $d$ is interpreted as
the length of a path from position (vertex) $i$ to position (vertex) $k$ on
which element $i$ is placed for a circular permutation $p\in\mathfrak{S}_{n}$,
where $d_{i}$ is signed positive if the path from position $i$ to position $k$
is clockwise, and signed negative if the path from position $i$ to position
$k$ is counterclockwise. We denote the corresponding path as
$\text{path}(d_{i})$, which is uniquely determined by $d_{i}$ of its circular
permutation $p\in\mathfrak{S}_{n}$. Let $d_{s}$ be the maximum-valued
component of a displacement vector $d$ of a circular permutation
$p\in\mathfrak{S}_{n}$, and $d_{t}$ be the minimum-valued component of the
displacement vector $d$. Since $\sum_{i=1}^{n}{d_{i}}=0$, $d_{s}$ is greater
than 0 and $d_{t}$ is less than 0 for any non-identity circular permutation
$p\in\mathfrak{S}_{n}$. If $d_{s}-d_{t}>n$ for each pair of indices $s$ and
$t$, then we renew $d_{s}$ as $d_{s}-n$ and $d_{t}$ as $d_{t}+n$,
respectively. This process is called _strictly contracting transformation_
[46]. If a displacment vector $d$ admits no strictly contracting
transformation, we say that a displacement vector $d$ is _stable_ , denoted
$\bar{d}$. For example, the maximum and minimum component values of
displacement vector $d^{\prime}=(7,4,0,0,-4,1,-5,-3)$ of the circular
permutation $5\,7\,3\,4\,8\,2\,6\,1\in\mathfrak{S}_{8}$ in Fig. 8(a) are
$d_{1}^{\prime}=7$ and $d_{7}^{\prime}=-5$, respectively. Since
$d_{1}^{\prime}-d_{7}^{\prime}=12>8$, we renew $d_{1}^{\prime}$ as
$d_{1}^{\prime}=7-8=-1$, and $d_{7}^{\prime}=-5+8=3$. This procedure continues
until we obtain a stable displacement vector $\bar{d}^{\prime}$, i.e., no pair
of maximum-valued component $d_{s}^{\prime}$ and the minimum-valued component
$d_{t}^{\prime}$ of $d^{\prime}$ satisfies $d_{s}^{\prime}-d_{t}^{\prime}>8$.
We leave it for the reader to verify that
$\bar{d}^{\prime}=(-1,4,0,0,-4,1,3,-3)$.
The value $\sum_{i=1}^{n}{|d_{i}|}$ of a stable displacement vector $\bar{d}$
of a circular permutation $p\in\mathfrak{S}_{n}$ is a key indicator of how
close a circular permutation $p\in\mathfrak{S}_{n}$ is to the identity
permutation in terms of a length using the generating set $S_{6}$. Note that
$\sum_{i=1}^{n}{|d_{i}|}$ is not zero for any non-identity permutation in
$\mathfrak{S}_{n}$, while $\sum_{i=1}^{n}{|d_{i}|}$ is 0 for the identity
permutation. By using a stable displacement vector $\bar{d}$, an inversion
number $I(\bar{d})$ is defined as
$I(\bar{d})=|\\{(i,j):(i+\bar{d_{i}}>j+\bar{d_{j}})\cup(i+\bar{d_{i}}+n<j+\bar{d_{j}}),\;1\leq
i<j\leq n\\}|$, which is the minimum length of a permutation factorization of
$p\in\mathfrak{S}_{n}$ using the generating set $S_{6}=\\{(i\,i+1):1\leq
i<n\\}\cup\\{(1\,n)\\}$ [29]. Now, at each step of sorting a circular
permutation $p\in\mathfrak{S}_{n}$ into $I$, we find an adjacent swapping to
reduce an inversion number by 1. Observe that if an adjacent pair of positions
(vertices) $v_{1}$ and $v_{2}$ have elements $s$ and $t$, respectively, such
that $\text{path}(s)$ and $\text{path}(t)$ are directed oppositely having an
intersection of edge $(v_{1},v_{2})$, then swapping elements $s$ and $t$ on
vertices $v_{1}$ and $v_{2}$ reduces an inversion number by 1. Observe also
the case where an adjacent pair of vertices $v_{1}$ and $v_{2}$ have elements
$s$ and $t$, respectively, such that $s$ is homed (i.e., $p(s)=s$) and $t$ is
not homed. If $\text{path}(t)$ crosses vertex $v_{1}$, then swapping elements
$s$ and $t$ on vertices $v_{1}$ and $v_{2}$ reduces an inversion number by 1.
Now, each step of the sorting procedure is to find an adjacent swapping that
reduces an inversion number by 1. As stated earlier in this section, if
$\bar{d}$ is a stable displacement vector of a circular permutation
$p\in\mathfrak{S}_{n}$, then the inversion number $I(\bar{d})$ is the minimum
length of a permutation factorization of $p$ using the generating set $S_{6}$.
Therefore, $I(\bar{d})$ is the minimum number of adjacent swappings required
for sorting a circular permutation $p\in\mathfrak{S}_{n}$ to the identity
permutation $I$ using the generating set $S_{6}$.
Input: A source and a target task assignment in a task swapping graph of ring
topology $\Gamma_{\pi_{1}}^{R(n)}$ and $\Gamma_{\pi_{2}}^{R(n)}$,
respectively.
Output: A minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{R(n)}$ to reach $\Gamma_{\pi_{2}}^{R(n)}$.
// Find a minimum-length permutation factorization of $\pi_{1}^{-1}\pi_{2}$ by
sorting $\pi_{2}^{-1}\pi_{1}$ to the identity permutation $I$ using the
generating set $S_{6}=\\{(i\,i+1):1\leq i<n\\}\cup\\{(1\,n)\\}$ begin
Let $\pi=\pi_{2}^{-1}\pi_{1}\in\mathfrak{S}_{n}$. Find a stable displacement
vector $\bar{d}$ of $\pi$, and calculate the inversion number $I(\bar{d})$;
while _$\pi\neq I$_ do
In a circular permutation $\pi\in\mathfrak{S}_{n}$ in $\Gamma_{\pi}^{R(n)}$,
find an adjacent pair of vertices $v_{1}$ and $v_{2}$ having elements $s$ and
$t$, respectively, such that $\text{path}(s)$ and $\text{path}(t)$ derived
from its stable displacement vector of $\pi$ are directed oppositely having an
intersection of edge $(v_{1},v_{2})$. If no such an adjacent pair exists, then
find an adjacent pair of vertices $v_{1}$ and $v_{2}$ having elements $s$ and
$t$, respectively, such that $s$ is homed (i.e., $\pi(s)=s$) and $t$ is not
homed in which $\text{path}(t)$ crosses vertex $v_{1}$;
Swap elements $s$ and $t$ on vertices $v_{1}$ and $v_{2}$, replacing $\pi$
with the resulting permutation. Compute a stable displacement vector of the
(updated) circular permutation $\pi$;
end while
Obtain a minimum-length sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{R(n)}$ to reach $\Gamma_{\pi_{2}}^{R(n)}$ by using the
above permutation factorization of $\pi_{1}^{-1}\pi_{2}$;
end
Algorithm 6 A task reassignment by using adjacent task swappings in a task
swapping graph of ring topology
Algorithm 6 describes the procedure of converting $\Gamma_{\pi_{1}}^{R(n)}$
into $\Gamma_{\pi_{2}}^{R(n)}$ by using a minimum-length sequence of adjacent
task swappings. By using Algorithm 6, we obtain a minimum-length sequence of
adjacent task swappings needed from a task swapping graph
$\Gamma_{p_{1}}^{R(8)}$ in Fig. 8(a) to reach a task swapping graph
$\Gamma_{p_{2}}^{R(8)}$ in Fig. 8(b), where
$p_{1}=5\,7\,3\,4\,8\,2\,6\,1\in\mathfrak{S}_{8}$ and
$p_{2}=3\,2\,8\,4\,7\,1\,5\,6\in\mathfrak{S}_{8}$. A direct computation shows
that $p_{2}^{-1}p_{1}$ is $7\,5\,1\,4\,3\,2\,8\,6$ and its stable displacement
vector is $(2,-4,2,0,-3,2,2,-1)$. An adjacent swapping between 7th position
(element 8) and 8th position (element 6) of $p_{2}^{-1}p_{1}$ reduces an
inversion number by 1. This sorting procedure is continued by using Algorithm
6 until arriving at the identity permutation (see below):
$7\,5\,1\,4\,3\,2\,8\,6\xrightarrow{<7,8>}7\,5\,1\,4\,3\,2\,6\,8$$\xrightarrow{<2,3>}7\,1\,5\,4\,3\,2\,6\,8\xrightarrow{<6,7>}7\,1\,5\,4\,3\,6\,2\,8$$\xrightarrow{<3,4>}7\,1\,4\,5\,3\,6\,2\,8\xrightarrow{<4,5>}7\,1\,4\,3\,5\,6\,2\,8$$\xrightarrow{<7,8>}7\,1\,4\,3\,5\,6\,8\,2\xrightarrow{<1,8>}2\,1\,4\,3\,5\,6\,8\,7$$\xrightarrow{<1,2>}1\,2\,4\,3\,5\,6\,8\,7\xrightarrow{<7,8>}1\,2\,4\,3\,5\,6\,7\,8$$\xrightarrow{<3,4>}1\,2\,3\,4\,5\,6\,7\,8$.
Now, we have the resulting minimum-length sequence of adjacent task swappings
needed from $\Gamma_{p_{1}}^{R(8)}$ to reach $\Gamma_{p_{2}}^{R(8)}$, which is
$((7\,8)$, $(2\,3)$, $(6\,7)$, $(3\,4)$, $(4\,5)$, $(7\,8)$, $(1\,8)$,
$(1\,2)$, $(7\,8)$, $(3\,4))$ of length 10.
Note that an upper bound of the number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{R(n)}$ to reach $\Gamma_{\pi_{2}}^{R(n)}$ is subject to the
diameter of the modified bubble-sort graph $MBS_{n}$ discussed in Section 3.
To the best of our knowledge, the formula of the diameter of $MBS_{n}$ is not
known [46, 27, 47]. Nevertheless, a minimum-length sequence of sorting an
aribtrary circular permutation $p\in\mathfrak{S}_{n}$ into the identity
permutation $I$ using the generating set $S_{6}=\\{(i\,i+1):1\leq
i<n\\}\cup\\{(1\,n)\\}$ in Algorithm 6 is obtained in polynomial time [29]. It
follows that Algorithm 6 runs in polynomial time as with other algorithms
involving permutation sortings discussed in this paper.
Input: A source and a target task assignment in a task swapping graph of an
arbitrary tree topology $\Gamma_{\pi_{1}}^{T(n)}$ and
$\Gamma_{\pi_{2}}^{T(n)}$, respectively.
Output: A sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{T(n)}$ to reach $\Gamma_{\pi_{2}}^{T(n)}$ in the number
$c(\pi)-n+\sum_{i=1}^{n}{d(i,\pi(i))}$ of steps, where
$\pi=\pi_{2}^{-1}\pi_{1}$, $c(\pi)$ is the number of cycles in $\pi$, and
$d(i,j)$ is the distance between agent $i$ and agent $j$ in
$\Gamma_{\pi}^{T(n)}$.
begin
Let $\pi=\pi_{2}^{-1}\pi_{1}$ and start the procedure of sorting
$\Gamma_{\pi}^{T(n)}$ to $\Gamma_{I}^{T(n)}$;
while _$\pi\neq I$_ do
In $\Gamma_{\pi}^{T(n)}$ find an adjacent pair of agents $a_{1}$ and $a_{2}$
such that their unhomed tasks $t_{1}$ and $t_{2}$, respectively, need to move
toward each other for their homed positions (i.e., $\pi(t_{k})=t_{k}$ for
$k=1$ and $k=2$). Or find an adjacent pair of agents $a_{1}$ and $a_{2}$ such
that its task $t_{1}$ is homed and its task $t_{2}$ is not homed,
respectively, in which task $t_{2}$ needs to move toward and cross agent
$a_{1}$ for its homed position (i.e., $\pi(t_{2})=t_{2}$). Then, swap task
$t_{1}$ on agent $a_{1}$ and task $t_{2}$ on agent $a_{2}$, replacing $\pi$
with the resulting permutation;
end while
Obtain a sequence of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{T(n)}$ to reach $\Gamma_{\pi_{2}}^{T(n)}$ by using the
above sorting procedure;
end
Algorithm 7 A task reassignment by using adjacent task swappings in a task
swapping graph of an arbitrary tree topology
Finally, we discuss a task swapping graph of an arbitrary tree topology that
has not been discussed in this section. A task swapping graph of a tree
topology having $n$ agents with their task assignment represented by
permutation $p\in\mathfrak{S}_{n}$ is denoted as $\Gamma_{p}^{T(n)}$. We
concern the procedure of converting a source task assignment in
$\Gamma_{\pi_{1}}^{T(n)}$ to a target task assignment in
$\Gamma_{\pi_{2}}^{T(n)}$ by using the minimum number of adjacent task
swappings.
Due to Corollary 3.1, we may not always obtain a tight upper bound for the
number of steps to convert $\Gamma_{\pi_{1}}^{T(n)}$ into
$\Gamma_{\pi_{2}}^{T(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ and
$\pi_{2}\in\mathfrak{S}_{n}$. However, we may apply Theorem 3.2 to convert
$\Gamma_{\pi_{1}}^{T(n)}$ into $\Gamma_{\pi_{2}}^{T(n)}$ in the number
$c(\pi)-n+\sum_{i=1}^{n}{d(i,\pi(i))}$ of steps, where
$\pi=\pi_{2}^{-1}\pi_{1}$, $c(\pi)$ is the number of cycles in $\pi$, and
$d(i,j)$ is the distance between position (vertex) $i$ and position (vertex)
$j$ in $\Gamma_{\pi}^{T(n)}$. Algorithm 7 describes the procedure of
converting $\Gamma_{\pi_{1}}^{T(n)}$ into $\Gamma_{\pi_{2}}^{T(n)}$ in the
above number of steps.
###### Proposition 4.5.
An upper bound for the number of adjacent task swappings needed from
$\Gamma_{\pi_{1}}^{T(n)}$ for $\pi_{1}\in\mathfrak{S}_{n}$ to reach
$\Gamma_{\pi_{2}}^{T(n)}$ for $\pi_{2}\in\mathfrak{S}_{n}$ is
$c(\pi)-n+\sum_{i=1}^{n}{d(i,\pi(i))}$, where $\pi=\pi_{2}^{-1}\pi_{1}$ and
$c(\pi)$ is the number of cycles in $\pi$.
###### Proof 5.
Reduction from Theorem 3.2, Corollary 3.1, and Algorithm 7. ∎
## 5 Conclusions
This paper presented task swapping networks of several basic topologies used
in distributed systems. Task swappings between adjacent agents in a network
topology are represented by task swappings of swapping distance 1 in the
corresponding task swapping graph. We considered the situation in which the
total cost of task migrations relies on the number of adjacent task swappings
involved in a given network topology. Minimum generator sequence algorithms
using several known generating sets for $\mathfrak{S}_{n}$ allow us to find a
minimum-length sequence of adjacent task swappings needed from a source task
assignment to reach a target task assignment in a task swapping graph of
several topologies, such as line, star, complete, complete bipartite, and
ring.
Task swapping graphs of the more complex topologies (e.g., 2D and 3D meshes,
hypercubes, etc) along with task swappings of swapping distance $k\geq 2$ have
not been discussed in this paper. We leave them to our future works.
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|
arxiv-papers
| 2011-07-10T11:35:16 |
2024-09-04T02:49:20.374885
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dohan Kim",
"submitter": "Dohan Kim",
"url": "https://arxiv.org/abs/1107.1851"
}
|
1107.1862
|
$Revision:62698$
$HeadURL:svn+ssh://svn.cern.ch/reps/tdr2/papers/XXX-08-000/trunk/XXX-08-000.tex$
$Id:XXX-08-000.tex626982011-06-2100:28:58Zalverson$ CR-2011/097
# Centrality and $\PT$ dependence of charged particle $\RAA$ in PbPb
collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 2.76 TeV
Andre S. Yoon for the CMS collaboration
###### Abstract
The transverse momentum ($p_{T}$) spectra of charged particles is measured by
CMS as a function of collision centrality in PbPb collisions at
$\sqrt{s_{{}_{NN}}}=2.76$ TeV. The results are compared to a pp reference
spectrum, constructed by interpolation between $\sqrt{s}$ = 0.9 and 7 TeV
measurements. The nuclear modification factor ($R_{AA}$) is constructed by
dividing the PbPb $p_{T}$ spectrum, normalized to the number of binary
collisions ($N_{coll}$), by the pp spectrum. Measured $R_{AA}$ in 0–5%
centrality bin is compared to several theoretical predictions.
Presented at QM2011: Quark Matter 2011
## 0.1 Introduction
The charged particle $\PT$ spectrum is an important observable for studying
the properties of the hot, dense medium produced in the collisions of heavy
nuclei. In particular, the modification of the $\PT$ spectrum compared to
nucleon-nucleon collisions at the same energy can shed light on the detailed
mechanism by which hard partons lose energy traversing the medium [1]. The
modification is typically expressed in terms of
$\RAA(\PT)=\frac{d^{2}N_{\mathrm{ch}}^{\mathrm{AA}}/d\PT
d\eta}{\left<T_{AA}\right>d^{2}\sigma_{\mathrm{ch}}^{\mathrm{NN}}/d\PT
d\eta},$ (1)
where $N_{\mathrm{ch}}^{\mathrm{AA}}$ and $\sigma_{\mathrm{ch}}^{\mathrm{NN}}$
represent the charged particle yield in nucleus-nucleus collisions and the
cross section in nucleon-nucleon collisions, respectively. The nuclear overlap
function $\left<T_{AA}\right>$ is the ratio of the number of binary nucleon-
nucleon collisions $\left<N_{coll}\right>$ calculated from a Glauber model of
the nuclear collision geometry. At RHIC, the factor of five suppression seen
in $\RAA$ was an early indication of strong final-state medium effects on
particle production. In this contribution, measured charged particle spectra
and $\RAA$ in bins of centrality for the $\PT$ range of 1–100 $\GeVc$ [2] are
presented.
## 0.2 Experimental Methods
This measurement is based on a data sample corresponding to an integrated
luminosity of 7 $\,\mu\text{b}^{\text{$-$1}}$ collected by the CMS experiment
[3] in 2.76 TeV PbPb collisions. A minimum bias event sample was collected
based on a trigger requiring a coincidence between signals in the +$z$ and
–$z$ sides of either the forward hadronic calorimeters (HF) or the beam
scintillator counters (BSC). To ensure a pure sample of inelastic hadronic
collision events, additional offline selections were performed. These include
a beam halo veto based on the BSC timing, an offline HF coincidence of at
least three towers ($E>3$ GeV) on each side of the interaction point, a
reconstructed vertex consisting of at least two pixel tracks of $\PT>75$
$\MeVc$, and a rejection of beam-scraping events based on the compatibility of
pixel cluster shapes with the reconstructed primary vertex. The collision
event centrality is determined on the total energy deposited in both HF
calorimeters.
In this analysis, data recorded by single-jet triggers with uncorrected
transverse energy thresholds of $E_{T}=35$ GeV (Jet35U) and 50 GeV (Jet50U)
are included in order to extend the statistical reach of the $\PT$ spectra.
The use of the jet-triggered data, which contributes dominantly to the
high-$\PT$ region ($\PT>50$ $\GeVc$), also allows to keep low
misidentification rates of the selected tracks being enforced by an implicit
track-calorimeter matching. The jet-energy thresholds in the trigger are
applied after subtracting the underlying event energy but without correcting
for calorimeter response. $E_{T}$ distributions of the most energetic
reconstructed jet with $|\eta|<2$, referred to as the leading jet, are
normalized per minimum bias event. Following the procedure introduced in the
analogous measurement of the charged particle spectra in 0.9 and 7 TeV pp
collisions [4], the spectra are calculated separately in three ranges of
leading-jet $E_{T}$, each corresponding to a fully efficient trigger path, and
then combined to obtain the final result.
Figure 1: (Left) Interpolated 2.76 TeV charged particle differential
transverse momentum cross section with ratios of combined interpolation to
various predictions and interpolation. (Right) Invariant differential yield in
bins of collision event centrality (symbols), compared to a pp reference
spectrum, scaled by the corresponding number of binary nucleon-nucleon
collisions (dashed lines). The systematic uncertainties on the PbPb
differential yields as a function of $\PT$ for the 0–5% and 5–90% are shown in
the bottom panel.
## 0.3 Results
The interpolated differential cross section in pp at 2.76 TeV [4] is shown in
the upper panel of Fig. 1, and its ratio with respect to various pythia tunes.
Also shown in the lower panel of Fig. 1 is the ratio of the predicted 2.76 TeV
cross section to that found by simply scaling the CMS measured 7 TeV result by
the expected 2.75 TeV to 7 TeV ratio from NLO calculations [5] and the
interpolation used in the recent ALICE publication [6].
The charged particle invariant differential yields are shown for the six
centrality bins in Fig. 1 and compared to the corresponding quantity taken
from the interpolated pp reference spectrum [4], normalized by the number of
binary collisions. The points have been scaled by the factors given in the
figure for easier viewing.
$\RAA$ is presented as a function of transverse momentum in Fig. 2 for each of
the six centrality bins. The shaded areas (boxes) around the points show the
systematic uncertainties including those from the interpolated pp reference
spectrum. An additional systematic uncertainty from the $T_{AA}$
normalization, common to all points is displayed as the shaded band around
unity in each plot. In the most-peripheral events (70–90%), a moderate
suppression of about two ($\RAA=0.5$) is observed at low $\PT$ with $\RAA$
rising gently with increasing transverse momentum. The suppression is
increasingly pronounced in the more-central collisions, as expected from the
longer average path lengths traversed by hard-scattered partons as they lose
energy via jet quenching. In the 0–5% bin, $\RAA$ reaches a minimum value of
0.13 around 6–7 GeVc. At higher $\PT$, the value of $\RAA$ rises and levels
off above 40 GeVc at a value of approximately 0.5.
Figure 2: (Left) $\RAA$ (filled circles) as a function of $\PT$ for six
centrality intervals. The error bars represent the statistical uncertainties,
and the yellow boxes the $\PT$-dependent systematic uncertainties on the
$\RAA$ measurements. (Right) Comparison to the measurements of $\RAA$ in
central heavy ion collisions at three different center-of-mass energies as a
function of $\PT$ for neutral pions and charged hadrons and to several
theoretical predictions [7, 8, 9].
Measured $\RAA$ in the most central PbPb events (0–5 %) is compared to a
number of model predictions, both for the LHC design energy of
$\sqrt{s_{{}_{\mathrm{NN}}}}$ = 5.5 TeV (PQM [7] and GLV [8]) and for the
actual 2010 collision energy of $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 2.76 TeV (ASW,
YaJEM, and an elastic scattering energy-loss model with parameterized escape
probability [9]). While most models predict the generally rising behavior that
is observed in the data at high $\PT$, the magnitude of the predicted slope
varies greatly depending on the details of the jet quenching implementation.
## 0.4 Summary
In this contribution, measurements of the charged particle transverse momentum
spectra have been presented for 2.76 TeV PbPb collisions in bins of collision
centrality. The results have been normalized to an interpolated 2.76 TeV
reference $\PT$ spectrum to construct $\RAA$. A dramatic suppression (2–6
$\GeVc$) and a significant rise in the higher $\PT$ region with leveling-off
around 40 $\GeVc$ of the charged particle spectrum have been observed for the
most-central PbPb events. The new CMS measurement should help constrain the
quenching parameters used in various models and further the understanding of
the energy-loss mechanism.
## References
* [1] D. d’Enterria, “Jet quenching”, Landolt-Boernstein, Springer-Verlag Vol. 1-23A (2010) 99, arXiv:0902.2011.
* [2] CMS Collaboration, “Nuclear modification factor for charged particle production at high $p_{T}$ in PbPb collisions a $\sqrt{s_{{}_{NN}}}=2.76$ TeV”, CMS Physics Analysis Summary CMS-PAS-HIN-10-005 (2011).
* [3] CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004. doi:10.1088/1748-0221/3/08/S08004.
* [4] CMS Collaboration, “Charged particle transverse momentum spectra in pp collisions at $\sqrt{s}$ = 0.9 and 7 TeV”, arXiv:arXiv/1104.3547.
* [5] F. Arleo, D. d’Enterria, and A. S. Yoon, “Single-inclusive production of large-pT charged particles in hadronic collisions at TeV energies and perturbative QCD predictions”, JHEP 06 (2010) 035, arXiv:1003.2963. doi:10.1007/JHEP06(2010)035.
* [6] ALICE Collaboration, “Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb–Pb Collisions at $\sqrt{s_{{}_{NN}}}=2.76$ TeV”, Phys. Lett. B696 (2011) 30, arXiv:1012.1004. doi:10.1016/j.physletb.2010.12.020.
* [7] A. Dainese, C. Loizides, and G. Paic, “Leading-particle suppression in high energy nucleus-nucleus collisions”, Eur. Phys. J. C38 (2005) 461, arXiv:hep-ph/0406201. doi:10.1140/epjc/s2004-02077-x.
* [8] I. Vitev and M. Gyulassy, “High $p_{T}$ tomography of $d$ \+ Au and Au+Au at SPS, RHIC, and LHC”, Phys. Rev. Lett. 89 (2002) 252301, arXiv:hep-ph/0209161. doi:10.1103/PhysRevLett.89.252301.
* [9] T. Renk et al., “Systematics of the charged-hadron $p_{T}$ spectrum and the nuclear suppression factor in heavy-ion collisions from $\sqrt{s}=200$ GeV to $\sqrt{s}=2.76$ TeV”, arXiv:arXiv/1103.5308.
|
arxiv-papers
| 2011-07-10T14:51:42 |
2024-09-04T02:49:20.385378
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andre S. Yoon, for CMS Collaboration",
"submitter": "Andre Yoon",
"url": "https://arxiv.org/abs/1107.1862"
}
|
1107.1866
|
# Priority-based task reassignments in hierarchical 2D mesh-connected systems
using tableaux
Dohan Kim dohankim1@gmail.com
###### Abstract
Task reassignments in 2D mesh-connected systems (2D-MSs) have been researched
and simulated for several decades. We propose a hierarchical 2D mesh-connected
system (2D-HMS) in order to exploit the regular nature of a 2D-MS. In our
approach priority-based task assignments and reassignments in a 2D-HMS are
represented by tableaux and their algorithms. We provide examples of priority-
based task reassignments in a 2D-HMS in which task relocations are simply
reduced to a jeu de taquin slide.
###### keywords:
Task relocation, Task reassignment, Young tableau, 2D mesh, Jeu de taquin
12pt
## 1 Introduction
A distributed system is a collection of processors connected by an arbitrary
interconnection network [1, 2]. Among various interconnection networks for
distributed systems, two-dimensional (2D) mesh has received extensive study
due to its simplicity, efficiency, and structural regularity [3, 4, 5, 6].
Some data structures, such as matrices and arrays, naturally fit into a 2D
mesh-connected system (2D-MS) [6]. Since tasks are often assigned to a submesh
in a 2D-MS, continuous submesh allocations and deallocations of different
sizes may cause _fragmentation_ [3, 4] in a 2D-MS. Task relocation is an
approach to decrease fragmentation by reassigning a running task to an idle
processor in a 2D-MS, which involves capturing and transferring the state of
the running task to the idle processor [3, 7]. Meanwhile, the main objectives
of task assignments and reassignments in distributed systems are their
applications to high performance computing [8, 7]. In a similar vein our
objective of task reassignments in a 2D-MS is to minimize task turnaround time
rather than reducing fragmentation.
For decades, a wide variety of ways to tackle task assignment and reassignment
problems have been researched, such as graph-theoretic [9, 10, 11, 12, 13,
14], mathematical programming [15], and heuristics [16, 17]. One of the common
methods to represent and solve a task assignment problem is a graph-theoretic
method, which uses a graph-matching algorithm between a task graph and a
processor graph [9, 10, 11]. To complement the graph-theoretic method, our
previous work [18] presented the _Young tableaux_ [19, 20, 21, 22] approach to
representing task assignments. In this paper we convert a 2D-MS into a 2D mesh
processor graph. Then, a 2D mesh processor graph is represented by a _Young
diagram_ [19, 21], and its task assignment is represented by a tableau. Our
greedy task relocation policy is based on a hierarchical 2D-MS in which task
relocations are performed systematically by using tableau algorithms.
The remainder of this paper is organized as follows. Section 2 presents the
problem statement along with its assumptions. We provide an introduction to
tableaux and their algorithms in Section 3. In Section 4 we present how 2D
mesh graphs are converted into tableaux. Section 5 shows how task relocations
under the greedy task relocation policy in a 2D-HMS are reduced to a jeu de
taquin slide, and how they are applied in a 2D-HMS. Section 6 gives an
application of our approach to priority-based task reassignments in a 2D-HMS.
Finally, we conclude in Section 7.
## 2 Priority-based task reassignment problem in a 2D-HMS
We first give the necessary definitions of task assignments in a heterogeneous
system. Then, we present our priority-based task reassignment problem in a
hierarchical 2D mesh-connected system along with its assumptions.
###### Definition 2.1 ([11]).
A _task graph_ $T=(V,E)$ is a directed acyclic graph, where each vertex in
$V=\\{v_{1},v_{2},\ldots,v_{n}\\}$ represents a task, and each edge
$(v_{i},\,v_{j})\in E\subset V\times V$ represents the precedence
relationships between tasks, i.e., $v_{j}$ cannot begin its execution before
$v_{i}$ completes its execution.
###### Definition 2.2 ([23, 24, 25]).
A heterogeneous system $P$ is a set of $m$ heterogeneous processors
$P=\\{p_{1},p_{2},\ldots,p_{m}\\}$ whose communications are described by a
network topology. If $T=(V,\,E)$ is a task graph, then the execution time of
$v_{i}\in V$ on $p_{j}\in P$ is the function $\omega:V\times
P\rightarrow\mathbb{Q}^{+}$. A heterogeneous system $P$ is said to be
_consistent_ if processor $p_{x}\in P$ executes a task $n$ times faster than
processor $p_{y}\in P$, then it executes all other tasks $n$ times faster than
processor $p_{y}$.
Let $T=(V,\,E)$ be a task graph and $P=\\{p_{1},p_{2},\ldots,p_{m}\\}$ be a
heterogeneous system. A startup cost of initiating a task on a processor is
assumed to be negligible. In a consistent system, the computation cost of task
$v_{s}$ on $p_{t}$ is defined by $\omega(v_{s},p_{t})=r(v_{s})/c(p_{t})$,
where $r(v_{s})$ is the computation or resource requirement of task $v_{s}$,
and $c(p_{t})$ is the execution rate or capacity of processor $p_{t}$ [24,
26]. In an inconsistent system, which is a generalization of a consistent
system, the computation cost of task $v_{s}$ on $p_{t}$ is defined by
$\omega(v_{s},p_{t})=w_{st}$ in which $w_{st}$ is the corresponding
$(s,t)^{\text{th}}$ entry in a $|V|\times|P|$ cost matrix $W$ [24]. In the
remainder of this paper, we assume that every 2D-MS is consistent.
###### Definition 2.3 ([27, 28]).
A 2D mesh-connected system consists of $m\times n$ processors structured as a
rectangular grid of height $m$ and width $n$. Each processor is addressed by
its coordinate $(i,j)$ for $1\leq i\leq m$ and $1\leq j\leq n$. An internal
processor $(x,y)$, where $1<x<m$ and $1<y<n$, is directly connected to its
four neighboring processors $(x-1,y),\;(x+1,y),\;(x,y-1),\;(x,y+1)$. A
processor in the four corners has two neighboring processors, while a
processor in the remaining boundary has three neighboring processors,
respectively (see Fig. 3(d) in Section 4).
###### Definition 2.4.
A _hierarchical 2D mesh-connected system_ of $m\times n$ heterogeneous
processors is a heterogeneous 2D mesh-connected system with the following
partial order:
$P(i-1,j)\prec_{p}P(i,j),\;\;P(i,j-1)\prec_{p}P(i,j),\;\;\;1<i\leq m,\;1<j\leq
n$,
where $P(a,b)\prec_{p}P(c,d)$ means that a processor addressed by $(a,b)$ has
a higher priority (or execution rate) than a processor addressed by $(c,d)$.
###### Definition 2.5 ([9, 10, 11]).
Let $T_{m}=\\{t_{1},t_{2},\ldots,t_{m}\\}$ be a set of $m$ tasks with or
without precedence constraints and $R_{n}=\\{p_{1},p_{2},\ldots,p_{n}\\}$ be a
set of $n$ processors. Let _A_ be a task assignment function between $T_{m}$
and $R_{n}$. Let $t_{p}^{e}(A)$ denote the total execution time of processor
$p$ for the task assignment $A$ and let $t_{p}^{i}(A)$ denote the total idle
time of processor $p$ for the task assignment $A$. The _turnaround time_ of
processor $p$ for the task assignment $A$ is the total time spent in the
processor $p$ for the task assignment $A$. Let
$t_{p}(A)\stackrel{{\scriptstyle\rm{def}}}{{=}}t_{p}^{i}(A)+t_{p}^{e}(A)$ and
$t(A)\stackrel{{\scriptstyle\rm{def}}}{{=}}\max_{p}t_{p}(A)$. We call $t(A)$
the task turnaround time of the task assignment $A$. An _optimal task
assignment_ is the task assignment $A^{\prime}$ that minimize the task
turnaround time:
$t(A^{\prime})=\displaystyle\min_{A}t(A)=\displaystyle\min_{A}\displaystyle\max_{p}t_{p}(A)$
.
We next define the necessary conditions for being an optimal task assignment
in a 2D-HMS.
###### Definition 2.6.
Let $T_{m}=\\{1,2,\ldots,m\\}$ be a set of $m$ tasks having priorities
represented by task IDs from 1 to $m$, where a lower task ID indicates a
higher priority111In some priority schemes [29, 24], a higher number indicates
a higher priority. Throughout this paper, it is assumed that a lower number
indicates a higher priority.. A total order relation $\prec_{t}$ is defined on
$T_{m}$ as follows. $t_{1}\prec_{t}t_{2}$ for any two tasks $t_{1}\in T_{m}$
and $t_{2}\in T_{m}$ means that $t_{1}$ has a higher priority than $t_{2}$.
Let $R_{n}$ be a 2D-HMS consist of $n\,(n\geq m)$ heterogeneous processors
whose priorities (or execution rates) of rows and columns are sorted in
descending order. Let $A$ be a bijective, priority-based task assignment
between $T_{m}$ and $m$ processors in $R_{n}$. The necessary conditions for
being an optimal task assignment between $T_{m}$ and $m$ processors in $R_{n}$
are defined as:
1. 1.
Priorities of the assigned tasks on processors strictly decrease in rows and
columns, i.e., $t_{1}\prec_{t}t_{2}$ whenever $A(t_{1})\prec_{p}A(t_{2})$ for
any two tasks $t_{1}\in T_{m}$ and $t_{2}\in T_{m}$.
2. 2.
Processors of $A(T_{m})$ in $R_{n}$ is left-justified, where the row sizes of
$A(T_{m})$ are weakly decreasing.
The first condition in Definition 2.6 ensures that a processor with a higher
priority executes a task with a higher priority. The second condition in
Definition 2.6 ensures that if one processor is idle and the other processor
is busy for two adjacent processors in a 2D-HMS, the processor with the lower
priority is chosen to be idle. We say that a task assignment or reassignment
$A$ in a 2D-HMS is _necessarily optimal_ if it satisfies the necessary
conditions for being an optimal task assignment in a 2D-HMS. However, these
conditions are not universal for task assignments in 2D-MSs in that they
depend on priority schemes and system characteristics. We focus on priority-
based task assignments in a 2D-HMS in which an optimal task assignment for
priority-based tasks satisfies the necessary conditions in Definition 2.6. Now
we define the priority-based task reassignment problem in a 2D-HMS.
The _priority-based task reassignment problem_ in a 2D-HMS is to find a
necessarily optimal task reassignment sequence $(A_{k})_{k=0}^{m-1}$ for the
given initial task assignment $A=A_{0}$ along with the task completion
sequence $(b_{i})_{i=1}^{m}$, where $b_{i}\in T_{m}$ for $1\leq i\leq m$. The
constraints and assumptions that we have made are as follows:
1. 1.
Both tasks and processors are heterogeneous.
2. 2.
Each processor can process at most one task at a time. Processors are
dedicated and consistent to the priority-based task assignment, where no other
task is processed when the priority-based task assignment is executed.
3. 3.
Each priority-based task reassignment is achieved by an iterative sequence of
task relocations, where each task relocation is allowed between two adjacent
processors in a 2D-HMS if one processor is in idle state and the other node is
in busy state.
4. 4.
Two task relocations do not occur simultaneously in a 2D-HMS.
5. 5.
Task relocation and communication costs are ignored.
6. 6.
The number of tasks are less than or equal to the number of the available
processors in a 2D-HMS. Otherwise, a _clustering_ [26] of tasks needs to be
employed as an intermediate step, which is beyond the scope of this paper.
Most of the traditional approaches [3, 4, 6, 27, 28] are based on the
assumptions that processors in a 2D-MS are homogeneous. Further, the topology
of the assigned tasks on a 2D-MS is restricted (e.g., rectangular or square-
mesh shape) and priority-based task assignments have not often been considered
[3, 4]. Thus, there is a lack of systematic mechanisms of task relocations in
a 2D-HMS. We use tableaux and their algorithms for the priority-based task
reassignment problem in a 2D-HMS. When representing a 2D-MS or a 2D-HMS, there
is also a tradeoff between using a Young diagram (or a tableau) versus a 2D
mesh graph. Since a Young diagram or a tableau is intended to represent a 2D
mesh graph in a compact manner, it does not efficiently express communication
costs in a 2D-HMS. If we ignore communication costs, a 2D mesh graph has an
equivalent form of a Young diagram or a tableau. In the next section we give
an introduction to tableaux and their algorithms.
## 3 Tableaux and their algorithms
A tableau is one of the essential tools to study combinatorics [21, 30],
representation theory [20, 22], symmetric functions [19, 31], etc. We first
introduce the required definitions of tableaux and their combinatorial
algorithms.
###### Definition 3.1 ([19]).
A _partition_ of _n_ is defined as a sequence
$\linebreak\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})$, where the
$\lambda_{k}$ are weakly decreasing and $\sum_{k=1}^{i}{\lambda_{k}}=n$. If
$\lambda$ is a partition of _n_ , then we write $\lambda\vdash n$.
###### Definition 3.2 ([19, 22, 18]).
Let $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{i})\vdash n$. A _Young
diagram_ (or _Ferrers diagram_) of shape $\lambda$ is a left-justified, finite
collection of cells, with row _j_ containing $\lambda_{j}$ cells for $1\leq
j\leq i$.
###### Definition 3.3 ([19]).
Let $\lambda$ be a partition. An _inner corner_ of the Young diagram of shape
$\lambda$ is a cell $(i,j)\in\lambda$ whose removal leaves the Young diagram
of a partition. An _outer corner_ of the Young diagram of shape $\lambda$ is a
cell $(i,j)\notin\lambda$ whose addition results in the Young diagram of a
partition.
###### Definition 3.4 ([19, 22, 32]).
Let $\lambda\vdash n$. A _tableau_ _t_ of shape $\lambda$ is a Young diagram
of shape $\lambda$ filled with a set of elements, often positive integers. A
_Young tableau_ _T_ of shape $\lambda$ is a tableau of shape $\lambda$ whose
entries are the numbers from 1 to $n$, each occurring once.
###### Definition 3.5 ([22]).
A _standard Young tableau_ is a Young tableau whose entries are strictly
increasing in rows and columns.
###### Definition 3.6 ([19]).
Let $\lambda\vdash n$. A _partial tableau_ $p$ of shape $\lambda$ is a tableau
whose entries are strictly increasing in rows and columns.
Note that a partial tableau $p$ in Definition 3.6 is the standard Young
tableau if the entries of $p$ are exactly $\\{1,2,\ldots,n\\}$.
###### Example.
Consider $t_{1}=$ $\young(135,24,6)$, $t_{2}=$ $\young(135,42,6)$. We see that
$t_{1}$ is standard, but $t_{2}$ is not.
The number of standard Young tableaux of the given shape $\lambda$ is obtained
from the _hook formula_.
###### Definition 3.7 ([20, 22]).
If $\nu=(i,j)$ is a cell in the Young diagram of shape $\lambda$, then the
_hook_ of $\nu$, denoted $H_{\nu}$, is the set of all cells directly to the
right of $\nu$ or directly below $\nu$, including $\nu$ itself:
$H_{\nu}=H_{i,j}=\\{(i,\,j^{\prime}):j^{\prime}\geq
j\\}\cup\\{(i^{\prime},\,j):i^{\prime}\geq i\\}.$
The _hook length_ of $\nu=(i,j)$, denoted $h_{i,j}$, is the number of cells in
its hook, i.e., $h_{i,j}=|H_{i,j}|$.
###### Theorem 3.1 ([33]).
If $\lambda\vdash n$, then the number $f^{\lambda}$ of standard Young tableaux
of shape $\lambda$ is
$f^{\lambda}=\dfrac{n!}{\prod_{(i,j)\in\lambda}h_{i,j}}$.∎
###### Example.
Labeling each cell with its hook length for the Young diagram of shape
$(3,2,1)$ and $(4,4,4,4)$ are given below:
$\young(531,31,1)$ , $\young(7654,6543,5432,4321)$ .
If the shape is $\lambda=(3,2,1)$, then $f^{(3,2,1)}=6!/(5\cdot 3^{2}\cdot
1^{3})=16$. If the shape is $\lambda=(4,4,4,4)$, then
$f^{(4,4,4,4)}=16!/(7\cdot 6^{2}\cdot 5^{3}\cdot 4^{4}\cdot 3^{3}\cdot
2^{2}\cdot 1)=24024$.
###### Definition 3.8 ([21, 19]).
Let $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{i})$ and
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{j})$ be partitions with
$\mu\subseteq\lambda$ (i.e., $i\leq j$ and $\mu_{k}\leq\lambda_{k}$ for $1\leq
k\leq i$). Then a _skew shape_ of $\lambda/\mu$ is the set of cells
$\lambda/\mu=\\{c:c\in\lambda$ and $c\notin\mu\\}$.
A skew shape of $\lambda/\mu$ is _normal_ if $\mu=\emptyset$. A tableau of
skew shape $\lambda/\mu$ is called a _skew tableau_ of shape $\lambda/\mu$. A
_partial skew tableau_ of shape $\lambda/\mu$ (or a partial tableau of skew
shape $\lambda/\mu$) is a skew tableau of shape $\lambda/\mu$ whose entries
are strictly increasing in rows and columns. A partial skew tableau is called
the _standard skew tableau_ if its entries are precisely $\\{1,2,\ldots,n\\}$.
###### Example.
Consider skew tableaux of shape $\lambda/\mu=(4,3,3,2)/(2,2)$ given by
$t_{1}=$ $\young(::17,::3,245,69)$, $t_{2}=$ $\young(::17,::5,243,69)$ .
We see that $t_{1}$ is a partial skew tableau, but $t_{2}$ is not.
###### Definition 3.9 ([34]).
A group $(G,\,\cdot\,)$ is a nonempty set _G_ , closed under a binary
operation $\cdot$ , such that the following axioms are satisfied:
1. 1.
$(a\cdot b)\cdot c=a\cdot(b\cdot c)$ for all $a,b,c\in G$,
2. 2.
There is an element _e_ $\in$ _G_ such that for all $x\in G,~{}e\cdot x=x\cdot
e=x$,
3. 3.
For each element $a\in G$, there is an element $a^{-1}\in G$ such that $a\cdot
a^{-1}=a^{-1}\cdot a=e$.
###### Definition 3.10 ([35]).
The group of all bijections $I_{n}\rightarrow I_{n}$, where
$I_{n}=\\{1,2,\ldots,n\\}$, is called the _symmetric group on n letters_ ,
denoted as $\mathfrak{S}_{n}$.
Since $\mathfrak{S}_{n}$ is the group of all permutations of
$I_{n}=\\{1,2,\ldots,n\\}$, the order of $\mathfrak{S}_{n}$ is $n!$ [34].
###### Definition 3.11 ([35, 34]).
Let $i_{1},i_{2},\ldots,i_{n}$ be distinct elements of
$I_{n}=\\{1,2,\ldots,n\\}$. Then $\begin{pmatrix}1&2&\cdots&n\\\
i_{1}&i_{2}&\cdots&i_{n}\end{pmatrix}$
$\stackrel{{\scriptstyle\rm{def}}}{{=}}[i_{1}\,i_{2}\,\cdots\,i_{n}]\in\mathfrak{S}_{n}$
denotes the permutation that maps $1\mapsto i_{1},2\mapsto
i_{2},\ldots,n\mapsto i_{n}$.
Now we discuss combinatorial algorithms of tableaux, including row insertion,
Robinson-Schensted, and jeu de taquin. We first describe the row insertion
algorithm. The _row insertion_ or _row bumping algorithm_ [20] takes a partial
tableau $P$, and a positive number $x$ that is not in $P$. Then, it constructs
the new partial tableau $P^{\prime}$. The procedure of row insertion is
described by Algorithm 1, which always yields a partial tableau.
$P=\Yboxdim 13pt\young(138\mbox{10},249,67,\mbox{11}\mbox{12})$
$\stackrel{{\scriptstyle\text{insert 5}}}{{\Rightarrow}}$ $\Yboxdim
13pt\young(13\bf{5}\mbox{10},249,67,\mbox{11}\mbox{12})$
$\overset{\bf{8}}{\leftarrow}$ $\Rightarrow$ $\Yboxdim
13pt\young(135\mbox{10},24\bf{8},67,\mbox{11}\mbox{12})$
$\overset{\bf{9}}{\swarrow}$ $\Rightarrow$ $\Yboxdim
13pt\young(135\mbox{10},248,67\bf{9},\mbox{11}\mbox{12})=P^{\prime}$.
Fig. 1: An example of row insertion.
Fig. 1 describes the procedure of row-inserting $x=5$ into $P$. Entries that
are bumped during the row insertion are indicated by boldface type in Fig. 1.
Input: a partial tableau $P$ and an input number $x$ that is not in $P$
Output: a partial tableau $P^{\prime}$ ($x$ is inserted into $P$)
begin
set $i:=1$ as the first row of $P$;
while _$x$ is less than some element of row $i$_ do
let $y$ be the smallest element of row $i$ that is greater than $x$; replace
$y$ by $x$; set $x:=y$ and $i:=i+1$;
end while
place $x$ (with its new cell) at the end row of $i$; // $x$ is the largest
element of row $i$
end
Algorithm 1 Row insertion (Row bumping) [20, 19]
The row insertion procedure is always reversible. If we have $P^{\prime}$
along with the address of the added cell in $P^{\prime}$, we can restore the
original partial tableau $P$. The procedure of _reverse bumping_ [20] is
described in Algorithm 2.
Input: a partial tableau $P^{\prime}$ and number $x=P^{\prime}_{i,j}$ in
$P^{\prime}$ (cell $(i,j)$ has to be an outer corner in the underlying Young
diagram of $P^{\prime}$).
Output: a partial tableau $P$
begin
set $x^{\prime}:=x$ and $k:=i-1$;
remove the cell of $x$ (including $x$ itself) in $P^{\prime}$;
while _$k\neq 0$_ do
let $y$ be the largest element of row $k$ that is smaller than $x^{\prime}$;
replace $y$ by $x^{\prime}$; set $x^{\prime}:=y$ and $k:=k-1$;
end while
end
Algorithm 2 Reverse bumping [20]
The address of the added cell in $P^{\prime}$ in Fig. 1 is $(3,3)$ having an
entry 9. Fig. 2 shows the procedure of reverse-bumping $x=9$ to $P^{\prime}$.
Note that it restores the original partial tableau $P$ in Fig. 1.
$P^{\prime}=\Yboxdim 13pt\young(135\mbox{10},248,67\bf{9},\mbox{11}\mbox{12})$
$\stackrel{{\scriptstyle\text{extract 9}}}{{\Rightarrow}}$ $\Yboxdim
13pt\young(135\mbox{10},24\bf{8},67,\mbox{11}\mbox{12})$
$\overset{\bf{9}}{\leftarrow}$ $\Rightarrow$ $\Yboxdim
13pt\young(13\bf{5}\mbox{10},249,67,\mbox{11}\mbox{12})$
$\overset{\bf{8}}{\nwarrow}$ $\Rightarrow$ $\Yboxdim
13pt\young(13\bf{8}\mbox{10},249,67,\mbox{11}\mbox{12})=P$.
Fig. 2: An example of reverse bumping.
We next describe the Robinson-Schensted algorithm. The _Robinson-Schensted
algorithm_ [36, 19] provides a bijection between elements of the symmetric
group $\mathfrak{S}_{n}$ and pairs of standard Young tableaux of the same
shape $\lambda\vdash n$. Recall that
$[i_{1}\,i_{2}\,\cdots\,i_{n}]\in\mathfrak{S}_{n}$ denotes the permutation
that maps $1\mapsto i_{1},2\mapsto i_{2},\ldots,n\mapsto i_{n}$. For each
$\pi=[i_{1}\,i_{2}\,\cdots\,i_{n}]\in\mathfrak{S}_{n}$, the Robinson-Schensted
algorithm constructs a sequence of Young tableaux pairs
$(\emptyset,\emptyset)=(P_{0},Q_{0}),(P_{1},Q_{1}),\ldots,(P_{n},Q_{n})=(P,Q)$
(see Algorithm 3). By the row insertion algorithm, $i_{1},i_{2},\ldots,i_{n}$
are inserted into the $P^{\prime}s$. Meanwhile, numbers $1,2,\ldots,n$ are
simply placed sequentially to the $Q^{\prime}s$ so that the shape of $P_{k}$
and the shape of $Q_{k}$ are the same for $1\leq k\leq n$ [19]. We call $P$
the _insertion tableau_ and $Q$ the _recording tableau_ of $\pi$, denoted
$P(\pi)$, and $Q(\pi)$, respectively [19, 21].
Input: $\pi=[x_{1}\,x_{2}\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$
Output: $P=P_{n}$ and $Q=Q_{n}$ (the insertion tableau and the recording
tableau of $\pi$, respectively)
begin
set $(P_{0},\,Q_{0}):=(\emptyset,\emptyset)$;
for _$k\leftarrow 1$ to $n$_ do
construct $P_{k}$ by inserting $x_{k}$ into $P_{k-1}$; place $k$ into
$Q_{k-1}$ so that the shape of $P_{k}$ and $Q_{k}$ are the same;
end for
return _$(P_{n},\,Q_{n})$_ ;
end
Algorithm 3 Robinson-Schensted algorithm ($\pi\rightarrow(P,Q))$ [36, 19]
Note that the insertion tableau $P$ and the recording tableau $Q$ of
$\pi=[i_{1}\,i_{2}\,\cdots\,i_{n}]\in\mathfrak{S}_{n}$ are both standard Young
tableaux of the same shape. The procedure of Algorithm 3 is reversible, i.e.,
$(P,Q)\rightarrow\pi$. Given $(P_{k},Q_{k})$, we retrieve the $k$th element of
$\pi$ (i.e., $i_{k}$) from $(P_{k},Q_{k})$, and find $P_{k-1}$ with the
$i_{k}$ removed by using the reverse bumping algorithm (see Algorithm 2). We
easily find $Q_{k-1}$ from $Q_{k}$ by removing $k$ (along with its cell) from
$Q_{k}$. Thus, given $(P,Q)=(P_{n},Q_{n})$, we construct a sequence of
standard Young tableaux pairs
$(P_{n},Q_{n}),(P_{n-1},Q_{n-1}),\ldots,(P_{0},Q_{0})=(\emptyset,\emptyset)$,
where all the elements of $\pi\in\mathfrak{S}_{n}$ are recovered in reverse
order [19]. The theorem about the Robinson-Schensted algorithm is as follows.
###### Theorem 3.2 ([36, 21]).
The Robinson-Schensted algorithm establishes a bijection between elements of
$\mathfrak{S}_{n}$ and pairs of standard Young tableaux of the same shape
$\lambda\vdash n$. In other words, $\sum_{\lambda\vdash
n}{(f^{\lambda})}^{2}=n!$, where $f^{\lambda}$ is the number of standard Young
tableaux of shape $\lambda\vdash n$. ∎.
Note that the bijection in the Robinson-Schensted algorithm is given by
$\pi\rightarrow(P,Q)$ rather than $\pi\rightarrow P$. Each insertion tableau
$P$ of $\pi=[x_{1}\,x_{2}\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$ corresponds to
an equivalence class of $\mathfrak{S}_{n}$, called a $P$-_equivalence class_.
###### Definition 3.12 ([19]).
Two permutations $\pi=[x_{1}\,x_{2}\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$ and
$\tau=[y_{1}\,y_{2}\,\cdots\,y_{n}]\in\mathfrak{S}_{n}$ are called _P-
equivalent_ if $P(\pi)=P(\tau)$, denoted $\pi\cong_{P}\tau$.
###### Example.
Both $[2\,1\,3]\in\mathfrak{S}_{3}$ and $[2\,3\,1]\in\mathfrak{S}_{3}$ have
the same insertion tableau, i.e., $[2\,1\,3]\cong_{P}[2\,3\,1]$. Similarly,
$[1\,3\,2]\in\mathfrak{S}_{3}$ and $[3\,1\,2]\in\mathfrak{S}_{3}$ have the
same insertion tableau, i.e., $[1\,3\,2]\cong_{P}[3\,1\,2]$.
Now we discuss _Knuth-equivalence classes_ of $\mathfrak{S}_{n}$ and their
relationship to $P$-equivalence classes of $\mathfrak{S}_{n}$.
###### Definition 3.13 ([21]).
Suppose $x<y<z$. A _Knuth transformation_ of a permutation
$\pi\in\mathfrak{S}_{n}$ is a transformation of $\pi\in\mathfrak{S}_{n}$ into
another permutation $\tau\in\mathfrak{S}_{n}$ that has one of the following
forms:
1. 1.
$\pi=[x_{1}\,\cdots\,y\,x\,z\,\cdots\,x_{n}]\in\mathfrak{S}_{n}\Longrightarrow\tau=[x_{1}\,\cdots\,y\,z\,x\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$,
2. 2.
$\pi=[x_{1}\,\cdots\,y\,z\,x\,\cdots\,x_{n}]\in\mathfrak{S}_{n}\Longrightarrow\tau=[x_{1}\,\cdots\,y\,x\,z\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$,
3. 3.
$\pi=[x_{1}\,\cdots\,x\,z\,y\,\cdots\,x_{n}]\in\mathfrak{S}_{n}\Longrightarrow\tau=[x_{1}\,\cdots\,z\,x\,y\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$,
4. 4.
$\pi=[x_{1}\,\cdots\,z\,x\,y\,\cdots\,x_{n}]\in\mathfrak{S}_{n}\Longrightarrow\tau=[x_{1}\,\cdots\,x\,z\,y\,\cdots\,x_{n}]\in\mathfrak{S}_{n}$.
Two permutations $\pi,\tau\in\mathfrak{S}_{n}$ are called _Knuth-equivalent_
if one of them can be obtained from the other by a sequence of Knuth
transformations, denoted $\pi\cong_{K}\tau$.
###### Example.
We see that $[2\,1\,3]\in\mathfrak{S}_{3}$ and $[2\,3\,1]\in\mathfrak{S}_{3}$
are Knuth-equivalent by (1) and (2) in Definition 3.13, written
$[2\,1\,3]\cong_{K}[2\,3\,1]$. Similarly, $[1\,3\,2]\in\mathfrak{S}_{3}$ and
$[3\,1\,2]\in\mathfrak{S}_{3}$ are Knuth-equivalent by (3) and (4) in
Definition 3.13, written $[1\,3\,2]\cong_{K}[3\,1\,2]$.
From the preceding examples, we observe that some permutations in
$\mathfrak{S}_{3}$ are Knuth-equivalent if they are $P$-equivalent, and vice
versa. More generally, $P$-equivalence classes and Knuth-equivalence classes
coincide in $\mathfrak{S}_{n}$.
###### Theorem 3.3 ([30, 21]).
Permutations in $\mathfrak{S}_{n}$ are Knuth-equivalent if and only if they
are P-equivalent. ∎
###### Definition 3.14 ([21, 19]).
Let $t$ be a tableau. The _reading word_ or _row word_ of $t$, denoted $r(t)$,
is the permutation of entries of $t$ obtained by concatenating the rows of $t$
from bottom to top, i.e., $r(t)=R_{k}R_{k-1}\ldots R_{1}$, where
$R_{1},\ldots,R_{k}$ are the rows of $t$.
###### Lemma 3.1 ([21]).
If $\pi$ is the reading word of a standard Young tableau $T$, then $T$ is
$P(\pi)$. ∎
The _jeu de taquin_ (or the ”teasing game”) of Scützenberger [37, 21] consists
of a set of rules for transforming _partial tableaux_ , while some properties
of partial tableaux are preserved during transformations. There are two kinds
of jeu de taquin slides, which are a forward and a backward slide. A forward
jeu de taquin slide is described in Algorithm 4. Note that the resulting
tableau of a forward jeu de taquin slide is still a partial tableau.
Input: a partial tableau $P$ of skew shape $\lambda/\mu$; an inner corner of
$\mu$
Output: a partial tableau $P^{\prime}$
begin
pick $x$ to be an inner corner of $\mu$;
while _$x$ is not an inner corner of $\lambda$_ do
if _$x=(i,j)$_ then
let $x^{\prime}$ be the cell of $\text{min}\\{P_{i+1,j}\,,\,P_{i,j+1}\\}$;
// if only one of $P_{i+1,j}$ and $P_{i,j+1}$ exists, then choose that value
as a minimum;
end if
slide $P_{x^{\prime}}$ into cell $x$ and set $x:=x^{\prime}$;
end while
return _the partial tableau $P^{\prime}$_;
end
Algorithm 4 A forward jeu de taquin slide [37, 19]
A backward jeu de taquin slide can be an invertible operation of a forward jeu
de taquin slide by performing the backward slide into the cell that was
vacated by the forward slide. A backward jeu de taquin slide is described in
Algorithm 5.
Input: a partial tableau $P$ of skew shape $\lambda/\mu$; an outer corner of
$\lambda$
Output: a partial tableau $P^{\prime}$
begin
pick $y$ to be an outer corner of $\lambda$;
while _$y$ is not an outer corner of $\mu$_ do
if _$y=(i,j)$_ then
let $y^{\prime}$ be the cell of $\text{max}\\{P_{i-1,j}\,,\,P_{i,j-1}\\}$;
// if only one of $P_{i-1,j}$ and $P_{i,j-1}$ exists, then choose that value
as a maximum;
end if
slide $P_{y^{\prime}}$ into cell $y$ and set $y:=y^{\prime}$;
end while
return _the partial tableau $P^{\prime}$_;
end
Algorithm 5 A backward jeu de taquin slide [37, 19]
###### Example.
Consider the following jeu de taquin slides for $x=(1,1)$ and $y=(2,4)$ in
$P$:
$P:$ $\Yboxdim 13pt\young(:359,248,67\mbox{10})$ , $\text{jdt}^{x}(P):$
$\Yboxdim 13pt\young(2359,478,6\mbox{10})$ , $\text{jdt}_{y}(P):$ $\Yboxdim
13pt\young(::35,2489,67\mbox{10})$ .
If we denote the forward jeu de taquin slide on $P$ from cell $x=(1,1)$ in $P$
as $\text{jdt}^{x}(P)$, and backward jeu de taquin slide on $P$ from cell
$y=(2,4)$ in $P$ as $\text{jdt}_{y}(P)$, then both $\text{jdt}^{x}(P)$ and
$\text{jdt}_{y}(P)$ are still partial tableaux. Furthermore, if we let cell
$(3,3)$ of $\text{jdt}^{x}(P)$ as $z$, we have
$\text{jdt}_{z}(\text{jdt}^{x}(P))=P$, which restores $P$.
###### Definition 3.15 ([21]).
Partial tableaux $P$ and $P^{\prime}$ are called _jeu de taquin equivalent_ ,
written $P\cong_{jdt}P^{\prime}$, if $P^{\prime}$ can be obtained from $P$ by
some sequence of jeu de taquin slides, or vice versa.
###### Theorem 3.4 ([37, 19]).
The jeu de taquin equivalence class of a given partial skew tableau $P$
contains exactly one partial tableau of normal shape, denoted $j(P)$. ∎
Theorem 3.4 says that jeu de taquin slides bring a partial tableau $P$ of skew
shape to a partial tableau $j(P)$ of normal shape. Recall that a skew shape
$\lambda/\mu$ is a normal shape if $\mu=\emptyset$. The following results show
that the jeu de taquin equivalence relation is closely connected to both the
$P$-equivalence and the Knuth-equivalence relation.
###### Lemma 3.2 ([21]).
Each jeu de taquin slide converts the reading word of a standard skew tableau
into a Knuth-equivalent one. ∎
###### Theorem 3.5 ([37, 21]).
Let $P$ and $P^{\prime}$ be standard skew tableaux. They are jeu de taquin
equivalent, i.e., $P\cong_{jdt}P^{\prime}$, if and only if their reading words
are Knuth-equivalent, i.e., $r(P)\cong_{K}r(P^{\prime})$.
###### Proof 1.
$P\cong_{jdt}P^{\prime}\Leftrightarrow j(P)=j(P^{\prime})\text{ (by Theorem
\ref{thm:jdt})}\Leftrightarrow r(j(P))\cong_{K}r(j(P^{\prime}))$ $\text{ (by
Lemma~{}\ref{lem:ReadingWordPequivalent} and
Theorem~{}\ref{thm:KnuthEquivalent}) }\Leftrightarrow
r(P)\cong_{K}r(P^{\prime})\text{ (by Lemma~{}\ref{lem:jdt})}.\qed$
## 4 2D mesh graphs and tableaux
Recall that $G^{\prime}=(V^{\prime},E^{\prime})$ is a subgraph of $G=(V,E)$ if
$V^{\prime}\subset V$ and $E^{\prime}\subset E$ [38]. A _vertex-induced
subgraph_ $G^{\prime}$ is the subgraph of $G$ that contains all edges of $G$
that join any two vertices in $V^{\prime}$ of $G^{\prime}$ [39, 38]. In this
section we present a certain type of vertex-induced subgraphs of 2D mesh
graphs and their relationship to tableaux.
###### Definition 4.1.
A _2D mesh graph $G$ of shape
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})\vdash n$_ is a left-
justified, a vertex-induced subgraph of a $k\times\lambda_{1}$ 2D mesh graph,
where each row contains $\lambda_{i}$ vertices for $1\leq i\leq k$. A 2D mesh
graph $G=(V,E)$ of shape $\lambda$ is _directed_ if all edges of $G$ are
directed in such a way that the horizontal edges are directed from left to
right and the vertical edges are directed from top to bottom.
Similarly to the traditional 2D mesh graphs, each vertex of 2D mesh graph
$G=(V,E)$ of shape $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})\vdash
n$ is represented by its coordinate. For example, a top-left corner vertex is
$(1,1)$ while a bottom-right corner vertex is $(k,\lambda_{k})$. If
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{k}$ for shape
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})\vdash n$, we say that
$\lambda$ is of _canonical shape_.
###### Definition 4.2.
Let $G_{1}=(V_{1},E_{1})$ be a 2D mesh graph of shape $\lambda$ and
$G_{2}=(V_{2},E_{2})$ be a 2D mesh graph of shape $\mu$, where
$\mu=(\mu_{1},\mu_{2},\ldots,\mu_{i})\subseteq\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{j})$,
i.e., $i\leq j$ and $\mu_{k}\leq\lambda_{k}$ for $1\leq k\leq i$,
respectively. A 2D mesh graph $G^{\prime}$ of skew shape $\lambda/\mu$ is the
vertex-induced subgraph of $G_{1}$, where the vertex set of $G^{\prime}$
consists of $V_{1}-V_{2}$.
Fig. 3: Conversion of a 2D mesh graph into a Young diagram or a Young tableau.
For example, the upper figures in Fig. 3(a), (b), and (c) show 2D mesh graphs
of shape $\lambda_{1}=(4,4,4,4)$, $\lambda_{2}=(4,3,2,2)$, and
$\lambda_{3}=\lambda/\mu$ where $\lambda=(4,4,4,4)$ and $\mu=(2,2)$,
respectively. Now we consider the relationships between 2D mesh graphs and
Young diagrams (or tableaux). For example, if we do not take communication
costs into account for a 2D mesh processor graph, it has the compact form
represented by a Young diagram or a tableau. In order to represent a 2D mesh
processor or task graph of shape $\lambda$ in a compact manner, we define a 2D
mesh Young diagram and a 2D mesh tableau.
###### Definition 4.3.
A _2D mesh Young diagram of shape_
$\lambda=(\lambda_{1},\,\lambda_{2},\ldots,\,\lambda_{k})\vdash n$ is a Young
diagram of shape $\lambda\vdash n$, where each cell $(i,j)$ corresponds to
each vertex $(i,j)$ in the 2D mesh graph $G=(V,E)$ of shape $\lambda\vdash n$.
###### Definition 4.4.
A _2D mesh Young diagram of skew shape_ $\lambda/\mu$ is a skew Young diagram
of shape $\lambda/\mu$ for $\mu\subseteq\lambda$. Similarly to Definition 4.3,
each cell $(i,j)$ corresponds to each vertex $(i,j)$ in the 2D mesh graph
$G=(V,E)$ of skew shape $\lambda/\mu$.
Fig. 3 shows how 2D mesh graphs are converted into 2D mesh Young diagrams.
Fig. 3(a) shows a 2D mesh Young diagram of canonical shape while Fig. 3(c)
shows a 2D mesh Young diagram of skew shape. We next define a 2D mesh tableau
in order to represent a labeled 2D mesh graph (see Fig. 3(d)).
###### Definition 4.5.
A _2D mesh tableau of shape_
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})\vdash n$ is a Young
tableau of shape $\lambda\vdash n$, denoted as $(\text{MT}_{i,j})$ of shape
$\lambda$ or $\lambda$-$(\text{MT}_{i,j})$, where each cell has a unique
number from 1 to $n$ and its underlying Young diagram is a 2D mesh Young
diagram of shape $\lambda\vdash n$.
###### Definition 4.6.
A _2D mesh tableau of skew shape_ $\lambda/\mu$ is a tableau of skew shape
$\lambda/\mu$, denoted as $(\text{MT}_{i,j})$ of shape $\lambda/\mu$, where
each cell has a unique number from the set $\\{1,2,\ldots,n\\}$ and its
underlying Young diagram is 2D mesh Young diagram of skew shape $\lambda/\mu$.
A graph-theoretic approach to Young diagrams or tableaux has already been
researched [40]. It focuses on graphs having the shape of a Young diagram or a
tableau, while this paper focuses on converting a 2D mesh graph into a Young
diagram or a tableau. The relationship between a certain labeling of directed
acyclic graphs and the number of standard Young tableaux has been discussed in
[41, 42]. The following propositions involve in counting aspects of
representing 2D mesh task and processor graphs, respectively.
###### Proposition 4.1.
Let $G=(V,E)$ be a directed 2D mesh graph of shape $\lambda\vdash n$. Let
$T=\\{1,2,\ldots,n\\}$ be a totally ordered set of $n$ tasks ordered by the
”less than” relation $<$. Suppose we assign tasks $T$ to vertices $V$
bijectively such that task ID of vertex $x$ is less than task ID of vertex $y$
for each edge $(x,y)\in E$. Then, the number of such assignments is the number
of standard Young tableaux of shape $\lambda\vdash n$. ∎
###### Proposition 4.2.
Let $G=(V,E)$ be an undirected 2D mesh graph of shape $\lambda\vdash n$. Let
$P=\\{1,2,\ldots,n\\}$ be a totally ordered set of $n$ processors ordered by
the ”less than” relation $<$. Suppose we assign processors $P$ to vertices $V$
bijectively such that processor IDs of vertices are increasing in rows and
columns. Then, the number of such assignments is the number of standard Young
tableaux of shape $\lambda\vdash n$. ∎
Proposition 4.1 and 4.2 directly follow from the definition of a standard
Young tableau of shape $\lambda\vdash n$. Each labeled 2D mesh task and
processor graph of shape $\lambda\vdash n$ corresponds to each 2D mesh tableau
of shape $\lambda\vdash n$. We call a 2D mesh task graph of shape
$\lambda\vdash n$ satisfying the hypothesis of Proposition 4.1 a _standard 2D
mesh task graph of shape $\lambda\vdash n$_, and a processor graph of shape
$\lambda\vdash n$ satisfying the hypothesis of Proposition 4.2 a _standard 2D
mesh processor graph of shape $\lambda\vdash n$_, respectively. Thus, counting
the number of standard 2D mesh task or processor graphs of shape
$\lambda\vdash n$ are reduced to counting the number of 2D mesh tableaux of
shape $\lambda\vdash n$ whose entries in rows and columns strictly increase,
i.e., $f^{\lambda}$ (see Theorem 3.1).
Now consider a task assignment between a standard 2D mesh task graph of shape
$\lambda\vdash n$ and a standard 2D mesh processor graph of the same shape by
their coordinates, i.e., each task addressed by $(i,j)$ of the 2D mesh task
graph is assigned to each processor addressed by $(i,j)$ of the 2D mesh
processor graph. Proposition 4.3 discusses the counting aspects of such task
assignments.
###### Proposition 4.3.
Let $\lambda\vdash n$. The number of bijective task assignments by their
coordinates between standard 2D mesh task graphs of shape $\lambda\vdash n$
and standard 2D mesh processor graphs of shape $\lambda\vdash n$ is
${(f^{\lambda})}^{2}$. Moreover, if we sum all the numbers of such task
assignments for all possible partitions (or shapes) of $n$, then the total
number is $\sum_{\lambda\vdash n}{(f^{\lambda})}^{2}=n!$.
###### Proof 2.
By Proposition 4.1 and 4.2, the number of standard 2D mesh task and processor
graphs of shape $\lambda\vdash n$ is both $f^{\lambda}$. Thus, for a given
shape $\lambda\vdash n$, the number of bijective task assignments by their
coordinates between standard 2D mesh task graphs of shape $\lambda\vdash n$
and standard 2D mesh processor graphs of shape $\lambda\vdash n$ is
${(f^{\lambda})}^{2}$. For a given $n$, if we consider all possible partitions
(or shapes) of $n$, then the total number of such task assignments is
$\sum_{\lambda\vdash n}{(f^{\lambda})}^{2}$, which is $n!$ by Theorem 3.2. ∎
In case a 2D mesh graph represents a 2D mesh processor graph, we say that the
associated 2D mesh Young diagram (respectively, 2D mesh tableau) is the _2D
mesh processor diagram_ (respectively, _2D mesh processor tableau_).
###### Definition 4.7.
A _2D mesh processor diagram of shape $\lambda\vdash n$_ is a 2D mesh Young
diagram of shape $\lambda\vdash n$, where each cell in the 2D mesh processor
diagram represents each processor in the heterogeneous system
$P=\\{p_{1},\,p_{2},\ldots,\,p_{n}\\}$. Each cell is assumed to have direct
communication links with its neighboring cells and have a routing capability,
allowing it to send and receive messages to and from any other cell in the
diagram.
To exploit the regular nature of a 2D mesh topology, we consider a
hierarchical 2D mesh processor diagram of canonical shape, where the rows and
columns of heterogeneous processors are sorted in descending order by their
priorities (or execution rates).
###### Definition 4.8.
A hierarchical 2D mesh processor diagram of canonical shape
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ for
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{k}$ is a 2D mesh processor diagram
consists of $k\times\lambda_{1}$ processors with the following partial order:
$P(i-1,j)\prec_{p}P(i,j),\;\;P(i,j-1)\prec_{p}P(i,j),\;\;\;1<i\leq
k,\;1<j\leq\lambda_{1}$,
where $P(a,b)\prec_{p}P(c,d)$ means that a processor addressed by $(a,b)$ has
a higher priority (or execution rate) than a processor addressed by $(c,d)$.
###### Proposition 4.4.
Let $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})\vdash n$ for
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{k}$, and let $P=\\{1,2,\ldots,n\\}$
be a totally ordered set of $k\times\lambda_{1}$ processors ordered by the
”less than” relation $<$. Each processor ID represents a processor priority,
where $u<v$ for any two processors $u\in P$ and $v\in P$ implies that
processor $u$ has a higher priority (or execution rate) than processor $v$.
Then, the number of ways to organize a hierarchical 2D mesh processor diagram
of shape $\lambda\vdash n$ by arranging $n$ processors in $P$ is
$f^{\lambda}$.
###### Proof 3.
It immediately follows from Definition 3.5 and 4.8.∎
We define a _hierarchical 2D mesh tableau_ in order to represent a priority-
based task assignment and its reassignments in a 2D-HMS.
###### Definition 4.9.
A _hierarchical 2D mesh tableau of shape_ $\lambda\vdash n$ is a tableau of
shape $\lambda$, denoted as $(\text{HMT}_{i,j})$ of shape $\lambda$ or
$\lambda$-$(\text{HMT}_{i,j})$, whose underlying 2D mesh Young diagram is a
hierarchical 2D mesh processor diagram of canonical shape $\lambda\vdash n$.
Entries of $\lambda$-$(\text{HMT}_{i,j})$ are task IDs from the set
$\\{1,2,\ldots,n\\}$ with the total order relation $\prec_{t}$ (see Definition
2.6). Empty entry of a cell is allowed in $\lambda$-$(\text{HMT}_{i,j})$,
while all non-empty entries along with their cells must form a tableau of
normal or skew shape, called a _maximally embedded tableau_ of
$\lambda$-$(\text{HMT}_{i,j})$.
###### Definition 4.10.
Let $\lambda$-$(\text{HMT}_{i,j})$ be a hierarchical 2D mesh tableau of shape
$\lambda\vdash n$. If the maximally embedded tableau of
$\lambda$-$(\text{HMT}_{i,j})$ is a tableau of normal shape, we say that
$\lambda$-$(\text{HMT}_{i,j})$ is of _normal shape_. Meanwhile, if the
maximally embedded tableau of $\lambda$-$(\text{HMT}_{i,j})$ is a tableau of
skew shape, we say that $\lambda$-$(\text{HMT}_{i,j})$ is of _skew shape_. If
the maximally embedded tableau of $\lambda$-$(\text{HMT}_{i,j})$ is a partial
tableau (i.e., a tableau whose entries are strictly increasing in rows and
columns), we say that $\lambda$-$(\text{HMT}_{i,j})$ is _standard_. Otherwise,
we say that $\lambda$-$(\text{HMT}_{i,j})$ is _generalized_.
In this paper $\lambda$-$(\text{HMT}_{i,j})$ is referred to as a standard
$\lambda$-$(\text{HMT}_{i,j})$ of normal shape unless otherwise stated. Each
entry in $\lambda$-$(\text{HMT}_{i,j})$ denotes a task, where a lower task ID
indicates a higher priority. Each cell represents each processor in the
hierarchical 2D mesh processor diagram. Therefore,
$\lambda$-$(\text{HMT}_{i,j})$ represents a priority-based task assignment in
a 2D mesh-connected system. Note that a priority-based task assignment
represented by $\lambda$-$(\text{HMT}_{i,j})$ satisfies the necessary
conditions for being an optimal task assignment in a 2D-HMS (see Definition
2.6).
###### Definition 4.11.
Let $\lambda$-$(\text{HMT}_{i,j})$ be a hierarchical 2D mesh tableau of shape
$\lambda\vdash n$. If the processor priority $P(a,b)$ is fixed for any cell
$(a,b)$ in $\lambda$-$(\text{HMT}_{i,j})$, we say that
$\lambda$-$(\text{HMT}_{i,j})$ is of the fixed processor priority.
Although the priorities of underlying processors in
$\lambda$-$(\text{HMT}_{i,j})$ have to decrease in rows and columns by
Definition 4.8, it is not required to specify the processor priority for each
cell (or processor) in $\lambda$-$(\text{HMT}_{i,j})$. Meanwhile,
$\lambda$-$(\text{HMT}_{i,j})$ of the fixed processor priority (see Fig. 7(a))
denotes the priorities of underlying processors in
$\lambda$-$(\text{HMT}_{i,j})$, where the label in the top-right corner of
each cell denotes the processor priority (cf. [18]).
## 5 Priority-based task reassignments in a 2D-HMS
Consider the priority-based task assignment in Fig. 4(a) represented by
$(\text{HMT}_{i,j})$ of shape $\lambda=(3,3,3)\vdash n$ for $n=9$. Assume
$\lambda$-$(\text{HMT}_{i,j})$ is not of the fixed processor priority.
(a) $A_{0}=\young(124,357,689)$ , (b) $\young(\bullet
24,357,689)\Rightarrow\young(2\bullet
4,357,689)\Rightarrow\young(24\bullet,357,689)\Rightarrow\young(247,35\bullet,689)\Rightarrow\young(247,359,68\bullet)$
.
(c) $A_{0}=\young(124,357,689)$ , $A_{1}=\young(247,359,68\hfil)$ ,
$A_{2}=\young(247,589,6\hfil\hfil)$ , $A_{3}=\young(479,58\hfil,6\hfil\hfil)$
, $A_{4}=\young(479,68\hfil,\hfil\hfil\hfil)$ ,
$A_{5}=\young(479,6\hfil\hfil,\hfil\hfil\hfil)$ ,
$A_{6}=\young(679,\hfil\hfil\hfil,\hfil\hfil\hfil)$ ,
$A_{7}=\young(79\hfil,\hfil\hfil\hfil,\hfil\hfil\hfil)$ ,
$A_{8}=\young(9\hfil\hfil,\hfil\hfil\hfil,\hfil\hfil\hfil)$ .
Fig. 4: A sequence of task reassignments for a task completion sequence (1, 3,
2, 5, 8, 4, 6, 7, 9).
Input: An initial task assignment $A_{0}$ represented by
$\lambda$-$(\text{HMT}_{i,j})$ of normal shape; a task completion sequence
$(b_{1},b_{2},\ldots,b_{m})$, where $m\geq 2$, in runtime
Output: A task reassignment sequence $(A_{k})_{k=0}^{m-1}$
begin
set $\lambda$-$(\text{HMT}^{(1)}_{i,j})$ :=$\lambda$-$(\text{HMT}_{i,j})$;
for _$k\leftarrow 1$ to $m-1$_ do
let $t$ be the maximally embedded tableau of
$\lambda$-$(\text{HMT}^{(k)}_{i,j})$ and $\mu$ be the shape of $t$; wait for a
completion of task $b_{k}$ in $\lambda$-$(\text{HMT}^{(k)}_{i,j})$; if task
$b_{k}$ is completed, then the cell (or processor) of task $b_{k}$ in
$\lambda$-$(\text{HMT}^{(k)}_{i,j})$ becomes vacated; set $x$ as the
corresponding cell in $\lambda$-$(\text{HMT}^{(k)}_{i,j})$;
while _$x$ is not an inner corner of $\mu$_ do
if _$x=(i,\,j)$_ then
let $x^{\prime}$ be the cell of
$\text{min}\\{\text{HMT}^{(k)}_{i+1,j}\,,\,\text{HMT}^{(k)}_{i,j+1}\\}$ // if
only one non-idle cell (or processor) exists in $(i+1,j)$ and $(i,j+1)$, then
choose that cell;
end if
relocate the task on cell $x^{\prime}$ into cell $x$ and set $x:=x^{\prime}$;
end while
set $A_{k}$:=$\lambda$-$(\text{HMT}^{(k)}_{i,j})$, where
$\lambda$-$(\text{HMT}^{(k)}_{i,j})$ is of normal shape;
set
$\lambda$-$(\text{HMT}^{(k+1)}_{i,j})$:=$\lambda$-$(\text{HMT}^{(k)}_{i,j})$;
end for
return _the task reassignment sequence $(A_{k})_{k=0}^{m-1}$_;
end
Algorithm 6 Task reassignments: $\lambda$-$(\text{HMT}_{i,j})$ of normal shape
If task 1 in Fig. 4(a) is completed first, the processor addressed by $(1,1)$
becomes idle. We mark the cell $(1,1)$ as $\bullet$ to show that the processor
addressed by $(1,1)$ is now in the idle state. Once a processor is in the idle
state, it seeks the right and below processor to perform task relocation.
Recall that task relocation is only allowed between two adjacent processors if
one is in the idle state and the other is in the busy state. If a processor is
in the idle state, it does not check the left and above processor to perform
task relocation. It is because the underlying 2D mesh processor diagram of
$\lambda$-$(\text{HMT}_{i,j})$ is hierarchical, it is not an optimal choice if
a task is to run on a processor with the lower execution rate. Thus, a
processor in the idle state always seeks both the right and below processor in
order to compare task priorities and to relocate a task. We see that task
$\text{HMT}_{1,2}$ has a higher priority than task $\text{HMT}_{2,1}$, i.e.,
$2\prec_{t}3$. Thus, task relocation involves the processor addressed by
$(1,1)$ and the processor addressed by $(1,2)$. Now task 2 has been relocated
and the processor addressed by $(1,2)$ becomes idle. If a processor becomes
idle, the choice for task relocation between the right and below processor is
always _greedy_ [43], allowing the task with the higher priority to occupy the
idle processor (see Fig. 4(b)). This process continues until no task
relocation is possible, which means that the right and below processor are
both idle or both not available. The final state of Fig. 4(b) is the task
reassignment $A_{1}$ for the completion of task 1. Given the initial task
assignment $A_{0}$ in Fig. 4(a), Fig. 4(c) shows the task reassignment
sequence $(A_{k})_{k=0}^{m-1}$ for $m=9$ corresponding to the task completion
sequence $(1,\,3,\,2,\,5,\,8,\,4,\,6,\,7,\,9)$. Algorithm 6 describes the
procedure in Fig. 4. Task reassignments take place in Algorithm 6 if a task
completion sequence is provided in run time. It turns out that the iterative
greedy task relocation mechanism in Algorithm 6 corresponds to the forward jeu
de taquin slide discussed in Algorithm 4, except that task relocation starts
with the cell that is indicated by the task completion sequence. Note that
each task assignment in a task reassignment sequence $(A_{k})_{k=0}^{m-1}$ in
Algorithm 6 satisfies the necessary conditions in Definition 2.6. We see that
each task assignment in $(A_{k})_{k=0}^{m-1}$ in Algorithm 6 is represented by
a hierarchical 2D mesh tableau of normal shape, where task IDs are increasing
in rows and columns on the underlying hierarchical 2D mesh processor diagram.
Therefore, $(A_{k})_{k=0}^{m-1}$ in Algorithm 6 is a necessarily optimal task
reassignment sequence.
Thus far, we have examined the case where the initial task assignment is
represented by $\lambda$-$(\text{HMT}_{i,j})$ of normal shape. Now we consider
the case where the initial task assignment is represented by
$\lambda$-$(\text{HMT}_{i,j})$ of skew shape.
$t_{1}$= $\young(\hfil\hfil 16,\hfil\hfil 4\hfil,235\hfil,78\hfil\hfil)$ ,
$t_{2}$= $\young(\hfil\hfil 16,\hfil 34\hfil,25\hfil\hfil,78\hfil\hfil)$ ,
$t_{3}$= $\young(::16,::4,235,78)$ , $t_{4}$= $\young(::16,:34,25,78)$ ,
$t_{5}$= $\young(1346,28\hfil\hfil,5\hfil\hfil\hfil,7\hfil\hfil\hfil)$ .
Fig. 5: Task reassignments by using forward jeu de taquin slides.
Input: An initial task assignment $A_{0}$ represented by
$\lambda$-$(\text{HMT}_{i,j})$ of skew shape
Output: A task reassignment sequence $(A_{k})_{k=0}^{m}$
begin
if $\beta$ is a partition of $m^{\prime}$ for the maximally embedded tableau
of $\lambda$-$(\text{HMT}_{i,j})$ of skew shape $\alpha/\beta$, then set $m$
as the value of $m^{\prime}$; set $\lambda$-$(\text{HMT}^{(1)}_{i,j})$
:=$\lambda$-$(\text{HMT}_{i,j})$; set $k:=1$;
while _the maximally embedded tableau of $\lambda$-$(\text{HMT}^{(k)}_{i,j})$
is not of normal shape_ do
if the maximally embedded tableau of $\lambda$-$(\text{HMT}^{(k)}_{i,j})$ is
of (skew) shape $\mu/\nu$, pick $x$ to be an inner corner of $\nu$;
while _$x$ is not an inner corner of $\mu$_ do
if _$x=(i,\,j)$_ then
let $x^{\prime}$ be the cell of
$\text{min}\\{\text{HMT}^{(k)}_{i+1,j}\,,\,\text{HMT}^{(k)}_{i,j+1}\\}$ // if
only one non-idle cell (or processor) exists in $(i+1,j)$ and $(i,j+1)$, then
choose that cell;
end if
relocate the task on cell $x^{\prime}$ into cell $x$ and set $x:=x^{\prime}$;
end while
set $A_{k}$:=$\lambda$-$(\text{HMT}^{(k)}_{i,j})$; set
$\lambda$-$(\text{HMT}^{(k+1)}_{i,j})$:=$\lambda$-$(\text{HMT}^{(k)}_{i,j})$;
$k:=k+1$;
end while
return _the task reassignment sequence $(A_{k})_{k=0}^{m}$_, where $A_{m}$ is
the task reassignment represented by $\lambda$-$(\text{HMT}^{(m)}_{i,j})$ of
normal shape;
end
Algorithm 7 Task reassignments: $\lambda$-$(\text{HMT}_{i,j})$ of skew shape
Consider a top-left corner of a hierarchical 2D mesh tableau $t_{1}$ or
$t_{2}$ in Fig. 5, where the processor with the highest priority is idle.
Thus, it is a natural choice to relocate a task from a processor with the
lower execution rate to a processor with the higher execution rate. Algorithm
7 describes the procedure in which the initial task assignment, represented by
a hierarchical 2D mesh tableau of skew shape, is converted into the task
reassignment, represented by a hierarchical 2D mesh tableau of normal shape.
As shown in Fig. 5, $t_{3}$ is the maximally embedded tableau of $t_{1}$, and
$t_{4}$ is the maximally embedded tableau of $t_{2}$, respectively. By
performing task relocations discussed in Algorithm 7 iteratively, the task
reassignment $t_{5}$ is obtained from the task assignment $t_{1}$ in Fig. 5.
Similarly, the task reassignment $t_{5}$ is also obtained from the task
assignment $t_{2}$ in Fig. 5. If $t_{2}$ represents an initial task assignment
$A_{0}$, then $t_{5}$ corresponds to the task reassignment $A_{3}$. According
to Algorithm 7, the initial choice of task relocation for $A_{1}$ must target
for either the cell $(1,2)$ or the cell $(2,1)$ in $t_{2}$. Note that the task
reassignment sequence $(A_{0},A_{1},A_{2},A_{3})$ is not uniquely determined,
depending on the choice of an initial task relocation. However, $A_{3}$ is
uniquely determined by Theorem 3.4, since the greedy-based task relocations on
$\lambda$-$(\text{HMT}_{i,j})$ follow the forward jeu de taquin slide rules.
Unlike a task reassignment represented by $\lambda$-$(\text{HMT}_{i,j})$ of
normal shape, a task reassignment represented by
$\lambda$-$(\text{HMT}_{i,j})$ of skew shape do not always satisfy the
necessary conditions for being an optimal task assignment. For example, non-
idle processors of $A_{0}$, $A_{1}$, and $A_{2}$ are not left-justified, which
implies that the processor with the higher execution rate is idle for some
pairs of adjacent processors in a 2D-HMS. However, the resulting task
reassignment $A_{3}$ satisfies the necessary conditions for being an optimal
task assignment in a 2D-HMS. Note that this process is reversible by using the
backward jeu de taquin slides as discussed in Algorithm 5. Now we define an
equivalence class of task reassignments up to task relocations under the
greedy task relocation policy.
###### Definition 5.1.
Two task assignments $t_{1}$, represented by a hierarchical 2D mesh tableau
$\lambda$-$(\text{HMT}^{1}_{i,j})$ of skew shape, and $t_{2}$, represented by
a hierarchical 2D mesh tableau $\lambda$-$(\text{HMT}^{2}_{i,j})$ of skew
shape, are called _task reassignment equivalent_ up to task relocations (under
the greedy task relocation policy described in Algorithm 7), denoted as
$t_{1}\cong_{t}t_{2}$, if they have the same resulting task reassignment,
represented by the same $\lambda$-$(\text{HMT}_{i,j})$ of normal shape.
###### Proposition 5.1.
Let $\lambda$-$(\text{HMT}^{1}_{i,j})$ and $\lambda$-$(\text{HMT}^{2}_{i,j})$
be hierarchical 2D mesh tableaux of skew shape whose maximally embedded
tableaux are both standard skew tableaux. If two task assignments $t_{1}$,
represented by $\lambda$-$(\text{HMT}^{1}_{i,j})$, and $t_{2}$, represented by
$\lambda$-$(\text{HMT}^{2}_{i,j})$, are task reassignment equivalent, then the
reading words of their maximally embedded tableaux are both Knuth-equivalent
and $P$-equivalent.
###### Proof 4.
Let $P$ be the maximally embedded tableau of
$\lambda$-$(\text{HMT}^{1}_{i,j})$ for the task assignment $t_{1}$, and $Q$ be
the maximally embedded tableau of $\lambda$-$(\text{HMT}^{2}_{i,j})$ for the
task assignment $t_{2}$. Since $t_{1}\cong_{t}t_{2}$ by hypothesis, $t_{1}$
and $t_{2}$ have the same resulting task reassignment represented by the same
$\lambda$-$(\text{HMT}_{i,j})$ of normal shape by Definition 5.1. Let $N$ be
the maximally embedded tableau of such $\lambda$-$(\text{HMT}_{i,j})$ of
normal shape. Each task relocation under the greedy task relocation policy
described in Algorithm 7 follows the forward jeu de taquin slide rules
described in Algorithm 4. Thus, $P\cong_{jdt}Q$ by Theorem 3.4. Since $P$ and
$Q$ are both standard skew tableaux by hypothesis satisfying $P\cong_{jdt}Q$,
we conclude that the reading words of $P$ and $Q$ are both Knuth-equivalent
and $P$-equivalent by Theorem 3.3 and 3.5. ∎
$\pi_{k}$: $\emptyset$ , $\young(7)$ , $\young(78)$ , $\young(28,7)$ ,
$\young(23,78)$ , $\young(235,78)$ , $\young(234,58,7)$ , $\young(134,28,5,7)$
, $\young(1346,28,5,7)$= $\pi$.
$\tau_{k}$: $\emptyset$ , $\young(7)$ , $\young(78)$ , $\young(28,7)$ ,
$\young(25,78)$ , $\young(23,58,7)$ , $\young(234,58,7)$ ,
$\young(134,28,5,7)$ , $\young(1346,28,5,7)$ = $\tau$.
Fig. 6: Insertion tableaux for the reading word of $t_{3}$ and the reading
word of $t_{4}$ in Fig. 5.
For example, the reading word of $t_{3}$ (i.e., the maximally embedded tableau
of $t_{1}$) in Fig. 5 is $\pi=[7\,8\,2\,3\,5\,4\,1\,6]\in\mathfrak{S}_{8}$.
Similarly, the reading word of $t_{4}$ (i.e., the maximally embedded tableau
of $t_{2}$) in Fig. 5 is $\tau=[7\,8\,2\,5\,3\,4\,1\,6]\in\mathfrak{S}_{8}$.
By Definition 3.13(3), we see that $\pi$ and $\tau$ are Knuth-equivalent by
the choice of $x=3$, $y=4$, and $z=5$. Fig. 6 shows the procedure for
constructing insertion tableaux for both $\pi$ and $\tau$. We also see that
$\pi$ and $\tau$ are $P$-equivalent.
## 6 Application
This section presents an application of priority-based task reassignments in a
2D-HMS. We first introduce the sequential task assignment problem in a 2D-HMS.
The sequential task assignment problem in a 2D-HMS is defined as follows. A
set of $m$ tasks $T_{m}=\\{1,2,\ldots,m\\}$ with the precedence relationship
$1\rightarrow 2\rightarrow\cdots\rightarrow m$ are to be assigned to a set of
$m$ heterogeneous processors in a 2D-HMS of shape $\lambda\vdash n$ $(m\leq
n)$ and executed sequentially without gaps. We assume that each task is
assigned to each processor in a 2D-HMS bijectively, where a 2D-HMS is
dedicated for a sequential task assignment. Tasks are heterogeneous, and their
priorities are assigned by their resource requirements in which the higher
task priority indicates the larger resource requirement. Find a bijective task
assignment between $T_{m}$ and a set of $m$ processors in the 2D-HMS of shape
$\lambda\vdash n$ $(m\leq n)$ in such a way that the task turnaround time is
minimized.
(a)$\Yboxdim{20pt}\young({\textrm{ }1}^{{}^{{}^{\textrm{ }\textrm{
}1}}}{\textrm{ }2}^{{}^{{}^{\textrm{ }\textrm{ }2}}}{\textrm{
}4}^{{}^{{}^{\textrm{ }\textrm{ }4}}}{\textrm{ }7}^{{}^{{}^{\textrm{ }\textrm{
}7}}},{\textrm{ }3}^{{}^{{}^{\textrm{ }\textrm{ }3}}}{\textrm{
}5}^{{}^{{}^{\textrm{ }\textrm{ }5}}}{\textrm{ }8}^{{}^{{}^{\textrm{ }\textrm{
}8}}}{11}^{{}^{{}^{11}}},{\textrm{ }6}^{{}^{{}^{\textrm{ }\textrm{
}6}}}{\textrm{ }9}^{{}^{{}^{\textrm{ }\textrm{
}9}}}{12}^{{}^{{}^{12}}}{14}^{{}^{{}^{14}}},{10}^{{}^{{}^{10}}}{13}^{{}^{{}^{13}}}{15}^{{}^{{}^{15}}}{16}^{{}^{{}^{16}}})$,
(b)$\Yboxdim
13pt\young(2134,5678,9\mbox{10}\mbox{11}\mbox{12},\mbox{13}\mbox{14}\mbox{15}\mbox{16})$,
(c)$\young(\hfil\hfil 15,\hfil\hfil 37,26\hfil\hfil,48\hfil\hfil)$,
(d)$\young(2615,4837,\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil)$,
(e)$\young(135\hfil,267\hfil,4\hfil\hfil\hfil,8\hfil\hfil\hfil)$ .
Fig. 7: Comparisons of task reassignments.
By Definition 2.6, a sequential task assignment represented by
$\lambda$-$(\text{HMT}_{i,j})$ of normal shape satisfies the necessary
conditions for being an optimal task assignment in a 2D-HMS. Further, an
optimal task assignment is represented by $\lambda$-$(\text{HMT}_{i,j})$ of
the fixed processor priority in which task ID and processor ID are the same
for each cell (see Fig. 7(a)). Unless a sequential task assignment is
represented by $\lambda$-$(\text{HMT}_{i,j})$ of the fixed processor priority,
we only concern the necessary conditions for being an optimal task assignment.
We define a _descent pair_ of a generalized $\lambda$-$(\text{HMT}_{i,j})$,
which breaks the first necessary condition for being an optimal task
assignment in a 2D-HMS (see Definition 2.6).
###### Definition 6.1.
Let $\lambda$-$(\text{HMT}_{i,j})$ be a generalized hierarchical 2D mesh
tableau. Suppose $\text{HMT}_{i,j}\prec_{t}\text{HMT}_{i-1,j}$ (respectively,
$\text{HMT}_{i,j}\prec_{t}\text{HMT}_{i,j-1}$). Then, $\\{(i-1,j),(i,j)\\}$
(respectively, $\\{(i,j-1),(i,j)\\}$) is called a _descent pair_ of a
generalized $\lambda$-$(\text{HMT}_{i,j})$.
If a descent pair occurs in a sequential task assignment represented by a
generalized $\lambda$-$(\text{HMT}_{i,j})$, it always has the better task
assignment in terms of task turnaround time. For example, swapping tasks on a
descent pair reduces task turnaround time for a sequential task assignment in
a 2D-HMS. It follows that if a sequential task assignment cannot be
represented by (a standard) $\lambda$-$(\text{HMT}_{i,j})$, it is not a
necessarily optimal task assignment. We see that the task assignment
represented by Fig. 7(b) has a descent pair in a generalized
$\lambda$-$(\text{HMT}_{i,j})$, i.e.,
$\text{HMT}_{1,2}\prec_{t}\text{HMT}_{1,1}$. Now consider priority-based task
reassignments for a sequential task assignment in a 2D-HMS. We see that the
task completion sequence is simply $(1,2,\ldots,m)$ for a sequential task
assignment of $m$ tasks. Note that a greedy task relocation policy keeps
$\lambda$-$(\text{HMT}_{i,j})$ from occurring any descent pair for a
sequential task reassignment. If we assume that every 2D-HMS is consistent
(see Definition 2.2) in which task relocations are cost-free, we have
Proposition 6.1.
###### Proposition 6.1.
Let $A_{0}$ be a sequential task assignment for $m\;(m\geq 2)$ tasks
represented by $\lambda$-$(\text{HMT}_{i,j})$ of normal shape. Let $T_{1}$ be
the task turnaround time for $A_{0}$ without task relocation, and $T_{2}$ be
the task turnaround time with the task reassignment sequence
$(A_{k})_{k=0}^{m-1}$ (see Algorithm 6). Then, $T_{2}$ is less than $T_{1}$,
i.e., $T_{2}<T_{1}$.
###### Proof 5.
It suffices to show that each task relocation allows each task to reduce its
task execution time for a sequential task assignment in a 2D-HMS. Let
$t_{i,x}$ be the task execution time of task $i$ on processor (or cell) $x$ in
$\lambda$-$(\text{HMT}_{i,j})$ before task relocation. After task relocation,
task $i$ is relocated from processor $x$ to its adjacent processor
$x^{\prime}$ such that $P(x^{\prime})\prec_{p}P(x)$, where
$P(x^{\prime})\prec_{p}P(x)$ means that an execution rate of processor
$x^{\prime}$ is higher than that of processor $x$. Thus,
$t_{i,x^{\prime}}<t_{i,x}$. Since each task reassignment consists of iterative
task relocations by Algorithm 6, we conclude that $T_{2}<T_{1}$. ∎
Now consider a sequential task assignment in a 2D-HMS represented by
$\lambda$-$(\text{HMT}_{i,j})$ of skew shape (see Fig. 7(c)). A traditional
task relocation mechanism [3] allows the submesh to slide up and become a new
task reassignment (see Fig. 7(d)). Note that a simple sliding mechanism does
not often result in the task reassignment having the form of a standard
hierarchical 2D mesh tableau. Rather, a descent pair may occur and become a
generalized hierarchical 2D mesh tableau. Recall that the task relocation
mechanism in Algorithm 7 results in the task reassignment represented by a
hierarchical 2D mesh tableau of normal shape (see Fig. 7(e)). For example, let
$A_{0}$ be an initial task assignment represented by
$\lambda$-$(\text{HMT}_{i,j})$ of skew shape in Fig. 7(c). Since $A_{0}$ has
four empty processors in the top-left corner of
$\lambda$-$(\text{HMT}_{i,j})$, the task reassignment sequence is
$(A_{0},A_{1},A_{2},A_{3},A_{4})$ by Algorithm 7. In the process of task
reassignments, the greedy task relocation policy keeps a hierarchical 2D mesh
tableau from occurring any descent pair. It follows that the resulting task
reassignment $A_{4}$ in Fig. 7(e) is a necessarily optimal task reassignment
for $A_{0}$ in Fig. 7(c).
## 7 Conclusions
In this paper we have presented a novel approach to representing priority-
based task reassignments in a hierarchical 2D mesh-connected system (2D-HMS).
A priority-based task reassignment in a 2D-HMS is represented by a
hierarchical 2D mesh tableau $\lambda$-$(\text{HMT}_{i,j})$ in which task
relocations under the greedy task relocation policy are reduced to a jeu de
taquin slide on $\lambda$-$(\text{HMT}_{i,j})$. In case we ignore task
relocation costs, the hierarchical nature of $\lambda$-$(\text{HMT}_{i,j})$
allows each task relocation to improve the task turnaround time. Some
comparisons are given between the traditional approach and our approach to
task relocations in a 2D-HMS. For a priority-based task assignment represented
by $\lambda$-$(\text{HMT}_{i,j})$, a traditional approach may generate a
descent pair for task relocations, which means that it is not a necessarily
optimal task reassignment. We have shown that our approach to task
reassignment keeps $\lambda$-$(\text{HMT}_{i,j})$ from occurring any descent
pair for task relocations.
To the best of our knowledge, this paper is the first attempt to study task
assignments and reassignments in 2D mesh-connected systems by using tableaux.
However, the assumptions that we have made about priority-based task
reassignments in a 2D-HMS still need to be relaxed. For example, we assumed
that communications and task relocations are cost-free in a 2D-HMS. Priority-
based task reassignments in a 2D-HMS with more relaxed assumptions (e.g., non-
zero task relocation costs) are left for future work.
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|
arxiv-papers
| 2011-07-10T15:48:34 |
2024-09-04T02:49:20.391144
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dohan Kim",
"submitter": "Dohan Kim",
"url": "https://arxiv.org/abs/1107.1866"
}
|
1107.1915
|
# Interface growth in the channel geometry and tripolar Loewner evolutions
Miguel A. Durán mdr1angel@df.ufpe.br Giovani L. Vasconcelos
giovani@df.ufpe.br Laboratório de Física Teórica e Computacional, Departamento
de Física, Universidade Federal de Pernambuco, 50670-901, Recife, Brazil.
###### Abstract
A class of Laplacian growth models in the channel geometry is studied using
the formalism of _tripolar_ Loewner evolutions, in which three points, namely,
the channel corners and infinity, are kept fixed. Initially, the problem of
fingered growth, where growth takes place only at the tips of slit-like
fingers, is revisited and a class of exact exact solutions of the
corresponding Loewner equation is presented for the case of stationary driving
functions. A model for interface growth is then formulated in terms of a
generalized tripolar Loewner equation and several examples are presented,
including interfaces with multiple tips as well as multiple growing
interfaces. The model exhibits interesting dynamical features, such as tip and
finger competition.
###### pacs:
68.70.+w, 05.65.+b, 61.43.Hv, 47.54.–r
## I Introduction
The Loewner equation describes a rather general class of growth processes in
two dimensions where a curve starts from a given point on the boundary of a
domain $\mathbb{P}$ in the complex $z$-plane and grows into the interior of
$\mathbb{P}$. More specifically, the Loewner equation loewner is a first-
order differential equation for the conformal mapping $g_{t}(z)$ from the
‘physical domain,’ consisting of the region $\mathbb{P}$ minus the curve, onto
a ‘mathematical domain’ represented by $\mathbb{P}$ itself. The specific form
of the Loewner equation depends both on the domain $\mathbb{P}$ and on the
point where the trace ends. The most studied cases are the chordal and radial
Loewner equations review3 , where in the former case the trace ends on a given
point on the boundary, while in the latter it ends on a point in the interior.
Other geometries such as the dipolar case BB2005 and the channel geometry
poloneses have also been considered in the literature. The Loewner equation
depends explicitly on a driving function, here denoted by $a(t)$, which is the
image of the growing tip under the mapping $g_{t}(z)$. An important
development in the field was the discovery by Schramm schramm that when
$a(t)$ is a Brownian motion the resulting Loewner evolution describes the
scaling limit of certain statistical mechanics models. This result spurred
great interest in the so-called stochastic Loewner equation (SLE) and the
subject has now been widely reviewed, see, e.g. review1 ; review2 ; cardy ; BB
.
Although the SLE has attracted most of the attention lately, the deterministic
version of the Loewner equation remains of considerable interest both in a
purely mathematical context kadanoff_jsp and in connection with applications
to growth processes selander ; makarov ; poloneses and integrable systems
theory zabrodin2 . Indeed, the deterministic Loewner equation has been used to
study Laplacian fingered-growth in both the chordal and radial cases selander
; makarov as well as in the channel geometry poloneses . In this class of
models, growth takes place only at the tips of slit-like fingers and the
driving function $a(t)$ has to follow a specific time evolution in order to
ensure that the tip grows along gradient lines of the corresponding Laplacian
field. Although this thin-finger model poloneses was able to reproduce some
of the qualitative behavior seen in experiments combustion , treating the
fingers as infinitesimally thin is a rather severe approximation.
More recently, the growth of “extended fingers” us_pre or “fat slits”
zabrodin1 , meaning a domain encircled by an interface with endpoints on the
real axis, was considered within the formalism of deterministic Loewner
evolutions in the upper half-plane. In particular, the interface growth model
considered in Ref. us_pre was shown to exhibit interesting dynamical features
akin to the finger competition mechanism observed in actual Laplacian growth
pelce . Many growth processes, however, take place in a bounded domain, e.g.,
within a channel pelce , for which a Loewner-equation approach was still
lacking. To develop a formalism based on Loewner evolutions to describe the
growth of an interface in the channel geometry is thus the main goal of the
present study. Because the Loewner function $g_{t}(z)$ fixes the points $z=\pm
1$ (the channel corners) and $z=i\infty$ (the channel ‘open’ end), we shall
refer to growth processes in the channel geometry as _tripolar_ Loewner
evolutions, to distinguish it from the dipolar case where two points are kept
fixed BB2005 .
We begin our analysis by revisiting in Sec. II the problem of fingered growth
in the channel geometry, where a curve (finger) grows from a point on the real
axis into a semi-infinite channel. We present a brief derivation of the
corresponding tripolar Loewner and report a new class of exact solutions for
the case of multiple fingers with stationary driving functions. In Sec. III we
then discuss a general class of models where a domain, bounded by an
interface, grows into a channel starting from a segment on the real axis. In
our model, the growth rate is specified at a certain number of special points
on the interface, referred to as _tips_ and _troughs_ , which then determine
the growth rate at the other points of the interface according to a specific
growth rule (formulated in terms of a polygonal curve in the mathematical
plane). A generalized tripolar Loewner equation for this problem is derived
and several examples are given in which the interface evolution is obtained by
a direct numerical integration of the Loewner equation. Our main results and
conclusions are summarized in Sec. IV.
## II Loewner Evolution for Fingers in the Channel
### II.1 Tripolar Loewner Equation
Figure 1: The physical $z$-plane and the mathematical planes at times $t$ and
$t+\tau$ for a single curve in the channel geometry. The Loewner function
$g_{t}(z)$ maps the physical domain in the $z$-plane onto the channel in the
mathematical $w$-plane, with the tip $\gamma(t)$ of the curve $\Gamma_{t}$
being mapped to the point $w=a(t)$. The portion (dashed line) of the curve
accrued during a subsequent infinitesimal time interval $\tau$ is mapped under
$g_{t}(z)$ to a vertical slit; see text.
In order to set the stage for the remainder of the paper and to establish the
relevant notation, we begin our discussion by considering the simplest Loewner
evolution in the channel geometry, namely, that in which a single curve grows
from a point on the real axis into a semi-infinite channel whose side walls
are placed at $x=\pm 1$ and the bottom wall is at $y=0$; see Fig. 1. Here the
relevant domain, $\mathbb{P}$, for the Loewner evolution is
$\mathbb{P}=\left\\{{z=x+\mbox{i}y\in\mathbb{C}:y>0,x\in]-1,1[}\right\\}.$
The curve at time $t$ is denoted by $\Gamma_{t}$ and its growing tip is
labeled by $\gamma(t)$. Now let $w=g_{t}(z)$ be the conformal mapping that
maps the ‘physical domain,’ corresponding to the upper half-channel
$\mathbb{P}$ in the $z$-plane minus the curve $\Gamma_{t}$, onto the upper
half-channel $\mathbb{P}$ in an auxiliary complex $w$-plane, referred to as
the ‘mathematical plane,’ i.e., we have
$g_{t}:\mathbb{P}\backslash\Gamma_{t}\rightarrow\mathbb{P},$
with the curve tip $\gamma(t)$ being mapped to a point $w=a(t)$ on the real
axis in the $w$-plane; see Fig. 1.
The mapping function $g_{t}(z)$ is required to satisfy the so-called
‘hydrodynamic condition’ at infinity, namely,
$g_{t}(z)=-iC(t)+z+O(\frac{1}{|z|}),\qquad{{\rm Im}(z)\rightarrow\infty},$ (1)
where $C(t)$ is a real-valued, monotonously increasing function of time
satisfying the condition $C(0)=0$. Condition (1), which implies
$g_{t}^{\prime}(i\infty)=1$, together with the requirement $g_{t}(\pm 1)=\pm
1$, uniquely determines (up to a reparametrization of the time $t$) the
mapping $g_{t}(z)$. The time parametrization is specified by fixing the
function $C(t)$, which in turn is related to the growing rate of the curve;
see below. The Loewner function $g_{t}(z)$ also satisfies the initial
condition
$g_{0}(z)=z,$ (2)
since we start with an empty channel.
We consider the growth process to be such that growth occurs only the tip of
the curve and in such a way that the accrued portion from time $t$ to time
$t+\tau$, where $\tau$ is an infinitesimal time interval, is mapped under
$g_{t}(z)$ to a vertical slit of (infinitesimal) height $h$ in the
mathematical $w$-plane; see Fig. 1. For convenience of notation, we shall
represent the mathematical plane at time $t+\tau$ as the complex
$\zeta$-plane, so that the mapping $\zeta=g_{t+\tau}(z)$ maps the physical
domain at time $t+\tau$ onto the upper half-channel $\mathbb{P}$ in the
$\zeta$-plane, with the new tip $\gamma(t+\tau)$ being mapped to the point
$\zeta=a(t+\tau)$.
In order to derive the Loewner equation for the problem formulated above it is
necessary to write $g_{t}$ in terms of $g_{t+\tau}$, i.e.,
$g_{t}=F(g_{t+\tau})$, where $w=F(\zeta)$ is the conformal mapping between the
mathematical $\zeta$\- and $w$\- planes, and then consider the limit $\tau\to
0$. Alternatively, one can transform the growth problem defined in the channel
geometry into a corresponding Loewner evolution in the upper half-plane. To do
that, we apply the following transformations:
$\tilde{z}=\phi(z),\qquad\tilde{w}=\phi(w),\qquad\tilde{\zeta}=\phi(\zeta),$
(3)
where
$\phi(z)=\sin\left(\frac{\pi}{2}z\right),$ (4)
so that the channels in the $z$-, $w$-, and $\zeta$\- planes are mapped to the
upper half-plane in the auxiliary complex $\tilde{z}$-, $\tilde{w}$-, and
$\tilde{\zeta}$\- planes, respectively. Let us also introduce introduce the
following notation:
$\tilde{g}_{t}=\phi\circ g_{t}\circ\phi^{-1}.$ (5)
Next, consider the mapping $\tilde{w}=\tilde{F}(\tilde{\zeta})$ from the upper
half-$\tilde{\zeta}$-plane onto the upper half-$\tilde{w}$-plane with a
vertical slit, so that one can write
$\tilde{g}_{t}(\tilde{z})=\tilde{F}(\tilde{g}_{t+\tau}(\tilde{z})).$ (6)
The slit mapping $\tilde{F}(\tilde{\zeta})$ can be easily computed (see Ref.
poloneses for details), and after taking the limit $\tau\to 0$, one obtains
the following Loewner equation
$\dot{\tilde{g}}_{t}(\tilde{z})=\frac{\pi^{2}}{4}d(t)\frac{1-\tilde{g}_{t}(\tilde{z})^{2}}{\tilde{g}_{t}(\tilde{z})-\tilde{a}},$
(7)
where
$\tilde{a}(t)=\sin\left[\frac{\pi}{2}a(t)\right].$ (8)
and the growth factor $d(t)$ is defined by
$d(t)=\lim_{\tau\to 0}\frac{h^{2}}{2\tau}.$ (9)
By taking the limit $\tilde{g}_{t}(z)\to\tilde{a}(t)$ in Eq. (7) in an
appropriate sense poloneses , one finds that the driving function $a(t)$ is
determined by
$\dot{\tilde{a}}=-\frac{\pi^{2}}{8}d(t)\tilde{a}.$ (10)
The Loewner evolution in the original channel geometry can then be obtained
from Eqs. (5) and (7), yielding
$\dot{g}_{t}(z)=\frac{\pi}{2}d(t)\frac{\cos\left[\frac{\pi}{2}g_{t}(z)\right]}{\sin\left[\frac{\pi}{2}g_{t}(z)\right]-\sin\left[\frac{\pi}{2}a(t)\right]},$
(11)
with
$\dot{a}=-\frac{\pi}{4}d(t)\tan\left(\frac{\pi}{2}a\right).$ (12)
From the boundary condition (1) it follows that
$C(t)=(\pi/2)\int_{0}^{t}d(t^{\prime})dt^{\prime}$. Without loss of generality
we shall take $C(t)=\pi t/2$ in this section, implying that $d(t)=1$. Note
that the Loewner equation (11) indeed fixes the points ${z}=\pm 1$ as well as
the point at infinity, in the sense of Eq. (1). Hence we shall refer to Eq.
(11) as the tripolar Loewner equation, in analogy with the dipolar case BB2005
where only the points $x_{\pm}=\pm 1$ are kept fixed. The tripolar Loewner
equation in the upper half-plane given in Eq. (7) also fixes the points
$\tilde{z}=\pm 1$ and the point at infinity. More specifically, the boundary
condition at infinity reads:
$\tilde{g}_{t}(\tilde{z})\approx\exp[-\pi^{2}t/4]\tilde{z}$, for
$|\tilde{z}|\to\infty$.
The Loewner equation given in Eq. (7) can be readily extended to the case of
multiple curves growing simultaneously in the channel, yielding
$\dot{\tilde{g}}_{t}=\frac{\pi^{2}}{4}(1-\tilde{g}_{t}^{2})\sum_{i=1}^{n}\frac{d_{i}(t)}{\tilde{g}_{t}-\tilde{a}_{i}(t)},$
(13)
where
$\dot{\tilde{a}}_{i}=-\frac{\pi^{2}}{8}d_{i}(t)\tilde{a}_{i}+\frac{\pi^{2}}{4}(1-\tilde{a}_{i}^{2})\sum_{\stackrel{{\scriptstyle
j=1}}{{j\neq i}}}^{n}\frac{d_{i}(t)}{\tilde{a}_{i}-\tilde{a}_{j}}.$ (14)
We shall assume for simplicity that the growth factors are all constant, i.e.,
$d_{i}(t)=d_{i}$, subjected to the condition $\sum_{i=1}^{n}d_{i}=1$, as
follows from our choice above for $C(t)$. Other growth models have been
considered poloneses where the normal velocity is of the form
$v_{n}\sim|\vec{\nabla}\phi|^{\eta}$, with the growth factors being given by
$d_{i}(t)=|f_{t}^{\prime\prime}(a_{i}(t))|^{-\eta/2-1}$, where $f_{t}(w)$ is
the inverse of $g_{t}(z)$. In this case, the analysis of the Loewner equation
is much more difficult because of the implicit dependence of the growth
factors $d_{i}(t)$ on the Loewner function $g_{t}(z)$. By considering the
simpler (but still very interesting) case where the growth factors are
constant in time, we are able both to find exact solutions for multifingers
and to easily integrate the Loewner equation (13) on the computer.
### II.2 Exact solutions
An exact solution for the Loewner function $g_{t}(z)$ for a single curve in
the channel geometry can be found explicitly for a stationary driving
function, i.e., $a(t)=0$, corresponding to a finger growing vertically along
the channel centerline poloneses . Here we extend such stationary solutions to
the case of multifingers. For simplicity we consider only the case of two
fingers, but the procedure outlined below applies, in principle, to any number
of fingers (although the calculations become increasingly more difficult).
For two fingers, Eq. (14) becomes
$\dot{\tilde{a}}_{1}=-\frac{\pi^{2}}{8}\left[d_{1}\tilde{a}_{1}-2d_{2}\frac{1-\tilde{a}_{1}^{2}}{\tilde{a}_{1}-\tilde{a}_{2}}\right],$
(15)
$\dot{\tilde{a}}_{2}=-\frac{\pi^{2}}{8}\left[d_{2}\tilde{a}_{2}-2d_{1}\frac{1-\tilde{a}_{2}^{2}}{\tilde{a}_{2}-\tilde{a}_{1}}\right],$
(16)
whose stationary solution (fixed point), $\tilde{a}_{1}^{0}$ and
$\tilde{a}_{2}^{0}$, are given by
$\tilde{a}_{1}^{0}=-\left[\frac{2d_{2}^{2}(1+d_{1})}{1+d_{2}}\right]^{1/2},\qquad\tilde{a}_{2}^{0}=\left[\frac{2d_{1}^{2}(1+d_{2})}{1+d_{1}}\right]^{1/2}.$
(17)
Note that if $d_{1}=d_{2}$, one has $\tilde{a}_{1}^{0}=-1/2$ and
$\tilde{a}_{2}^{0}=1/2$, and the solution corresponds to two identical fingers
growing vertically. This symmetrical two-finger solution is, of course,
related to the single-finger solution described in Ref. poloneses by a simple
transformation, since in this case we can view each finger as growing along
the centerline of a channel with half the width of the original channel. Thus,
we are mainly interested here in asymmetrical solutions where $d_{1}\neq
d_{2}$. In the remainder of this section we shall omit, for convenience of
notation, the upperscripts from $\tilde{a}_{1}^{0}$ and $\tilde{a}_{2}^{0}$.
The Loewner equation (13) for two fingers reads
$\dot{\tilde{g}}_{t}=\frac{\pi^{2}}{4}\frac{(1-\tilde{g}_{t}^{2})[\tilde{g}_{t}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})]}{(\tilde{g}_{t}-\tilde{a}_{1})(\tilde{g}_{t}-\tilde{a}_{2})},$
(18)
where we have used the fact that $d_{1}+d_{2}=1$. This equation can be easily
integrated, yielding an implicit solution for the mapping $\tilde{g}_{t}$ in
the form
$\left[\frac{\tilde{g}_{t}+1}{\tilde{z}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{g}_{t}-1}{\tilde{z}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{g}_{t}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{\tilde{z}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}\right]^{\alpha_{3}}=e^{-\frac{\pi^{2}}{2}t},$
(19)
where
$\alpha_{1}=\frac{(1+\tilde{a}_{1})(1+\tilde{a}_{2})}{1+(d_{2}\tilde{a}_{1}+d_{1}\tilde{a}_{2})},\qquad\alpha_{2}=\frac{(1-\tilde{a}_{1})(1-\tilde{a}_{2})}{1-(d_{2}\tilde{a}_{1}+d_{1}\tilde{a}_{2})},$
(20)
and $\alpha_{3}=2-\alpha_{1}-\alpha_{2}$.
The tip trajectories $\gamma_{i}(t)$, for $i=1,2$, can now be obtained by
setting $\tilde{z}=\tilde{\gamma}_{i}$ and
$\tilde{g}(\tilde{\gamma}_{i})=\tilde{a}_{i}$ in Eq. (19), where
$\tilde{\gamma}_{i}=\sin(\pi\gamma_{i}/2)$. One then obtains
$\left[\frac{\tilde{\gamma_{1}}+1}{\tilde{a}_{1}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{\gamma_{1}}-1}{\tilde{a}_{1}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{\gamma_{1}}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{d_{1}(\tilde{a}_{1}-\tilde{a}_{2})}\right]^{\alpha_{3}}=e^{\frac{\pi^{2}}{2}t},$
(21)
$\left[\frac{\tilde{\gamma_{2}}+1}{\tilde{a}_{2}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{\gamma_{2}}-1}{\tilde{a}_{2}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{\gamma_{2}}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{d_{2}(\tilde{a}_{2}-\tilde{a}_{1})}\right]^{\alpha_{3}}=e^{\frac{\pi^{2}}{2}t},$
(22)
where the roots for $\tilde{\gamma}_{1}$ and $\tilde{\gamma}_{2}$ must be
chosen in the upper half-plane. The asymptotic trajectories in the limit
$t\to\infty$ can be found by writing $\gamma_{i}=x_{i}+iy_{i}$ and considering
the limit $y_{i}\to\infty$. After a straightforward calculation one obtains
$\gamma_{1}(t)\approx\frac{\alpha_{1}-\alpha_{2}-\alpha_{3}}{2}+i\frac{\pi}{2}t,\qquad\gamma_{2}(t)\approx\frac{\alpha_{1}-\alpha_{2}+\alpha_{3}}{2}+i\frac{\pi}{2}t,\qquad
t\to\infty.$ (23)
We thus see that for large times the two fingers approach the vertical axes
$x_{1,2}=(\alpha_{1}-\alpha_{2}\mp\alpha_{3})/2,$ (24)
respectively, and both of them move asymptotically with the constant speed
$v=\pi/2$.
An example of such a solution is shown in Fig. 2(a), where we used $d_{1}=2/3$
and $d_{2}=1/3$, which from (17) implies $a_{1}^{0}=-0.3534$ and
$a_{2}^{0}=0.6387$. In this case, the fingers should approach the vertical
axes $x_{1}=-0.3712$ and $x_{2}=0.6099$, as predicted by (23), and this is
indeed observed in the figure. Note that this asymptotic behavior holds for
any initial conditions $a_{1}(0)$ and $a_{2}(0)$, on account of the fact that
the fixed point given in (17) is stable. This is illustrated in Fig. 2(b)
where we show a solution with initial conditions $a_{1}(0)=-0.6$ and
$a_{2}(0)=0.4$. To generate the curves shown in Fig. 2 we integrated the
Loewner equation given in Eq. (13) using the numerical scheme described in
us_pre .
(a) (b)
Figure 2: Evolution of two fingers with $d_{1}=2/3$ and $d_{2}=1/3$. The
initial conditions are: $a_{1}(0)=a_{1}^{0}=-0.3534$ and
$a_{2}(0)=a_{2}^{0}=0.6387$ in (a), and $a_{1}(0)=-0.6$ and $a_{2}(0)=0.6$ in
(b). In both cases the trajectories approach the axes $x_{1}=-0.3712$ and
$x_{2}=0.6099$.
### II.3 Tripolar Loewner Chains
In the spirit of chordal Loewner chains BB , it is clear that the tripolar
Loewner equation (13) can be extended to describe the growth of more general
domains in the channel geometry, not restricted to a curve or a set of curves.
The corresponding tripolar Loewner chain thus reads
$\dot{g}_{t}(z)=\left(1-g_{t}(z)^{2}\right)\int_{-1}^{1}\frac{\rho_{t}(x)dx}{g_{t}(z)-x}\,,$
(25)
where the density of singularities $\rho_{t}(x)$ can be viewed as a measure of
the growth rate at a point $z$ at the boundary of the growing set that is the
preimage of $x$ under $g_{t}(z)$. Since the shape of the growing domain is
fully encoded in the map $g_{t}(z)$, equation (25) specifies the growth model
once the density $\rho_{t}(x)$ is known. As discussed elsewhere us_pre , the
formalism of Loewner chains is of limited practical use, except when the
density is a sum of Dirac $\delta$-functions, in which case we are back to
multiple growing curves. In this context, it is important to consider more
explicit models to describe the growth of a domain bounded by an interface.
One such model was first introduced in Ref. us_pre for the upper half-plane,
and in the next section it is extended to the channel geometry.
## III Loewner Evolutions for Interfaces in the Channel
Figure 3: The physical and mathematical planes for a growing interface in the
channel geometry. The mapping $g_{t}(z)$ maps the interface $\Gamma_{t}$ to a
segment on the real axis, while the new interface $\Gamma_{t+\tau}$ is mapped
to a tent-like shape.
### III.1 Tripolar Loewner Equation for Interfaces
Here we consider the problem of an interface starting from a segment
$[z_{1},z_{2}]\subset(-1,1)$ on the real axis in the $z$-plane and growing
into the upper half-channel, as indicated in Fig. 3. In our growth model we
assume, for simplicity, that the growing interface has certain special points,
referred to as _tips_ and _troughs_ , where the growth rate is a local maximum
and a local minimum, respectively, while the interface endpoints $z_{1}$ and
$z_{2}$ remain fixed. Let us denote by $\Gamma_{t}$ the interface at time $t$
and by $K_{t}$ the growing region delimited by $\Gamma_{t}$ and the real axis.
It is assumed that the curve $\Gamma_{t}$ is simple so that the physical
domain $\mathbb{P}\backslash K_{t}$ is simply connected, hence $K_{t}$ is a
hull. As before, the Loewner function $g_{t}(z)$ maps the physical domain in
the $z$-plane onto the upper half-channel $\mathbb{P}$ in the mathematical
plane $w$-plane, that is,
$g_{t}:\mathbb{P}\backslash K_{t}\rightarrow\mathbb{P},$
Under the action of $g_{t}(z)$, the interface $\Gamma_{t}$ is mapped to an
interval on the real axis in the $w$-plane, with the images of the tips,
troughs and end points being denoted by $a_{i}(t)$, $i=1,...,N$, where $N$ is
the total number of such special points; see Fig. 3. The mapping function
$g_{t}(z)$ is again required to satisfy satisfy the hydrodynamic condition (1)
and the initial condition (2). Here, however, we shall not fix the function
$C(t)$ a priori.
Figure 4: The transformed physical and the mathematical planes for a growing
interface. The ‘tilded’ planes are obtained from the original domains shown in
Fig. 3 by applying the transformation
$\phi(z)=\sin\left(\frac{\pi}{2}z\right)$; see text.
The growth dynamics in our model is specified by requiring that tips and
troughs grow along gradient lines in such a way that the interface
$\Gamma_{t+\tau}$ at time $t+\tau$, for infinitesimal $\tau$, is mapped under
$g_{t}(z)$ to a polygonal curve, as shown in Fig. 3. The exterior angle at the
$i$-th vertex of our polygonal curve is denoted by $\pi(1-\alpha_{i})$, with
the convention that if the angle is greater than $\pi$ the corresponding
parameter $\alpha_{i}$ is negative. In other words, the parameter
$\alpha_{i}$’s are negative for tips and positive for troughs and end points.
From a trivial trigonometric relation it follows that
$\sum_{i=1}^{N}\alpha_{i}=0.$ (26)
If we denote by ${h}_{i}$ the heigth of the $i$th vertex of the polygonal
curve in the $w$-plane, then as $\tau\to 0$ the parameters $h_{i}$ and
$\alpha_{i}$ all go to zero, and in this limit one finds the following
relation:
$\sum_{i=1}^{N}a_{i}\alpha_{i}=0.$ (27)
As before, the mathematical plane at time $t+\tau$ is represented by the
complex $\zeta$-plane, so that under the mapping $\zeta=g_{t+\tau}(z)$, the
interface tips, troughs, and end points at time $t+\tau$ are mapped to the
points $\zeta=a_{i}(t+\tau)$, as indicated in Fig. 3. To derive the tripolar
Loewner equation for the growing interface we first need to obtain the mapping
$w=F(\zeta)$, so that we can write $g_{t}(z)=F(g_{t+\tau}(z))$, and then take
the limit $\tau\to 0$. As discussed in Sec. II, it is more convenient however
to work with the tripolar Loewner equation in the upper half-plane by applying
the transformations given in Eq. (3). The corresponding transformed domains
are shown in Fig. 4. In the $\tilde{w}$-plane, we obtain (in the limit of
$\tau\to 0$) a polygonal curve whose vertices are located at the points
$\tilde{a}_{i}+i\tilde{h}_{i}$, where $\tilde{a}_{i}=\sin(\pi a_{i}/2)$ and
the heigth $\tilde{h}_{i}$ in the $\tilde{w}$-plane is given by
$\tilde{h}_{i}=\frac{\pi}{2}\cos\big{(}\frac{\pi}{2}a_{i}\big{)}h_{i},$ (28)
and the exterior angle at the $i$th vertex is denoted by
$\pi(1-\tilde{\alpha}_{i})$. Relations analogous to (26) and (27) are
obviously valid for $\tilde{\alpha}_{i}$ and $\tilde{a}_{i}$.
Next consider the mapping, $\tilde{w}=\tilde{F}(\tilde{\zeta})$, from the
upper half-$\tilde{\zeta}$-plane onto the upper half-$\tilde{w}$-plane with a
polygonal cutout region; see Fig. 4. Since the domain in the $\tilde{w}$-plane
can be viewed as a degenerate polygon, the mapping function $\tilde{F}$ can be
easily obtained from the Schwarz-Christoffel transformation CKP , which yields
$\tilde{g}_{t}=\tilde{F}(\tilde{g}_{t+\tau})=K\varphi(\tilde{g}_{t+\tau})+C,$
(29)
where
$\varphi(\tilde{\zeta})=\int^{\tilde{\zeta}}\prod_{i=1}^{N}{[z-\tilde{a}_{i}(t+\tau)]^{-\tilde{\alpha}_{i}}}\,dz.$
(30)
The constants $K$ and $C$ are determined so as to guarantee that
$\tilde{F}(\pm 1)=\pm 1$, and hence we can rewrite Eq. (29) as
$\tilde{g}_{t}=\frac{2\varphi(\tilde{g}_{t+\tau})-[\varphi(1)+\varphi(-1)]}{\varphi(1)-\varphi(-1)}.$
(31)
The integral in Eq. (30) cannot, in general, be performed explicitly. It is
thus convenient to expand the integrand in powers of the infinitesimal
quantities $\tilde{\alpha}_{i}$ and then proceed with the integration term by
term. Doing this up to the first order in the $\tilde{\alpha}_{i}$’s, one
finds
$\varphi(\zeta)=\zeta-\sum_{i=1}^{N}\tilde{\alpha}_{i}(t)[\zeta-\tilde{a}_{i}(t+\tau)]\ln[\zeta-\tilde{a}_{i}(t+\tau)].$
(32)
If one now inserts Eq. (32) into Eq. (31), expand the resulting expression up
to first order in $\tau$, and then take $\tau\to 0$, one obtains the following
Loewner equation
$\dot{\tilde{g}}_{t}(z)=\sum_{i=1}^{N}\tilde{d}_{i}(t)\left\\{[\tilde{g}_{t}-\tilde{a}_{i}(t)]\ln[\tilde{g}_{t}-\tilde{a}_{i}(t)]-A_{i}(t)\tilde{g}_{t}+B_{i}(t)\right\\},$
(33)
with
$A_{i}(t)=\frac{1}{2}\left\\{[1+\tilde{a}_{i}(t)]\ln[1+\tilde{a}_{i}(t)]+[1-\tilde{a}_{i}(t)]\ln[1-\tilde{a}_{i}(t)]\right\\},$
(34)
$B_{i}(t)=\frac{1}{2}\left\\{[1+\tilde{a}_{i}(t)]\ln[1+\tilde{a}_{i}(t)]-[1-\tilde{a}_{i}(t)]\ln[1-\tilde{a}_{i}(t)]\right\\},$
(35)
where the ‘tilded’ growth factors $\tilde{d}_{i}(t)$ are defined by
$\tilde{d}_{i}(t)=\lim_{\tau\to 0}\frac{\tilde{\alpha}_{i}}{\tau}.$ (36)
The time evolution of the driving functions $\tilde{a}_{i}$ is determined by
requiring that the points $\tilde{a}_{i}(t+\tau)$ be taken under the mapping
$F$ to the points $\tilde{a}_{i}(t)$. In other words, we impose that
$\tilde{a}_{i}(t)=F(\tilde{a}_{i}(t+\tau)).$ (37)
Applying the procedure used above to derive the Loewner equation Eq. (33), it
immediately follows that the differential equations governing the dynamics of
the driving functions $\tilde{a}_{i}$ can be obtained by simply setting
$\tilde{g}_{t}=\tilde{a}_{i}$ in Eq. (33), which yields
$\dot{\tilde{a}}_{i}=\sum^{N}_{j=1}\tilde{d}_{j}(t)\left\\{[\tilde{a}_{i}-\tilde{a}_{j}(t)]\ln\left|\tilde{a}_{i}-\tilde{a}_{j}(t)\right|-A_{j}(t)\tilde{a}_{i}+B_{j}(t)\right\\}.$
(38)
Figure 5: Loewner evolution for a symmetric interface growing in the channel
geometry. The solid curves show the interface at various times $t$, starting
from $t=0.5$ up to $t=2.0$ with a time separation of $\Delta t=0.3$ between
successive curves.
The tilded growth factors $\tilde{d}_{i}$ defined in Eq. (36) can be related
to the growth factors, ${d}_{i}(t)=\lim_{\tau\to
0}\left({{\alpha}_{i}}/{\tau}\right)$, defined for the original channel
geometry by virtue of trivial trigonometric relations. After some calculation,
one finds
$\tilde{d}_{i}=\frac{{C}_{i+1}-{C}_{i}}{\tilde{a}_{i+1}-\tilde{a}_{i}}+\frac{{C}_{i-1}-{C}_{i}}{\tilde{a}_{i}-\tilde{a}_{i-1}},\qquad
2\leq i\leq N-1,$ (39)
where
$C_{i}=\frac{\pi}{2}\cos\left(\frac{\pi}{2}a_{i}\right)\sum_{j=1}^{i-1}d_{i-j}(a_{i}-a_{i-j}).$
(40)
From the Eqs. (26) and (27) it also follows that the growth factors $d_{i}$
satisfy the following relations
$\sum_{i=1}^{N}d_{i}=0,$ (41) $\sum_{i=1}^{N}a_{i}d_{i}=0.$ (42)
In view of the relations above, we can express the parameters $d_{1}(t)$ and
$d_{N}(t)$ associated with the end points in terms of the other growing
factors $d_{i}(t)$, so that our growth model is completely specified by
prescribing the functions $d_{i}(t)$ at the tips and troughs of the interface.
Thus, once the growth factors $d_{i}$ are given, one can compute the tilded
growth factors $\tilde{d}_{i}$ from Eq. (39), proceed with the integration of
the Loewner equation given in Eq. (33), and then invert the transformation (5)
to obtain the Loewner mapping $g_{t}(z)$. Next we discuss some examples of
interface growth described by the above model.
### III.2 Examples
#### III.2.1 Single Tip
For an interface with a single tip (i.e., $N=3$), the factors $d_{1}(t)$ and
$d_{3}(t)$ can be written in terms of $d_{2}(t)$, so that we can set
$d_{2}=-1$ without loss of generality, since this amount to a mere rescaling
of the time coordinate. Let us consider first the case in which the interface
is symmetric with respect to the channel centerline: $a_{2}(t)=0$ and
$a_{3}(t)=-a_{1}(t)\equiv a(t)$. This imply, in particular, that
$\tilde{d}_{1}=\tilde{d}_{3}=-\tilde{d}_{2}/2$. Introducing the parameter
$\tilde{d}(t)=|\tilde{d}_{2}(t)|$, the Loewner equation (33) then becomes
$\dot{\tilde{g}}_{t}(z)=\frac{1}{2}\tilde{d}(t)\left\\{[\tilde{g}_{t}+\tilde{a}(t)]\ln[\tilde{g}_{t}+\tilde{a}(t)]+[\tilde{g}_{t}-\tilde{a}(t)]\ln[\tilde{g}_{t}-\tilde{a}(t)]-2\tilde{g}_{t}\ln\tilde{g}_{t}-A_{s}(t)\tilde{g}_{t}\right\\},$
(43)
where
$A_{s}(t)=[1+\tilde{a}(t)]\ln[1+\tilde{a}(t)]+[1-\tilde{a}(t)]\ln[1-\tilde{a}(t)],$
(44)
with the dynamics for $\tilde{a}(t)$ being given by
$\dot{\tilde{a}}(t)=\tilde{d}(t)\left(2\ln 2-A_{s}\right)\tilde{a},$ (45)
In this case, one can show from Eq. (39) that $\tilde{d}(t)$ becomes
$\tilde{d}(t)=\frac{\pi}{2}\tan\left[\frac{\pi}{2}a(t)\right]=\frac{\pi}{2}\frac{\tilde{a}(t)}{\sqrt{1-\tilde{a}^{2}(t)}},$
(46)
where we have used the fact that $|d_{2}|=1$.
Figure 6: Loewner evolution for an asymmetric interface growing in the channel
geometry. The solid curves show the interface at several times from $t=0.5$ up
to $t=1.7$, with $\Delta t=0.3$ between successive curves, while the dashed
curve represents the path traced by the tip.
An example of a symmetric interface in a channel is shown in Fig. 5. In this
figure, the solid curves represent the interface at various times $t$,
starting from $t=0.5$ up to $t=2.0$ with a time separation of $\Delta t=0.3$
between successive curves. To generate the curves shown in Fig. 5 we
integrated the Loewner equation (43) backwards in time from ‘terminal
conditions’ $\tilde{g}_{t}=\tilde{w}$, for
$\tilde{w}\in[-\tilde{a}(t),\tilde{a}(t)]$, to obtain the respective initial
values $\tilde{g}_{0}$ from which we determine the corresponding points $z$ on
the interface: $z=({2}/{\pi})\arcsin\left(\tilde{g}_{0}\right)$. One sees from
the figure that as time proceeds the interface expands and tends to occupy the
entire channel. It is worth mentioning, however, that it is hard to go past
the latest time shown in Fig. 5 because the function $a(t)$ becomes very close
to unity, rendering the numerical integration of the Loewner equation very
difficult after this point.
In the case of an asymmetric interface, we need to integrate the full Loewner
equation given in Eq. (33) with $N=3$. In Fig. 6 we show a solution where the
solid curves represent the interface at various times $t$ (see caption for
details), while the dashed curve indicates the path traced by the tip
$\gamma_{t}=g^{-1}_{t}(a_{2}(t))$. As seen in Fig. 6, the tip is “repelled” by
its image with respect to the closest channel wall, so that an asymptotically
symmetrical shape is expected for sufficiently long time. (It is difficult,
however, to integrate the Loewner equation for larger times for reasons
already mentioned.)
Figure 7: Symmetrical growing interface with two tips in the channel for
$d_{2}=d_{4}=-2$ and $d_{3}=1.5$. The solid curves represent the interface at
times from $t=0.5$ to $t=1.7$, with $\Delta t=0.3$ between successive curves,
whereas the dashed lines indicate the trajectories of the tips and the trough.
#### III.2.2 Multiple Tips
Figure 8: Asymmetrical growing interface with two tips in the channel. The
growth factors are the same as in Fig. 7 but the interface is initially
located off center.
Let us now illustrate some of the patterns that emerge from our growth model
for interfaces with two tips and one trough (i.e., $N=5$), in which case tip
competition can arise. Here we shall consider for simplicity only cases in
which $d_{i}(t)=d_{i}={\rm const.}$ We expect nevertheless that the
qualitative behavior seen in this simpler situation (see below) should also be
valid in more general cases. As a first example, we show in Fig. 7 a
symmetrical growing interface, where the dashed lines represent the
trajectories of the tips and the trough. In this figure, we started with
symmetrical initial conditions, namely, $a_{5}(0)=-a_{1}(0)=0.3$,
$a_{4}(0)=-a_{2}(0)=0.2$, and $a_{3}(0)=0$, and chose the growth factors of
the two tips to be the same, $d_{2}=d_{4}=-1$, so as to preserve the initial
symmetry, with the trough growth factor being $d_{3}(t)=0.6$.
If we break the symmetry, either by choosing an asymmetric initial condition
or using different growth factors for the tips, then inevitably of the tips
will grow faster and “screen” the other tip. For instance, in Fig. 8 we show a
situation where the growth factors are the same as in Fig. 7 but the initial
segment from which the interface grows is off-center, namely, $a_{1}(0)=0.2$,
$a_{2}(0)=0.3$, $a_{3}(0)=0.5$, $a_{4}(0)=0.7$, and $a_{5}(0)=0.8$. In this
case, the tip closest to the right channel wall grows slower and falls behind
the other tip. Here, however, the screening is partial in the sense that the
slower tip continues to grow but with a velocity that is a fraction of that of
the faster tip. (Total screening, whereby the slowest tip eventually stops
growing altogether, can be observed if the faster tip has a sufficiently large
growth factor.)
#### III.2.3 Multiples Interfaces
Figure 9: Loewner evolution for two symmetrical surfaces growing in the the
channel geometry for $d(t)=1$ for each interface. The interfaces start
respectively at $[-3,-1]$ and $[1,3]$. The different curves represent the
interfaces at time intervals of $\Delta t=0.3$.
The generic Loewner equation given in Eq. (33) can also describe the problem
of multiple growing interfaces in the channel geometry. In this case each
interface $\Gamma^{i}_{t}$, for $i=1,...,n$, where $n$ is the number of
distinct interfaces, will be mapped under $g_{t}(z)$ to a corresponding
interval on the real axis in the $w$-plane. Similarly, each advanced interface
$\Gamma^{i}_{t+\tau}$ is mapped by $g_{t}(z)$ to a polygonal curve in the
$w$-plane. One can readily convince oneself that the generic Loewner evolution
defined by (33) and (38) applies to this case as well, where $N$ is now the
total number of vertices corresponding to the sum of the number of vertices
for each interface. Note that for each interface we have conditions analogous
to (41) and (42), but involving only the growth factors $d_{i}$ of the
respective interface.
Figure 10: Loewner evolution for two asymmetric growing fingers. Here the
growth factors are the same as in Fig. 9 but the initial segments from which
the interface starts to grow were shifted to the right.
In Fig. 9 we show numerical solutions for two growing symmetrical interfaces,
with one tip each, where the interfaces are the images of one another with
respect to the $y$ axis. Here the interfaces start to grow from the intervals
$[-3,-1]$ and $[1,3]$, respectively, with the corresponding tips starting at
symmetrical points $a_{4}(0)=-a_{2}(0)=2$. (The interfaces have the same
growing factors, meaning that $d_{2}=d_{4}=-1$.) Notice that, as time goes by,
the inner sides of the two interfaces move towards one another leaving a
narrow channel between them. For sufficiently large time, the width of such a
channel becomes infinitesimally small so that for all practical purposes the
resulting evolution will look like a single, symmetrical interface.
If the symmetry between the two interfaces is broken, then one of them will
eventually move ahead of the other one. For instance, in Fig. 10 we show the
case in which the growth factors are the same as in Fig. 9 but the initial
positions of the interfaces are no longer symmetric with respect to the
channel centerline. In this case, the interface closer to the centerline is
the “winner,” in the sense that its tips moves ahead of the tip of the second
interface. This competition between the two growing interfaces resembles the
so-called “shadowing effect” in fingering phenomena, whereby the longer
fingers grow faster and hinder the growth of the shorter ones in their
vicinities.
## IV Conclusions
We have discussed a class of growth models in the channel geometry where the
growth dynamics is described in terms of tripolar Loewner evolutions. In the
tripolar geometry, the Loewner mapping fixes the points $z=\pm 1$ and the
point at infinity. The growing domain starts from a point (in the case of a
curve) or from an interval (in the case of a domain bounded by an interface)
on the real axis, ending at infinity (for infinite time). In particular, a
novel class of exact solutions of the tripolar Loewner equation for multiple
curves was obtained for the case of stationary driving functions. As for the
more difficult problem of interface growth in the channel geometry, we
introduced a generalized tripolar Loewner equation and presented several
numerical solutions, including the case of an interface with multiple tips as
well as the case of multiple interfaces. It was argued that the behavior seen
in the model is reminiscent of the phenomenon of tip/finger competition
observed in fingering experiments.
A natural extension of the work presented here would be to consider the
problem of tripolar stochastic Loewner equation (SLE) and investigate its
connection with statistical mechanics models and conformal field theory.
Although stochastic growth processes were not considered in this paper, it is
nonetheless hoped that a better understanding of deterministic Loewner
evolutions in the channel geometry, which was the main thrust of our study,
may be of some help in tackling the more difficult problem of tripolar SLEs.
Another possible direction for future work would be to consider more general
geometries, such as “quadripolar” Loewner evolutions where four points are
kept fixed. This geometry would be of interest in connection, for instance,
with growth processes in doubly periodic domains.
###### Acknowledgements.
This work was supported in part by the Brazilian agencies FINEP, CNPq, and
FACEPE and by the special programs PRONEX and CT-PETRO.
## References
* (1) Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89, 103–121 (1923).
* (2) Lawler, G.: Introduction to the stochastic Loewner evolution. In: Random Walks and Geometry, pp. 261–294. Walter de Gruyter, Berlin (2004).
* (3) Bauer, M., Bernard, D.: Dipolar stochastic Loewner evolutions, J. Stat. Mech. P03001 (2005). ArXiv:math-ph/0411038.
* (4) Gubiec, T., Szymczak, P.: Fingered growth in the channel geometry: A Loewner-equation approach. Phys. Rev. E 77, 041602-1–041602-12 (2008).
* (5) Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000).
* (6) Gruzberg, I. A., Kadanoff, L. P.: The Loewner equation: Maps and shapes. J. Stat. Phys. 114, 1183–1198 (2004).
* (7) Kager, W., Nienhuis, B.: A guide to stochastic Loewner evolution and its applications. J. Stat. Phys. 115, 1149–1229 (2004).
* (8) Cardy, J.: SLE for theoretical physicists. Ann. Phys. 318, 81–118 (2005).
* (9) Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432, 115–221 (2006).
* (10) Kager, W., Nienhuis, B., Kadanoff, L. P.: Exact solutions for Loewner evolutions. J. Stat. Phys. 115, 805–822 (2004).
* (11) Selander, G: Two deterministic growth models related to diffusionn limited aggregation. Ph.D. thesis, Royal Institute of Technology, Stockholm (1999).
* (12) Carleson, L., Makarov, N.: Laplacian path models. J. Anal. Math. 87, 103–150 (2002).
* (13) Zabrodin, A.: Growth processes related to the dispersionless Lax equations. Physica D 235, 101–108 (2007).
* (14) Zik, O., Moses, E.: Fingering instability in combustion: An extended view. Phys. Rev. E 60, 518–531 (1999).
* (15) Durán, M. A., Vasconcelos, G. L.: Interface growth in two dimensions: A Loewner-equation approach. Phys. Rev. E 82, 031601-1–03161-8 (2010).
* (16) Zabrodin, A.: Growth of fat slits and dispersionless KP hierarchy. J. Phys. A: Math. Theor. 42, 085206 (2009).
* (17) P. Pelcé, Dynamics of curved fronts (Academic Press, San Diego, 1988).
* (18) G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a complex variable: theory and technique (Hod Books, Ithaca, 1983).
|
arxiv-papers
| 2011-07-11T01:10:09 |
2024-09-04T02:49:20.402386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Miguel A. Dur\\'an and Giovani L. Vasconcelos",
"submitter": "Miguel Angel Duran Roa",
"url": "https://arxiv.org/abs/1107.1915"
}
|
1107.2057
|
# the associated map of the nonabelian Gauss-Manin connection
Ting Chen
###### Abstract.
The Gauss-Manin connection for nonabelian cohomology spaces is the
isomonodromy flow. We write down explicitly the vector fields of the
isomonodromy flow and calculate its induced vector fields on the associated
graded space of the nonabelian Hogde filtration. The result turns out to be
intimately related to the quadratic part of the Hitchin map.
## 1\. introduction
The variation of Hodge structures for families of complex Kähler manifolds has
been a much studied subject. Let $\pi:\mathscr{X}\to S$ be a proper
holomorphic submersion of connected complex manifolds. Ehresmann’s Lemma says
that it is a locally trivial fiber bundle with respect to its underlying
differentiable structure. In particular all the fibers of $\pi$ are
diffeomorphic. So if $s\in S$ and $X_{s}$ the fiber of $\pi$ over $s$,
$\mathscr{X}\to S$ can be viewed as a variation over $S$ of complex structures
on the underlying differentiable manifold of $X_{s}$.
Let $\pi^{k}:\mathscr{V}\to S$ be the corresponding vector bundle of
cohomologies whose fiber at $s\in S$ is $H^{k}(X_{s},\mathbb{C})$,
$k\in\mathbb{N}$. Since $\mathscr{X}\to S$ is locally trivial differentiably
(and therefore topologically), there is an induced local identification of
fibers of $\mathscr{V}\to S$. In another word, there is a flat connection on
the vector bundle $\mathscr{V}\to S$. This connection is called the Gauss-
Manin connection for the cohomologies of the family of complex Kähler
manifolds $\mathscr{X}\to S$.
From Hodge theory we know there is a natural Hodge filtration on the vector
bundle $\mathscr{V}\to S$: $\mathscr{V}=F^{0}\supset F^{1}\supset
F^{2}\ldots\supset F^{k}$. Let $\nabla$ be the Gauss-Manin connection,
Griffiths transversality theorem says that $\nabla(F^{p})\subset
F^{p-1}\otimes\Omega^{1}_{S}$, $1\leq p\leq k$. So if $gr\mathscr{V}$ is the
associated graded vector bundle of the filtered bundle $\mathscr{V}$, then the
induced map $gr\nabla$ of $\nabla$ on $gr\mathscr{V}$ will be
$\mathscr{O}_{S}$-linear. In fact, $gr\nabla$ is equal to a certain Kodaira-
Spencer map[3]. We call $gr\nabla$ the _associated map of the Gauss-Manin
connection_.
The above has a nonabelian analogue. Let $G$ be a complex algebraic group, $X$
a smooth algebraic curve over $\mathbb{C}$ of genus $g$. Let $Conn_{X}$ be the
moduli space of principal $G$-bundles over $X$ equipped with a flat
connection. If we denote as $H^{1}(X,G)$ the first Čech cohomology of $X$ with
coefficients the constant sheaf in $G$, then $Conn_{X}$ can be naturally
identified with $H^{1}(X,G)$, by considering the gluing data of flat
$G$-bundles. Since the group $G$ can be nonabelian, we call $Conn_{X}$ the
nonabelian cohomology space of $X$.
Let $\mathscr{M}_{g}$ be the moduli space of genus $g$ complex algebraic
curves. The universal curve $\mathscr{X}\to\mathscr{M}_{g}$ is (roughly) a
variation of complex structures of the underlying real surface, and the
universal moduli space of connections $\mathscr{C}onn\to\mathscr{M}_{g}$ is
the corresponding bundle of nonabelian cohomologies. For the same reason as
before there is a Gauss-Manin connection on the bundle
$\mathscr{C}onn\to\mathscr{M}_{g}$. The local trivialization that defines it
is often called the isomonodromy deformation, or the isomonodromy flow of
$\mathscr{C}onn$ over $\mathscr{M}_{g}$.
There is also a nonabelian analogue of Hodge filtration which was determined
by Carlos Simpson[8], using a generalized definition of filtration of spaces.
A vector space with filtration is equivalent, by the Rees construction, to a
locally free sheaf over $\mathbb{C}$ with a $\mathbb{C}^{\ast}$-action, and
with the fiber over $1$ being the vector space itself. To define the
nonabelian Hodge “filtration” on the space $\mathscr{C}onn$ therefore, it
would be reasonable to find a family of spaces over $\mathbb{C}$ whose fiber
over $1$ is $\mathscr{C}onn$, together with a $\mathbb{C}^{\ast}$-action on
the family. The way to do it in this case is to introduce the notion of
$\lambda$-connections on a principal $G$-bundle on $X$, for any
$\lambda\in\mathbb{C}$. It is a generalization of the notion of connections on
a $G$-bundle. In particular a $1$-connection is an ordinary connection, and a
$0$-connection is a so called _Higgs field_ , which is an object of much
interest to people in complex geometry and high energy physics. The moduli
space of principal $G$-bundles over $X$ together with a Higgs field is called
the Higgs moduli space over $X$, and denoted as $Higgs_{X}$. Simpson’s
definition of nonabelian Hodge filtration immediately implies that the
associated graded space of $Conn_{X}$ is $Higgs_{X}$. Then a question arises:
what is the associated map of the nonabelian Gauss-Manin connection on the
associated graded space? The answer is: it is a lifting111Here the word
lifting has a slightly more general meaning: it means a map of tangent vectors
in the opposite direction of the pushforward, without requiring its
composition with pushforward being identity. In fact, this lifting here
composed with pushforward is zero. of tangent vectors on the relative Higgs
moduli space $\mathscr{H}iggs\to\mathscr{M}_{g}$. On the other hand there is a
well-known Hitchin map from $Higgs_{X}$ to some vector spaces. The quadratic
part of the Hitchin maps also induce a lifting of tangent vectors on
$\mathscr{H}iggs\to\mathscr{M}_{g}$. The fact that the two liftings agree is
the content of our theorem.
###### Theorem 1.1.
The lifting of tangent vectors on $\mathscr{H}iggs\to\mathscr{M}_{g}$
representing the associated map of the nonabelian Gauss-Manin connection is
equal up to a constant multiple to the lifting of tangent vectors induced from
the quadratic Hitchin map.
Closely related results have been obtained in [1], where the authors apply
localization for vertex algebras to the Segal-Sugawara construction of an
internal action of the Virasoro algebra on affine Kac-Moody algebras to lift
twisted differential operators from the moduli of curves to the moduli of
curves with bundles. Their construction gives a uniform approach to several
phenomena describing the geometry of the moduli spaces of bundles over varying
curves, including a Hamiltonian description of the isomonodromy equations in
terms of the quadratic part of Hitchin’s system. Our result and proof are much
more elementary, avoiding the need for the vertex algebra machinery.
The organization of the paper is as follows. In section 2 we give a detailed
definition of all the objects concerned and a precise statement of the
theorem. The rest of the sections are devoted to the proof. In section 3 we
recall the definition of Atiyah bundles and some of its properties that will
be useful in the proof. In section 4 we use deformation theory to write the
tangent spaces to $\mathscr{C}onn$ as certain hypercohomology spaces. Section
5 gives an explicit description of the lifting of tangent vector on
$\mathscr{C}onn\to\mathscr{M}_{g}$ given by the isomonodromy flow. Section 6
extend the isomonodromy lifting to the moduli space of $\lambda$-connections
for any $\lambda\neq 0$. Finally section 7 takes the limit of the lifting at
$\lambda=0$, which is precisely the associated map of the nonabelian Gauss-
Manin connection, and shows that it is equal up to a constant to the quadratic
Hitchin lifting of tangent vectors.
I would like to thank my advisor Ron Donagi for introducing me to the subject
and for many invaluable discussions.
## 2\. definitions and statement of the theorem
All objects and morphisms in this paper will be algebraic over $\mathbb{C}$,
unless otherwise mentioned.
### 2.1. Moduli space of connections and isomonodromy flow
Let $g$ be a natural number greater or equal to 2, so that a generic curve of
genus $g$ has no automorphisms. The moduli space of all genus $g$ curves is a
smooth Deligne-Mumford stack, but if we restrict to the curves that has no
automorphisms, the moduli space is actually a smooth scheme. Let
$\mathscr{M}_{g}$ be this scheme. In this paper we will ignore all the special
loci of the moduli spaces (as explained below) and focus on local behaviors
around generic points.
Let $G$ be a semisimple Lie group, $X$ a smooth curve of genus $g$. Let
$Bun_{X}$ be the coarse moduli space of regular stable $G$-bundles on $X$.
$Bun_{X}$ is also a smooth scheme[7]. The total space of the cotangent bundle
$T^{\ast}Bun_{X}$ is an open subscheme of the Higgs moduli space over $X$ [4].
However since we are only concerned with generic situations, we will use
$Higgs_{X}$ to denote the open subscheme $T^{\ast}Bun_{X}$.
Let $Conn_{X}$ be the moduli space of pairs $(P,\nabla)$, where $P$ is a
stable $G$-bundle on $X$, and $\nabla$ is a connection on $P$. $\nabla$ is
necessarily flat as the dimension of $X$ is equal to 1. $Conn_{X}$ is an
affine bundle on $Bun_{X}$ whose fiber over $P\in Bun_{X}$ is a torsor for
$T_{P}^{\ast}Bun_{X}$. So it is also a smooth scheme.
Let $\mathscr{C}onn\to\mathscr{M}_{g}$ be the relative moduli space of pairs
whose fiber at $X\in\mathscr{M}_{g}$ is $Conn_{X}$. Let $Irrep_{X}$ be the
space of all irreducible group homomorphisms $\pi_{1}(X)\to G$, $Irrep_{X}$ is
a smooth scheme[5]. There is also the relative space
$\mathscr{I}rrep\to\mathscr{M}_{g}$. The Riemann-Hilbert correspondence
$RH:Conn_{X}\to Irrep_{X}$ taking a flat connection to its monodromy is an
analytic(and therefore differentiable) inclusion. Let
$S\subset\mathscr{M}_{g}$ be a small neighborhood of $X$ in analytic topology.
By Ehresmann’s Lemma the family of curves over $S$ is a trivial family with
respect to the differentiable structure. This implies that the restriction of
$\mathscr{I}rrep$ over $S$ is a differentiable trivial family. The trivial
sections or trivial flows induce a flow on the restriction of $\mathscr{C}onn$
over $S$, by the Riemann-Hilbert correspondence. This flow is called the
isomonodromy flow of $\mathscr{C}onn$ over $\mathscr{M}_{g}$.
### 2.2. $\lambda$-connections and nonabelian Hodge filtration
As explained in the last section, $Conn_{X}$ is the nonabelian cohomology
space of $X$ with in coefficient $G$, and the isomonodromy flow on
$\mathscr{C}onn\to\mathscr{M}_{g}$ is the nonabelian Gauss-Manin connection.
To define Hodge filtration on $Conn_{X}$ one need to generalize the definition
of a filtration. A filtration on a vector space $V$ is equivalent, by the Rees
construction[4], to a locally free sheaf $W$ on $\mathbb{C}$ whose fiber at
$1\in\mathbb{C}$ is isomorphic to $V$, together with a
$\mathbb{C}^{\ast}$-action on $W$ compatible with the usual
$\mathbb{C}^{\ast}$-action on $\mathbb{C}$. The fiber of $W$ at
$0\in\mathbb{C}$ will be isomorphic to the associated graded vector space of
$V$.
This sheaf definition of filtrations can be generalized in an obvious way to
define filtrations on a space that is not a vector space. In our case the
space is $Conn_{X}$, and its Hodge filtration is constructed as follows.
$Conn_{X}$ parametrizes pairs ($P$,$\nabla$). Let $P$ also denote the sheaf of
sections of $P$ on $X$, $adP$ be the adjoint bundle of $P$ as well as the
sheaf of its sections, and $\mathfrak{g}$ the Lie algebra of $G$. A connection
$\nabla$ is a map of sheaves
$\nabla:P\to adP\otimes\Omega_{X}^{1}$
that after choosing local coordinates for $X$ and local trivialization for $P$
can be written as
$(\frac{\partial}{\partial x}+[A(x),\ ])\otimes dx$
where $A(x)$ is a $\mathfrak{g}$-valued function and the bracket means the
right multiplication action of $G$ on $\mathfrak{g}$. A $\lambda$-connection
on $P$ is defined to be a map of sheaves $\nabla_{\lambda}:P\to
adP\otimes\Omega_{X}^{1}$ that in local coordinates can be written as
$(\lambda\frac{\partial}{\partial x}+[A(x),\ ])\otimes dx$. Let the moduli
space of $\lambda$-connections be denoted as $\lambda Conn_{X}$. For
$\lambda\neq 0$, $\nabla\leftrightarrow\lambda\cdot\nabla$ is a bijection
between $Conn_{X}$ and $\lambda Conn_{X}$. For $\lambda=0$, the definition of
a $0$-connection agrees with that of a Higgs field. So $0Conn_{X}$ is just
$Higgs_{X}$.
Let $\mathcal{T}_{X}$ be the moduli space of all $\lambda$-connections for all
$\lambda\in\mathbb{C}$. There is a natural map $\mathcal{T}_{X}\to\mathbb{C}$
taking a $\lambda$-connection to $\lambda$, whose preimage at $1\in\mathbb{C}$
is $Conn_{X}$. In fact, Simpson showed that the nonabelian Hodge filtration of
$Conn_{X}$ is precisely the sheaf of sections of this map, with the
$\mathbb{C}^{\ast}$-action given by multiplication by $\lambda$ for
$\lambda\in\mathbb{C}^{\ast}$[4]. The $\mathbb{C}^{\ast}$-action is algebraic
and induces an isomorphism of $Conn_{X}$ and $\lambda Conn_{X}$.
In the ordinary Hodge theory, if one uses the sheaf definition of filtrations,
then the associated map of the Gauss-Manin connection is obtained as follows.
Start with the Gauss-Manin connection on $\mathscr{V}\to S$, the local
trivialization by the flat sections gives a lifting of tangent vectors
$L:T_{s}S\to T_{v}\mathscr{V}$
for $s\in S$ and $v\in\mathscr{V}$ s.t. $\pi^{k}(v)=s$. The lifting $L$ is a
spliting of $\pi^{k}_{\ast}$, i.e. it satisfies
$\pi^{k}_{\ast}\circ L=id_{T_{s}S}$
Let $\mathscr{W}\to\mathbb{C}$ be the sheaf associated to the Hodge filtration
on $\mathscr{V}\to S$. The fiber of $\mathscr{W}$ at $1$ is $\mathscr{V}\to
S$, and denote the fiber over $\lambda$ as
$\pi^{k}_{\lambda}:\mathscr{V}_{\lambda}\to S$. The action of
$\lambda\in\mathbb{C}^{\ast}$ induces an isomorphism of $\mathscr{V}$ and
$\mathscr{V}_{\lambda}$. So the local trivialization of $\mathscr{V}\to S$
induces a local trialization of $\mathscr{V}_{\lambda}\to S$ via this
isomorphism. Let $L_{\lambda}:T_{s}S\to T_{v_{\lambda}}\mathscr{V}_{\lambda}$
be the induced lifting on $\mathscr{V}_{\lambda}\to S$ _multiplied by
$\lambda$_. $L_{\lambda}$ satisfies
$\pi^{k}_{\lambda\ast}\circ L_{\lambda}=\lambda\cdot id_{T_{s}S}$
$L_{\lambda}$ is defined for all $\lambda\neq 0$. For a fixed vector
$\vec{t}\in T_{s}S$, the images of $\vec{t}$ under all the $L_{\lambda}$,
$\lambda\neq 0$ gives a vector field on the total space of $\mathscr{W}$ away
from $\mathscr{V}_{0}$, which is the fiber over $0\in\mathbb{C}$. The
continuous limit of that vector field on $\mathscr{V}_{0}$ exist, and
therefore defines a lifting $L_{0}:T_{s}S\to T_{v_{0}}\mathscr{V}_{0}$ on
$\mathscr{V}_{0}\to S$. $L_{0}$ satisfies
$\pi^{k}_{0\ast}\circ L_{0}=0$
i.e. the images of $\vec{t}\in T_{s}S$ under $L_{0}$ is a vectors field on the
_fiber_ $V_{0,s}$ of $\mathscr{V}_{0}$ over $s$. This vector field is in fact
linear and defines a linear map on $V_{0,s}$. Also $\mathscr{V}_{0}$ is
identified with $gr\mathscr{V}$. From these we see $L_{0}$ really gives a
vector bundle map $gr\mathscr{V}\to gr\mathscr{V}\otimes\Omega^{1}_{S}$, and
that map is the associated map of the Gauss-Manin that we started with.
So in nonabelian Hodge theory, in order to calculate the associated map of the
nonabelian Gauss-Manin connection, we will start with the lifting $L$ induced
from the isomonodromy flow on $\mathscr{C}onn\to\mathscr{M}_{g}$(by a slight
abuse of notation we will use the same notations for the liftings, the meaning
should be clear from the context), and try to find the associated limit
lifting $L_{0}$. Specifically, let $\mathscr{T}\to\mathscr{M}_{g}$ be the
relative moduli space whose fiber at $X\in\mathscr{M}_{g}$ is
$\mathcal{T}_{X}$. $\mathscr{T}$ maps to $\mathbb{C}$ and the fiber at
$\lambda$ is the relative moduli space of $\lambda$-connections, which is
denoted $\lambda\mathscr{C}onn$. There is clearly also a
$\mathbb{C}^{\ast}$-action on $\mathscr{T}$ compatible with the
$\mathbb{C}^{\ast}$-action on $\mathbb{C}$. Let $L_{\lambda}$ be analogously
the lifting on $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ induced by the
lifting $L$ via the $\mathbb{C}^{\ast}$-action and multiplied by $\lambda$.
Then the limit lifting $L_{0}$ will be the associated map that we want to
calculate. It will again be a vertical lifting, i.e. the images of $L_{0}$
will be vectors tangent to the fibers $Higgs_{X}$ of
$\mathscr{H}iggs\to\mathscr{M}_{g}$, $X\in\mathscr{M}_{g}$.
### 2.3. Quadratic Hitchin map and statement of the theorem
$Higgs_{X}$ has a symplectic structure as it is equal to $T^{\ast}Bun_{X}$.
Let $<\ ,\ >$ be the Killing form on the Lie algebra $\mathfrak{g}$ of $G$,
the quadratic Hitchin map is
$qh:Higgs_{X}\to H^{0}(X,\Omega^{\otimes 2})$
$(P,\theta)\mapsto<\theta,\theta>$
where $\theta\in H^{0}(X,adP\otimes\Omega^{1}_{X})$ is a 0-connection or a
Higgs field. One can define a lifting of tangent vectors associated to $qh$
$L_{qh}:T_{X}\mathscr{M}_{g}\to T_{(P,\theta)}Higgs_{X}$ $f\mapsto
H_{qh^{\ast}f}|_{(P,\theta)}$
where $f\in T_{X}\mathscr{M}_{g}\cong H^{1}(X,TX)$ is viewed as a linear
function on $H^{0}(X,\Omega^{\otimes 2})$ by Serre duality, and
$H_{qh^{\ast}f}$ is the Hamiltonian vector field of $qh^{\ast}f$ on
$Higgs_{X}$.
The theorem can now be more precisely stated as:
###### Theorem 2.1 (precise version of Theorem 1.1).
The limit lifting of tangent vectors $L_{0}$ associated to the isomonodromy
lifting $L$ is equal to $\frac{1}{2}L_{qh}$.
## 3\. Atiyah bundles
Before starting to prove the theorem, we recall here some facts about Atiyah
bundles which will be used later. As before let $X$ be a smooth curve of genus
$g$, $G$ a semisimple Lie group, $p:P\to X$ a principal $G$-bundle over $X$.
### 3.1. Atiyah bundle and its sections
Let $TP$ be the tangent bundle over $P$. $G$ acts on $P$ and has an induced
action on $TP$. The action is free and compatible with the vector bundle
structure of $TP\to P$, so the quotient will be a vector bundle $TP/G\to
P/G=X$. This vector bundle over $X$ is called the Atiyah bundle associated to
$P$, and denoted as $A_{P}$.
In fact, $TP$ is isomorphic to the fiber product of $P$ and $A_{P}$ over $X$.
So any section $t$ of $A_{P}$ over $X$ has a unique lift $\tilde{t}$ that
makes the diagram commute Dashto dashdash¿
The lift $\tilde{t}$ can be viewed as a vector field on $P$ which is
$G$-invariant. Conversely, any $G$-invariant vector field on $P$ defines a
section $t$ in the quotient bundle. Therefore sections of $A_{P}$ over $X$ are
the same as $G$-invariant vector fields on $P$.
### 3.2. Atiyah sequence
The sequence of tangent bundles associated to $P\to X$ is:
$0\to T_{P/X}\to TP\to p^{\ast}TX\to 0$
$G$ acts on the sequence, and the quotient is
$0\to adP\to A_{P}\to TX\to 0$
This quotient sequence is called the Atiyah sequence of $A_{P}$. We will
denote the map $A_{P}\to TX$ also as $p_{\ast}$.
### 3.3. Relation to connections
If $\nabla$ is a connection on $P$, then $\nabla$ must be flat since the
dimension of $X$ is 1. So over a small open subset $U\subset X$, there is a
natural trivialization of $P$ associated to $\nabla$
$\tau:U\times F\longrightarrow P|_{U}$
given by the flat sections of $\nabla$. Here F denotes a torsor for $G$.
The local trivialization gives a local section $\tilde{s}_{U}:p^{\ast}TU\to
TP|_{U}$, which is the composition
(1)
$p^{\ast}TU\xrightarrow{\tau^{-1}_{\ast}}p_{U}^{\ast}TU\xrightarrow{(id,0)}p_{U}^{\ast}TU\oplus
p_{F}^{\ast}TF\xrightarrow{\tau_{\ast}}TP|_{U}$
where $p_{U}$ and $p_{F}$ are the projections of $U\times F$ to $U$ and $F$.
Since $\tilde{s}_{U}$ is canonically associated to $\nabla$, so for two such
open subsets $U,V$, $\tilde{s}_{U}$ and $\tilde{s}_{V}$ agree on their
intersection. So there is a well-defined map $\tilde{s}:p^{\ast}TU\to TP$.
Since $\tau$ is $G$-invariant and the map $(id,0)$ is obviously $G$-invariant,
$\tilde{s}_{U}$ is $G$-invariant. So $\tilde{s}$ is $G$-invariant, and gives a
map $s:TX\to A_{P}$. The map $(id,0)$ in the definition of $\tilde{s}_{U}$
implies that $s$ is a splitting of $p_{\ast}:A_{P}\to TX$, i.e. $p_{\ast}\circ
s=id_{TX}$. We can also say that $s$ is a splitting of the Atiyah sequence.
To summarize, for any connection $\nabla$ on $P$ there is uniquely associated
a splitting $s$ of the Atiyah sequence of $P$. $s$ is locally defined as the
splitting $(id,0)$ with $P$(and therefore $A_{P}$) locally trivialized by
$\nabla$.
## 4\. tangent spaces
Now we start to prove the theorem. In this section we will identify the
tangent spaces of $\mathscr{C}onn$ and more generally $\lambda\mathscr{C}onn$
as some hypercohomology spaces, so that we may write down the isomonodromy
lifting $L$ and the extened liftings $L_{\lambda}$ explicitly in the next two
sections.
The tangent space to a moduli space at a regular point is identified with the
infinitesimal deformations of the object corresponding to that point. So we
are really looking at infinitesimal deformations of the objects parametrized
by $\mathscr{C}onn$, which are triples ($X$,$P$,$\nabla$). We start with
deformations of pairs ($X$,$P$).
### 4.1. Deformation of pairs
From Deformation Theory, the following two propositions are well-known.
###### Proposition 4.1.1.
The tangent space to $\mathscr{M}_{g}$ at a point $X$ is naturally isomorphic
to $H^{1}(X,TX)$.
###### Proposition 4.1.2.
The tangent space to $Bun_{X}$ at a point $P$ is naturally isomorphic to
$H^{1}(X,adP)$.
Let $\mathscr{B}un$ be the moduli space of pairs ($X$,$P$). We expect that
generically the tangent space at a point ($X$,$P$) would satisfy
$0\to H^{1}(X,adP)\to T_{(X,P)}\mathscr{B}un\to H^{1}(X,TX)\to 0$
On the other hand since the Atiyah sequence of $P$ $0\to adP\to A_{P}\to TX\to
0$ induces
$0\to H^{1}(X,adP)\to H^{1}(X,A_{P})\to H^{1}(X,TX)\to 0$
It is natural to guess that
###### Proposition 4.1.3.
$T_{(X,P)}\mathscr{B}un$ is naturally isomorphic to $H^{1}(X,A_{P})$.
###### Proof.
the proof is a combination of the usual proofs for Proposition 4.1.1 and
Proposition 4.1.2. Let $\\{U_{i}\\}_{i\in I}$ be an Čech covering of $X$,
$P_{\epsilon}\to X_{\epsilon}$ a family of principal $G$-bundles over
$D_{\epsilon}=\mathbb{C}[\epsilon]/(\epsilon^{2})$, which restrict to $P\to X$
over the closed point. Over each $U_{i}$, let
$\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}\ \ \ \ \ \
(\phi_{i}^{\vee}:\mathscr{O}_{P|_{U_{i}}}\otimes\mathbb{C}[\epsilon]/(\epsilon^{2})\leftarrow\mathscr{O}_{P_{\epsilon}|_{U_{i}}})$
be an isomorphism of $G$-bundles. So it is compatible with the $G$-actions and
descends to an isomorphism
$\iota_{i}:U_{i}\times D_{\epsilon}\to X_{\epsilon}|_{U_{i}}\ \ \ \ \ \
(\iota_{i}^{\vee}:\mathscr{O}_{U_{i}}\otimes\mathbb{C}[\epsilon]/(\epsilon^{2})\leftarrow\mathscr{O}_{X_{\epsilon}|_{U_{i}}})$
Over $U_{ij}=U_{i}\cap U_{j}$, the transition functions are related as in the
commutative diagram
Let $\xi_{ij}\in\Gamma(U_{ij},TX)$ be the vector field on $U_{ij}$ such that
$(\iota_{j}^{-1}\circ\iota_{i})^{\vee}=Id+\epsilon\xi_{ij}$, and
$\eta_{ij}\in\Gamma(P|_{U_{ij}},TP)$ be the vector field on $P|_{U_{ij}}$ such
that $(\phi_{j}^{-1}\circ\phi_{i})^{\vee}=Id+\epsilon\eta_{ij}$. Because
$\phi_{i}$ is $G$-invariant, $\eta_{ij}$ is $G$-invariant. So one can view it
as $\eta_{ij}\in\Gamma(U_{ij},A_{P})$. $(\eta_{ij})_{i,j\in I}$ form a Čech
1-cochain on $X$ with coefficients in $A_{P}$.
$(\eta_{ij})_{i,j\in I}$ is closed because it comes from transition functions
$\phi_{j}^{-1}\circ\phi_{i}$. Any closed cochain $(\eta_{ij})_{i,j\in I}$
comes from some $D_{\epsilon}$ family of pairs. Also for a fixed
$D_{\epsilon}$ family of pairs, a different choice of $\phi_{i}$’s will result
in a cocycle differing from $(\eta_{ij})_{i,j\in I}$ by an exact cocycle. And
any exact cocycle is the result of different choices of $\phi_{i}$’s.
Therefore the infinitesimal deformations of ($X$,$P$) are in natural
correspondence with $H^{1}(X,A_{P})$, which proves the proposition. ∎
### 4.2. Deformation of triples
Now we come to the infinitesimal deformations of a triple ($X$,$P$,$\nabla$).
First a notation related to the connection $\nabla$. As discussed in section
3.3, a connection $\nabla$ on $P$ is equivalent to a splitting of the Atiyah
sequence Let $\hat{s}\in H^{0}(X,A_{P}\otimes\Omega_{X}^{1})$ denote the
global section associated to the splitting map $s$. We see that
$\hat{s}\mapsto 1$ under the map $H^{0}(X,A_{P}\otimes\Omega_{X}^{1})\to
H^{0}(X,TX\otimes\Omega_{X}^{1})\cong H^{0}(X,\mathscr{O}_{X})$.
To find the deformation of the triple ($X$,$P$,$\nabla$), let
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) be a family of triples
over $D_{\epsilon}$ starting with it. Let $s_{\epsilon}$ be the family of
sections corresponding to $\nabla_{\epsilon}$. As in the proof of Proposition
4.1.3, let $\\{U_{i}\\}_{i\in I}$ again be an Čech covering of $X$, and
$\phi_{i}$, $\iota_{i}$, $i\in I$ defined in the same way. Let
$s_{i}:TU_{i}\to A_{P}|_{U_{i}}$ and $\sigma_{i}:TU_{i}\to adP|_{U_{i}}$ be
sections such that the following diagram commutate: The target space of
$\sigma_{i}$ is $adP$ instead of $A_{P}$, because $p_{\ast}\circ s=id$ for all
$s$, so $\sigma_{i}$, being the derivative of $s$ (locally on $U_{i}$, under
the trivialization of the family $\phi_{i}$), projects to $0$ under
$p_{\ast}$.
A deformation of the triple should contain the information about the
deformation of the pair ($X$,$P$) as well as the deformation of $\nabla$. So
the data associated to the infinitesimal family
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) should be the pair:
$(\eta_{ij})_{i,j\in I},(\sigma_{i})_{i\in I}$
where $(\eta_{ij})_{i,j\in I}$ is defined in section 4.1.3 and shown to
characterize the deformation of the pair ($X$,$P$), and $(\sigma_{i})_{i\in
I}$ describe the deformation of $\nabla$.
The data $((\eta_{ij})_{i,j\in I},(\sigma_{i})_{i\in I})$ looks like a
1-cocycle in defining the hypercohomology of some complex of sheaves. Recall
that the tangent space to $Higgs_{X}$ at a point $(P,\theta)$ is
$\mathbb{H}^{1}(X,adP\xrightarrow{[\ ,\theta]}adP\otimes\Omega^{1}_{X})$. We
will prove an analogous result about the tangent spaces to $\mathscr{C}onn$.
On $U_{ij}$, the transition relations are expressed in the following diagram:
Since $(\iota_{j}^{-1}\circ\iota_{i})^{\vee}=Id+\epsilon\xi_{ij}$ and
$(\phi_{j}^{-1}\circ\phi_{i})^{\vee}=Id+\epsilon\eta_{ij}$, we can write down
the two horizontal maps more explicitly. $\forall\ Y+\epsilon Y_{1}\in
TU_{ij}\times D_{\epsilon}$, its image $Y^{\prime}+\epsilon Y_{1}^{\prime}$
under $d(\iota_{j}^{-1}\circ\iota_{i})$ is determined by: for any function $f$
on $U_{ij}$,
$(Y^{\prime}+\epsilon Y_{1}^{\prime})(f)=(I+\epsilon\xi_{ij})(Y+\epsilon
Y_{1})(I-\epsilon\xi_{ij})(f)$
After simplification we get $Y^{\prime}=Y,Y_{1}^{\prime}=Y_{1}+[\xi_{ij},Y]$,
where the bracket is the Lie bracket of vector fields on $U_{ij}$. Similarly
$\forall\ Z+\epsilon Z_{1}\in A_{P}|_{U_{ij}}\times D_{\epsilon}$ (by section
3.1 it can be viewed as a $G$-invariant vector field on $P|_{U_{ij}}$), we get
$d(\phi_{j}^{-1}\circ\phi_{i})(Z+\epsilon
Z_{1})=Z+\epsilon(Z_{1}+[\eta_{ij},Z])$, where the bracket is the Lie bracket
of ($G$-invariant) vector fields on $P|_{U_{ij}}$.
The diagram is commutative, i.e. $\forall\ Y+\epsilon Y_{1}\in TU_{ij}\times
D_{\epsilon}$
$d(\phi_{j}^{-1}\circ\phi_{i})\circ(s_{i}+\epsilon\sigma_{i})(Y+\epsilon
Y_{1})=(s_{j}+\epsilon\sigma_{j})\circ
d(\iota_{j}^{-1}\circ\iota_{i})(Y+\epsilon Y_{1})$
After simplification we get
$s_{i}(Y)=s_{j}(Y)$ (2)
$(\sigma_{j}-\sigma_{i})(Y)=[\eta_{ij},s_{i}(Y)]-s_{j}([\xi_{ij},Y])$
So if we use $\hat{\sigma}_{i}\in H^{0}(X,adP\otimes\Omega_{X}^{1})$ to denote
the global section associated to $\sigma_{i}$, the pair
$((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$
is a hyper Čech 1-cochain on $X$ with coefficients in
$A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X}$
where the map $[\ ,\hat{s}]$ is defined as: if
$\hat{s}=s^{\prime}\otimes\omega$, where $s^{\prime}\in
H^{0}(X,A_{P}),\omega\in H^{0}(X,\Omega_{X}^{1})$, then $[\ ,\hat{s}]:=[\
,s^{\prime}]\otimes\omega-s^{\prime}\otimes[p_{\ast}(\ ),\omega]$.
###### Proposition 4.2.1.
$T_{(X,P,\nabla)}\mathscr{C}onn$ is naturally isomorphic to
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X})$.
###### Proof.
To any $D_{\epsilon}$ family of triples
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) is associated a hyper
1-cochain $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ by the above
discussion. It is closed because of three facts: first, $(\eta_{ij})_{i,j\in
I}$ is a closed Čech 1-cochain with coefficients in $A_{P}$ \- it’s closed
again because it comes from the transition function
$\phi_{j}^{-1}\circ\phi_{i}$; second, because of (2); third, the complex
$A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X}$ has only two nonzero
terms. These three facts imply that $((\eta_{ij})_{i,j\in
I},(\hat{\sigma}_{i})_{i\in I})$ is closed. Any closed hyper 1-cochain comes
from some $D_{\epsilon}$ family of triples. Also for a fixed $D_{\epsilon}$
family of triples, a different choice of the $\phi_{i}$’s will result in a
hyper cocycle differing from $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in
I})$ by an exact hyper cocycle. And any exact hyper cocycle is the result of
different choices of the $\phi_{i}$’s. Therefore the infinitesimal
deformations of ($X$,$P$,$\nabla$) are in natural correspondence with
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\ ,\hat{s}]}adP\otimes\Omega^{1}_{X})$,
which is what we need to prove. ∎
### 4.3. Tangent spaces to $\lambda\mathscr{C}onn$
Let $\lambda\in\mathbb{C}$ be a fixed complex number. For the moduli space
$\lambda\mathscr{C}onn$ of triples ($X$,$P$,$\nabla_{\lambda}$) where
$\nabla_{\lambda}$ is a $\lambda$-connection, the statement about its tangent
spaces is completely analogous to that when $\lambda=1$.
For a $\lambda$-connection $\nabla_{\lambda}$ on $P$, $\lambda\neq 0$,
$\frac{1}{\lambda}\nabla_{\lambda}$ is an ordinary connection, therefore
corresponds to a splitting $s_{\frac{1}{\lambda}\nabla_{\lambda}}$ of the
Atiyah sequence of $P$. Let $s_{\lambda}=\lambda\cdot
s_{\frac{1}{\lambda}\nabla_{\lambda}}$, so $s_{\lambda}$ is a
“$\lambda$-splitting” of the Atiyah sequence of $P$, i.e. $p_{\ast}\circ
s_{\lambda}=\lambda\cdot id_{TX}$. Therefore to any $\lambda$-connection
$\nabla_{\lambda}$($\lambda\neq 0$) is associated a $\lambda$-splitting of the
Atiyah bundle. Notice that this is true for $\lambda=0$ as well, as a
0-splitting of the Atiyah bundle of $P$ is exactly a Higgs field on $P$.
Let $\hat{s}_{\lambda}\in H^{0}(X,A_{P}\otimes\Omega_{X}^{1})$ be the global
section associated to $s_{\lambda}$, we see $\hat{s}_{\lambda}\mapsto\lambda$
under the map $H^{0}(X,A_{P}\otimes\Omega_{X}^{1})\to
H^{0}(X,TX\otimes\Omega_{X}^{1})\cong H^{0}(X,\mathscr{O}_{X})$. The arguments
in the last subsection can be repeated with slight changes (replace 1 by
$\lambda$ at appropriate places) to give the following statement.
###### Proposition 4.3.1.
$T_{(X,P,\nabla_{\lambda})}\lambda\mathscr{C}onn$ is naturally isomorphic to
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{\lambda}]}adP\otimes\Omega^{1}_{X})$, $\forall\lambda\in\mathbb{C}$
### Remark
When $\lambda=0$, the result agrees with the previous results about tangent
spaces to the Higgs moduli space.
## 5\. isomonodromy vector field
The nonabelian Gauss-Manin connection on $\mathscr{C}onn\to\mathscr{M}_{g}$ is
the isomonodromy flow. The local trivialization of
$\mathscr{C}onn\to\mathscr{M}_{g}$ given by the flow induces a lifting of
tangent vectors $L:T_{X}\mathscr{M}_{g}\to T_{(X,P,\nabla)}\mathscr{C}onn$. We
have identified these tangent spaces as (hyper)cohomology spaces in the last
section, now we will write down the map $L$ as a map of cohomology spaces. We
start with a useful fact about an isomonodromy family of connections.
### 5.1. Universal connection of an isomonodromy family
In [5] Inaba et al. constructed the moduli space of triples
($X$,$P$,$\nabla$), and a universal $G$-bundle on the universal curve with a
universal connection. Though they did it for a special case(rank 2 parabolic
vector bundle on $\mathbb{P}^{1}$ with 4 points), the more general case can be
done similarly. The universal connection, when restricted to an isomonodromy
family of triples, has the following important property.
###### Proposition.
If ($X_{t}$,$P_{t}$,$\nabla_{t}$) is an isomonodromy family of triples over a
complex line $D=Spec(\mathbb{C}[t])$, then the restriction of the universal
connection on $P_{t}$(viewed as a $G$-bundle over the total space of $X_{t}$)
is flat.
###### Proof.
If we only look at the underlying differentiable structure, the isomonodromy
family over $D=\mathbb{C}[t]$ is a trivial family of triples. The trivial
family structure gives a flat connection on $P_{t}$, which must be equal to
the restriction of the universal connection on $P_{t}$ since they are equal on
each fiber of the family. ∎
### 5.2. Isomonodromy lifting of tangent vectors
For $\forall\lambda\in\mathbb{C}$, let $\pi_{\lambda}$ be the projection:
$\pi_{\lambda}:\lambda\mathscr{C}onn\to\mathscr{M}_{g}$
$(X,P,\nabla_{\lambda})\mapsto X$
From the proof of Proposition 4.1.3 and the discussions in front of
Proposition 4.2.1 it is not hard to see that the differential of
$\pi_{\lambda}$ is induced from the map $(p_{\ast},0)$ of complexes of sheaves
The lifting of tangent vectors induced from the isomonodromy flow is a
splitting of the map $\pi_{1\ast}$ Notice that the splitting map $s:TX\to
A_{P}$ associated to $\nabla$ gives a map of the complexes The diagram is
commutative because $[\ ,\hat{s}]\circ s$ is basically bracketing $\hat{s}$
with itself and therefore equal to 0. The map of complexes $(s,0)$ is
obviously a splitting of the map $(p_{\ast},0)$.
The map $(s,0)$ of the complexes of sheaves induce a map on the first
hypercohomology, which we denote as $H^{1}(s)$.
###### Proposition 5.2.1.
The isomonodromy lifting $L$ is equal to
$H^{1}(s):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}]}adP\otimes\Omega^{1}_{X})$
###### Proof.
At a point ($X$,$P$,$\nabla$) of $\mathscr{C}onn$, let
($X_{\epsilon}$,$P_{\epsilon}$,$\nabla_{\epsilon}$) be an isomonodromy family
of triples over $D_{\epsilon}$ starting with it. Again let $\\{U_{i}\\}_{i\in
I}$ be an Čech covering of $X$.
Over $U_{i}$, Let
$\tau_{i,\epsilon}:X_{\epsilon}|_{U_{i}}\times F\to P_{\epsilon}|_{U_{i}}$
be the trivialization of $P_{\epsilon}|_{U_{i}}$ over $X_{\epsilon}|_{U_{i}}$
determined by the flat universal connection (see section 5.1) on
$P_{\epsilon}|_{U_{i}}$, and $\tau_{i}$ be its restriction at $\epsilon=0$.
Let
$\iota_{i}:U_{i}\times D_{\epsilon}\to X_{\epsilon}|_{U_{i}}$
be an isomorphism and define
$\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$
as the composition
$P|_{U_{i}}\times
D_{\epsilon}\xrightarrow{(\tau_{i}^{-1},id_{D_{\epsilon}})}U_{i}\times
D_{\epsilon}\times
F\xrightarrow{(\iota_{i},id_{F})}X_{\epsilon}|_{U_{i}}\times
F\xrightarrow{\tau_{i,\epsilon}}P_{\epsilon}|_{U_{i}}$
Let $\xi_{ij}$, $\eta_{ij}$, $s_{\epsilon}$, $s_{i}$ and $\sigma_{i}$ be all
defined as before in the proofs of proposition 4.1.3 and section 4.2. Notice
that since the local trivializations of the $G$-bundles are canonically given
by the flat universal connection, $\tau_{i,\epsilon}$ and $\tau_{j,\epsilon}$
agree on $U_{ij}$, i.e. on $U_{ij}$
$\tau_{i,\epsilon}=\tau_{j,\epsilon}$ $\tau_{i}=\tau_{j}$
Therefore over $U_{ij}$, the transition map $\phi_{j}^{-1}\circ\phi_{i}$ fits
in the diagram In another word with the local trivializations
$(\tau_{i}^{-1},id_{D_{\epsilon}})$ and $(\tau_{j}^{-1},id_{D_{\epsilon}})$,
the transition map $\phi_{j}^{-1}\circ\phi_{i}$ corresponds to
$(\iota_{j}^{-1}\circ\iota_{i},id_{F})$. Let
$(\phi_{j}^{-1}\circ\phi_{i})^{\prime}$ and $\eta_{ij}^{\prime}$ be
$(\phi_{j}^{-1}\circ\phi_{i})$ and $\eta_{ij}$ under the local
trivializations, then
$(\phi_{j}^{-1}\circ\phi_{i})^{\prime}=(\iota_{j}^{-1}\circ\iota_{i},id_{F})$
and therefore
$Id+\epsilon\eta_{ij}^{\prime}=(Id+\epsilon\xi_{ij},Id_{F})$
Comparing the coefficients of $\epsilon$ we get
$\eta_{ij}^{\prime}=(\xi_{ij},0)$
According to the last paragraph in section 3.3, we see this means precisely
that $\eta_{ij}=s(\xi_{ij})$.
With $\phi_{i}:P|_{U_{i}}\times D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$ defined
as above, $s_{\epsilon}|_{U_{i}}:TX_{\epsilon}|_{U_{i}}\to
A_{P_{\epsilon}}|_{U_{i}}$ correspond to the section $s_{i}:TU_{i}\times
D_{\epsilon}\to A_{P}|_{U_{i}}\times D_{\epsilon}$ constant along
$D_{\epsilon}$, i.e. $\sigma_{i}=0$.
Therefore $\hat{\sigma}_{i}=0$, and the pair
$(\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I}$
is exactly the hyper 1-cocycle which is the image of $(\xi_{ij},0)$ under the
map $H^{1}(s)$, which finishes the proof. ∎
## 6\. extended isomonodromy lifting
The associated lifting $L_{\lambda}$ is obtained by extending the isomonodromy
lifting $L$ to $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ by the
$\mathbb{C}^{\ast}$-action, and multiplying by $\lambda$. For a fixed
$\lambda$, $\lambda\neq 0$, the $\mathbb{C}^{\ast}$-action gives an
isomorphism
$\mathscr{C}onn\leftrightarrow\lambda\mathscr{C}onn$
$\nabla\leftrightarrow\lambda\cdot\nabla$
The induced lifting on $\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ by $L$ via
the isomorphism, called the extended isomonodromy lifting, can be written very
similarly as $L$. In the same way that the splitting map $s$ associated to a
connection $\nabla$ induces a map $H^{1}(s)$ of hypercohomologies, the
$\lambda$-splitting map $s_{\lambda}$ associated to a $\lambda$-connection
$\nabla_{\lambda}$ induces a map of the corresponding hypercohomology spaces,
which will be denoted as $H^{1}(s_{\lambda})$.
###### Proposition 6.1.
The extended isomonodromy lifting of tangent vector on
$\lambda\mathscr{C}onn\to\mathscr{M}_{g}$ is given by:
$\frac{1}{\lambda}H^{1}(s_{\lambda}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{\lambda}]}adP\otimes\Omega^{1}_{X})$
###### Proof.
Since the map of moduli spaces is $\nabla\mapsto\lambda\cdot\nabla$ (or
$s\mapsto\lambda s$, $\hat{s}\mapsto\lambda\hat{s}$), the induced map on the
tangent spaces $T_{(X,P,\nabla)}\mathscr{C}onn\to
T_{(X,P,\lambda\nabla)}\lambda\mathscr{C}onn$ is
$\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}]}adP\otimes\Omega^{1}_{X})\xrightarrow{H^{1}(id,\lambda)}\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\lambda\hat{s}]}adP\otimes\Omega^{1}_{X})$
where $(id,\lambda)$ is the map of complexes of sheaves and
$H^{1}(id,\lambda)$ is the induced map on hypercohomology.
So to get the corresponding lifting on $\lambda\mathscr{C}onn$, i.e. to make
the following diagram commutate, the vertical map on the right must be
$\frac{1}{\lambda}H^{1}(\lambda s)$. ∎
Since $L_{\lambda}$ is the extended isomonodromy lifting multiplied by
$\lambda$, $L_{\lambda}=H^{1}(s_{\lambda})$. $L_{\lambda}$ is a
$\lambda$-lifting of tangent vectors.
## 7\. limit lifting at $\lambda=0$
The continuous limit of $L_{\lambda}$ at $\lambda=0$ is a 0-lifting
$L_{0}:T_{X}\mathscr{M}_{g}\to T_{(X,P,\nabla_{0})}\mathscr{H}iggs$. Since
$L_{\lambda}=H^{1}(s_{\lambda})$, by continuity $L_{0}$ is equal to
$H^{1}(s_{0}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,A_{P}\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$
where $s_{0}$ is the 0-splitting of the Atiyah bundle of $P$ associated to the
0-connection(or Higgs field) $\nabla_{0}$ on $P$. Because $\pi_{0\ast}\circ
H^{1}(s_{0})=0$, so in fact $H^{1}(s_{0})$ can be written as
$H^{1}(s_{0}):H^{1}(X,TX)\longrightarrow\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$
The images of a vector in $\vec{t}\in T_{X}\mathscr{M}_{g}$ under
$H^{1}(s_{0})$ form a vector field on the fiber $Higgs_{X}$ of $\pi_{0}$.
Recall that the quadratic Hitchin map on $Higgs_{X}$ is
$qh:Higgs_{X}\to H^{0}(X,\Omega^{\otimes 2})$
$(P,s_{0})\mapsto<\hat{s}_{0},\hat{s}_{0}>$
and its associated lifting of tangent vectors is
$L_{qh}:H^{1}(X,TX)\to\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$ $f\mapsto
H_{qh^{\ast}f}|_{(P,s_{0})}$
The main theorem (Theorem 2.1) is that $H^{1}(s_{0})$ is equal to
$\frac{1}{2}L_{qh}$. To prove it we need two lemmas. For the first lemma, Let
$((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ be a representative of
an arbitrary element $v\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$. Because on $U_{ij}$,
$<\hat{s}_{0},\hat{\sigma}_{j}-\hat{\sigma}_{i}>=<\hat{s}_{0},[\eta_{ij},\hat{s}_{0}]>=-<[\hat{s}_{0},\hat{s}_{0}],\eta_{ij}>=0$,
therefore
$<\hat{s}_{0},\hat{\sigma}_{i}>=<\hat{s}_{0},\hat{\sigma}_{j}>$
Let $<\hat{s}_{0},\hat{\sigma}>\in H^{0}(X,\Omega^{\otimes 2})$ denote the
resulting global quadratic differential form.
###### Lemma 7.1.
Using the above notations, the differential of the map $qh$ is equal to:
$qh_{\ast}:\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})\to H^{0}(X,\Omega^{\otimes 2})$
$v\mapsto 2<\hat{s}_{0},\hat{\sigma}>$
###### Proof.
Let $\\{U_{i}\\}_{i\in I}$ be the Čech covering of the curve $X$,
$(P_{\epsilon},s_{\epsilon})$ the family of Higgs bundles over $D_{\epsilon}$
that correspond to $v$, i.e. for some $\phi_{i}:P|_{U_{i}}\times
D_{\epsilon}\to P_{\epsilon}|_{U_{i}}$, some $s_{i}:TU_{i}\to adP|_{U_{i}}$
and the given $\sigma_{i}:TU_{i}\to adP|_{U_{i}}$, the diagram is commutative.
Because
$qh:(P_{\epsilon},s_{\epsilon})\mapsto<\hat{s}_{\epsilon},\hat{s}_{\epsilon}>$,
and that over $U_{i}$,
$<\hat{s}_{\epsilon},\hat{s}_{\epsilon}>=<\hat{s}_{i}+\epsilon\hat{\sigma}_{i},\hat{s}_{i}+\epsilon\hat{\sigma}_{i}>=<\hat{s}_{i},\hat{s}_{i}>+2<\hat{s}_{i},\hat{\sigma}_{i}>\epsilon$,
so
$qh:(P_{\epsilon},s_{\epsilon})\mapsto<\hat{s}_{0},\hat{s}_{0}>+2<\hat{s}_{0},\hat{\sigma}>\epsilon$.
Taking the coefficient of $\epsilon$, we see that $qh_{\ast}$ maps $v$ to
$2<\hat{s}_{0},\hat{\sigma}>$. ∎
For the second lemma, let $\omega_{H}$ be the symplectic 2-form on
$Higgs_{X}$, $((\eta_{ij})_{i,j\in I},(\hat{\sigma}_{i})_{i\in I})$ and
$((\eta^{\prime}_{ij})_{i,j\in I},(\hat{\sigma}^{\prime}_{i})_{i\in I})$
representatives of two vectors
$v,v^{\prime}\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$.
###### Lemma 7.2.
Let $\int:H^{1}(X,\Omega^{1}_{X})\to\mathbb{C}$ be the canonical map, then
$\omega_{H}(v,v^{\prime})=\int(\eta_{ij}\sqcup\hat{\sigma}^{\prime}_{i}+\eta^{\prime}_{ij}\sqcup\hat{\sigma}_{i})$
where $\sqcup$ means the cup product $\cup$ of Čech cochains composed with the
Killing form $<\ ,\ >$.
###### Proof.
see [6] Proposition 7.12. ∎
###### Theorem.
$H^{1}(s_{0})$ is equal to $\frac{1}{2}L_{qh}$.
###### Proof.
$\forall f\in H^{1}(X,TX)$, we want to show that $L_{qh}(f)=2H^{1}(s_{0})(f)$.
Let $((\eta^{\prime}_{ij})_{i,j\in I},(\hat{\sigma}^{\prime}_{i})_{i\in I})$
be a representative of an element $v\in\mathbb{H}^{1}(X,adP\xrightarrow{[\
,\hat{s}_{0}]}adP\otimes\Omega^{1}_{X})$. Using Lemma 7.1,
$\omega_{H}(L_{qh}(f),v)=d(qh^{\ast}f)(v)=df(qh_{\ast}v)=df(2<\hat{s}_{0},\sigma>)=f(2<\hat{s}_{0},\sigma>)$
Using Lemma 7.2,
$\omega_{H}(H^{1}(s_{0})(f),v)=\omega_{H}((s_{0}(f),0),(\eta_{ij},\sigma_{i}))=f(<\hat{s}_{0},\sigma>)$
So $L_{qh}(f)=2H^{1}(s_{0})(f)$, $\forall f\in H^{1}(X,TX)$. Therefore
$H^{1}(s_{0})=\frac{1}{2}L_{qh}$. ∎
## References
* [1] Ben-Zvi, David; Frenkel, Edward Geometric realization of the Segal-Sugawara construction. Topology, geometry and quantum field theory, 46 C97, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004
* [2] José Bertin, Introduction to Hodge Theory, AMS Bookstore, 2002.
* [3] P. Griffiths, Periods of integrals on algebraic manifolds II (Local Study of the Period Mapping) Amer. J. Math. , 90 (1968) pp. 808 C865
* [4] Hitchin, N. J. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59 C126.
* [5] M. Inaba, K. Iwasaki and M.-H. Saito, Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlev e equation of type V I, Part I , Publ. Res. Inst. Math. Sci. 42, no. 4 (2006), 987 C1089. (math.AG/0309342).
* [6] Markman, Eyal Spectral curves and integrable systems. Compositio Math. 93 (1994), no. 3, 255 C290.
* [7] Mumford, David Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962)
* [8] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry Santa Cruz 1995, Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997), 217-281.
|
arxiv-papers
| 2011-07-11T15:26:27 |
2024-09-04T02:49:20.416661
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ting Chen",
"submitter": "Ting Chen",
"url": "https://arxiv.org/abs/1107.2057"
}
|
1107.2147
|
# AGN Unification at $z\sim 1$: $u-R$ Colors and Gradients in X-ray AGN Hosts
S. Mark Ammons11affiliation: present address: Steward Observatory, University
of Arizona, 933 Cherry Ave., Tucson, AZ 85721, ammons@as.arizona.edu †
†affiliationmark: , David J. V. Rosario22affiliation: present address: Max-
Planck-Institut f r extraterrestrische Physik (MPE), Giessenbachstr.1, 85748
Garching, Germany, rosario@ucolick.org , David C. Koo33affiliation: present
address: University of California, Santa Cruz, 1156 High St., Santa Cruz, CA
95064, koo, max, mmozena, kocevski, mcgrath, magee@ucolick.org , Aaron A.
Dutton44affiliation: present address: Dept. of Physics and Astronomy,
University of Victoria, Victoria, BC, V8P 5C2, Canada, dutton@uvic.ca , Jason
Melbourne55affiliation: present address: California Institute of Technology,
MS 301-17, Pasadena, CA, 91125, jmel@caltech.edu , Claire E. Max33affiliation:
present address: University of California, Santa Cruz, 1156 High St., Santa
Cruz, CA 95064, koo, max, mmozena, kocevski, mcgrath, magee@ucolick.org , Mark
Mozena33affiliation: present address: University of California, Santa Cruz,
1156 High St., Santa Cruz, CA 95064, koo, max, mmozena, kocevski, mcgrath,
magee@ucolick.org , Dale D. Kocevski33affiliation: present address: University
of California, Santa Cruz, 1156 High St., Santa Cruz, CA 95064, koo, max,
mmozena, kocevski, mcgrath, magee@ucolick.org , Elizabeth J.
McGrath33affiliation: present address: University of California, Santa Cruz,
1156 High St., Santa Cruz, CA 95064, koo, max, mmozena, kocevski, mcgrath,
magee@ucolick.org , Rychard J. Bouwens66affiliation: present address: Leiden
Observatory, Leiden University, NL-2300 RA Leiden, The Netherlands,
bouwens@ucolick.org , and Daniel K. Magee33affiliation: present address:
University of California, Santa Cruz, 1156 High St., Santa Cruz, CA 95064,
koo, max, mmozena, kocevski, mcgrath, magee@ucolick.org
###### Abstract
We present uncontaminated rest-frame $u-R$ colors of 78 X-ray-selected AGN
hosts at $0.5<z<1.5$ in the Chandra Deep Fields measured with HST/ACS/NICMOS
and VLT/ISAAC imaging. We also present spatially-resolved $NUV-R$ color
gradients for a subsample of AGN hosts imaged by HST/WFC3. Integrated,
uncorrected photometry is not reliable for comparing the mean properties of
soft and hard AGN host galaxies at $z\sim 1$ due to color contamination from
point-source AGN emission. We use a cloning simulation to develop a
calibration between concentration and this color contamination and use this to
correct host galaxy colors.
The mean $u-R$ color of the unobscured/soft hosts beyond $\sim 6$ kpc is
statistically equivalent to that of the obscured/hard hosts (the soft sources
are $0.09\pm 0.16$ magnitudes bluer). Furthermore, the rest-frame $V-J$ colors
of the obscured and unobscured hosts beyond $\sim 6$ kpc are statistically
equivalent, suggesting that the two populations have similar distributions of
dust extinction. For the WFC3/IR sample, the mean $NUV-R$ color gradients of
unobscured and obscured sources differ by less than $\sim 0.5$ magnitudes for
$r>1.1$ kpc. These three observations imply that AGN obscuration is
uncorrelated with the star formation rate beyond $\sim 1$ kpc.
These observations favor a unification scenario for intermediate-luminosity
AGNs in which obscuration is determined geometrically. Scenarios in which the
majority of intermediate-luminosity AGN at $z\sim 1$ are undergoing rapid,
galaxy-wide quenching due to AGN-driven feedback processes are disfavored.
††affiliationtext: Hubble Fellow
## 1 INTRODUCTION
The physical connection between embedded supermassive black holes and their
host galaxies remains an active topic of research. The discovery of
correlations between bulge properties and black hole mass (e.g., stellar mass
and velocity dispersion, Magorrian et al., 1998; Ferrarese & Merritt, 2000;
Gebhardt et al., 2000) imply that this connection spans an astonishing range
of spatial scales; the AU-sized accretion disk whose characteristics determine
the black hole growth rate must receive information about the conditions of
the galaxy on scales of kiloparsecs, or eight orders of magnitude larger.
These correlations between black hole mass and various bulge properties hint
at simultaneous evolution of AGNs and their host galaxies.
The processes that are involved in the maintenance of the $M-\sigma$ relation
may also govern the evolution of host galaxy morphology and color. In the
local universe, galaxies exhibit bimodal distributions of several critical
parameters, including rest-frame optical color, star formation rate, and gas
fraction (Strateva et al., 2001; Kauffmann et al., 2003; Baldry et al., 2004,
etc). Star formation must be suppressed significantly to explain the low star
formation rates and the redness of early-type, passively evolving galaxies
today. There is circumstantial evidence for the participation of AGN in this
suppression of star formation, or “quenching,” including their presence in the
green valley of the galaxy color-magnitude diagram (CMD), intermediate between
the blue cloud and red sequence (Kauffmann et al., 2003; Nandra et al., 2007;
Schawinski et al., 2009; Cardamone et al., 2010), although the AGN fraction
does not appear to be higher in the green valley (Xue et al., 2010).
Major mergers of gas-rich galaxies can result in the rapid growth of central
black holes. In this picture, the collision of gas clouds saps angular
momentum, efficiently sending gas to the central regions of the galaxy
(Hernquist, 1989), where it is largely consumed in star formation and
partially accreted onto one or more BHs. In several models, the “M-$\sigma$”
relation is maintained through self-regulating processes involved in galaxy
and supermassive black hole (SMBH) growth (Hopkins et al., 2005; Di Matteo et
al., 2005; Hopkins et al., 2009, and references within). This self-regulation
may be modulated by star formation, in which momentum injection by Type Ia
supernovae and radiation pressure from O stars disrupt further star formation
(Murray et al., 2005). However, the immense energies associated with
gravitational accretion onto a SMBH dominate the energy budget of the galaxy;
this suggests that the AGN itself may provide sufficient energy to remove gas
and limit the growth of the BH (Silk & Rees, 1998; Haehnelt et al., 1998;
King, 2003; Di Matteo et al., 2005; Hopkins et al., 2005; Springel et al.,
2005; Hopkins et al., 2006; Menci et al., 2008). A further consequence of the
most energetic AGN feedback processes may be the expulsion of a majority of
the galactic gas (Sanders et al., 1988; Di Matteo et al., 2005; Hopkins et
al., 2006). This quenching of star formation may be a mechanism by which AGN
activity controls the evolution of galaxies from the blue cloud to the red
sequence.
One feature of a model that includes rapid, AGN-induced quenching is an
evolution between obscured and unobscured AGN states (Hopkins et al., 2006;
Menci et al., 2008). The ejection of gas/dust in the central core of the
galaxy is quickly followed by gas removal on kpc-scales. This process
naturally produces a positive correlation between AGN obscuration and the mass
fraction of young stars (a consequence of sufficient gas supply) in the host
galaxies (Menci et al., 2008). Unobscured AGN are expected to reside in
relatively gas-free host galaxies, cleared by a previous blowout phase. These
hosts are expected to be redder than the hosts of obscured AGN, which would
have higher star-formation rates.
It is not clear how this theoretical evolution between obscured and unobscured
states is consistent with the observed AGN unification paradigm. AGN
unification schemes attempt to explain the properties of the many observed
types of AGN with a common physical model. The most successful unification
picture postulates that the differences between Seyfert types in the local
universe are due to variation of the orientation of the observer relative to a
centralized dusty torus (Antonucci, 1993; Urry et al., 1995). Since
obscuration is solely determined by geometry in a unified scheme, there is no
reason to expect differences between the properties of the host galaxies of
obscured and unobscured AGN.
Many comparisons of obscured and unobscured AGNs have been performed at low
and high redshift, using both optical- and X-ray-based diagnostics of AGN
obscuration and various measures of star formation rate or gas content.
Locally, obscured Seyferts are associated with significant starforming
activity, in some cases more than comparable unobscured hosts (Cid Fernandes &
Terlevich, 1995; Gonzalez Delgado et al., 2001). Lacy et al. (2007) finds that
obscured Type II QSOs at redshifts of $0.3<z<0.8$ are associated with
significant star forming activity. However, using HI measurements, unobscured
broad-line AGN hosts at low-redshift are observed to be gas-rich (Ho et al.,
2008). Schawinski et al. (2009) find no dependence of host galaxy color on
obscuring column for a small sample of SWIFT BAT AGN (Gehrels et al., 2004;
Tueller et al., 2008) in the local universe. Page et al. (2004) find stronger
submillimeter emission in an X-ray obscured sample compared to an X-ray
unobscured sample, implying a higher star formation rate in the obscured
sample. Gilli et al. (2009) find that the clustering lengths of $z\sim 1$
X-ray selected AGNs with broad-lines and those that lack broad lines are
similar. They also note that the clustering length of the X-ray hard sources
is statistically similar to that of the soft sources, although the errors are
dominated by number statistics. Shao et al. (2010) measure the star formation
rates of X-ray selected AGNs in GOODS-North with Herschel far-infrared fluxes,
finding no correlation with X-ray absorbing column.
Pierce et al. (2010b), CP10 hereafter, uses optical colors of X-ray selected
AGNs at $z\sim 0.7$ in AEGIS to compare hard and soft sources. Using colors
measured with both integrated and extended apertures, they find that
unobscured sources are bluer than obscured sources in rest-frame $U-V.$
### 1.1 Comparing the Colors and Color Gradients of Obscured and Unobscured
AGN
Generally, understanding the relationship between AGN host galaxy properties
on kpc-scales and the properties of the AGN on smaller scales can be helpful
in constraining the role of AGN in the evolution of galaxies. We focus on
comparing the host properties of obscured and unobscured sources, given the
hypothesis that unobscured sources will reside in redder hosts in a feedback
scenario involving AGN-driven blowouts. We adopt a two-pronged approach: We
utilize two imaging datasets of different spatial resolution, allowing us to
probe host properties on two spatial scales. We first measure the rest-frame
$u-R$ colors of a large sample of AGN host galaxies at high galactic radii (r
$\sim 6-12$ kpc) and search for correlations of color with AGN obscuration or
hardness ratio. The combination of optical and infrared colors is a powerful
diagnostic of the presence of young stars (Gil de Paz & Madore, 2002),
particularly rest-frame $u-R$, or observed $V-J$ at $z\sim 0.8$. This sample
is limited in spatial resolution by the infrared imaging data ($0\farcs 3$ \-
$0\farcs 6$ for ISAAC imaging and $\sim 0\farcs 25$ for NIC3).
We then turn to a smaller sample of AGN host galaxies imaged by the Wide Field
Camera 3 Infrared Channel (WFC3/IR) on HST as part of the Early Release
Science (ERS) demonstration program (Windhorst et al., 2010). The improved
spatial resolution of $\sim 0\farcs 2$ enabled by WFC3 allows us to probe
rest-frame $NUV-R$ color closer to the central regions of host galaxies at
$r\sim 1$ kpc, where AGN-induced winds may be the most energetic and color
effects potentially more drastic.
Section 2 identifies the VLT/ISAAC and ACS data we use in the Great
Observatories Origins Deep Survey (GOODS) field South (Giavalisco et al.,
2004) and the sample selection. Section 3 details the methodology used to
obtain uncontaminated aperture photometry of AGN host galaxies in these
fields. Section 4 describes a suite of cloning simulations used to check the
analysis and derive error bars. Section 5 presents results for all samples and
Section 6 concludes. We assume a flat cosmology throughout, with $H_{0}=70$
km/s/Mpc, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$.
## 2 DATA
This paper focuses on a set of $\sim 78$ X-ray selected sources in the Great
Observatories Origins Deep Survey (GOODS) field South (Giavalisco et al.,
2004) and North (Alexander et al., 2003), overlapping the Chandra Deep Fields
(CDF). The deep, 1 Megasecond Chandra exposure in the CDF-S and the 2 Ms
exposure in the CDF-North have revealed hundreds of AGNs at $0.5<z<1.5$
(Giacconi et al., 2002; Barger et al., 2003; Szokoly et al., 2004). Deep
optical and near-infrared imaging from the HST Advanced Camera for Surveys
(ACS) is available in four filters, F435W (B), F606W (V), F775W (i), and
F850LP (z). This paper makes use of deep J,H, and Ks imaging in the CDFS using
the ISAAC instrument at the VLT. The data reduction and depths are described
in Retzlaff et al. (2010b).
We utilize deep $F110W$ and $F160W$ imaging publicly available in the GOODS-
North and South fields. These NIC-3 (Thompson et al., 1998) pointings have
been acquired through a variety of programs, including Guest Observer, DDT and
Parallel observing programs, totaling 1400 orbits of $F110W$ and $F160W$
imaging. As part of an HST archival program, Magee et al. (2007) have reduced
and mosaiced all NIC-3 imaging in GOODS. This data reduction procedure is
described fully in Magee et al. (2007). Further details regarding the use of
this NIC-3 data set are given in section 4.2.1 of Ammons et al. (2009).
The near-ultraviolet (NUV) filter is defined as used for the GALEX mission
(Martin et al., 2005; Morrissey et al., 2005, 2007). The $NUV-R$ color is
defined as in Salim et al. (2007). All magnitudes are in AB units unless
explicitly stated otherwise. Selection criteria are enumerated in section 2.4.
### 2.1 Near-Infrared ISAAC Imaging
The ISAAC instrument on the VLT has been used to mosaic the ACS region of
GOODS-South to median $5\sigma$ point source depths of $J_{AB}=25.2$,
$H_{AB}=24.7$, and $Ks_{AB}=24.7$ (Retzlaff et al., 2010b). The spatial
coverage in each of the bands is $172.4$, $159.6$, and $173.1$ arcmin2 in J,
H, and Ks respectively. The exposure times varied from 3-8 hours in each band,
totalling 1.3 Megaseconds. This program was queue-scheduled to optimize image
quality; the image FWHM’s range from $0\farcs 3$ to $0\farcs 65.$ Fully
reduced and photometrically calibrated data is available from the GOODS ISAAC
webpage (Retzlaff et al., 2010a).
### 2.2 Near-Infrared NICMOS Imaging
In this study, we use nearly 100 square arcminutes of NIC-3 $F110W$ and
$F160W$ imaging in GOODS North and South. These pointings, which have been
acquired through a variety of observing programs and are publicly available,
have been fully reduced and mosaiced by Magee et al. (2007). This reduction
procedure includes basic calibration (removing the instrumental signature),
cosmic-ray removal, treatment for post-SAA cosmic ray persistence and
electronic ghosts, sky subtraction, non-linear count-rate correction, artifact
masking, robust alignment and registration for large mosaics, weight map
generation, and drizzling onto a final image frame (Magee et al., 2007).
Sensitivities vary across the GOODS fields due to differing levels of
exposure.
### 2.3 WFC3/IR ERS Imaging
We use public WFC3/IR imaging in the ERS region with the $F125W$ and $F160W$
filters. The data were reduced using the standard calwf3 pipeline. After
pipeline calibration, a residual gain ”pedestal” was noticed (see Windhorst et
al., 2010) and corrected through a multiplicative scaling of each quadrant.
Images were corrected for geometric distortion and drizzled onto a 90mas grid
using the MultiDrizzle package (Koekoemer et al., 2002, 2007). Individual
pointings were registered to a single astrometric grid using SWarp (Bertin,
2002), which corrected for a small rotation offset in the astrometric solution
from MultiDrizzle. A final mosaic of the data was created with SWarp and a
small residual sky gradient was removed. The compact source sensitivity is
27.5 and 27.2 magnitudes AB for a $5\sigma$ detection in the $F125W$ and
$F160W$ bands, respectively.
### 2.4 Sample Selection
The overall sample is composed of X-ray selected sources in the GOODS ACS
regions with redshifts, as described below. The overall sample is divided into
subsamples for further analysis with cuts in the X-ray hardness ratio and
X-ray luminosity. The samples and selection criteria are enumerated in Table
1.
This sample is X-ray selected using the 1 Ms Chandra Deep Field South (CDFS)
exposure as described in Giacconi et al. (2002) and the 2 Ms CDF-N exposure
(Alexander et al., 2003). All GOODS-South X-ray sources were chosen with a
SExtractor (Bertin & Arnouts, 1996) S/N detection threshold of 2.1. GOODS-
North X-ray point sources are identified with WAVDETECT (Freeman et al., 2002)
using a false-positive probability threshold of $10^{-7}$ in one of seven
X-ray bands ranging from $0.5$ keV to $8.0$ keV. This threshold is comparable
to the GOODS-South S/N detection threshold of 2.1.
Of 339 X-ray sources detected in GOODS-South (Giacconi et al., 2002), 191
unique sources are identified as extragalactic (non-stellar), lie in the ACS
and ISAAC tiled region, and have photometric redshifts from COMBO-17 (Wolf et
al., 2004; Zheng et al., 2004) or Luo et al. (2010) or spectroscopic redshifts
from Szokoly et al. (2004) or Silverman et al. (2010). Sources with extended
X-ray emission are omitted from the sample. A redshift cut isolating sources
with $0.5<z<1.5$ and an optical magnitude cut of $R<24$ trim the sample to 56
objects.
Of the 503 X-ray point sources detected in GOODS-North (Alexander et al.,
2003), 284 have reliable spectroscopic redshifts in Barger et al. (2003). 22
sources possess spectroscopic redshifts, fall in the redshift range
$0.5<z<1.5,$ appear within the NIC-3 $F160W$ tiled region in GOODS-North, have
$L_{X,0.5-8keV}>10^{42}$ ergs s-1, and have $R<24.$ The median X-ray detection
S/N between 0.5 and 8.0 keV for the remaining sample is 11.95 and the minimum
is 4.91, above the detection threshold for the GOODS-South catalog. Including
AGN from both northern and southern fields, 78 AGN remain in the sample. The
identification numbers, redshifts, X-ray luminosities, hardness ratios, $u-R$
colors, and $U-B$ colors are given in Table LABEL:tab:data.
92% of the sample of 78 AGN hosts have secure spectroscopic redshifts. We
divide the sample at a hardness ratio of $-0.2$ to isolate obscured and
unobscured sources. Sources with $HR<-0.2$ are termed unobscured or soft and
sources with $HR>-0.2$ are termed obscured or hard. These labels are used to
refer to these types of sources throughout this paper. We use the hardness
ratio as defined as $HR=(H-S)/(H+R)$ in Giacconi et al. (2002) and Szokoly et
al. (2004), where H and S are the net instrument count rates in the hard (2-10
keV) and the soft (0.5-2 keV) bands, respectively.
Hardness ratio is expected to be an imperfect measure of AGN obscuration due
to K-correction effects. We assess this using a subsample of GOODS-South X-ray
sources with full X-ray spectral fits available in Tozzi et al. (2006). A
hardness ratio of $-0.2$ at $z\sim 1$ corresponds to a neutral hydrogen column
density of $2\times 10^{22}$ cm-2. The neutral hydrogen column corresponding
to HR = $-0.2$ shifts by a factor of $\sim 2.5$ from $z\sim 0.5$ to $z\sim
1.5$ due to the k-correction (Tozzi et al., 2006). The range of hardness
ratios that correspond to $2\times 10^{22}$ cm-2 over the redshift range
$0.5<z<1.5$ is $-0.3<$ HR $<-0.1$. 25% of soft sources selected with HR
$<-0.2$ have $N_{H}>2\times 10^{22}$ cm-2; 18% of hard sources selected with
HR $>-0.2$ have $N_{H}<2\times 10^{22}$ cm-2. This low level of sample mixing
does not affect the conclusions of this paper.
The final redshift ranges, mean redshifts, and numbers of sources in each
subsample are indicated in Table 1. Table 1 lists three groups, each of which
are further subdivided into hard and soft subsamples: “all”, “lum”, and “ERS.”
The “all” sample is selected according to the criteria listed above. The “lum”
group is a subsample of the “all” group and only includes objects more
luminous than $L_{X,0.5-10keV}=10^{43.5}$ ergs s${}^{-1}.$ The “ERS” subsample
only includes sources in the ERS region sampled by HST WFC3 imaging.
To enable unbiased comparison of obscured and unobscured host properties, we
cut the redshift ranges of the hard and soft subsamples to impose the mean
redshifts of these two samples to match. The redshift distribution of soft
sources is generally weighted towards higher redshifts, so we trim the highest
redshift sources from the soft subsamples. Similarly, we trim low redshift
sources from the hard samples. As a result, both the mean redshifts and the
mean X-ray luminosities of the soft and hard subsamples within each group
match closely. As shown in Figure 1, the trimmed redshift histograms of the
two populations are broadly similar. Note in Table 1 that both the mean
redshift and mean X-ray luminosity of the “lum” group is larger than the “all”
group, as expected for a subsample of more luminous objects. The mean redshift
of the “ERS” group is significantly lower than that of the “all” group because
the ERS region overlaps with a known galaxy overdensity at $z\sim 0.73.$ The
mean X-ray luminosity of the “ERS-hard” subsample and the “ERS-soft” subsample
are significantly different, $10^{43.02}$ ergs s-1 and $10^{43.54}$ ergs s-1,
respectively; this is due to two particularly X-ray luminous sources in the
small “ERS-soft” subsample. If these are removed, the mean colors of the “ERS-
soft” sample do not change more than the $1\sigma$ measurement error.
Figure 1: Redshift histograms for soft (blue line) and hard (red line) sources. Soft and hard sources are divided at a hardness ratio of $-0.2.$ The mean redshifts of the two populations are shown as dashed lines. The mean redshifts of the samples match when the redshift limits are trimmed as described in Section 2.4. The redshift distributions of the soft and hard sources are broadly similar, both displaying peaks at known redshifts of $z\sim 0.7$ and $z\sim 1.0$. Table 1: Table of sample sizes, redshift ranges, mean redshift, mean X-ray luminosities, summarized color information, and selection criteria for all subsamples. Sample | # | $z$ range | $\langle z\rangle$ | $\langle L_{X}\rangle$ | $\langle(u-r)_{int}\rangle$ | $\langle(u-r)_{outer}\rangle$ | Selection
---|---|---|---|---|---|---|---
all | 78 | 0.5-1.5 | 0.877 | 43.11 | 1.65$\pm$0.09 | 1.69$\pm$0.10 | $L_{X}>42$
all-hard | 42 | 0.55-1.5 | 0.869 | 43.10 | 1.81$\pm$0.10 | 1.74$\pm$0.11 | $L_{X}>42$, $HR>-0.2$
all-soft | 31 | 0.5-1.3 | 0.864 | 43.12 | 1.45$\pm$0.11 | 1.65$\pm$0.12 | $L_{X}>42$, $HR<-0.2$
lum-all | 19 | 0.5-1.5 | 1.018 | 43.92 | 1.31$\pm$0.12 | 1.65$\pm$0.14 | $L_{X}>43.5$
lum-hard | 8 | 0.5-1.5 | 0.994 | 43.85 | 1.94$\pm$0.17 | 1.76$\pm$0.18 | $L_{X}>43.5$, $HR>-0.2$
lum-soft | 10 | 0.5-1.3795 | 1.001 | 44.01 | 0.78$\pm$0.16 | 1.61$\pm$0.17 | $L_{X}>43.5$, $HR<-0.2$
ERS-all | 24 | 0.5-1.5 | 0.788 | 43.21 | 1.61$\pm$0.11 | 1.78$\pm$0.13 | $L_{X}>42$
ERS-hard | 15 | 0.5-1.5 | 0.793 | 43.02 | 1.85$\pm$0.15 | 1.74$\pm$0.16 | $L_{X}>42$, $HR>-0.2$
ERS-soft | 9 | 0.5-1.5 | 0.781 | 43.54 | 1.20$\pm$0.16 | 1.85$\pm$0.17 | $L_{X}>42$, $HR<-0.2$
Table 2: Tabulated redshifts, X-ray luminosities, hardness ratios, and measured colors for individual sources. Sources identified with “LUO” are GOODS-South sources as numbered in Luo et al. (2010). Sources identified with “XID” are GOODS-South sources as numbered in Szokoly et al. (2004) but not found in Luo et al. (2010). Sources identified with “ABB” are GOODS-North sources numbered in Alexander et al. (2003) with spectroscopic redshifts from Barger et al. (2003). XID | z | type | $L_{X}$ | HR | $(u-r)_{outer}$ | $(U-B)_{outer}$ | $(u-r)_{int}$ | $(U-B)_{int}$
---|---|---|---|---|---|---|---|---
LUO 408 | 1.23 | 1 | 43.79 | -0.45 | 1.66 | 0.564 | 1.70 | 0.564
LUO 370 | 1.02 | 1 | 43.48 | 0.12 | 1.87 | 1.04 | 1.71 | 0.694
LUO 335 | 1.22 | 1 | 43.46 | -0.2 | 2.18 | 0.695 | 0.341 | 0.0121
LUO 324 | 0.84 | 1 | 44.03 | -0.53 | 2.07 | 0.628 | 0.683 | 0.0259
LUO 319 | 0.66 | 1 | 43.04 | -0.49 | 2.48 | 0.716 | 1.39 | 0.435
LUO 316 | 0.67 | 1 | 43.36 | -0.4 | 1.93 | 0.723 | 1.99 | 0.665
LUO 302 | 0.84 | 1 | 43.21 | -0.32 | 1.09 | 0.559 | 1.46 | 0.529
LUO 288 | 1.03 | 0.5 | 43.44 | -0.37 | 1.48 | 0.521 | 1.78 | 0.582
LUO 283 | 0.96 | 0.5 | 43 | -0.23 | 0.944 | 0.225 | 1.28 | 0.341
LUO 273 | 0.74 | 1 | 43.52 | -0.56 | 1.64 | 0.679 | 0.722 | 0.241
LUO 267 | 1.22 | 1 | 44.19 | -0.47 | 1.83 | 0.669 | 0.459 | 0.0749
LUO 249 | 0.67 | 1 | 43.29 | 0.52 | 1.77 | 0.566 | 1.82 | 0.669
LUO 247 | 0.73 | 1 | 44.48 | -0.54 | 1.83 | 0.669 | 0.380 | 0.0914
LUO 246 | 0.73 | 1 | 43 | 0.06 | 2.00 | 0.669 | 2.01 | 0.662
LUO 242 | 1.03 | 1 | 44.23 | -0.63 | 1.08 | 0.454 | 0.386 | 0.135
LUO 224 | 0.73 | 1 | 43.23 | 0.44 | 2.07 | 0.771 | 1.67 | 0.609
LUO 216 | 0.53 | 1 | 42.78 | -0.47 | 2.23 | 0.657 | 2.38 | 0.704
LUO 168 | 1.1 | 1 | 44.21 | 0.61 | 1.65 | 0.755 | 1.91 | 0.698
LUO 167 | 0.57 | 1 | 43.17 | -0.55 | 1.91 | 0.588 | 1.74 | 0.546
LUO 145 | 0.67 | 1 | 42.92 | -0.45 | 2.24 | 0.794 | 2.20 | 0.783
LUO 127 | 0.61 | 1 | 43.42 | 0.11 | 1.88 | 0.589 | 1.45 | 0.470
LUO 49 | 1.04 | 1 | 43.73 | -0.44 | 2.22 | 0.789 | 0.620 | 0.168
LUO 270 | 0.96 | 1 | 43.32 | -0.54 | 1.43 | 0.644 | 1.45 | 0.458
LUO 311 | 1.31 | 1 | 42.64 | -1 | 0.316 | 0.0924 | 0.668 | 0.168
LUO 118 | 1.03 | 1 | 42.8 | 0.1 | 1.89 | 0.715 | 1.74 | 0.613
LUO 228 | 1.09 | 1 | 43.56 | 1 | 1.90 | 0.764 | 1.99 | 0.752
LUO 189 | 0.6 | 1 | 43.3 | 1 | 1.86 | 0.643 | 2.28 | 0.679
LUO 90 | 0.55 | 1 | 42.49 | 0.16 | 1.61 | 0.604 | 1.78 | 0.583
LUO 381 | 0.66 | 1 | 42.32 | -0.17 | 2.07 | 0.719 | 2.20 | 0.752
LUO 304 | 0.84 | 1 | 42.84 | 0.01 | 1.13 | 0.271 | 1.57 | 0.333
LUO 415 | 1.14 | 1 | 42.42 | -1 | 1.92 | 0.712 | 2.11 | 0.757
LUO 393 | 0.67 | 1 | 43.9 | 0.39 | 2.39 | 0.833 | 2.27 | 0.774
LUO 109 | 0.93 | 0.9 | 42.83 | 0.13 | 1.35 | 0.531 | 1.67 | 0.619
LUO 204 | 0.73 | 1 | 42.38 | 1 | 2.09 | 0.766 | 2.46 | 0.796
LUO 310 | 0.73 | 1 | 43.22 | 1 | 2.09 | 0.748 | 2.05 | 0.717
LUO 211 | 1.22 | 1 | 42.65 | -1 | 1.08 | 0.374 | 1.45 | 0.462
LUO 254 | 0.74 | 1 | 42.19 | -1 | 1.99 | 0.654 | 2.29 | 0.804
LUO 115 | 0.76 | 0.6 | 42.17 | -0.47 | 1.44 | 0.331 | 2.03 | 0.451
LUO 386 | 1.17 | 1 | 43.43 | 0.52 | 1.11 | 0.462 | 1.71 | 0.579
LUO 226 | 1.04 | 1 | 43.23 | 0.23 | 1.52 | 0.746 | 1.27 | 0.554
LUO 260 | 1.32 | 1 | 43.45 | 0.6 | 1.08 | 0.289 | 0.834 | 0.244
LUO 131 | 0.74 | 1 | 43.54 | 1 | 1.61 | 0.599 | 1.89 | 0.631
LUO 72 | 0.72 | 1 | 43.39 | 1 | 1.78 | 0.377 | 1.75 | 0.357
XID 511 | 0.77 | 1 | 42.01 | -0.24 | 0.868 | 0.379 | 0.952 | 0.411
LUO 298 | 0.67 | 1 | 42 | -0.14 | 1.98 | 0.694 | 2.13 | 0.743
XID 516 | 0.67 | 1 | 42.36 | -0.26 | 0.726 | 0.330 | 1.02 | 0.371
LUO 235 | 1.03 | 1 | 42.62 | -0.03 | 0.930 | 0.452 | 1.05 | 0.419
LUO 178 | 0.96 | 1 | 42.46 | 0.18 | 1.93 | 0.667 | 1.85 | 0.642
LUO 117 | 0.68 | 1 | 42.49 | 0.34 | 2.14 | 0.748 | 2.17 | 0.774
LUO 114 | 0.58 | 1 | 42.41 | -0.01 | 1.72 | 0.498 | 2.00 | 0.632
XID 580 | 0.66 | 1 | 42.32 | -1 | 1.90 | 0.730 | 2.21 | 0.726
LUO 129 | 1.33 | 1 | 43.72 | 1 | 1.61 | 0.472 | 1.95 | 0.551
LUO 177 | 0.74 | 1 | 44.37 | 1 | 2.03 | 0.821 | 2.28 | 0.839
LUO 195 | 0.67 | 1 | 43.3 | 1 | 2.01 | 0.694 | 2.28 | 0.777
XID 611 | 0.98 | 1 | 43.41 | 1 | 1.38 | 0.646 | 2.00 | 0.635
XID 612 | 0.74 | 1 | 43.3 | 1 | 1.49 | 0.527 | 1.68 | 0.570
ABB 40 | 1.379 | 1 | 44.42 | -0.43 | 1.42 | 0.299 | 1.24 | 0.228
ABB 55 | 0.64 | 1 | 42.04 | -0.1 | 1.67 | 0.567 | 1.71 | 0.576
ABB 82 | 0.68 | 1 | 42.6 | 0.34 | 1.73 | 0.642 | 1.93 | 0.723
ABB 90 | 1.14 | 1 | 43.42 | 0.5 | 1.57 | 0.575 | 1.72 | 0.504
ABB 113 | 0.85 | 1 | 43.2 | -0.39 | 2.01 | 0.783 | 2.19 | 0.670
ABB 115 | 0.68 | 1 | 43.49 | -0.34 | 1.71 | 0.663 | 1.19 | 0.423
ABB 121 | 0.52 | 1 | 42.35 | 0.2 | 1.73 | 0.476 | 1.92 | 0.524
ABB 142 | 0.75 | 1 | 42.61 | -0.07 | 2.21 | 0.814 | 2.22 | 0.775
ABB 150 | 0.76 | 1 | 42.89 | 0.49 | 1.50 | 0.582 | 1.79 | 0.667
ABB 157 | 1.26 | 1 | 43.99 | 0.38 | 1.15 | 0.341 | 1.31 | 0.365
ABB 158 | 1.01 | 1 | 43.11 | 0.34 | 2.09 | 0.725 | 2.23 | 0.737
ABB 177 | 1.02 | 1 | 42.66 | -0.32 | 1.19 | 0.432 | 1.54 | 0.504
ABB 193 | 0.96 | 1 | 43.67 | -0.55 | 0.878 | 0.362 | 0.742 | 0.279
ABB 201 | 1.02 | 1 | 42.8 | 0.64 | 1.91 | 0.786 | 1.73 | 0.601
ABB 205 | 1.38 | 0.5 | 43.51 | -0.33 | 1.21 | 0.259 | 1.55 | 0.314
ABB 212 | 0.94 | 1 | 42.26 | -0.37 | 2.19 | 1.08 | 1.80 | 0.756
ABB 217 | 0.52 | 1 | 42.2 | 0.68 | 2.63 | 0.768 | 2.51 | 0.721
ABB 222 | 0.86 | 1 | 42.93 | -0.22 | 1.90 | 0.654 | 1.96 | 0.664
ABB 351 | 0.94 | 1 | 42.14 | -0.3 | 1.77 | 0.650 | 1.99 | 0.680
ABB 352 | 0.94 | 1 | 42.55 | 0.53 | 1.49 | 0.547 | 1.61 | 0.587
ABB 384 | 1.02 | 1 | 43.52 | 0.77 | 1.74 | 0.648 | 1.95 | 0.675
ABB 451 | 0.84 | 1 | 44.04 | -0.42 | 1.44 | 0.568 | 0.893 | 0.260
#### 2.4.1 Color Magnitude Diagram
Figure 2 plots the rest-frame $u-R$ vs $M_{R}$ color magnitude diagrams (CMD)
of the “all” sample. The rest-frame photometry is obtained by performing
$3\farcs$ filled-aperture photometry on all optical/NIR bands for all AGN,
interpolating linearly through the SED, and sampling the de-redshifted SED at
the appropriate filter wavelengths. For comparison, rest-frame $u-R$ and
$M_{R}$ measurements for the underlying galaxy population are plotted as small
black circles for qualitative comparison. This sample of $535$ control
galaxies is selected from the GOODS MUSIC catalog (Santini et al., 2009)
identically to the AGN sample (i.e., spectroscopic redshifts with $0.5<z<1.5$
and $R<24$) except that X-ray sources with $L_{X}>10^{42}$ ergs s-1 are
excluded. Rest-frame colors are computed via interpolation through observed
ACS/ISAAC bvizJHK photometry, as described in Section 3. A $\sim 0.15$
magnitude correction is added to the IR bands to correct for a difference in
photometric aperture size between this sample and the MUSIC sample.
In Figure 2, it is apparent that the AGN tend to reside in the most luminous
systems. Both red sequence galaxies ($u-R\sim 2.0$) and blue cloud galaxies
($u-R\sim 1.0$) are present in this sample (Salim et al., 2007). The mean loci
of the soft and hard sources are shown as blue and red error bars,
respectively. Using integrated photometry, it immediately appears that the
soft sources are significantly bluer and more optically luminous than the hard
sources. However, integrated photometry is likely contaminated by the
nonthermal emission produced by the AGN engine. The next section introduces
our method of measuring uncontaminated rest-frame host colors.
Figure 2: Rest frame AB $u-R$ vs. $M_{R}$ color-magnitude diagram of the “all”
sample, using integrated photometry. Large circles denote strong
($L_{X,0.5-10keV}>10^{43}$ ergs s-1) sources and small circles denote weak
($L_{X,0.5-10keV}<10^{43}$ ergs s-1) sources. Hard, obscured sources (HR
$>-0.2$) are shown with overplotted squares and red colors. Soft sources (HR
$<-0.2$) are shown with blue colors. Rest-frame values are estimated from
observed photometry as in the text. Both $u-R$ colors and $M_{R}$ are measured
using a filled $3\farcs$ diameter circular aperture. Colors and absolute
magnitudes are uncorrected for central point source contamination. Blue and
red error bars denote the mean colors and luminosities for the soft and hard
sources, respectively, with errors calculated as in Section 5.4. Small black
circles denote the colors and magnitudes of the underlying galaxy population,
selected as explained in the text.
## 3 APERTURE PHOTOMETRY
We now measure aperture photometry with an elliptical annular aperture with
inner de-projected radius of $0\farcs 75$ and outer de-projected radius of
$1\farcs 5$, hereafter labeled the “outer” aperture. We select a large inner
radius of $0\farcs 75$ that minimizes contamination from central point
sources. This dimension is oversized with respect to the spatial resolution of
the near-IR imaging, which is set by ground-based seeing. The outer radius of
$1\farcs 5$ is chosen to maximize the total signal-to-noise ratio over all
bands for typical galaxy profiles of sersic index $n=2$.
The axis ratios and position angles of the annuli are determined for each AGN
host separately by manually fitting to isophotes in the ACS z-band image. This
is performed by lining up ellipses with major axis lengths of $3\farcs 0$ with
variable axis ratios and position angles until the nearest isophote is
visually matched. Only one annular aperture is used for each AGN host.
Rest-frame colors are computed from observed photometry with spline
interpolations using IDL’s interpol function and known filter transmission
functions. The spline-fit SED is constrained to match the observed photometry
upon integration with the appropriate filter curves. Rest-frame magnitudes in
$u$, $R$, and other bands are computed by integrating the de-redshifted SED
with these filter functions. We avoid using galaxy template spectra for
fitting photometry, which can be problematic with the addition of non-thermal
AGN continuum emission. We test our technique by comparing rest-frame $u-R$
colors measured in a filled $2\farcs 5$ diameter aperture with those computed
by Taylor et al. (2009) using SED-fitting techniques. For $44$ isolated
sources in GOODS-South that overlap the two samples, the $u-R$ colors match
with a symmetric scatter of $\sigma=0.15$ magnitudes and a minor systematic
offset of $0.015$ magnitudes. The scatter shrinks to $\sigma=0.11$ magnitudes
for sources not obviously dominated by central point sources, similar to the
average level of measurement error in the sample. We conclude that our
technique is appropriate for measuring the average rest-frame colors of
subsamples of galaxies.
The “outer” aperture is expected to be robust to contamination from central
point sources associated with nonstellar AGN continua. However, light from the
central AGN may be scattered into the “outer” aperture by telescope
diffraction, or the effect of a non-ideal Point Spread Function (PSF). PSF
contamination can be non-trivial for cases in which the AGN emission is
dominant over the light from the host galaxy. These cases require either a
correction to the aperture photometry or special image processing to reveal
the host galaxy. We present two techniques for correcting aperture photometry
for the effect of scattered AGN contamination. Firstly, Section 3.1 describes
a method of estimating contamination from the PSF and subtracting its
contribution from measured photometry. Secondly, Section 3.2 describes the
implementation of the CLEAN (Hogbom, 1974; Keel et al., 1991) deconvolution
algorithm to remove contamination directly from the images before measuring
photometry. Both methods requires excellent constraint of the PSF.
### 3.1 Correcting “Outer” Aperture Colors for PSF Contamination
Central embedded point sources contribute flux to annular photometric
apertures via PSF scattering. We approximate the contaminating flux
$F_{contam}$ for a given source in a single band $S$ as
$F_{contam}\;=\;\frac{\displaystyle\sum_{outer}\;F_{PSF,S}}{\displaystyle\sum_{core}\;F_{PSF,S}}\times\displaystyle\sum_{core}\;F_{image,S}$
where $F_{image,S}$ is the flux distribution of the source in band $S$ and
$F_{PSF,S}$ is the flux distribution of the centered PSF in band $S.$
$\Sigma_{outer}$ denotes a sum of fluxes over the “outer” annular aperture.
$\Sigma_{core}$ here denotes a sum over a filled, circular aperture with
radius $0\farcs 18$. Note that the first fraction is similar to a
concentration measurement, except that wider aperture is elliptical and
annular. The decontaminated flux is then given by
$F_{decontam}\;=\;\left(\displaystyle\sum_{outer}\;F_{image,S}\right)\;-F_{contam}$
This method corrects for all PSF-scattered flux originating from within the
central core aperture, no matter the source. PSF-scattered flux originating
from outside of the core aperture is not removed, but it is assumed that this
light is intrinsically stellar.
### 3.2 CLEAN Method
This section describes the use of the CLEAN algorithm to deconvolve galaxy
images and allow estimation of intrinsic, uncontaminated photometry in a
variety of photometric apertures.
The CLEAN algorithm was originally designed to remove confusing sidelobe
patterns in synthesized-beam radio interferometry at high resolution (Hogbom,
1974) but has been modified to perform deconvolution of direct imaging (Keel
et al., 1991). CLEAN iteratively constructs a model of the instrinsic light
distribution, returning both the model and the image residual. In each
iteration, CLEAN adds “points” (delta functions) to the model that correspond
to the location of the maximum residual in the original image. The intensity
of the point is a fraction (gain) of the intensity of the maximum residual.
The convolution of the PSF with that point is subtracted from the original
image. The loop repeats until all of the flux from a specified region in the
measured image has been removed or after a maximum number of iterations. CLEAN
preserves the photometric scale of the original image (Keel et al., 1991).
For this study, CLEAN is performed over a $4\farcs 5$ box in both ACS and
ISAAC images. The chosen number of maximum iterations is 500 and the gain is
$5\%.$ The algorithm is modified to find peaks using a cross-correlation of
the PSF against the image rather than subtracting at the maximum pixel
location.
The deconvolved image is the sum of the residual image and the CLEAN model.
Galaxy fluxes uncontaminated by the wings of any central point source can then
be measured by performing aperture photometry directly on the deconvolved
image with a specified aperture.
### 3.3 Stacking Analyses
Stacking sky-subtracted, reduced images of galaxies in various bands is
helpful for revealing the mean fluxes and colors of various sub-populations,
and serving as a check on other methods. In this study, we stack images that
have been masked to remove contaminating background sources and deconvolved
with CLEAN. We use observed $V$ and $J$ colors, which correspond to nearly $u$
and $R$ at $z\sim 0.8$. The gain in S/N realized by stacking is tempered by
two major disadvantages: (1) Observed-frame colors do not represent the mean
rest-frame colors, as the sources lie at a variety of redshifts; and (2)
Stacking requires the use of a single photometric aperture on a stacked image
rather than a variety of apertures that correspond to the position angle and
axis ratio of individual sources. With these caveats in mind, we use a
circular, annular photometric aperture of inner radius $0\farcs 75$ and outer
radius $1\farcs 5$ to compute observed $V-J$ colors.
### 3.4 Sky Subtraction
Meaurements of surface brightness at high galactic radii are highly sensitive
to errors in sky subtraction. The sky value for each band is measured by
manually masking 3-sigma sources and computing the geometric mean of the
values in a circular annulus with inner radius $4\farcs 0$ and outer radius
$6\farcs 0.$ These radii were determined empirically by selecting empty
regions in ACS and ISAAC fields and attempting to predict the sky value in an
inner circle of $3\farcs 0$ diameter; this combination of sky radii minimized
the errors in all ACS and ISAAC bands. It is important to fix the annular
radii across bands so as not to measure artificial colors from galactic
contamination.
Other methods of measuring sky were tested, including using sky measurements
at varying radii to extrapolate the sky value inwards to zero radius. The
simple masked mean is superior to extrapolation.
### 3.5 PSF Selection
PSF reference stars are used to correct the annular photometry for the
presence of bright, centralized point sources. It is important that these PSF
references be reliable to enable good recovery of underlying galaxy
photometry.
The ACS, NIC3, and WFC3 point spread functions are spatially and temporally
stable compared to ground-based PSFs. For each ACS, NIC3, or WFC3 band, images
of bright unsaturated stars are taken from locations in the field. The radial
profiles of the stars are compared to TinyTim PSFs to verify that the sources
are unresolved. The PSFs are chosen to be in empty, isolated regions; any
sources in the area are masked with replicated boxes of pixels taken from
nearby empty regions with the area. A sky subtraction is performed using the
masked mean method described in the above section. The PSFs are centered by
sub-pixel interpolation.
For ISAAC PSF reference stars, the sources are verified as unresolved by
comparing to ACS imaging. In each ISAAC field, the brightest, isolated star
with pixel values less than 10 times saturation level is selected. Sky
subtraction and centering are performed on these PSFs as described above. The
ACS and WFC3 PSF sizes are set to $3\farcs\times 3\farcs$, or the same size as
the annulus used in aperture photometry, and the ISAAC and NIC3 PSFs are set
to $7\farcs 5\times 7\farcs 5$ to capture any light scattered to high radii.
### 3.6 Computing $NUV-R$ Color Gradients for ERS sample
The selection of the ERS sample of 34 X-ray sources is described in Section
2.4. We select PSFs as described in Section 3.5 and use the CLEAN algorithm to
deconvolve the images as in Section 3.2. To measure color gradients for
individual sources, we first compute surface brightnesses in all observed
bands with the CLEANed images for five concentric elliptical apertures of mean
radii $0\farcs 16$, $0\farcs 33$, $0\farcs 59$, $0\farcs 88$, and $1\farcs
27$. The smallest of these apertures is filled and the larger four are
annular. The axis ratio and position angle of the ellipses is chosen as
described in the first part of Section 3. For each aperture, we interpolate
through observed $B$, $V$, $i$, $z$, $F125W$, and $F160W$ surface brightnesses
to compute a rest-frame $NUV-R$ color. For each source, we convert the
apparent angles of arcseconds into physical units of kpc and interpolate the
radius onto five points: 1.3, 2.67, 4.78, 7.13, and 10.3 kpc.
## 4 CHARACTERIZING MEASUREMENT ERROR WITH SIMULATIONS
Multiple sources of error come into play in the measurements of the colors of
AGN host galaxies at $z\sim 1.$ First, host colors may be contaminated by the
non-stellar colors of AGN emission. In the case of integrated colors,
photometric measurements with filled apertures directly sum AGN-related
emission and stellar light. In the case of “outer” colors measured with
annular apertures, emission from the central AGN may still fall in the
photometric aperture due to PSF scattering. Second, our attempted method to
correct for PSF contamination (described in 3.1) will be sensitive to PSF
variation across the field. Third, flux measurements in large annular
apertures may be sensitive to sky subtraction errors, depending on the exact
dimensions of the aperture.
These potential errors motivate a detailed approach to estimating measurement
errors. Using a suite of cloning simulations, we characterize measurement
error as function of the strength of a central point source and the intrinsic
surface brightness of the galaxy host. In these simulations, we modify images
of local galaxies to approximate their appearances as if they were located at
higher redshift. One powerful advantage of cloning simulations is that we are
able to measure the intrinsic surface brightness of an individual cloned
galaxy with high S/N before PSF convolution. As this information is difficult
to obtain for high-redshift galaxies, cloning will allow us to measure errors
at fainter host magnitudes than the alternative approach of adding point
sources to neighboring galaxies at comparable redshifts to the AGN sample.
### 4.1 Implementing Cloning
Sloan Digital Sky Survey (SDSS, York et al., 2000) images of ten local
galaxies were downloaded from the SDSS website in u’g’r’i’z’ filters,
mosaiced, and sky subtracted. These ten galaxies were chosen because their
distribution of inclination angle, half-light radius, and concentration are
representative of the “all” sample. The natural colors of these galaxies are
not representative of the “all” sample, however, so the colors are manipulated
to match the mean colors of the “all” sample during the cloning process. The
final reduced image sizes range from 27 kpc to 67 kpc, covering physical
regions larger than the diameter of the most expansive photometric aperture at
high redshift for this sample (25.4 kpc at $z=1.5$). The seeing in physical
units ($\sim 20-50$ pc) is several orders of magnitude smaller than the
effective image quality expected at high redshift ($\sim 500-4000$ pc).
Ultraviolet images were downloaded from the GALEX public website for these
four galaxies in both near-UV (NUV) and far-UV (FUV) filters.
Images of cloned galaxies are generated by inserting scaled copies of these
SDSS and GALEX images into empty regions of ACS $B,V,i,z$, ISAAC $J,H,Ks$, and
NIC3 $F110W$ and $F160W$ images. The change in scales from the local universe
to high redshift is calculated assuming a flat cosmology with $H_{0}=70$
km/s/Mpc, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$ with the IDL “lumdist”
function. For an individual cloned instance of a galaxy, the target redshift
is generated randomly over the range $0.5<z<1.5$ and used to assign GALEX/SDSS
filters to the ACS/ISAAC filters into which they redshift. For a given
redshift, the filter closest in wavelength is chosen. At the lowest redshift
of $z=0.5$, the SDSS $z^{\prime}$ image is far bluer than the wavelength
required to fill the ISAAC $Ks$ band image ($\sim 1.5\;\mu$m). However, the
morphological K-correction between rest-frame $\sim 0.9\;\mu m$ and $\sim
1.5\;\mu m$ is expected to be small.
Once source filters are assigned to target filters, the scaled copies are
convolved with point spread functions drawn from the target images. Each of 10
empty target locations in the ACS/ISAAC images is chosen nearby a reference
star, which is used for convolution. The reference star is not used for
simulated observation in later steps. The images and PSFs are supersampled by
a factor of 3 for this convolution and brought back to the target pixel
resolution. The ISAAC point spread functions possess considerable noise, so
2-D Moffat fits of these PSFs are used for convolutions. The radial profiles
of the Moffat fits match the radial profiles of the fitted ISAAC PSFs to
within 10% at all radii less than $3\farcs 0$ for all 10 target locations.
For each cloned galaxy, the observed z-band surface brightness in the $0\farcs
75<r<1\farcs 5$ elliptical aperture is set randomly between $21$ and $31$
magnitudes per square arcsecond. The amplitudes of the images in each of the
other filters are chosen according to the mean rest-frame SED of the actual
host galaxy population as measured from this data set ($(u-R)^{0}\sim 1.6$;
$(R-J)^{0}\sim 1.1$). The intrinsic host surface brightnesses (i.e., before
convolution) in the elliptical aperture are computed and recorded for
comparison later.
To test the robustness of wide aperture photometry against contamination from
AGN emission, a point source of random amplitude is added to the host galaxy
center. The Point Source Fraction, or ratio between the flux in the point
source and the flux in the entire host galaxy, ranges from $0.1$ to $1000$.
### 4.2 Calculation of Critical Errors
The images of ten local galaxies are cloned 1600 times in each of 10 insertion
locations in the ACS and ISAAC fields, 800 with no central point source, and
800 with central simulated point sources. The redshifts and bolometric galaxy
luminosity vary as described above. For each clone, full aperture photometry
is performed as described in Section 3.1, using an aperture correction to
compute uncontaminated surface brightnesses. We compare the measured surface
brightness with the true, instrinsic surface brightness in the correct
aperture before convolution with the PSF.
#### 4.2.1 Galaxies without Central Point Sources
Here, we estimate photometric error in the “outer” aperture as a function of
intrinsic host surface brightness using cloned galaxies that lack central
point sources. Given the measured surface brightnesses for the “all” sample,
we estimate photometric errors and evaluate the reliability of the “outer”
photometry for this sample.
Figure 3 plots the difference between the true surface brightness and measured
surface brightness as a function of true surface brightness for 800 cloned
galaxies in all observed bands. The differences are plotted in relative flux
units, or $(f_{meas}-f_{true})/f_{meas}.$ Shaded regions denote the $1\sigma$
random error inferred by fitting Gaussian functions to the relative flux
difference distributions. The red vertical lines denote the surface brightness
limit where the implied random photometric error exceeds $0.4$ magnitudes. We
choose this limit for two reasons: First, the expected number of sources with
negative flux measurement (a catastrophic failure) is limited to a few percent
of the subsample whose random errors are of order $\sim$40% in flux units.
Second, $0.4$ magnitudes of random error will average down to our expected
level of systematic error ($\sim 0.1$ mag) in samples with $\sim 16-32$
sources.
Histograms in all panels of Figure 3 show the distribution of measured surface
brightnesses in each band for the “all” sample. These histograms indicate that
only a few sources are fainter than the limit corresponding to $0.4$
magnitudes of error (red line) in all observed bands redder than $B$. In the
$B$-band panel, a blue histogram denotes the distribution of $B$ surface
brightnesses for sources in the “all” subsample with $z<0.638$. Sources below
this redshift require the $B$-band to interpolate a rest-frame $u$ flux;
sources above this redshift do not require $B$-band measurements. Of all the
observed bands, the highest photometric errors are seen in the $B$-band for
the “all” sample. However, none of the sources for which $B$-band is necessary
to compute a rest-frame $u$ flux have a photometric error greater than $0.4$
magnitudes.
We conclude that, for the “all” sample as selected, rest-frame $u-R$ color can
be reliably constrained for all sources that lack central point sources.
Figure 3: Plots of relative flux differences $(f_{meas}-f_{true})/f_{meas}$ in
ACS and ISAAC bands for cloned galaxies without central point sources, plotted
as a function of true galaxy surface brightness in the “outer” aperture in
units of AB magnitudes per square arcsecond. Each filled circle denotes a
separate cloned instance of a local galaxy. Different colors identify the
original local galaxy. Gray dashed lines indicate the mean inferred systematic
error as a function of surface brightness for these simulations. Shaded
regions denote the $1\sigma$ random error inferred from these simulations. Red
vertical lines plot the surface brightness at which the total inferred
photometric error exceeds 0.4 magnitudes. Solid black lines plot histograms of
measured surface brightness for the “all” sample in each band. The blue line
in the upper left plot shows the histogram of measured $B$-band surface
brightness for galaxies with redshifts $z<0.638$, below which $B$-band
measurements are required to compute a rest-frame $u$ flux. Note that very few
galaxies exceed $0.4$ magnitudes of error in any band, except for in the
$B$-band. However, for those galaxies that require an observed $B$ flux to
compute a rest-frame $u$ flux, none exceed $0.4$ magnitudes of photometric
error.
#### 4.2.2 Galaxies with Central Point Sources
It is expected that the presence of central point sources will cause error in
surface brightness measurements, although the use of a wide annulus (inner
radius $0\farcs 75$) will reduce this effect. As nonthermal AGN emission tends
to be bluer (rest-frame $u-R\sim 0.3$) compared to galaxy light (rest-frame
$u-R\sim 1-2$), this error is likely to be larger in the observed $B$ and
$V$-band for a sample with $0.5<z<1.5$. We now quantify the photometric error
in each band as a function of central point source strength through our
cloning analysis.
For this analysis, the cloning simulation of section 4.2.1 is repeated, but
with artificial point sources added of varying amplitudes. The ratio of the
total amplitude of the point source to the total ampltitude of the galaxy
(measured over a $3\farcs$ diameter circle), or Point Source Fraction, is
varied between 0.1 and 1000. Aperture photometry is performed on each of the
cloned galaxies and corrected for contamination as described in Section 3.1.
We then compare the corrected surface brightnesses to the intrinsic host
surface brightnesses to evaluate photometric error as a function of Point
Source Fraction.
The relative flux differences in all bands are shown in Figure 4, as in Figure
3, but now plotted against the logarithm of the Point Source Fraction. Shaded
regions denote the $1\sigma$ random error inferred by fitting Gaussian
functions to the relative flux difference distributions. The red vertical
lines denote the Point Source Fraction at which the random photometric error
exceeds $0.4$ magnitudes. There are clear trends between the reconstructed
surface brightness and the amplitude of the central point source in all bands.
The photometric error is less than $0.4$ magnitudes for sources with Point
Source Fraction less than $\sim 25-50$ in each band. Notice that the HST $V$
and $i$ bands have the least photometric error for a given Point Source
Fraction and the $B$ and $K$ bands have the most.
Figure 4: Plots of relative flux differences $(f_{meas}-f_{true})/f_{meas}$ in
ACS and ISAAC bands for cloned galaxies with central point sources, plotted as
a function of the Point Source Fraction. Each panel plots simulated points for
a different observed band. All symbols are defined as in Figure 3.
Figure 4 indicates that “outer” aperture flux measurements will be reliable
for sources with Point Source Fraction less than 30 in all bands. While the
Point Source Fraction is difficult to measure for non-simulated data, a more
easily-measured proxy can be used. Here we use the ratio between the amplitude
in a $0\farcs 18$ aperture and a circular, annular aperture of inner radius
$0\farcs 18$ and outer radius $1\farcs$, or $C_{0.18/1.0}$. We define the
Normalized Concentration $C_{0.18/1.0,norm}$ as
$C_{0.18/1.0,norm}=\frac{C_{0.18/1.0,image}}{C_{0.18/1.0,PSF}}.$
This parameter will correct for PSF variation in a field. Figure 5 plots the
relationship between $C_{0.18/1.0,norm}$ and the known Point Source Fraction
as measured for the cloned galaxies above in observed $V$. The tightness of
the correlation suggests that measurements of $C_{0.18/1.0,norm}$ in actual
galaxies can be used to look up the Point Source Fraction for a given band.
Figure 5: Plot of measured log($C_{0.18/1.0,norm}$) as a function of input
Point Source Fraction for the 800 cloned galaxies in this simulation. Dashed
lines mark fitted spline functions. The tightness of the correlation suggests
that the easily measured $C_{0.18/1.0}$ is a good proxy for point source
contamination.
### 4.3 Correction for Contamination in Integrated Fluxes
Constructing CMDs that are corrected for point source contamination requires
an unbiased estimate of the host galaxy’s absolute magnitude. In practice,
this is more difficult than estimating uncontaminated color in the “outer”
aperture, as it requires measurement of the integrated galaxy flux beneath a
central point source. Here, we describe our method of using imaging to
estimate point source contamination for a given band.
The tightness of the correlation in Figure 5 suggests that $C_{0.18/1.0,norm}$
may be used as a proxy for point source contamination. To estimate the point
source contamination, we first measure $C_{0.18/1.0}$ in all bands. Next, for
each band, we fit separate spline functions to the $C_{0.18/1.0,norm}$ / Point
Source Fraction relations in the simulated galaxies (e.g., Figure 5). These
spline functions are inverted to create monotonic look-up tables. Finally, the
$C_{0.18/1.0,norm}$ values are used to look up the Point Source Fraction in
each band. The Point Source Fraction is used to calculate and subtract a
correcting factor from the measured integrated photometry.
We estimate rest-frame $R$-band absolute magnitudes by performing this
procedure in the observed $JHK$ bands. These are displayed in the next section
in corrected CMDs for the “all” sample.
### 4.4 Implications for Observations
Our cloning studies have constrained systematic and random measurement errors
as a function of intrinsic surface brightness and Point Source Fraction in all
bands. We use these relations to calculate systematic and random errors for
each AGN host in the “all” sample. We estimate Point Source Fraction as
described in Section 4.3. We note that using measured instead of intrinsic
surface brightnesses to look up errors will underestimate error bars for some
of the faintest sources. However, the original sample cut of observed $R<24$
(total galaxy magnitude) has largely removed intrinsically faint sources from
the sample, and very few of the remaining sources have photometric errors
larger than $0.4$ magnitudes.
## 5 RESULTS
### 5.1 Corrected Color-Magnitude Diagrams
Figure 6 plot “outer” $u-R$ colors against corrected $M_{R}$ for the “all” AGN
sample. Rest-frame $M_{R}$ is estimated from observed $JHK$ integrated
photometry ($3\farcs 0$ filled aperture) via interpolation. Corrections for
point source contamination are measured using source concentration as
described in Section 4.3 above. Figure 6 shows uncorrected $u-R$ and $M_{R}$
as light blue and light red points. These uncorrected loci are linked with the
corrected photometry with vectors.
The simulations above have shown that photometric measurements using annular
apertures avoid color contamination due to central point sources reliably.
However, estimating uncontaminated absolute magnitude is more difficult, as it
involves estimating the flux directly beneath the AGN point source. For large
point source contamination (Point Source Fraction $>$ 30), this method
produces uncertain results. Therefore, we make no conclusions about the AGN
samples involving the mean or individual values of $M_{R}.$
Figure 6: Rest frame $u-R$ vs. $M_{R}$ color-magnitude diagram, corrected for
point source contamination. Symbols are plotted as in Figure 2. $u-R$ colors
have been measured with annular “outer” photometric apertures. $M_{R}$
magnitudes are measured with integrated apertures. Sources judged to be
marginally contaminated by a central point source associated with AGN emission
(Point Source Fraction in observed B greater than 0.35) are corrected for this
contamination in $M_{R}$. For these contaminated sources, uncorrected,
integrated measurements are shown as faint, empty points. Uncorrected,
integrated loci are connected to corrected, “outer” loci with dashed lines.
Rest-frame values are estimated from observed photometry as in the text. Blue
and red error bars denote the mean colors and luminosities for the soft and
hard sources, respectively, with errors calculated as in Section 5.4. Small
black circles denote the colors and magnitudes of the underlying galaxy
population, selected as explained in Section 2.4.1. Note that photometry is
measured with $3\farcs$ filled apertures for this control sample, not “outer”
apertures. The soft source at the bluest extreme of the diagram (XID 100) has
a significantly bluer “outer” color than its integrated color, unlike the
remainder of the “all” sample; it appears to be composed of a blue point
source with a bluer star-forming extension to the NE.
It is clear in Figure 6 that the difference in mean rest-frame $u-R$ color
between soft and hard sources seen in Figure 2 is drastically reduced by using
annular photometric apertures and correcting for contamination from AGN
emission. The mean colors are also listed in Table 1; note that the use of
“outer” apertures reduces the soft-hard color difference by a factor of nearly
three. Given the measurement errors, the mean rest-frame $u-R$ color of the
“all-soft” and “all-hard” subsamples are statistically equivalent.
Note that the mean color and magnitude of the obscured population is different
in Figure 2 and 6, despite the lack of point sources in this population. The
corrected “outer” magnitudes of the obscured population are $\sim 0.1$ bluer
than the uncorrected, integrated magnitudes. Because of the negative color
gradients in the host galaxies, the outer regions are bluer than the nuclear
regions. For the unobscured sources, this effect is overwhelmed by the removal
of the central point source contamination, which causes the corrected “outer”
magnitudes to be redder than the integrated magnitudes.
### 5.2 Mean SED Comparison
The surface brightnesses of the AGN sample are plotted as de-redshifted
spectral energy distributions (SED) in Figure 7. Surface brightnesses are
plotted in absolute AB magnitudes, modified to account for a de-projection of
inclined disks (i.e., fluxes are multiplied by $b/a$, where $b$ is the
semiminor axis and $a$ is the semimajor axis). Figure 7 plots the logarithmic
mean (or geometric mean of magnitudes) of these SEDs for the “all-soft” and
“all-hard” subsamples. SEDs of Bruzual & Charlot (2003) Single Stellar
Population (SSP) bursts are shown for comparison. The SEDs of the soft and
hard sources appear to be similar, although the soft sources are $\sim 0.3$
magnitudes brighter than the hard sources at all wavelengths.
The mean SEDs of the two populations are inconsistent with a starburst
population of age $\sim 25$ Myr. The SEDs more resemble those of intermediate
and old stellar populations, although the rise at near-infrared wavelengths
compared to these SSP models implies that dust extinction is present. Young
stellar populations and their accompanying ultraviolet emission may be hidden
by this dust.
Figure 7: Logarithmic mean of surface brightnesses for all-soft sample and
all-hard sample. The dashed lines represent the one-sigma uncertainties on the
mean SEDs. Red lines denote measurements for soft AGN and blue lines denote
measurements for hard AGN. Error bars are measured from full cloning
simulations and a bootstrap procedure to estimate error due to finite
sampling. SEDs of Bruzual & Charlot (2003) Single Stellar Population (SSP)
bursts of varying ages are shown for comparison.
### 5.3 Mean Color Gradients in ERS sample
The principal advantage of using the “outer” aperture is the avoidance of
central point sources; the primary disadvantage is that is does not probe the
central regions of host galaxies. The inner radius of the “outer” aperture was
set by the image quality of the ISAAC JHK imaging, approximately $0\farcs 65$
in the worst case. We now utilize $F125W$ and $F160W$ imaging in GOODS-South
from the WFC3/IR imager aboard HST, with a spatial resolution of $\sim 0\farcs
2$, to constrain host colors at smaller radii. Although the spatial coverage
of the WFC3 infrared imaging is less than 50 square arcminutes, the improved
depth relative to ISAAC ($H_{AB}=27.0$ for point sources, $5\sigma$) permits
extending the sample to fainter magnitudes.
We compute $NUV-R$ color gradients for each host galaxy as described in
Section 3.6. Figure 8 displays the mean $NUV-R$ color gradients for the “ERS-
soft” and “ERS-hard” subsamples. The shaded regions denote the 68% confidence
intervals as computed in Section 5.4 below. Two curves are displayed for soft
sources; the blue curve shows the mean color gradient for all 15 soft sources
and the hatched curve shows the mean color gradient for the 10 soft sources
whose Point Source Fraction in V-band is less than 10. Point Source Fraction
is measured in the $V$-band as described in Section 4.3.
Figure 8: Mean color gradients for three populations in the WFC3 ERS sample:
Obscured sources (HR $>-0.2$), all unobscured sources (HR $<-0.2$), and only
unobscured sources with little point source contamination (V Point Source
Fraction $<$ 10). Filled regions are 68% confidence intervals. With the
removal of sources strongly contaminated by central point sources, the mean
color gradients of the obscured and unobscured sources are similar to $\sim
0.5$ magnitudes at all radii greater than 1 kpc.
Note that because the images are CLEANed and the central aperture is filled,
contaminating point sources will affect the color of the innermost point only
($r=1.3$ kpc) in Figure 8. The innermost point of the mean color gradient of
the full soft sample, denoted by the blue curve, is significantly bluer than
all other points. However, when sources with significant point source
contamination (Point Source Fraction $>$ 10) are removed from the soft sample,
as shown with the hatched curve, the innermost point returns to a color that
is similar to the colors at all other radii. The removal of these contaminated
sources does not affect the mean colors of the hosts beyond $r\sim 2$ kpc.
The colors of the hard and soft sources are statistically identical at all
radii beyond $r\sim 2$ kpc whether objects with significant point source
contamination have been removed or not. Removing objects with central point
sources from the soft sample allows us to make an unbiased color comparison
between soft and hard sources for the central $r=1.3$ kpc point; when these
objects are removed, the remaining soft sample and the hard sample are
statistically equivalent in color at $r=1.3$ kpc. Assuming that the point
source contamination in the objects with Point Source Fraction $>$ 10 is
caused by nonstellar emission associated with the AGN, we may conclude that
the colors of the soft and hard sources for this subsample are similar at all
radii greater than $r\sim 1$ kpc.
### 5.4 Error Considerations
The method of averaging can possibly affect the mean photometric properties of
a sample. When the averaging is performed in flux space, with equal weights on
each photometric point, the principal outcomes of the comparisons between the
various types of AGN in this sample are unchanged. Similarly, when the
averaging is performed in flux space with stronger weights towards fainter
fluxes, the comparison is unchanged.
Systematic measurement errors are seen in the cloning simulation; we estimate
these for individual galaxies and subtract them. The random errors on the mean
SEDs at a particular wavelength and on the mean colors of subsamples are the
sum of two components: The first is a random error caused by sky-subtraction
errors, photon error in the photometric measurement, and other errors
associated with basic aperture photometry. The second major error, dominant
over photometric error, is due to small-number statistics in these sub-
samples. We estimate this error with a Monte Carlo bootstrap method, as
outlined in Press et al. (1992). In this method, new sub-samples are
repeatedly drawn (with replacement) from the overall data set, analyzed, and
used to populate a distribution. The random error due to the finite size of
the sample is then given by the standard deviation of critical parameters
(e.g., mean rest-frame $u$ magnitude) in this distribution.
The result of this analysis is that the bootstrap error in $u-R$ is 0.11
magnitudes for the all-soft subsample and 0.10 magnitudes for the all-hard
subsample. This can be understood in terms of the intrinsic brightness
distribution and color scatter in the samples. The intrinsic variation in
$u-R$ color for this sample is $\sigma\sim 0.7$ magnitudes. The error on the
mean of some parameter due to incomplete sampling of a population goes as
$\sigma/\sqrt{N},$ where $\sigma$ is the standard deviation of the parameter
for the parent population and $N$ is the size of the sample. With $N\sim
31-42$ for the all-soft and all-hard subsamples, respectively, the error on
the mean is $\sim 0.13$ and $\sim 0.11$ magnitudes for $u-R$ color,
respectively. These alternate estimates are nearly the same as that estimated
with a Monte Carlo bootstrap procedure.
### 5.5 Are Obscured AGN Hosts More Dusty than Unobscured AGN Hosts?
$u-R$ and $NUV-R$ colors are sensitive probes of low levels of star formation
(Salim et al., 2007), but these colors can be affected by dust extinction.
Here we search for systematic differences in galactic dust extinction between
the obscured and unobscured AGN host populations. In a rest-frame $U-V$, $V-J$
color-color diagram, the effects of dust extinction can be separated to some
extent from the effects of stellar age (see Figure 9). $V-J$ color is more
sensitive to dust extinction than stellar age (Wuyts et al., 2007).
We compute $V-J$ “outer” colors to estimate the differential effects of dust
extinction. Rest-frame $J$ band magnitudes are calculated from observed
BVizJHK photometry using linear interpolations as described in Section 3. Half
of the sample has $z>0.85$, beyond which rest-frame $J$ shifts redder than
observed $Ks$, the reddest band available in this sample. Color extrapolations
are required beyond this redshift. We estimate the error in this extrapolation
by measuring the natural dispersion in rest-frame $z-J$ “outer” color
(observed $H-Ks$) at a single rest-frame $R-z$ (observed $J-H$) color slice
for sources with $z<0.85$. This dispersion is 0.10 magnitudes, implying that
the extrapolation error is less than 0.1 magnitudes for all sources in the
sample. The error on the mean of the samples will be much smaller than this.
The bottom panel of Figure 9 plots the “outer” colors of the “all” sample in a
rest-frame $U-V$, $V-J$ color-color diagram. Only sources for which observed
$Ks$ is measured are included, i.e., only sources with ISAAC imaging in the
southern GOODS field. The mean rest-frame $U-V$ color of the two populations
are discrepant by 0.16 magnitudes, which is consistent with the results shown
in Table 1 within the errors. The mean $V-J$ color of the obscured population
is redder than the unobscured population by $0.17$ magnitudes. These
differences are not statistically significant, suggesting that the two
populations are similar in mean galactic dust extinction. The two populations
do not occupy clearly separated regions in the diagram. Approximately one-half
of the sample falls in a region corresponding to $A_{V}>1$, suggesting that a
significant fraction of the sources are dusty, which is consistent with other
studies using similar data and methodology (Cardamone et al., 2010).
Figure 9: Top panels: Rest-frame $U-V$ vs. $V-J$ color-color diagrams of all
galaxies in a mid-IR selected sample with $L_{V}>5\times 10^{9}L_{\odot}$
(included with permission from Wuyts et al. (2007)). SDSS+2MASS galaxies
(small gray dots) are plotted as a local reference. Top left panel: Galaxies
are color-coded by dust extinction. Dust extinction affects rest-frame $V-J$
color more than $U-V$. Top right panel: Galaxies are color-coded by mean
stellar age. Stellar age affects rest-frame $U-V$ color more than $V-J.$
Bottom panel: Rest-frame $U-V$ vs. $V-J$ color-color diagram using “outer”
aperture for AGN with observed $Ks$ available. Small black circles denote the
colors of the underlying galaxy population, selected as explained in Section
2.4.1. Note that photometry is measured with $3\farcs$ filled apertures for
this control sample, not “outer” apertures. The mean colors of the AGN sample
are shown in the center of the diagram; the obscured locus is close to the
unobscured locus in $V-J$, suggesting that the populations have similar mean
extinctions.
### 5.6 $u-R$ Properties of AGN Hosts
As seen in Figure 7 and Table 1, the mean SEDs and mean rest-frame $u-R$
colors of AGN hosts beyond $r\sim 6$ kpc are largely similar across X-ray
obscuration. The hosts of the soft AGN, on average, are $\sim 0.11$ magnitudes
bluer in $u-R$ and $\sim 0.2$ magnitudes bluer in $NUV-R$ than the hosts of
the hard AGN beyond $r\sim 6$ kpc. This effect is weak and not statistically
significant. When this analysis is repeated for luminous objects with
$L_{X}>10^{43}$ ergs s-1, the $u-R$ color of the soft sources is statistically
equivalent to that of the hard sources. The soft sources are $0.17$ magnitudes
bluer than the hard sources, but this effect is not statistically significant.
We have checked this result with a simple stacking exercise, in which aperture
photometry is performed on a stack of the processed images of a particular
population. For this procedure, we use sky-subtracted images whose ACS/ISAAC
alignment is good to $\sim 0\farcs 1$ that have been deconvolved with CLEAN as
described in section 3.2. This stacking method simulates the measurement of
flux-averaged colors, which effectively weights brighter sources higher than
fainter sources relative to a logarithmic mean. We use observed $V$ and $J$
colors, which correspond to nearly $u$ and $R$ at $z\sim 0.8$. Using a single
circular, annular aperture of $0\farcs 75$ inner radius and $1\farcs 5$ outer
radius, the observed $V_{AB}-J_{AB}$ colors of the stacked soft and hard
samples are $0.81$ and $0.95$ magnitudes, respectively, suggesting that the
soft sources are $0.14$ magnitudes bluer than the hard sources beyond $r\sim
6$ kpc. Although this value is larger than the difference in $u-R$ colors
measured via SED averaging, the similarity is reasonable considering that (1)
the bands are in the observed frame and not the rest-frame and (2) a single
aperture is being used for a variety of host morphologies.
We have seen that the mean rest-frame $V-J$ color is also similar for the soft
and hard sources (Figure 9). This implies that the mean rest-frame $u-R$
colors of the soft and hard subsamples are affected by dust extinction at
similar levels. Given both the similarity of the rest-frame $u-R$ colors and
the similarity of the mean dust extinction in the soft and hard subsamples, it
follows that the mean extinction-corrected rest-frame $u-R$ colors of the soft
and hard sources would also be similar.
## 6 DISCUSSION AND CONCLUSION
### 6.1 Comparison with Previous Work
This work is the first to measure the UV/optical colors of a large sample of
AGN host galaxies at $z\sim 1$ while quantitatively, reliably correcting for
point source contamination in individual galaxies. Ignoring or insufficiently
correcting for point source contamination results in systematic error in
comparing the colors of hard and soft X-ray sources (see Figure 6).
Direct comparisons of our results with those obtained for Seyfert galaxies in
the local universe are problematic, as numerous AGN selection techniques are
commonly used in the literature. At low redshift, the effects of point source
contamination on host galaxy color measurements are negligible because the
galaxies are well-resolved. Locally, Schawinski et al. (2009) find little host
color difference between obscured and unobscured X-ray-selected AGN, in
agreement with our results. Although the sample is small, their selection
techniques and use of X-ray-derived obscuration measurements are more
analogous to those utilized in this study. Given the similarity of our
selection criteria, the agreement of our results suggests that the mean color
offset between soft and hard sources has not evolved since $z\sim 1$.
Ho et al. (2008) find that unobscured broad-line AGN hosts at low-redshift are
observed to possess large reservoirs of HI gas. This result is interpreted as
being evidence against a feedback scenario, in which unobscured AGNs are
thought to be caught in the act of expelling gas (Ho et al., 2008). No
comparison is made between Type I and II sources. Considering the use of
different selection methodologies and measurements, there is no inconsistency
between this study and our own. For a sample of mid-infrared Type II QSOs at
$0.3<z<0.8$, Lacy et al. (2007) find that the host galaxies exhibit
significant star formation rates ($3-90\;M_{\sun}$ yr-1) and that disk
inclination correlates with silicate features, implying that at least some of
the reddening arises from the host galaxy. However, it is not clear how these
star formation rates would compare to similarly selected Type I QSOs at
similar redshift. Moreover, the AGN selection techniques, selected luminosity
range, and redshift range are different from those in the present study,
preventing direct comparison.
Page et al. (2004) compares the $850\;\mu$m fluxes of luminous ($L_{X}\sim
10^{45}$ ergs s-1), X-ray-selected AGN at $1<z<3$, finding that obscured
sources have significantly more submillimeter emission than unobscured
sources. As a function of redshift, the mean flux difference between obscured
and unobscured sources is consistent with zero at redshifts below $z\sim 1.5$
and rises significantly at higher redshifts. Contrastingly, Shao et al. (2010)
measures the far-infrared-derived star formation rates of X-ray selected
intermediate-luminosity AGNs ($L_{X}\sim 10^{43}$ ergs s-1) at $z\sim 1$ in
GOODS-North, finding no dependence of star formation rate on X-ray absorbing
column density. It appears that the mean star formation rate of obscured
sources diverges from that of unobscured sources at either the highest X-ray
luminosities ($L_{X}\sim 10^{45}$ ergs s-1), at higher redshifts beyond $z\sim
2$, or in both of these regimes.
CP10 measures the rest-frame U-V colors of intermediate-luminosity, X-ray-
selected AGN in the Extended Groth Strip (Pierce et al., 2010b). They find
that soft sources are systematically bluer than hard sources in both nuclear
($r<0\farcs 2$), extended ($0\farcs 2<r<1\farcs 0$), and integrated apertures.
The mean difference in extended color between soft and hard populations in our
GOODS sample differs from the CP10 value by $2.5\sigma$ ($0.3$ mag in $u-R$).
Our color offsets are consistent to within $1\sigma$ ($0.1$ mag in $u-R$) when
the CP10 sample is trimmed of all sources that they identify visually as
having “definite” or “possible” point sources. In addition, color offsets are
similar when no correction for point source contamination is used in our
sample ($0.3$ mag in $u-R$; see Figure 2 and Table 1). In CP10’s Figure 11d it
is clear that sources visually identified as possessing “definite” or
“possible” point sources display anomalous color gradients (i.e., blue nuclear
regions). These lines of evidence point to some amount of point source
contamination in CP10’s extended colors.
Cardamone et al. (2010) find that X-ray selected AGN hosts at $z\sim 1$ are
frequently in dust-enshrouded galaxies. Approximately one-half of our sample
occupy a region in the rest-frame $U-V$ vs. $V-J$ diagram corresponding to
$A_{V}>1$, so our results are not inconsistent. Although the authors do not
explicitly compare the locations of obscured and unobscured sources in this
plot, it is clear from their Figure 2 that the mean locations of these
populations are similar, as in our sample.
### 6.2 Implications for AGN Unification and Galaxy Evolution Scenarios
In this paper, we have assembled an X-ray-selected AGN sample and measured the
mean $u-R$ colors beyond $r\sim 6$ kpc in the host galaxies. We found that the
mean rest-frame SEDs were similar for soft and hard sources both for objects
of intermediate X-ray luminosities ($L_{X}>10^{42}$ ergs s-1) and objects of
higher luminosities ($L_{X}>10^{43.5}$ ergs s-1, $\langle L_{X}\rangle\sim
10^{43.9}$ ergs s-1). We also found that the mean loci of the soft and hard
populations in a rest-frame $U-V$ vs. $V-J$ were similar, implying that the
mean properties of galactic dust extinction of the two populations were
similar. In Section 5.3, we used high spatial resolution WFC3/IR imaging to
compute the mean rest-frame $NUV-R$ color gradients of soft and hard sources.
The CLEAN deconvolution algorithm was used to remove PSF contamination effects
before computing colors. It was seen that the colors of soft and hard sources
were significantly different only in the nucleus ($r<1.3$ kpc), and when
sources with strong point sources in the observed $V$-band were removed from
the soft sample, the mean colors of the soft and hard sources were similar
with respect to the error bars at all radii. The limiting spatial resolution
of the WFC3 imaging prevents us from probing colors at radii smaller than
$r\sim 1$ kpc. We conclude that the host colors of soft and hard sources are
statistically equivalent at all radii larger than $r\sim 1$ kpc. These
observations indicate that unobscured AGN are not redder than obscured AGN at
$z\sim 1$ at intermediate X-ray luminosities.
In our sample, galaxy-wide dust extinction is weakly or not correlated with
nuclear AGN obscuration as probed by X-ray hardness ratio. Taken with the
evidence that galaxy color is weakly or not correlated with X-ray hardness
ratio, this implies that the conditions of star formation and dust extinction
are uncorrelated with the conditions of neutral hydrogen obscuration on
nuclear scales. These observations favor AGN unification scenarios (Antonucci,
1993; Urry et al., 1995), in which AGN obscuration is determined by the
orientation of a torus on parsec scales with respect to the observer. This
conclusion appears to be independent of X-ray luminosity in our sample for
luminosities lower than $10^{44.5}$ ergs s${}^{-1}.$
Recently, several models of galaxy formation have been able to explain the
bimodality of galaxy colors through AGN feedback processes (e.g., Di Matteo et
al., 2005; Lapi et al., 2006; Hopkins et al., 2006; Menci et al., 2008). These
models imply that unobscured AGN would be associated with a quenching of star
formation, resulting in a reddening of the stellar population, even in the
outer regions of the host (beyond $r\sim 6$ kpc) in the most energetic cases
(Hopkins et al., 2006; Menci et al., 2008). The color of a single stellar
population (SSP) reddens by $\sim 0.7$ magnitudes in $u-R$ over a quasar
lifetime following the ceasing of star formation (solar metallicity from age
10 Myr to age 100 Myr, Maraston (2005)), and thus the difference in color
resulting from quenching in unobscured hosts should have been detectable in
this study.
Our observations instead suggest that the majority of intermediate luminosity
AGN at $z\sim 1$ are not undergoing nor have recently experienced rapid,
galaxy-wide quenching due to AGN-driven feedback processes. A second possible
interpretation is that if AGN-related blowout events have occurred in the
population of intermediate luminosity AGNs at $z\sim 1$, the radial extent of
quenched stellar populations must be restricted to $\sim 1$ kpc on average or
the color signature of the quenched stellar population must have since been
rewritten by fresh gas accretion and resulting star formation.
## 7 ACKNOWLEDGEMENTS
This work has been supported in part by the NSF Science and Technology Center
for Adaptive Optics, managed by the University of California (UC) at Santa
Cruz under the cooperative agreement No. AST-9876783.
Some of the data presented in this paper were obtained from the Multimission
Archive at the Space Telescope Science Institute (MAST). STScI is operated by
the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA
Office of Space Science via grant NAG5-7584 and by other grants and contracts.
Support for Program number HST-HF-51250.01-A was provided by NASA through a
Hubble Fellowship grant from the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Incorporated, under NASA contract NAS5-26555.
S.M.A acknowledges fellowship support by the Allen family through UC
Observatories/Lick Observatory. This work is based in part on observations
made with the European Southern Observatory telescopes obtained from the
ESO/ST-ECF Science Archive Facility. Observations have been carried out using
the Very Large Telescope at the ESO Paranal Observatory under Program ID(s):
LP168.A-0485.
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|
arxiv-papers
| 2011-07-11T21:04:39 |
2024-09-04T02:49:20.427463
|
{
"license": "Public Domain",
"authors": "S. Mark Ammons (University of Arizona), David J. V. Rosario (MPE),\n David C. Koo (UCSC), Aaron A. Dutton (University of Victoria), Jason\n Melbourne (Caltech), Claire E. Max (UCSC), Mark Mozena (UCSC), Dale D.\n Kocevski (UCSC), Elizabeth J. McGrath (UCSC), Rychard J. Bouwens (Leiden\n Observatory), and Daniel K. Magee (UCSC)",
"submitter": "S. Mark Ammons",
"url": "https://arxiv.org/abs/1107.2147"
}
|
1107.2328
|
# Recent LHCb Results
Giacomo Graziani, on behalf of the LHCb collaboration
###### Abstract
The LHCb experiment started its physics program with the 37 pb-1 of pp
collisions at $\sqrt{s}$=7 TeV delivered by the LHC during 2010. The
performances and capability of the experiment, conceived for precision
measurements in the heavy flavour sector, are illustrated through the first
results from the experimental core program. A rich set of production studies
provide precision QCD and EW tests in the unique high rapidity region covered
by LHCb. Notably, results for W and Z production are very encouraging for
setting constraints on the parton PDFs.
###### Keywords:
LHCb, LHC, B physics, forward physics
###### :
12.38.Qk, 13.85.Ni, 13.85.Qk, 13.25.Hw, 13.20.He, 14.40.Pq
## 1 The Concept of LHCb
Figure 1: The LHCb detector
The LHCb experimentAlves et al. (2008) is searching for new physics through
non standard virtual contributions to CP violating and rare decays of heavy
hadrons. It is implemented as a single arm spectrometer covering the
pseudorapidity region $1.9<\eta<4.9$, where the largest production rate of
$b\overline{b}$ pairs per solid angle is expected at LHC energies. This
coverage is unique at the LHC and complementary to the general purpose
detectors ATLAS and CMS. The detector design, sketched in fig. 1, emphasizes
proper time resolution, to identify the short lived $b$ hadrons and resolve
the fast $B_{s}^{0}$ oscillations, while excellent invariant mass resolution
and particle identification (PID) capabilities are needed to disentangle
decays of heavy flavours from the dominant hadronic background. These detector
features also make LHCb a powerful tool for heavy flavour spectroscopy and
production studies in its unique rapidity range, providing precision QCD tests
at the unprecedented energies reached by the LHC.
The geometry of the Vertex Locator (VELO) is optimized for detecting the
decays of $b$ hadrons in the forward direction. Its Si $\mu$-strips sensors
are perpendicular to the beam and are located on two retractable half
stations. When stable beam is declared, the two halves are closed and the
inner border of the active area is only 8 mm away from the beam axis. Tracking
is completed by a set of stations equipped with Si $\mu$-strips (for the inner
part) and straw tubes (for the outer part), located upstream and downstream a
warm dipole magnet providing an integrated field of 4 Tm. The combined
performances of two RICH detectors result in excellent $\pi$/K/p separation in
the 1–100 GeV/c momentum range. The calorimetric system provides
e/$\gamma$/hadrons separation through a scintillating pad detector, a
preshower, electromagnetic (shashlik) and hadronic (Fe/scint. tiles)
calorimeters. Muons are identified by five stations equipped with multi–wire
proportional chambers (or GEM chambers for the most inner part), interspaced
by iron absobsers. A fast, flexible and efficient trigger is essential to
extract the interesting events from the nominal 40 MHz collision rate. It is
implemented in two levels: the initial decision comes from an hardware level
(L0), selecting particles of high transverse momentum using the informations
from the calorimetric and muon systems, which can operate at 40 MHz with a
relatively low $p_{T}$ threshold ($\sim$ 1 GeV/c). A software level (HLT),
running on a massive computer farm, performs an online event reconstruction,
reducing the event rate from 1 MHz to 2–3 kHz through inclusive and exclusive
selections.
The experiment was designed to work at a luminosity of $2\times 10^{32}$
cm-2s-1, 50 times lower than the LHC design luminosity, in order to minimize
the pile–up.
## 2 The 2010 run
During its spectacular startup in 2010, the LHC machine gradually increased
luminosity over 5 orders of magnitude, reaching the nominal LHCb value already
by the end of the run. Due to the limited number of colliding bunches in this
phase, LHCb acquired events with up to 2.6 visible collisions per crossing,
six times the design value. The flexibility of trigger and DAQ systems allowed
the experiment to cope with this high pile–up while keeping high efficiency
for key channels, thereby maintaining the statistics for physics studies. The
detector was fully working, with a negligible amount of dead channels in all
subsystems, and the overall DAQ efficiency was 90%. Eventually, LHCb recorded
an integrated luminosity of 37 pb-1, similar to ATLAS and CMS. Though being a
small sample compared to the nominal 2 fb-1 per year expected in the longer
term, this dataset corresponds already to about 5 billion $b$ events in the
detector acceptance, sufficient statistic for the first competitive physics
results, as well as the validation of detector performances. We will present
in this contribution a selection of results from $b$ decays, illustrating the
potentiality of the experiment, followed by an overview of QCD and EW studies
at LHCb.
## 3 Proper Time Reconstruction and the $B_{s}^{0}$
The VELO vertex resolution was measured on real data by comparing the
reconstructed vertexes from randomly chosen subgroups of tracks in the same
primary vertex. For a typical vertex of 25 tracks, a resolution of 16 (76)
$\mu$m was obtained for the transverse (longitudinal) direction. The impact
parameter resolution was measured to be 13 + (26 GeV/c)/$p_{T}$ $\mu$m,
corresponding to a typical proper time resolution for $B$ decays of 50 fs.
Preliminary results(LHCb preliminary results (2011), LHCb-CONF-2011-001) for
the lifetime measurements of $B^{0},B^{+},B^{0}_{s}$ and $\Lambda_{b}$, using
final states containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, are
shown on table 1. The excellent agreement with PDG averages demonstrates the
level of control on the lifetime scale.
Channel | event yield | lifetime (ps), with stat. and syst. error | PDG2010 (ps)
---|---|---|---
$B^{+}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ | 6741 $\pm$ 85 | 1.689 $\pm$ 0.022 $\pm$ 0.047 | 1.638 $\pm$ 0.011
$B^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ | 2668 $\pm$ 58 | 1.512 $\pm$ 0.032 $\pm$ 0.042 | 1.525 $\pm$ 0.009
$B^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K_{S}$ | 838 $\pm$ 31 | 1.558 $\pm$ 0.056 $\pm$ 0.022 | 1.525 $\pm$ 0.009
$B_{s}^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ | 570 $\pm$ 24 | 1.447 $\pm$ 0.064 $\pm$ 0.056 | 1.477 $\pm$ 0.046
$\Lambda_{b}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\Lambda$ | 187 $\pm$ 16 | 1.353 $\pm$ 0.108 $\pm$ 0.035 | 1.391${}^{+0.038}_{-0.037}$
Table 1: Event yields and lifetimes obtained from several $b$ hadron decays to
exclusive states containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
in the range 0.3 $<$ t $<$ 14 ps. Events are triggered using dimuon candidates
with invariant mass compatible with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ and minimum $p_{T}$ of 500 MeV/c, without any bias on lifetime. The
current PDG values for the lifetimes are shown for comparison. Figure 2:
Invariant mass (left) and proper time t (right) distributions for the
$B_{s}^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ candidates with
t¿0.3 ps. The signal (green dashed line), background (red dashed line) and
total (blue solid line) contributions obtained from a two-dimensional fit are
shown on the plots.
$B_{s}^{0}$ mesons, largely unexplored at the B factories, provide some of the
most promising channels for the emergence of new physics. From the “golden
mode” $B_{s}^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ the
CP–violating oscillation phase $\phi_{s}$, for which an accurate (error $\sim$
2 mrad) SM prediction exists, can be measured. Figure 2 shows the invariant
mass and lifetime distributions for the candidate events from lifetime
unbiased triggers. The combination of impact parameter, invariant mass
resolution (7 MeV/c2, with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mass constrained to the PDG value) and PID performances results in a very
clean event sample. As can be seen from table 1, despite the limited
statistics, the accuracy of the lifetime measurement in this mode is already
comparable with the Tevatron results, obtained from $\sim$ 100 times more
integrated luminosity.
Flavour oscillations of $B_{s}^{0}$ mesons were searched for in the more
abundant $D_{s}\pi$ and $D_{s}3\pi$ modes. Neural network based tagging
algorithm were developed and calibrated on real data using self–tagging modes.
The limited statistics of the control modes is presently the main source of
uncertainty and prevents the use of the powerful same side tagger for
$B_{s}^{0}$ with this dataset. However, a tagging power of $3.8\pm 2.1\%$ was
obtained, allowing $B_{s}^{0}$ oscillations to be observed, with a frequency
measured to be(LHCb preliminary results (2011), LHCb-CONF-2011-005):
$\Delta m_{s}=17.63\pm 0.11(stat)\pm 0.04(syst)~{}\textrm{ps}^{-1}$
as illustrated in figure 3. The measurement agrees well with the world’s best
published result to date from CDF ($17.77\pm 0.10\pm
0.07~{}\textrm{ps}^{-1}$).
Figure 3: Preliminary results of the $B_{s}^{0}$ oscillations analysis. On
the left: the flavour asymmetry for $B_{s}^{0}$ candidates as a function of
proper time modulo 2$\pi/\Delta m_{s}$. The fitted asymmetry is superimposed.
On the right: fitted oscillation amplitude as a function of $\Delta m_{s}$.
A time–dependent tagged analysis was performed for
$B_{s}^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$, leading to a
first loose bound on $\phi_{s}$ (LHCb preliminary results (2011), LHCb-
CONF-2011-006). We expect to reach an accuracy of 35 mrad, comparable to the
SM predicted value, with the first inverse fb of data expected in 2011.
Another key $B_{s}^{0}$ channel is the ultra-rare $\mu^{+}\mu^{-}$ decay, with
a predicted BR of $(3.2\pm 0.2)\cdot 10^{-9}$ in the SM, that could receive
significant contributions from new pseudo–scalar intermediate states. The
channel is also very clean experimentally, with no significant peaking
background. The analysis of the full 37 pb-1 data sample showed no candidate
events, leading to a 95% CL limit of Aaij et al. (2011a)
$BR(B_{s}^{0}\to\mu^{+}\mu^{-})<5.6\cdot 10^{-8}$
which is very close to the best limit from CDF ($4.3\cdot 10^{-8}$), obtained
from 3.7 fb-1. The sensitivity is expected to attain the SM level in LHCb with
about 2 fb-1.
The competitiveness of the experiment for $B_{s}^{0}$ physics is also
demonstrated by the first observation of several decay channels: $B_{s}^{0}\to
D_{s2}X\mu\nu$, $B_{s}^{0}\to K^{*0}\overline{K^{*0}}$, $B_{s}^{0}\to
D^{0}K^{*}$, $B_{s}^{0}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}$
Aaij et al. (2011b, c); LHCb preliminary results (2011). The latter, a CP
eigenstate, can provide an interesting contribution to the measurement of
$\phi_{s}$.
## 4 Particle ID at work
The most spectacular demonstration of the RICH’s performances is provided by
the analysis of charmless $B\to hh$ decays, where $\pi/K/p$ discrimination is
essential to disentangle the $B^{0}\to\pi^{+}\pi^{-},B^{0}\to
K^{+}\pi^{-},B_{s}^{0}\to K^{+}K^{-},B_{s}^{0}\to\pi^{+}K^{-},\Lambda_{b}\to
pK^{-}$ and $\Lambda_{b}\to p\pi^{-}$ modes. PID performances are calibrated
on real data using the abundant $D^{*}$ and $\Lambda$ decays. The $B\to hh$
yields are then fitted simultaneously taking into account the expected cross
feeds. The $B^{0}\to\pi^{+}\pi^{-}$ and $B_{s}^{0}\to K^{+}K^{-}$ modes will
allow for a measurement of the CKM angle $\gamma$ from loop diagrams. From the
2010 data sample we extract promising yields of 275 $\pm$ 24
$B^{0}\to\pi^{+}\pi^{-}$ and 333 $\pm$ 21 $B_{s}^{0}\to K^{+}K^{-}$
candidates, corresponding to about one fourth of what obtained by CDF from 1
fb-1.
Although more statistics is needed for a competitive $\gamma$ measurement,
with 2010 data we already reached the sensitivity to confirm the direct CP
violation in the $B^{0}\to K^{+}\pi^{-}$ mode from the time integrated CP
asymmetry, as shown in figure 4. The possible biases on the measured raw
asymmetry were constrained from real data: the detector asymmetry was found to
be $A_{D}=-0.004\pm 0.004$ from $D,D^{*}\to K\pi$ decays, while the production
asymmetry was estimated to be $A_{p}=0.009\pm 0.008$ from
$B^{\pm}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$. The results,
compared with world averages from HFAG, are (LHCb preliminary results (2011),
LHCb-CONF-2011-023):
mode | LHCb (preliminary) | HFAG average
---|---|---
$A_{CP}(B^{0}\to K^{+}\pi^{-})$ | $-0.077\pm 0.033\pm 0.007$ | $-0.098^{+0.012}_{-0.011}$
$A_{CP}(B_{s}^{0}\to\pi^{+}K^{-})$ | $0.15\pm 0.19\pm 0.02$ | $0.39\pm 0.17$
The 2.3$\sigma$ effect for $B^{0}$ is the first hint for CP violation measured
at LHC. Also in this case, the measurement for $B_{s}^{0}$ is already
competitive with the world average.
Figure 4: Invariant mass distributions for (left) $B^{0}\to K^{+}\pi^{-}$ and
(right) $\overline{B^{0}}\to K^{-}\pi^{+}$ candidates. The dots represent the
data, while the curves show the result of the unbinned ML analysis fit: total
(blue), signal (red), combinatorial background (gray) and cross–feed
components from the other $B\to hh$ modes.
Performances of the calorimeters were also excellent, with a resolution of 7.2
MeV/c2 for the $\pi^{0}$ mass from unconverted photons and a response
uniformity within 2%. Despite the harsh hadronic environment and the high
pile–up experienced in 2010, clean samples of exclusive radiative decays as
$B^{0}\to K^{*}\gamma$ and $\chi_{c}\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\gamma$ could be reconstructed, implying nice prospects for the physics
with the rare $b\to s\gamma$ modes.
## 5 LHCb as a general purpose forward detector
Particle production rates can be studied by LHCb in a rapidity range
complementary to the general purpose detectors. The excellent PID and
vertexing capabilities, combined with the low $p_{T}$ thresholds of the
trigger, allow to select a wide range of exclusive states with good
efficiency. LHCb can contribute in particular to heavy flavour spectroscopy,
studying of X(3872) and the other unexpected states recently observed, the
$B_{c}$ and the other double heavy flavour states. We summarize here some of
these studies, that are discussed in more details in the other LHCb
contributions to these Proceedings.
The production cross section of $b$ hadrons is of obvious importance for LHCb,
but also as a QCD testbench and as a crucial input for background
determination in many key channels at LHC, notably the search for the Higgs
boson. It was measured using the inclusive $b\to D^{0}X\mu^{-}\nu$ mode from
just the first 15 nb-1 of data Aaij et al. (2010). A measurement of similar
accuracy was possible with 5 pb-1 using events with a delayed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decaying to $\mu^{+}\mu^{-}$
Aaij et al. (2011d). The results, compared to some theoretical predictions,
are showed in fig. 5. Extrapolating to the full rapidity range, we get
$\displaystyle\sigma(p\overline{p}\to b\overline{b}X)|_{\textrm{s=7 TeV}}$
$\displaystyle=(284\pm 20\pm 49)~{}\mu b~{}~{}~{}~{}~{}~{}$
$\displaystyle(b\to D^{0}X\mu^{-}\nu)$ $\displaystyle=(288\pm~{}~{}4\pm
48)~{}\mu b~{}~{}~{}~{}~{}~{}$
$\displaystyle(b\to{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X),$
in good agreement with the production models tuned on the Tevatron results.
Figure 5: Results of $b\overline{b}$ cross section measurements from charm
events (left, as a function of pseudorapidity) and from delayed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ (right, as a function of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse momentum), compared
to theoretical predictions.
The first low–luminosity data allowed LHCb to allocate a relevant fraction of
the trigger bandwidth to charm physics. Production of all open charm states
was measured using minimum–bias trigger, resulting in valuable QCD tests
(notably from the $D^{+}/D^{+}_{s}$ production ratio) and reducing
uncertainties for backgrounds in many CPV measurements.
A rich quarkonium physics program has begun by measuring the production rates
of prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\psi(2S)$,
$\chi_{c}$, $\Upsilon$(1S). Figure 6 shows the results for bottomonium,
compared with the CMS values at lower $\eta$, nicely illustrating the
complementarity of LHC experiments. Data also cover production at very low
transverse momentum, down to less than 1 GeV/c, well below the typical
threshold of theoretical predictions!
Double ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production was also
observed, as well as $B_{c}$ production in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ channel (LHCb
preliminary results (2011), LHCb-CONF-2011-009 and LHCb-CONF-2011-017), paving
the way for the exploration of states from double heavy flavour production.
Figure 6: The cross section for $\Upsilon$(1S) production is shown as a
function of transverse momentum (left, compared with theory) and
pseudorapidity (right, compared to CMS in the central region).
## 6 Scrutinizing parton PDFs
The LHCb rapidity region allows the study of deep inelastic scattering in the
unique unexplored high–$Q^{2}$ regions at very low $x$. Drell–Yan production
of muon pairs can be observed for $Q^{2}>5$ GeV2, probing parton PDFs down to
$x\sim 10^{-6}$. Production of $W$ and $Z$ bosons, corresponding to $x\sim
10^{-4},Q^{2}=M_{W,Z}^{2}$, has been observed already with the first 17 pb-1
of data (LHCb preliminary results (2011), LHCb-CONF-2011-012).
The cross sections are predicted with accuracy varying with $\eta$ from 3 to
10%, the uncertainty on parton PDFs contributing to a large extent.
Constraints to the PDFs can thus be obtained, particularly from the charge
asymmetry of $W$ production, since many systematics cancel and its $\eta$
dependence is very sensitive on PDFs.
A clean sample of $Z\to\mu^{+}\mu^{-}$ decays can be easily selected using the
invariant mass constraint on muons having $p_{T}>$ 20 GeV/c, with almost no
background expected. With an estimated 69% efficiency, 833 candidate events
survive the selection (see fig. 7).
The signature for $W$ decays is momentum imbalance in the transverse plane,
with an isolated high–$p_{T}$ ($>$ 20 GeV/c) muon and little other activity in
the event to suppress background from QCD jets. The muon is also required to
be compatible with the primary vertex to suppress backgrounds from $b,c$
decays. For the resulting 7624 $W^{+}$ and 5732 $W^{-}$ candidates we estimate
a selection efficiency of 30% and a signal to background ratio of about 1.6.
The resulting cross sections are
$\displaystyle\sigma_{Z}({\scriptstyle 81<m_{Z}<101\rm{~{}GeV/c}^{2}})\times
BR(Z\to\mu^{+}\mu^{-},{\scriptstyle
2<\eta_{\mu}<4.5,p_{T\mu}>20\rm{~{}GeV/c}})=$ $\displaystyle 73\pm 4\pm
7\textrm{~{}pb}$ $\displaystyle\sigma_{W^{+}}\times
BR(W\to\mu\nu,{\scriptstyle 2<\eta_{\mu}<4.5,p_{T\mu}>20\rm{~{}GeV/c}})=$
$\displaystyle 1007\pm 48\pm 100\textrm{~{}pb}$
$\displaystyle\sigma_{W^{-}}\times BR(W\to\mu\nu,{\scriptstyle
2<\eta_{\mu}<4.5,p_{T\mu}>20\rm{~{}GeV/c}})=$ $\displaystyle~{}682\pm
40\pm~{}68\textrm{~{}pb}$
The systematic errors are dominated by the uncertainty on luminosity, a
component which cancels in the ratios. As shown in fig. 7, the error on the
$W$ charge asymmetry is already comparable to the uncertainty due to the PDFs.
With the much larger statistics expected for 2011, LHCb will start
contributing to the determination of the proton structure.
Figure 7: On the left, invariant mass distribution for the
$Z\to\mu^{+}\mu^{-}$ candidates. On the right, charge asymmetry of the $W$
production as a function of $\eta_{\mu}$, compared with the prediction of the
NLO MCFM generator. The shaded area shows the uncertainty due to the MSTW08
PDF set used in the model.
## 7 Conclusions
The first data sample acquired in 2010 provided a proof of principle of the
LHCb concept. With only 37 pb-1, all the necessary ingredients for the key $B$
physics measurements, namely proper time resolution, background suppression
and tagging capabilities, have been demonstrated. The first world class
results show that LHCb is already picking up the baton from the successful
Tevatron $B$ physics program, confirming how high–precision measurements in
the heavy flavour sector can be achieved at hadron colliders.
LHCb physics is also expanding well beyond the core program, with many
original results on production studies in the unique forward region covered by
the experiment, as documented by the other 8 LHCb contributions to this
conference.
According to the LHC schedule, we expect to collect our first fb-1 within the
2011 run. A wide range of unexplored flavour territories is opening out in
front of us.
## References
* Alves et al. (2008) A. Alves, et al., _JINST_ 3, S08005 (2008).
* LHCb preliminary results (2011) LHCb preliminary results (2011), reports submitted to conferences are available on ”http://cdsweb.cern.ch/collection/LHCb Conference Contributions”.
* Aaij et al. (2011a) R. Aaij, et al., _Phys.Lett._ B699, 330–340 (2011a), 1103.2465.
* Aaij et al. (2011b) R. Aaij, et al., _Phys.Lett._ B698, 14–20 (2011b), 1102.0348.
* Aaij et al. (2011c) R. Aaij, et al., _Phys.Lett._ B698, 115–122 (2011c), 1102.0206.
* Aaij et al. (2010) R. Aaij, et al., _Phys.Lett._ B694, 209–216 (2010), 1009.2731.
* Aaij et al. (2011d) R. Aaij, et al., _Eur.Phys.J._ C71, 1645 (2011d), 1103.0423.
|
arxiv-papers
| 2011-07-12T15:40:29 |
2024-09-04T02:49:20.444329
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Giacomo Graziani (on behalf of the LHCb collaboration)",
"submitter": "Giacomo Graziani",
"url": "https://arxiv.org/abs/1107.2328"
}
|
1107.2419
|
# Scaling and intermittency in incoherent $\alpha$–shear dynamo
Dhrubaditya Mitra1 and Axel Brandenburg1,2
1NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE-10691
Stockholm, Sweden
2Department of Astronomy, AlbaNova University Center, Stockholm University,
SE-10691 Stockholm, Sweden E-mail: dhruba.mitra@gmail.comE-mail:
brandenb@nordita.org
(, Revision: 1.144 )
###### Abstract
We consider mean-field dynamo models with fluctuating $\alpha$ effect, both
with and without large-scale shear. The $\alpha$ effect is chosen to be
Gaussian white noise with zero mean and a given covariance. In the presence of
shear, we show analytically that (in infinitely large domains) the mean-
squared magnetic field shows exponential growth. The growth rate of the
fastest growing mode is proportional to the shear rate. This result agrees
with earlier numerical results of Yousef et al. (2008) and the recent
analytical treatment by Heinemann et al. (2011) who use a method different
from ours. In the absence of shear, an incoherent $\alpha^{2}$ dynamo may also
be possible. We further show by explicit calculation of the growth rate of
third and fourth order moments of the magnetic field that the probability
density function of the mean magnetic field generated by this dynamo is non-
Gaussian.
###### keywords:
Dynamo – magnetic fields – MHD – turbulence
## 1 Introduction
The dynamo mechanism that generates large-scale magnetic fields in
astrophysical objects is a topic of active research. Almost all astrophysical
bodies, e.g., the Galaxy, or the Sun, show presence of large-scale shear
(differential rotation). It is now well established that this shear is an
essential ingredient in the dynamo mechanism. This view is also supported by
direct numerical simulations (DNS). In particular, DNS of convective flows
have been able to generate large-scale dynamos predominantly in the presence
of shear (Käpylä et al., 2008; Hughes & Proctor, 2009), while non-shearing
large-scale dynamos are only possible at very high rotation rates (Käpylä et
al., 2009). The other vital constituent of the large-scale dynamo mechanism is
helicity of the flow which is often described by the $\alpha$ effect. Indeed
most dynamos, including the early model by Parker (1955), are the result of an
$\alpha$ effect combined with shear. Of these two ingredients, shear is
typically constant over the time scale of generation of the dynamo. For the
case of the solar dynamo, shear or differential rotation are constrained by
helioseismology. By contrast, measuring the $\alpha$ effect is a non-trivial
exercise. Whenever it has been obtained from DNS studies, it was found to have
large fluctuations in space and time; see e.g., Brandenburg et al. (2008) for
the PDF of different components of the $\alpha$ tensor and Cattaneo & Hughes
(1996) for the fluctuating time series of the total electromotive force, which
is however different from the $\alpha$ effect. What effect do these
fluctuations have on the properties of the $\alpha$–shear dynamo? In
particular, can a fluctuating $\alpha$ effect about a zero mean drive a large-
scale dynamo in conjunction with shear? Recent DNS studies (Brandenburg,
2005a; Yousef et al., 2008; Brandenburg et al., 2008, hereafter referred to as
BRRK) suggest that the answer to this question is yes. Yousef et al. (2008)
have further provided compelling evidence that the growth rate of the large-
scale magnetic energy in such dynamos scales linearly with the shear rate, and
the wavenumber of the fastest growing mode scales as the square root of the
shear rate. It has been proposed that such a dynamo can also emerge by an
alternate mechanism (which has nothing to do with fluctuations of the $\alpha$
effect) involving the interaction between shear and mean current density. This
mechanism is called the shear–current effect (Rogachevskii & Kleeorin, 2003).
However recent DNS studies of BRRK as well as those of Brandenburg (2005b)
have not found evidence in support of it, and analytical works (Rädler &
Stepanov, 2006; Rüdiger & Kitchatinov, 2006; Sridhar & Subramanian, 2009), for
small magnetic Reynolds number, have doubted its existence. Furthermore, the
shear–current effect does not produce the observed scaling, namely, growth-
rate scales with the shear rate squared, and the wavenumber of the fastest
growing mode scales linearly with the shear rate; see Section 4.2 of BRRK.
It behooves us then to consider the interaction between fluctuating $\alpha$
effect and shear as a possible dynamo mechanism. Kraichnan (1976) was first to
propose that fluctuations of kinetic helicity can give rise to negative
turbulent diffusivities, thereby giving rise to a dynamo that is effective
only in small scales (Moffatt, 1978). However, more relevant in the present
context is the work of Vishniac & Brandenburg (1997), who have investigated
(using numerical tools) an one-dimensional model and (with analytical
techniques) a simple mean-field model in a one-mode truncation with shear and
fluctuating $\alpha$ effect. These models they referred to as incoherent
$\alpha$–$\Omega$ dynamos. By dividing the poloidal field component by the
toroidal one, they turned the mean-field dynamo equations with multiplicative
noise into one with additive noise of Langevin type. They drew an analogy
between the behaviour of the mean magnetic field and Brownian motion in the
sense that the mean field does not grow although the mean-square field grows
with a growth rate that scales with shear rate to the $2/3$ power.
Over the last decade, the problem has been studied using a variety of
different approaches (Sokolov, 1997; Silant’ev, 2000; Fedotov et al., 2006;
Proctor, 2007; Kleeorin & Rogachevskii, 2008; Sur & Subramanian, 2009). Models
with fluctuating $\alpha$ effect, which depend on both space and time, can be
divided into two categories, one in which $\alpha$ is inhomogeneous
(Silant’ev, 2000; Proctor, 2007; Kleeorin & Rogachevskii, 2008) and the other
in which it is a constant in space. (There are other technical differences in
the two approaches as well.) In this paper we shall confine ourselves to the
second class of models. For the first category, using a multiscale expansion,
Proctor (2007) found a quadratic dependence of growth rate on shear rate. This
result is contradicted by Kleeorin & Rogachevskii (2008) who found that the
mean field averaged over $\alpha$ does not show a dynamo and that the
fluctuating $\alpha$ effect adds to the diagonal components of the turbulent
magnetic diffusivity tensor. Heinemann et al. (2011) have recently proposed a
simple analytically tractable model, in which they consider the first and
second moments (calculated over the distribution of $\alpha$) of the
horizontally averaged mean field. The first moment shows the absence of dynamo
effect, but the second moment shows exponential growth with the same scaling
behaviour observed by Yousef et al. (2008). Motivated by their study, we solve
here a similar model using a technique known as Gaussian integration by parts
to obtain the same results. We further show that, even in the absence of
shear, it may be possible for incoherent $\alpha^{2}$ dynamos to operate. For
the model of Vishniac & Brandenburg (1997) we can also calculate the higher
(third and fourth) order moments of the magnetic field to demonstrate that the
probability distribution function (PDF) of the mean magnetic field generated
by an incoherent $\alpha$–shear dynamo is non-Gaussian.
## 2 Mean-field model
Our mean-field model is designed to describe the simulations of Yousef et al.
(2008) and BRRK. In particular, there is a large-scale velocity given by
${\overline{\bm{U}}}=Sx{\bm{e}}_{y}$. The turbulence is generated by an
isotropic external random force with zero net helicity. The mean fields are
defined by averaging over two coordinate directions $x$ and $y$. By assuming
shearing-periodic boundary conditions, as done in the simulations, this
averaging obeys the Reynolds rules. By virtue of the divergence-less property
of the mean magnetic field, v.i.z., $\partial_{j}\bar{B}_{j}=0$, and in the
absence of an imposed mean field, $\bar{B}_{z}$ can be set to zero. The
resultant mean-field equations then have the following form (BRRK)
$\partial_{t}\bar{B}_{x}=-\alpha_{yx}\partial_{z}\bar{B}_{x}-\alpha_{yy}\partial_{z}\bar{B}_{y}-\eta_{yx}\partial^{2}_{z}\bar{B}_{y}+\eta_{yy}\partial^{2}_{z}\bar{B}_{x},$
(1)
$\partial_{t}\bar{B}_{y}=S\bar{B}_{x}+\alpha_{xx}\partial_{z}\bar{B}_{x}+\alpha_{xy}\partial_{z}\bar{B}_{y}-\eta_{xy}\partial^{2}_{z}\bar{B}_{x}+\eta_{xx}\partial^{2}_{z}\bar{B}_{y}.\;$
(2)
Here, $\bar{B}_{x}$ and $\bar{B}_{y}$ are the $x$ and $y$ components of the
mean (averaged over $x$ and $y$ coordinate directions) magnetic field,
$\eta_{ij}$ are the four relevant components of the turbulent magnetic
diffusivity tensor and $\alpha_{ij}$ are the four relevant components of the
$\alpha$ tensor. The shear–current effect works via a non-zero $\eta_{yx}$,
provided its sign is the same as that of $S$. In this paper we do not consider
the possibility of the shear–current effect; hence we set $\eta_{yx}=0$. Here
we have ignored the molecular diffusivity, as we are interested in the limit
of very high magnetic Reynolds numbers.
The fluctuating $\alpha$ effect is modelled by choosing each component of the
$\alpha$ tensor to be an independent Gaussian random number with zero mean and
the following covariance (no summation over repeated indices is assumed):
$\langle\alpha_{ij}(t)\alpha_{kl}(t^{\prime})\rangle=D_{ij}\delta_{ik}\delta_{jl}\delta(t-t^{\prime}).$
(3)
We also assume that the $\alpha_{ij}$s are constants in space. Note that DNS
studies of BRRK have shown that the coefficients of turbulent diffusivity also
show fluctuations in time, but we have ignored that in this paper. To make our
notation fully transparent, let us clearly distinguish between two different
kinds of averaging we need to perform. The mean fields themselves are
constructed by Reynolds averaging, which in our case is horizontal averaging,
and is denoted by an overbar. As we are dealing with mean field models, the
quantities appearing in our equations have already been Reynolds averaged.
But, as the $\alpha_{ij}$ in our mean field equation are stochastic, we study
the properties of our model by writing down evolution equations for different
moments of the mean magnetic field averaged over the statistics of
$\alpha_{ij}$. This is denoted by the symbol $\langle\cdot\rangle$. To give an
example, the first moment (mean over statistics of $\alpha$) of the $x$
component of the mean magnetic field is denoted by $\langle\bar{B}_{x}\rangle$
and the second moment (mean square) is denoted by
$\langle\bar{B}_{x}^{2}\rangle$.
In numerical studies, such averaging has to be performed by averaging over
sufficiently many simulations with independent realisations of noise, as was
done by Sur & Subramanian (2009). In controlled experimental setup such
ensemble averaging is done by executing the experiment several times with
idential initial conditions. In nature we expect the noise to have a finite
correlation length. Averaging over different realisation of the noise can then
be done by averaging over spatial domains of size larger than the correlation
length of the noise. Note further that in the present case the components of
$\bar{\bm{B}}$ undergo a random walk and are hence non-stationary. Therefore
we cannot replace ensemble averages by time averages.
## 3 Results
### 3.1 First and second moments
It is convenient to study the possibility of a dynamo effect in Fourier space.
Under Fourier transform,
$\hat{B}_{x}=\int\bar{B}_{x}e^{-ikz}dk,\quad\hat{B}_{y}=\int\bar{B}_{y}e^{-ikz}dk.$
(4)
Equations (1) and (2) transform to,
$\partial_{t}\hat{B}_{x}=-\alpha_{yx}ik\hat{B}_{x}-\alpha_{yy}ik\hat{B}_{y}+\eta_{yx}k^{2}\hat{B}_{y}-\eta_{yy}k^{2}\hat{B}_{x},$
(5)
$\partial_{t}\hat{B}_{y}=S\hat{B}_{x}+\alpha_{xx}ik\bar{B}_{x}+\alpha_{xy}ik\hat{B}_{y}+\eta_{xy}k^{2}\hat{B}_{x}-\eta_{xx}k^{2}\hat{B}_{y}.\;$
(6)
Equations (5) and (6) are a set of coupled stochastic differential equations
(SDEs) with multiplicative noise.
We now want to write down an evolution equation for
$\bm{C}^{1}=(\langle\hat{B}_{x}\rangle,\langle\hat{B}_{y}\rangle)$, which is
the mean field averaged over the statistics of $\alpha_{ij}$. Note that
$\bm{C}^{1}$ is the first order moment of the probability distribution
function (PDF) of the magnetic field. As the $\alpha$ effect is taken to be
Gaussian and white-in-time it is possible to obtain closed equations of the
form
$\partial_{t}\bm{C}^{1}={\sf\bm{N}_{1}}\bm{C}^{1}$ (7)
with
${\sf\bm{N}_{1}}=\left[\begin{array}[]{cc}-k^{2}(\eta_{yy}+D_{yx})&k^{2}\eta_{yx}\\\
S+k^{2}\eta_{xy}&-k^{2}(\eta_{xx}+D_{xy})\end{array}\right].$ (8)
If there is no shear–current effect, i.e. $\eta_{yx}=0$, there is no dynamo.
The fluctuations of the $\alpha$ effect actually enhance the diagonal
components of the turbulent magnetic diffusivity. This result agrees with
Heinemann et al. (2011) who used a different model and found that the mean
magnetic field does not grow. Vishniac & Brandenburg (1997) also found the
same for their simplified zero-dimensional model. However, if one assumes that
different components of the $\alpha$ tensor are correlated, a different result
can be obtained; see Section 3.3.
We have used two different techniques to derive the matrix ${\sf\bm{N}_{1}}$
in (8). In Appendix A, following Brissaud & Frisch (1974), we have used a
perturbation expansion in powers of noise strength. This expansion works even
when $\alpha$ is not white in time. However, for white-in-time $\alpha$ it is
enough to retain only the leading order term. In Appendix B we have used
Gaussian integration by parts which works because of the white-in-time nature
of the $\alpha$ effect. This method can be easily applied to study higher
order moments of the PDF of the magnetic field too. Hence we shall use it
extensively in the rest of this paper.
Although the mean magnetic field does not grow, the mean-squared magnetic
field can still show growth. To study this we now write a set of equations for
the time evolution of the second moment (covariance) of the mean magnetic
field averaged over $\alpha_{ij}$. We emphasise that we are not considering
here the covariance of the actual magnetic field that would include also the
small-scale magnetic fluctuations, which are relevant to the small-scale
dynamo (Kazantsev, 1968). The covariance of the actual magnetic field was
later also considered by Hoyng (1987) in connection with $\alpha$–$\Omega$
dynamos. Following Heinemann et al. (2011), we define a covariance vector,
$\bm{C}^{2}\equiv(\langle{\hat{B}}_{x}{\hat{B}}^{\ast}_{x}\rangle,\langle{\hat{B}}_{y}{\hat{B}}^{\ast}_{y}\rangle,\langle{\hat{B}}_{x}{\hat{B}}^{\ast}_{y}\rangle+\langle{\hat{B}}_{x}^{\ast}{\hat{B}}_{y}\rangle).$
(9)
The evolution equation for the covariance vector is given by
$\partial_{t}\bm{C}^{2}={\sf\bm{N}_{2}}\bm{C}^{2},$ (10)
where ${\sf\bm{N}_{2}}$ is given by
$\displaystyle\left[\begin{array}[]{ccc}-2k^{2}\eta_{yy}&2k^{2}D_{yy}&k^{2}\eta_{yx}\\\
2k^{2}D_{xx}&-2k^{2}\eta_{xx}&S+k^{2}\eta_{xy}\\\
2(S+k^{2}\eta_{xy})&2k^{2}\eta_{yx}&-k^{2}(D_{yx}+D_{xy}+2\eta_{xx})\end{array}\right].$
(14)
The only non-trivial terms in the derivation of the above equation are the
terms which are products of components of $\alpha$ effect and two components
of the magnetic field. We evaluate them by using the same technique used to
obtain (7) and (8); see Appendix B for details.
The characteristic equation of the matrix ${\sf\bm{N}}_{2}$ is a third order
equation, the solutions of which gives the three solutions for the growth rate
$2\gamma$. For simplicity let us also choose
$\eta_{xx}=\eta_{yy}\equiv\eta_{\rm t}$. In other words, we take the turbulent
magnetic diffusivity tensor to be diagonal and isotropic. We further note that
$D_{yx}$ and $D_{xy}$ contribute only in enhancing the turbulent magnetic
diffusivity of $C_{3}$, so we can safely ignore them compared to $\eta_{xx}$.
With these simplifying assumptions the equation for the growth rate reduces to
$\xi^{3}-4k^{2}D_{yy}\left[S^{2}+k^{2}D_{xx}\xi\right]=0,$ (15)
where $\xi=2(k^{2}\eta_{\rm t}+\gamma)$. For large enough $S$ we can always
ignore the second term inside the parenthesis of (15). This gives the three
roots of $\gamma$ as
$\gamma=-k^{2}\eta_{\rm
t}+\left(\frac{1}{2}k^{2}D_{yy}S^{2}\right)^{1/3}(1,\omega,\omega^{2}),$ (16)
where $(1,\omega,\omega^{2})$ are the three cube roots of unity, of which
$\omega$ and $\omega^{2}$ have negative real parts. The same dispersion
relation is obtained by Heinemann et al. (2011). The wavenumber of the fastest
growing mode, $k^{\rm peak}$, is given by
$k^{\rm peak}=|S|^{1/2}\left(\frac{D_{yy}}{54\eta_{\rm t}^{3}}\right)^{1/4}.$
(17)
The growth rate of the fastest growing mode is given by
$\gamma=\frac{2^{1/3}}{6}\left(1-\frac{2^{1/6}}{\sqrt{3}}\right)\left(\frac{D_{yy}}{\eta_{\rm
t}}\right)^{1/2}|S|.$ (18)
This is the same scaling numerically obtained by Yousef et al. (2008).
### 3.2 Incoherent $\alpha^{2}$ dynamo
Let us now consider a different case where shear is zero. In that case, (15)
becomes a quadratic equation in $\xi$,
$\xi^{2}-4k^{4}D_{xx}D_{yy}=0,$ (19)
with solutions,
$\gamma=k^{2}\left(-\eta_{\rm t}\pm\sqrt{D_{xx}D_{yy}}\right).$ (20)
Hence, it may be possible for fluctuations of $\alpha$ to drive a large-scale
dynamo (in the mean-square sense) even in the absence of velocity shear.
To summarise there are two possible dynamo mechanisms in our dynamo model. In
both of them the magnetic field grows in the mean-square sense. The first one
is an incoherent $\alpha$–shear dynamo. For large enough shear this is the
fastest growing mode. However, this dynamo has no oscillating modes because
the modes for which $\gamma$ have a non-zero imaginary part have negative real
part. An incoherent $\alpha^{2}$ dynamo mechanism also exists in this model.
The condition for excitation of a fluctuating $\alpha$–shear dynamo is
$\frac{(k^{2}D_{yy}S^{2}/2)^{1/3}}{k^{2}\eta_{\rm t}}>1,$ (21)
and the condition for excitation of a fluctuating $\alpha^{2}$ dynamo is
$\frac{\sqrt{D_{xx}D_{yy}}}{\eta_{\rm t}}>1.$ (22)
The condition that a fluctuating $\alpha^{2}$ dynamo is preferred compared to
an $\alpha$–shear one is
$\frac{4k^{8}D^{3}_{xx}D_{yy}}{S^{4}}>1,$ (23)
or $\sqrt{2}k^{2}D_{xx}/|S|>1$ for $D_{xx}=D_{yy}$.
To compare with DNS we need to use some estimates of $\eta_{\rm t}$, $D_{xx}$
and $D_{yy}$. We use $\eta_{\rm t}=u_{\rm rms}/3k_{\rm f}$, as obtained by Sur
et al. (2008) without shear. A slightly larger value was found by BRRK in the
presence of shear. We further use $D_{xx}=D_{yy}=u_{\rm rms}^{2}/9$. For this
choice of parameters the incoherent $\alpha^{2}$ dynamo does not grow. Here,
$u_{\rm rms}$ is the mean-squared velocity and $k_{\rm f}$ corresponds to the
characteristic Fourier mode of the forcing if the turbulence has been
maintained by an external force, as done by Yousef et al. (2008) or BRRK. For
turbulence maintained by convection, $k_{\rm f}$ should be replaced by the
Fourier mode corresponding to the integral scale of the turbulence. Typically,
mean-field theory applies for modes with $k<k_{\rm f}$. Lengths are measured
in units of $1/k_{\rm f}$ and velocity is measured in the unit of $u_{\rm
rms}$. This makes $1/u_{\rm rms}k_{\rm f}$ the unit of time. The two
dispersion relations are plotted in Fig. 1 for different values of $S$.
Figure 1: Sketch of the dispersion relation, (15) for different values of
velocity shear $S$ (continuous lines with difference colours/grey shades from
bottom to top $S=0.5,1.,1.5,2.,2.5$), and (20) (broken line) at the very
bottom. Velocity shear and $\gamma$ have dimensions of inverse time and are
measured in the units of $u_{\rm rms}k_{\rm f}$.
### 3.3 Effects of mutual correlations between components of the $\alpha$
tensor
So far we have assumed that only the self-correlations of the components of
the $\alpha$ tensor are non-zero and the mutual correlations zero. Let us now
generalise (3) to
$\langle\alpha_{ij}(t)\alpha_{kl}(t^{\prime})\rangle={\mathcal{D}}^{ij}_{kl}\delta(t-t^{\prime}).$
(24)
Obviously, ${\mathcal{D}}^{ij}_{kl}={\mathcal{D}}^{kl}_{ij}$ and
${\mathcal{D}}^{ij}_{ij}=D_{ij}$. It is again possible to write a closed
equation for the first moment of the magnetic field in the form
$\partial_{t}\bm{C}^{1}={\sf\bm{N}_{1}}\bm{C}^{1}$ with
${\sf\bm{N}_{1}}=\left[\begin{array}[]{cc}-k^{2}(\eta_{\rm
t}+{\tilde{\eta}}_{yy})&k^{2}{\tilde{\eta}}_{yx}\\\
S+k^{2}{\tilde{\eta}}_{xy}&-k^{2}(\eta_{\rm
t}+{\tilde{\eta}}_{xx})\end{array}\right]\\!,$ (25)
where
$\displaystyle{\tilde{\eta}}_{yy}={\mathcal{D}}^{yx}_{yx}-{\mathcal{D}}^{yy}_{xx},$
$\displaystyle{\tilde{\eta}}_{yx}=-{\mathcal{D}}^{yx}_{yy}+{\mathcal{D}}^{yy}_{xy},$
(26)
$\displaystyle{\tilde{\eta}}_{xy}=-{\mathcal{D}}^{xy}_{xx}+{\mathcal{D}}^{xx}_{yx},$
$\displaystyle{\tilde{\eta}}_{xx}={\mathcal{D}}^{xy}_{xy}-{\mathcal{D}}^{xx}_{yy}.$
(27)
This is a generalization of (8). Here, for simplicity, we have assumed
$\eta_{xx}=\eta_{yy}=\eta_{\rm t}$ and $\eta_{yx}=\eta_{xy}=0$; in other words
the conventional shear–current effect is taken to be zero.
Note that, unlike the self-correlation terms, i.e. ${\mathcal{D}}^{ij}_{ij}$,
which must be positive, the mutual correlation terms (e.g.,
${\mathcal{D}}^{yy}_{xx}$, or ${\mathcal{D}}^{yx}_{yy}$) can have either sign.
Hence, it is a-priori not clear from (25) whether a dynamo is possible or not.
However, we note two interesting possibilities below. First, the mutual
correlations of the fluctuating $\alpha$ now contribute to off-diagonal
components of the turbulent magnetic diffusivity tensor. A dynamo is possible
if
$\frac{S{\tilde{\eta}}_{yx}}{k^{2}\eta_{\rm t}}>1.$ (28)
Such a dynamo might look deceptively similar to the shear–current dynamo, but
is actually not so because in the regular shear–current effect $\eta_{yx}$
emerges due to the presence of shear and hence must be proportional to $S$ for
small $S$. Hence for a regular shear–current dynamo we would have $\gamma\sim
S^{2}$ and $k^{\rm peak}\sim S$; see BRRK. But here ${\tilde{\eta}}_{xy}$
emerges due to fluctuations of $\alpha$ and may be independent of $S$, at
least for small $S$. This would imply that $\gamma\sim S$ and $k^{\rm
peak}\sim\sqrt{S}$ but this time even for the first moment of the magnetic
field. The growth rate of such a dynamo may however be quite small as it is
proportional to ${\tilde{\eta}}_{yx}$ which is the difference between two
terms each of which are correlations between different components of the
$\alpha$ tensor. Secondly, if
${\mathcal{D}}^{yy}_{xx}>{\mathcal{D}}^{yx}_{yx}$ the fluctuating $\alpha$
effect gives negative contributions to even the diagonal components of
turbulent diffusivity, which is reminiscent of the result of Kraichnan (1976).
### 3.4 Scaling in a simpler zero-dimensional model
The essential physics of (1) and (2) can be captured by an even simpler mean-
field model in a one-mode truncation, but with fluctuating $\alpha$ effect,
introduced by Vishniac & Brandenburg (1997). Their model, rewritten in our
notation and setting all $k$ factors to unity is
$\displaystyle\partial_{t}\bar{B}_{x}$ $\displaystyle=$
$\displaystyle\alpha\bar{B}_{y}-\eta_{\rm t}\bar{B}_{x},$ (29)
$\displaystyle\partial_{t}\bar{B}_{y}$ $\displaystyle=$
$\displaystyle-S\bar{B}_{x}-\eta_{\rm t}\bar{B}_{y}.$ (30)
This model can be analysed in exactly similar ways. By construction, this
model does not have a fluctuating $\alpha^{2}$ effect, and $\alpha$ is a
Gaussian random variable with zero mean and covariance
$\langle\alpha(t)\alpha(t^{\prime})\rangle=D\delta(t-t^{\prime}).$ (31)
For this model we adopt a more general framework and define the growth-rate of
the $p$-th order moment of the magnetic field (the first order is the mean and
the second order is the covariance ) to be $p\gamma_{p}$. Explicit
calculations, shown in Appendix C, give
$\gamma_{1}=-\eta_{\rm t},\quad\gamma_{2}=-\eta_{\rm
t}+\left(\frac{4DS^{2}}{8}\right)^{1/3}\sim S^{2/3}.$ (32)
We show in Appendix C that this two-third scaling with shear rate in this
zero-dimensional model is equivalent to $\gamma\sim S$ scaling for (1) and
(2).
### 3.5 Possibility of intermittency
We note that (5) and (6) can be considered as coupled stochastic differential
equations of the Langevin type but with multiplicative noise. We have taken
the probability distribution function (PDF) of the noise to be Gaussian. But,
by virtue of multiplicative noise, the PDF of the magnetic field may be non-
Gaussian. We have already calculated the first and second moments of this PDF.
To probe non–Gaussianity we need to calculate the higher order moments. For
(5) and (6) this is a formidable problem. But it is far simpler for the model
of Vishniac & Brandenburg (1997). Sokolov (1997) has already argued that the
statistics of the magnetic field in the model of Vishniac & Brandenburg (1997)
is intermittent; see also Sur & Subramanian (2009). In Appendix C we show that
the growth rate for the third and fourth order moments are given by
$\gamma_{3}=-\eta_{\rm
t}+\left(\frac{18DS^{2}}{27}\right)^{1/3}\\!\\!,\;\;\gamma_{4}=-\eta_{\rm
t}+\left(\frac{84DS^{2}}{64}\right)^{1/3}\\!\\!.\;\;$ (33)
Clearly, $\gamma_{p}$ has the same scaling dependence on $D$ and $S$,
independent of $p$, but nevertheless they are different, i.e., the PDF is non-
Gaussian. This non-Gaussianity is best described by plotting
$\zeta_{p}=\frac{\gamma_{p}+\eta_{\rm t}}{(DS^{2})^{1/3}}$ (34)
versus $p$ in Fig. 2.
Let us now conjecture that as $p\to\infty$, $\zeta_{p}$ remains finite.
Remembering that $\zeta_{1}=0$, the general form would then be
$\zeta_{p}=\left(\frac{(p-1)(a_{0}+a_{1}p+a_{2}p^{2})}{p^{3}}\right)^{1/3}.$
(35)
Substituting the form back in (32) and (33) we find $a_{0}=36$, $a_{1}=-30$,
and $a_{2}=7$. This formula is also plotted in Fig. 2.
Figure 2: $\zeta_{p}$ versus $p$ as obtained from (34). If the magnetic field
had obeyed Gaussian statistics, $\zeta_{p}$ versus $p$ would have been
constant.
## 4 Conclusions
In this paper we have analytically solved a mean-field dynamo model with
fluctuating $\alpha$ effect to find self-excited solutions. We have studied
the growth rate of different moments (calculated over the statistics of
$\alpha$) of the magnetic field. There are three crucial aspects in which our
results, the DNS of Yousef et al. (2008), and the analytical results by
Heinemann et al. (2011) agree: (a) There is no dynamo for the first moment of
the magnetic field, (b) the second moment (mean-square) of the magnetic field
shows dynamo action, and (c) the fastest growing mode has a growth rate
$\gamma\sim S$ at Fourier mode $k^{\rm peak}\sim\sqrt{S}$. We have further
shown that these aspects of our results can even be reproduced by a simpler
zero-dimensional mean field model due to Vishniac & Brandenburg (1997). For
this simpler model we have also calculated the growth rate for third and
fourth order moments and we have explicitly demonstrated the non-Gaussian
nature of the PDF of the magnetic field. Given that the incoherent
$\alpha$–shear dynamo (often with an additional coherent part) is the most
common dynamo mechanism our results provide a qualitative reasoning of why
large-scale magnetic fields in the universe may be intermittent. However note
also that we have merely shown that the growth rates of the different moments
of the magnetic field are different. The eventual nature of the PDF of the
magnetic field will also be influenced by the saturation of this dynamo which
is outside the realm of this paper. We have also shown that it is possible to
find growth of the first moment (mean) of the magnetic field if mutual
correlations between different components of the $\alpha$ tensor are assumed
to have a certain form (Section 3.3). It will thus be important to check such
assumptions from future DNS.
As our paper has been inspired by Heinemann et al. (2011) it is appropriate
that we compare and contrast our model and techniques with theirs. Their model
consists of the equations of magnetohydrodynamics (MHD) with an external
Gaussian, white-in-time force (in the evolution equation for velocity) with
the additional assumption that the non-linear term in the velocity equation is
omitted. The model thus applies in the limit of Reynolds number $\mbox{Re}\ll
1$. They perform averaging over $xy$-coordinates to obtain mean field
equations with an $\alpha$ effect which depends on the helicity averaged over
coordinate directions. Our mean field model is derived by first averaging over
coordinate directions (standard Reynolds averaging) with the additional
assumptions on the statistics (Gaussian, white-in-time) of $\alpha$. Our
results are thus not limited by the smallness of the Reynolds number, although
all the usual limitations of mean-field theory apply. The assumption of the
Gaussian nature of $\alpha$ is well supported by numerical evidence; see Fig.
10 of BRRK. Heinemann et al. (2011) have further used a quasi-two-dimensional
velocity field, but this we feel is not an important limitation. They average
the first and second moment of the magnetic field over the realisations of
force by using cumulant expansion in powers of the Kubo number. As they
truncate the expansion at the lowest order in Kubo number it applies to the
case of small Kubo number. The most restrictive assumption in our model is the
assumption of the white-in-time nature of the $\alpha$ effect. This assumption
however allows us to obtain closed equations for all the moments of the
magnetic field. The results of Heinemann et al. (2011) is not limited by this
assumption. It is interesting to note that, even under the assumption of the
white-in-time nature of the $\alpha$ effect, we obtain the same scaling
behaviour as Heinemann et al. (2011) and the DNS studies of Yousef et al.
(2008).
Here, let us also mention that Kolokolov, Lebedev and Sizov (2011) have
recently applied similar techniques to study small-scale kinematic dynamos in
a smooth delta-correlated velocity field in the presence of shear to find
$\gamma_{n}=\frac{3}{2^{5/3}}n^{4/3}D^{1/3}S^{2/3}\sim\lambda n^{4/3},$ (36)
where $\lambda$ is the expression for the largest Lyapunov exponent describing
the divergence of two initially close fluid particles, This Lyapunov exponent
was earlier obtained for such flows by Turitsyn (2007). Interestingly, this is
exactly the same scaling with shear as in Vishniac & Brandenburg (1997). In
the absence of shear the small-scale dynamo can still operate (Chertkov,
Kolokolov and Vergassola, 1997) with $\gamma_{n}\sim n^{2}$.
Proctor (2007) have also considered a model similar to ours, although somewhat
simpler and more relevant to the solar dynamo, using multiscale expansions.
After averaging over the fluctuating $\alpha$ effect, he still finds an
effective $\alpha$ effect from which he obtains a dynamo which grows in the
mean (as opposed to mean–square in our case). His results give the scaling,
$k^{\rm peak}\sim S$ and $\gamma\sim S^{2}$ which disagree with the DNS
results of Yousef et al. (2008).
## Acknowledgements
We thank Tobi Heinemann, Matthias Rheinhardt, and Alex Schekochihin for useful
discussions on the incoherent $\alpha$–shear dynamo. Financial support from
the European Research Council under the AstroDyn Research Project 227952 is
gratefully acknowledged.
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* Zinn-Justin (1999) Zinn-Justin J., 1999, Quantum Field Theory and Critical Phenomenon. Oxford University Press, Oxford
## Appendix A Derivation of equations (7) and (8)
It is possible to derive (8) via a method described by Brissaud & Frisch
(1974). This method is superior to the one described in Appendix B in the
sense that this can be applied even when $\alpha$ is not necessarily white in
time, but has (small) non-zero correlation time. On the other hand it is more
cumbersome to apply this method to calculate the higher order moments of the
PDF of the magnetic field. For the sake of completeness we reproduce below the
calculations of Brissaud & Frisch (1974) as applied to our problem.
Let us write symbolically the evolution equations for the mean-field in
Fourier space in the following way,
$\partial_{t}{\mathcal{B}}=\left[{\sf\bm{M}^{\rm
d}}+{\sf\bm{M}}\right]{\mathcal{B}}.$ (37)
Here, ${\mathcal{B}}=(\hat{B}_{x},\hat{B}_{y})$ is a column vector,
${\sf\bm{M}^{\rm d}}$ is the deterministic part of the evolution equations
(i.e., the part that depends on $\eta_{ij}$),
${\sf\bm{M}^{\rm d}}=\left[\begin{array}[]{cc}-k^{2}\eta_{yy}&0\\\
S&-k^{2}\eta_{yy}\end{array}\right],$ (38)
which is also independent of time, and ${\sf\bm{M}}$ is the random part (i.e.,
the part that depends on $\alpha_{ij}$), with
${\sf\bm{M}}=ik\left[\begin{array}[]{cc}-\alpha_{yx}&-\alpha_{yy}\\\
\alpha_{xx}&\alpha_{xy}\end{array}\right]\equiv ik{\sf\bm{A}}.$ (39)
To begin, we do not assume that $\alpha$ is white-in-time but that it has
finite correlation time $T_{\rm corr}$. Later we shall take the limit of
$T_{\rm corr}\to 0$ in a specific way to reach the white-noise-limit. This is
equivalent to the regularization in Appendix B
In this section, for simplicity, let us choose our units such that at $t=0$,
${\mathcal{B}}=(1,1)$. In that case the solution to (37) can be easily recast
in the integral form
${\mathcal{B}}(t)=e^{{\sf\bm{M}^{\rm
d}}t}+\int_{0}^{t}dt^{\prime}e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime}){\mathcal{B}}(t^{\prime}).$ (40)
Note that we have $\langle{\sf\bm{M}}\rangle=0$ because we have assumed all
the components of $\alpha$ to have zero mean. Iterating this equation, we
obtain
$\displaystyle\\!\\!\\!{\mathcal{B}}(t)=e^{{\sf\bm{M}^{\rm
d}}t}+\int_{0}^{t}dt^{\prime}e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime})e^{{\sf\bm{M}^{\rm d}}t^{\prime}}$
(41)
$\displaystyle\\!\\!\\!+\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime})e^{{\sf\bm{M}^{\rm
d}}(t^{\prime}-t^{\prime\prime})}{\sf\bm{M}}(t^{\prime\prime}){\mathcal{B}}(t^{\prime\prime})$
$+\mbox{higher order terms}$. Let us also assume that the strength of the
fluctuations of $\alpha$ are finite and bounded by $\sigma$. Equation (41) is
then an expansion in powers of $\sigma t$. To obtain closed equations for the
first moment of the magnetic field, we average (41) over the statistics of
$\alpha$ and then take the derivative with respect to $t$. Remembering our
earlier notation ${\bm{C}}^{1}\equiv\langle{\mathcal{B}}\rangle$, we obtain
$\partial_{t}{\bm{C}}^{1}(t)={\sf\bm{M}^{\rm
d}}{\bm{C}}^{1}(t)+\int_{0}^{t}dt^{\prime}\langle{\sf\bm{M}}(t)e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime}){\mathcal{B}}(t^{\prime})\rangle.$
(42)
This obviously is not yet a closed equation. To obtain a closure, note that
for
$\sigma(t^{\prime}-s)\ll 1,$ (43)
from (40), we have
${\mathcal{B}}(t^{\prime})\approx e^{{\sf\bm{M}^{\rm
d}}(t^{\prime}-s)}{\mathcal{B}}(s)+O(\sigma(t^{\prime}-s)).$ (44)
Substituting (44) in the integrand of the double integral in (42) we obtain
the factorization,
$\displaystyle\langle{\sf\bm{M}}(t)e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime})e^{{\sf\bm{M}^{\rm
d}}(t^{\prime}-s)}{\mathcal{B}}(s)\rangle\approx$
$\displaystyle\langle{\sf\bm{M}}(t)e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime})\rangle\langle e^{{\sf\bm{M}^{\rm
d}}(t^{\prime}-s)}{\mathcal{B}}(s)\rangle,$ (45)
if we assume that
$t^{\prime}-s\gg T_{\rm corr}.$ (46)
Equations (43) and (46) can both hold only for small Kubo number,
$K=\sigma T_{\rm corr}\ll 1.$ (47)
Substituting (45) in (42) and using again (44) we obtain
$\partial_{t}{\bm{C}}^{1}(t)={\sf\bm{M}^{\rm
d}}{\bm{C}}^{1}(t)+\int_{0}^{t}dt^{\prime}\langle{\sf\bm{M}}(t)e^{{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\sf\bm{M}}(t^{\prime})\rangle{\bm{C}}^{1}(t^{\prime}),$
(48)
Following Brissaud & Frisch (1974), we shall call this equation the Bourret
equation. For small Kubo number we can invert (44) to have
${\mathcal{B}}(t^{\prime})\approx e^{-{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\mathcal{B}}(t).$ (49)
Averaging (49) over the noise we obtain
${\bm{C}}^{1}(t^{\prime})\approx e^{-{\sf\bm{M}^{\rm
d}}(t-t^{\prime})}{\bm{C}}^{1}(t).$ (50)
Substitute this back into (48), noting in addition that for short-correlated
$\alpha$ and $t\gg T_{\rm corr}$, we can replace $\tau\equiv t-t^{\prime}$ in
(48) and extend the integral from zero to infinity to obtain
$\partial_{t}{\bm{C}}^{1}={\sf\bm{M}^{\rm
d}}{\bm{C}}^{1}+\int_{0}^{\infty}d\tau\langle{\sf\bm{M}}(\tau)e^{{\sf\bm{M}^{\rm
d}}\tau}{\sf\bm{M}}(0)e^{-{\sf\bm{M}^{\rm d}}\tau}\rangle{\bm{C}}^{1},$ (51)
where we have omitted the $t$ argument on all ${\bm{C}}^{1}(t)$ for brevity.
To get this result, remember that the integral above gives negligible
contribution for $\tau\gg T_{\rm corr}$ and the matrices ${\sf\bm{M}^{\rm d}}$
and ${\sf\bm{M}}(\tau)$ do not necessarily commute. To go to the white-in-time
limit we need to take the limit $\sigma\to\infty$, $T_{\rm corr}\to 0$ in such
a way that the product $\sigma^{2}T_{\rm corr}$ remains finite. In this limit
the Kubo number goes to zero and the various approximations made above become
exact. In this limit the integral in (51) reduces to
$\displaystyle\partial_{t}{\bm{C}}^{1}(t)$ $\displaystyle=$
$\displaystyle{\sf\bm{M}^{\rm
d}}{\bm{C}}^{1}(t)+\langle{\sf\bm{M}}{\sf\bm{M}}\rangle{\bm{C}}^{1}(t)$ (52)
$\displaystyle=$ $\displaystyle\left[{\sf\bm{M}^{\rm
d}}-k^{2}\langle{\sf\bm{A}}{\sf\bm{A}}\rangle\right]{\bm{C}}^{1}(t).$
The correlator on the right hand side of (52) can be obtained by using (39)
together with (3). This reproduces (8). If instead of (3), (24) is used, (25)
can be obtained.
Finally, note that for an $\alpha$ effect which has finite-time-correlations
(instead of white-in-time) higher order terms in the expansion in (41) are
needed. In such cases it may not even be possible to obtain closed equations
like (7).
## Appendix B Averaging over Gaussian noise
We explain here another technique used to derive (7) and (8). Let us begin by
considering Gaussian vector-valued noise $\nu_{j}(t)$ (not necessarily white-
in-time) and an arbitrary functional of that, $F(\bm{\nu})$. Then,
$\langle F(\bm{\nu})\nu_{j}(t)\rangle=\int
dt^{\prime}\langle\nu_{j}(t)\nu_{k}(t^{\prime})\rangle\left\langle\frac{\delta
F}{\delta\nu_{k}(t^{\prime})}\right\rangle,$ (53)
where the average factorises by virtue of the Gaussian property of the noise.
Here the operator $\delta(\cdot)/\delta\nu_{k}$ is the functional derivative
with respect to $\bm{\nu}$. This useful identity often goes by the name
Gaussian integration by parts; see, e.g., Zinn-Justin (1999), Section 4.2 for
a proof; see also e.g., Frisch (1996), Frisch & Wirth (1997), or Mitra &
Pandit (2004), where this method has been used to derive closed moment
equations for the Kraichnan model of passive scalar advection (Kraichnan,
1968).
To obtain an evolution equation for
$\bm{C}^{1}=(\langle\hat{B}_{x}\rangle,\langle\hat{B}_{y}\rangle)$ we average
each term of (5) and (6) over the statistics of $\alpha_{ij}$. Terms which are
product of components of $\alpha_{ij}$ and $\hat{B}_{x}$ or $\hat{B}_{y}$ can
be evaluated by using the identity in (53). In particular,
$\displaystyle\langle\alpha_{yx}\hat{B}_{x}\rangle$ $\displaystyle=$
$\displaystyle\int
dt^{\prime}\langle\alpha_{yx}(t)\alpha_{kl}(t^{\prime})\rangle\left\langle\frac{\delta\hat{B}_{x}(t)}{\delta\alpha_{kl}(t^{\prime})}\right\rangle$
(54) $\displaystyle=$ $\displaystyle
D_{yx}\frac{\delta\hat{B}_{x}(t)}{\delta\alpha_{kl}(t)}.$
Here we have considered the magnetic field to be a functional of
$\alpha_{ij}$. Substituting the covariance of $\alpha_{ij}$ from (3),
integrating the $\delta$ function over time and contracting over the Kronecker
deltas, we obtain the last equality in (54).
The functional derivatives of the components of the magnetic field with
respect to $\alpha_{ij}$ can be obtained by first formally integrating (5) and
(6) to obtain $\hat{B}_{x}(t)$ and $\hat{B}_{y}(t)$, respectively, and then
calculating their functional derivatives with respect to $\alpha_{ij}$. We
actually need the functional derivative
$\delta(\hat{B}_{x}(t))/\delta\alpha_{kl}(t^{\prime})$ and then take the limit
$t\to t^{\prime}$. This is a non-trivial step due to the singular nature of
the correlation function of
$\langle\alpha_{kl}(t)\alpha_{kl}(t^{\prime})\rangle$ as $t\to t^{\prime}$. To
get around the difficulty it is possible to replace the Dirac delta function
in (3) with a regularised even function and then take limits. We refer the
reader to Zinn-Justin (1999), Section 4.2, for a detailed discussion. This
regularization is equivalent to using the Stratanovich prescription for the
set of coupled SDEs (5) and (6); see, e.g., Gardiner (1994). For reference,
all the non-zero functional derivatives needed are given below:
$\displaystyle\frac{\delta\hat{B}_{x}(t)}{\delta\alpha_{yx}(t)}=-ik\hat{B}_{x},$
$\displaystyle\frac{\delta\hat{B}_{x}(t)}{\delta\alpha_{yy}(t)}=-ik\hat{B}_{y},$
$\displaystyle\frac{\delta\hat{B}_{y}(t)}{\delta\alpha_{xx}(t)}=ik\hat{B}_{x},$
$\displaystyle\frac{\delta\hat{B}_{y}(t)}{\delta\alpha_{xy}(t)}=ik\hat{B}_{y}.$
(55)
In particular, since there is no $\alpha_{yy}$ term in (6), we have
$\delta(\hat{B}_{y}(t))/\delta\alpha_{yy}(t)=0$, so
$\left\langle\alpha_{yy}\hat{B}_{y}\right\rangle=D_{yy}\left\langle\frac{\delta\hat{B}_{y}}{\delta\alpha_{yy}}\right\rangle=0.$
Putting everything together we can now average (5) over the statistics of
$\alpha$ to obtain,
$\displaystyle\partial_{t}\langle\hat{B}_{x}\rangle$ $\displaystyle=$
$\displaystyle-\langle ik\alpha_{yx}\hat{B}_{x}\rangle-\langle
ik\alpha_{yy}\hat{B}_{y}\rangle$ (56)
$\displaystyle+\eta_{yx}k^{2}\langle\hat{B}_{y}\rangle-\eta_{yy}k^{2}\langle\hat{B}_{x}\rangle$
$\displaystyle=$ $\displaystyle-ik(-ik)D_{yx}\langle\hat{B}_{x}\rangle$
$\displaystyle+\eta_{yx}k^{2}\langle\hat{B}_{y}\rangle-\eta_{yy}k^{2}\langle\hat{B}_{x}\rangle$
$\displaystyle=$
$\displaystyle-(\eta_{yy}+D_{yx})k^{2}\langle\hat{B}_{x}\rangle+\eta_{yx}k^{2}\langle\hat{B}_{y}\rangle.$
This gives us the first row of the matrix ${\sf\bm{N}_{1}}$ in (8). Applying
the same technique to (6), the second row can be obtained.
The same technique can be applied to obtain the matrix ${\sf\bm{N}_{2}}$. Here
we end up with evaluating terms of the general form
$\langle\alpha_{ij}\hat{B}_{k}\hat{B}^{\ast}_{l}\rangle=D_{ij}\left\langle\frac{\delta\hat{B}_{k}}{\delta\alpha_{ij}}\hat{B}^{\ast}_{l}+\hat{B}_{k}\frac{\delta\hat{B}^{\ast}_{l}}{\delta\alpha_{ij}}\right\rangle.$
(57)
The only non-zero functional derivatives are given in (55). We show the
calculation explicitly for the two following examples:
$\displaystyle\langle\alpha_{yx}{\hat{B}}_{x}{\hat{B}}^{\ast}_{x}\rangle$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
D_{yx}\left\langle\frac{\delta\hat{B}_{x}}{\delta\alpha_{yx}}\hat{B}^{\ast}_{x}+\frac{\delta\hat{B}^{\ast}_{x}}{\delta\alpha_{yx}}\hat{B}_{x}\right\rangle$
(58) $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
D_{yx}\left[-ik\langle{\hat{B}}_{x}{\hat{B}}^{\ast}_{x}\rangle+ik\langle{\hat{B}}_{x}{\hat{B}}^{\ast}_{x}\rangle\right]=0,$
and
$\displaystyle\langle\alpha_{yy}{\hat{B}}_{x}^{\ast}{\hat{B}}_{y}\rangle$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle
D_{yy}\left\langle\frac{\delta\hat{B}_{y}}{\delta\alpha_{yy}}\hat{B}^{\ast}_{x}+\frac{\delta\hat{B}^{\ast}_{x}}{\delta\alpha_{yy}}\hat{B}_{y}\right\rangle$
(59) $\displaystyle\\!\\!=\\!\\!$ $\displaystyle
ikD_{yy}\langle{\hat{B}}_{y}{\hat{B}}^{\ast}_{y}\rangle.$
Note further that, instead of using the functional calculus above, the same
evolution equations for ${\bm{C}}^{1}$ can be obtained by using the technique
due to Brissaud & Frisch (1974). This is demonstrated in Appendix A.
## Appendix C A zero-dimensional mean-field model with fluctuating $\alpha$
A simpler mean-field model in a one-mode truncation, but with fluctuating
$\alpha$ effect, was introduced by Vishniac & Brandenburg (1997); see (29) and
(30). For this model we define the following moments of successive orders,
$\displaystyle\bm{C}^{1}$ $\displaystyle\equiv$
$\displaystyle(\langle\bar{B}_{x}\rangle,\langle\bar{B}_{y}\rangle),$ (60)
$\displaystyle\bm{C}^{2}$ $\displaystyle\equiv$
$\displaystyle(\langle\bar{B}_{x}^{2}\rangle,\langle\bar{B}_{y}^{2}\rangle,\langle\bar{B}_{x}\bar{B}_{y}\rangle),$
(61) $\displaystyle\bm{C}^{3}$ $\displaystyle\equiv$
$\displaystyle(\langle\bar{B}_{x}^{3}\rangle,\langle\bar{B}_{x}^{2}\bar{B}_{y}\rangle,\langle\bar{B}_{x}\bar{B}_{y}^{2}\rangle,\langle\bar{B}_{y}^{3}\rangle),$
(62) $\displaystyle\bm{C}^{4}$ $\displaystyle\equiv$
$\displaystyle(\langle\bar{B}_{x}^{4}\rangle,\langle\bar{B}_{x}^{3}\bar{B}_{y}\rangle,\langle\bar{B}_{x}^{2}\bar{B}_{y}^{2}\rangle,\langle\bar{B}_{x}\bar{B}_{y}^{3}\rangle,\langle\bar{B}_{y}^{4}\rangle).$
(63)
Each of these moments satisfies a closed equation of the form
$\partial_{t}\bm{C}^{p}={\sf\bm{N}_{p}}\bm{C}^{p}.$ (64)
The matrices ${\sf\bm{N}_{p}}$ can be found by applying the technique
described in Appendix B and by using the covariance of $\alpha$ given in (31).
We give below the first four matrices:
${\sf\bm{N}_{1}}=\left[\begin{array}[]{cc}-\eta_{\rm t}&0\\\ -S&-\eta_{\rm
t}\end{array}\right],$ (65)
${\sf\bm{N}_{2}}=\left[\begin{array}[]{ccc}-2\eta_{\rm t}&2D&0\\\
0&-2\eta_{\rm t}&-2S\\\ -S&0&-2\eta_{\rm t}\end{array}\right],$ (66)
${\sf\bm{N}_{3}}=\left[\begin{array}[]{cccc}-3\eta_{\rm t}&0&6D&0\\\
-S&-3\eta_{\rm t}&0&D\\\ 0&-2S&-3\eta_{\rm t}&0\\\ 0&0&-3S&-3\eta_{\rm
t}\end{array}\right],$ (67)
${\sf\bm{N}_{4}}=\left[\begin{array}[]{ccccc}-4\eta_{\rm t}&0&12D&0&0\\\
-S&-4\eta_{\rm t}&0&6D&0\\\ 0&-2S&-4\eta_{\rm t}&0&2D\\\ 0&0&-3S&-4\eta_{\rm
t}&0\\\ 0&0&0&-4S&-4\eta_{\rm t}\end{array}\right].$ (68)
The growth rate at order $p$ is defined to be
$\bm{C}^{p}\sim\exp(p\gamma_{p}t)$. This gives $\gamma_{1}=-\eta_{\rm t}$,
i.e., there is no dynamo. But this also gives dynamo modes with positive
eigenvalues given by
$\gamma_{p}\sim S^{2/3}D_{xx}^{1/3},\quad p=2,3,\ldots$ (69)
The same result was obtained by Vishniac & Brandenburg (1997) for $\gamma_{2}$
by using a different method.
Note the striking similarity between matrix ${\sf\bm{N}_{2}}$ in (66) and
matrix ${\sf\bm{N}_{2}}$ in (10). A trivial way of generalising (66) to one
spatial dimension is to replace $D$ and $\eta_{\rm t}$ in (66) by $k^{2}D$ and
$k^{2}\eta_{\rm t}$, respectively. The solution of the resultant eigenvalue
problem gives the scaling, $\gamma\sim S$ and $k^{\rm peak}\sim\sqrt{S}$.
Thus, (66) for this zero dimensional model is equivalent to (10) in the
space–time model.
|
arxiv-papers
| 2011-07-12T21:10:19 |
2024-09-04T02:49:20.452310
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dhrubaditya Mitra (NORDITA), Axel Brandenburg (NORDITA)",
"submitter": "Dhrubaditya Mitra",
"url": "https://arxiv.org/abs/1107.2419"
}
|
1107.2470
|
# An identity on the $2m$-th power mean value of the generalized Gauss
sums111This work was supported by the National Natural Science Foundation of
China, Grant No. 11071121.
Feng Liu and Quan-Hui Yang
School of Mathematical Sciences, Nanjing Normal University,
Nanjing 210046, P. R. China Corresponding author. E-mail addresses:
fliu_1986@126.com; yangquanhui01@163.com.
Abstract. In this paper, using combinatorial and analytic methods, we prove an
exact calculating formula on the $2m$-th power mean value of the generalized
quadratic Gauss sums for $m\geq 2$. This solves a conjecture of He and Zhang
[‘On the $2k$-th power mean value of the generalized quadratic Gauss sums’,
Bull. Korean Math. Soc. 48 (2011), No.1, 9-15]. 2010 Mathematics Subject
Classification: Primary 11M20.
Keywords and phrases: $2m$-th power mean, exact calculating formula,
generalized quadratic Gauss sums.
## 1 Introduction
Let $q\geq 2$ be an integer and $\chi$ be a Dirichlet character modulo $q$.
For any integer $n$, the classical quadratic Gauss sums $G(n;q)$ and the
generalized quadratic Gauss sums $G(n,\chi;q)$ are defined respectively by
$G(n;q)=\sum^{q}_{a=1}e\left(\frac{na^{2}}{q}\right),$
and
$G(n,\chi;q)=\sum^{q}_{a=1}\chi(a)e\left(\frac{na^{2}}{q}\right),$
where $e(x)=e^{2\pi ix}$.
The study of $G(n,\chi;q)$ is important in number theory, since it is a
generation of $G(n,q)$. In [5], Weil proved that if $p\geq 3$ is a prime, then
$|G(n,\chi;p)|\leq 2\sqrt{p}.$
In fact, Cochrane and Zheng [2] generalized this result to any integer. That
is, for any integer $n$ with $(n,q)=1,$ we have
$|G(n,\chi;q)|\leq 2^{\omega}(q)\sqrt{q},$
where $\omega(q)$ is the number of all distinct prime divisors of $q$.
Beside the upper bound of $G(n,\chi;q)$, the power mean value of
$|G(n,\chi;q)|$ had also been studied by some authors. W. Zhang (see [6])
proved that if $p$ is an odd prime and $n$ is an integer with $(n,p)=1$, then
$\sum_{\chi\mod
p}|G(n,\chi;p)|^{4}=\left\\{\begin{array}[]{ll}(p-1)[3p^{2}-6p-1+4\left(\frac{n}{p}\right)\sqrt{p}]~{},&p\equiv
1\mod~{}4~{};\\\ ~{}~{}~{}~{}~{}~{}~{}(p-1)(3p^{2}-6p-1)~{},&p\equiv
3\mod~{}4~{}.\end{array}\right.$
and
$\sum_{\chi\mod
p}|G(n,\chi;p)|^{6}=(p-1)(10p^{3}-25p^{2}-4p-1),~{}~{}\text{if}~{}~{}p\equiv
3\mod 4,$
where $(\frac{n}{p})$ is the Legendre symbol. For $p\equiv 1\mod 4$, it is
still an open question to calculate the exact value of $\sum_{\chi\mod
p}|G(n,\chi;p)|^{6}$.
In 2005, W. Zhang and H. Liu [7] proved that if $q\geq 3$ is a square-full
number, then for any integer $n,k$ with $(nk,q)=1,k\geq 1$, we have
$\sum_{\chi\mod q}|G(n,\chi;q)|^{4}=q\cdot\phi^{2}(q)\underset{p\mid
q}{\prod}(k,p-1)^{2}\cdot\prod_{\begin{subarray}{c}p\mid q\\\
(k,p-1)=1\end{subarray}}\frac{\phi(p-1)}{p-1},$
where $\phi(q)$ is the Euler funtion.
Recently, Y. He and W. Zhang [3] proved the following result.
Let odd number $q>1$ be a square-full number. Then for any integer $n$ with
$(n,q)=1$ and $k$=3 or 4, we have the identity
$\sum_{\chi\mod q}{|G(n,\chi;q)|^{2k}}=4^{(k-1)\omega(q)}\cdot
q^{k-1}\cdot\phi^{2}{(q)}.$
Besides, they conjectured the above identity also holds for $k\geq 5$.
In this paper, we prove this conjecture in the following.
###### Theorem 1.
Let odd number $q>1$ be a square-full number, $m\geq 2$ be an integer. Then
for any integer $n$ with $(n,q)=1$, we have the identity
$\sum_{\chi\mod q}{|G(n,\chi;q)|^{2m}}=4^{(m-1)\omega(q)}\cdot
q^{m-1}\cdot\phi^{2}{(q)}.$
## 2 Proofs
Let $p\geq 3$ be a prime, and let $k,n,a$ be three integers with $1\leq k\leq
n$. Write
$T_{p}(n,k,a)=\underset{x_{1}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{k}}{p}\right).$
In order to prove Theorem 1, we need some lemmas on the value of
$T_{p}(n,k,a)$.
###### Lemma 1.
(See [4, Theorem 8.2].) Let $p\geq 3$ be a prime. Then for any integer $a$, we
have
$\sum^{p-1}_{x=1}\left(\frac{x^{2}+ax}{p}\right)=\left\\{\begin{array}[]{ll}-1,&\text{if}~{}~{}p\nmid
a;\\\ p-1,&\text{if}~{}~{}p\mid a.\end{array}\right.$
This is a basic lemma which we will use to calculate the value of
$T_{p}(n,k,a)$.
###### Lemma 2.
Let $p\geq 3$ be a prime. Then for any integer $n\geq 1$, we have
$T_{p}(n,n,0)=\left\\{\begin{array}[]{ll}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0~{}\mbox{,}&\text{if}~{}~{}2\nmid
n\mbox{;}\\\
p^{(n-2)/2}(p-1)\left(\frac{-1}{p}\right)^{n/2}~{}\mbox{,}&\text{if}~{}~{}2\mid
n.\end{array}\right.$
###### Proof.
By the definition of $T_{p}(n,k,a)$ and Lemma 1, for $n\geq 3,$ we have
$\displaystyle T_{p}(n,n,0)$ $\displaystyle=$
$\displaystyle\underset{x_{1}+\cdots+x_{n}\equiv 0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}\sum^{p-1}_{x_{n}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-1}x_{n}}{p}\right)$ $\displaystyle=$
$\displaystyle~{}~{}\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}\left(\frac{x_{1}x_{2}\cdots
x_{n-1}(-x_{1}-\cdots-x_{n-1})}{p}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{-1}{p}\right)\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\cdot\sum^{p-1}_{x_{n-1}=1}\left(\frac{x_{n-1}^{2}+(x_{1}+\cdots+x_{n-2})x_{n-1}}{p}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{-1}{p}\right)\underset{x_{1}+\cdots+x_{n-2}\not\equiv
0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)\cdot(-1)$
$\displaystyle+\left(\frac{-1}{p}\right)\underset{x_{1}+\cdots+x_{n-2}\equiv
0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)\cdot(p-1)$ $\displaystyle=$
$\displaystyle\left(\frac{-1}{p}\right)(-1)\
{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)$
$\displaystyle-\left(\frac{-1}{p}\right)(-1)\underset{x_{1}+\cdots+x_{n-2}\equiv
0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)$
$\displaystyle+\left(\frac{-1}{p}\right)(p-1)\underset{x_{1}+\cdots+x_{n-2}\equiv
0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{-1}{p}\right)p\underset{x_{1}+\cdots+x_{n-2}\equiv
0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-2}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n-2}}{p}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{-1}{p}\right)p\cdot T_{p}(n-2,n-2,0).$
It is easy to calculate $T_{p}(1,1,0)$ and $T_{p}(2,2,0)$.
$\displaystyle
T_{p}(1,1,0)=\sum^{p-1}_{x=1}\left(\frac{x}{p}\right)=0~{}\mbox{,}$
$\displaystyle T_{p}(2,2,0)=\underset{x_{1}+x_{2}\equiv 0\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}}\left(\frac{x_{1}x_{2}}{p}\right)=\sum^{p-1}_{x_{1}=1}\left(\frac{-x_{1}^{2}}{p}\right)=\left(\frac{-1}{p}\right)(p-1).$
Therefore, we have
$\displaystyle
T_{p}(2k+1,2k+1,0)=\left(\left(\frac{-1}{p}\right)p\right)^{k}T_{p}(1,1,0)=0~{}\mbox{,}$
$\displaystyle
T_{p}(2k,2k,0)=\left(\left(\frac{-1}{p}\right)p\right)^{k-1}T_{p}(2,2,0)=\left(\frac{-1}{p}\right)^{k}p^{k-1}(p-1).$
This completes the proof of Lemma 2. ∎
###### Lemma 3.
Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Then for any integer $a$
with $(a,p)=1$, we have
$T_{p}(n,n,a)=\left\\{\begin{array}[]{ll}\left(\frac{a}{p}\right)p^{(n-1)/2}\left(\frac{-1}{p}\right)^{(n-1)/2},&\text{if}~{}~{}2\nmid
n;\\\ -p^{(n-2)/2}\left(\frac{-1}{p}\right)^{n/2},&\text{if}~{}~{}2\mid
n.\end{array}\right.$
###### Proof.
Since $(a,p)=1,$ we have
$\displaystyle T_{p}(n,n,a)$ $\displaystyle=$
$\displaystyle\underset{x_{1}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n}}{p}\right)$ $\displaystyle=$
$\displaystyle\underset{ax_{1}+\cdots+ax_{n}\equiv a\mod
p}{\sum^{p-1}_{ax_{1}=1}\sum^{p-1}_{ax_{2}=1}\cdots\sum^{p-1}_{ax_{n}=1}}\left(\frac{ax_{1}ax_{2}\cdots
ax_{n}}{p}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{a}{p}\right)^{n}\underset{x_{1}+\cdots+x_{n}\equiv
1\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{n}}{p}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{a}{p}\right)^{n}T_{p}(n,n,1).$
The calculation of $T_{p}(n,n,1)$ is very similar to that of $T_{p}(n,n,0)$ in
Lemma 2, so we directly give the result here.
$T_{p}(n,n,1)=\left\\{\begin{array}[]{ll}p^{(n-1)/2}\left(\frac{-1}{p}\right)^{(n-1)/2}\mbox{,}&\text{if}~{}~{}2\nmid
n;\\\ -p^{(n-2)/2}\left(\frac{-1}{p}\right)^{n/2}\mbox{,}&\text{if}~{}~{}2\mid
n.\end{array}\right.$
Hence,
$T_{p}(n,n,a)=\left(\frac{a}{p}\right)^{n}T_{p}(n,n,1)=\left\\{\begin{array}[]{ll}\left(\frac{a}{p}\right)p^{(n-1)/2}\left(\frac{-1}{p}\right)^{(n-1)/2},&\text{if}~{}~{}2\nmid
n\mbox{;}\\\ -p^{(n-2)/2}\left(\frac{-1}{p}\right)^{n/2},&\text{if}~{}~{}2\mid
n.\end{array}\right.$
This completes the proof of Lemma 3. ∎
###### Lemma 4.
Let $p\geq 3$ be a prime, and let $k,n,a$ be three integers with $1\leq k\leq
n$. Then we have
$None$
$T_{p}(n,k,a)=\left\\{\begin{array}[]{ll}(-1)^{n-k}\left(\frac{a}{p}\right)p^{(k-1)/2}\left(\frac{-1}{p}\right)^{(k-1)/2}~{}\mbox{,}&\text{if}~{}~{}2\nmid
k~{}\mbox{and}~{}p\nmid a;\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0~{}\mbox{,}&\text{if}~{}~{}2\nmid
k~{}\mbox{and}~{}p~{}|~{}a;\\\
(-1)^{n+1-k}\left(\frac{-1}{p}\right)^{k/2}p^{(k-2)/2}~{}\mbox{,}&\text{if}~{}~{}2\mid
k~{}\mbox{and}~{}p\nmid a;\\\
(-1)^{n-k}(p-1)\left(\frac{-1}{p}\right)^{k/2}p^{(k-2)/2}~{}\mbox{,}&\text{if}~{}~{}2\mid
k~{}\mbox{and}~{}p~{}|~{}a.\end{array}\right.$
###### Proof..
For $k\leq n-1$, we have
$\displaystyle T_{p}(n,k,a)$ $\displaystyle=$
$\displaystyle\underset{x_{1}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{k}}{p}\right)$ $\displaystyle=$
$\displaystyle\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}\left(\frac{x_{1}x_{2}\cdots
x_{k}}{p}\right)-\underset{x_{1}+\cdots+x_{n-1}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}}\left(\frac{x_{1}x_{2}\cdots
x_{k}}{p}\right)$ $\displaystyle=$ $\displaystyle-T_{p}(n-1,k,a).$
Then by induction on $n$ we have
$T_{p}(n,k,a)=(-1)^{n-k}T_{p}(k,k,a)$
for all $n\geq k$. By Lemma 2 and Lemma 3, we obtain equation (1), which
completes the proof of Lemma 4.
∎
###### Lemma 5.
Let $p\geq 3$ be a prime, $\alpha\geq 2$, $a$ and $n$ be three integers with
$1\leq a\leq p^{\alpha}-1$ and $(n,p)=1$. If $p^{\alpha-1}\parallel{a^{2}-1}$,
then we write $a=rp^{\alpha-1}+\varepsilon$, where $1\leq r\leq p-1$ and
$\varepsilon=\pm 1$. Then we have
$\underset{b=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{nb^{2}(a^{2}-1)}{p^{\alpha}}\right)=\left\\{\begin{array}[]{lll}0~{}\mbox{,}&\text{if}~{}~{}p^{\alpha-1}\nmid
a^{2}-1;\\\ p^{\alpha-1}\big{[}\big{(}\frac{2\varepsilon
rn}{p}\big{)}G(1;p)-1\big{]}~{}\mbox{,}&\text{if}~{}~{}p^{\alpha-1}\parallel
a^{2}-1;\\\
\phi(p^{\alpha})~{}\mbox{,}&\text{if}~{}~{}p^{\alpha}~{}|~{}a^{2}-1~{}.\end{array}\right.$
###### Proof..
See the proof of Lemma 4 of [3].
∎
###### Lemma 6.
Let $p\geq 3$ be a prime. Then for any two integers $n\geq 1$ and $a$, we have
$\underset{x_{1}+x_{2}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}1=\left\\{\begin{array}[]{ll}\big{(}(p-1)^{n}-(-1)^{n}\big{)}\big{/}p~{}\mbox{,}&\text{if}~{}~{}p\nmid
a;\\\
\big{(}(p-1)^{n}+(p-1)(-1)^{n}\big{)}\big{/}p~{}\mbox{,}&\text{if}~{}~{}p~{}|~{}a.\end{array}\right.$
###### Proof..
$\displaystyle\underset{x_{1}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}1$
$\displaystyle=$
$\displaystyle\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}1-\underset{x_{1}+\cdots+x_{n-1}\equiv
a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}}1$
$\displaystyle=$
$\displaystyle(p-1)^{n-1}-\underset{x_{1}+\cdots+x_{n-1}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n-1}=1}}1.$
Then by induction on $n$, we have
$\displaystyle\underset{x_{1}+\cdots+x_{n}\equiv a\mod
p}{\sum^{p-1}_{x_{1}=1}\sum^{p-1}_{x_{2}=1}\cdots\sum^{p-1}_{x_{n}=1}}1$
$\displaystyle=$
$\displaystyle\sum^{n-2}_{k=1}(-1)^{k+1}(p-1)^{n-k}+(-1)^{n-2}\underset{x_{1}+x_{2}\equiv
a\mod p}{\sum_{x_{1}=1}^{p-1}\sum^{p-1}_{x_{2}=1}}1$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\big{(}(p-1)^{n}-(-1)^{n}\big{)}\big{/}p\mbox{,}&\text{if}~{}~{}p\nmid
a;\\\
\big{(}(p-1)^{n}+(p-1)(-1)^{n}\big{)}\big{/}p\mbox{,}&\text{if}~{}~{}p~{}|~{}a.\end{array}\right.$
This completes the proof of Lemma 6. ∎
###### Lemma 7.
(See [1, Theorem 9.13].) For any odd prime $p$, we have
$G^{2}(1;p)=\left(\frac{-1}{p}\right)p~{}.$
###### Lemma 8.
(See [7, Lemma 6].) Let $m,n\geq 2$ and $u$ be three integers with $(m,n)=1$
and $(u,mn)=1$. Then for any character $\chi=\chi_{1}\chi_{2}$ with $\chi_{1}$
mod $m$ and $\chi_{2}$ mod $n$, we have the identity
$G(u,\chi;mn)=\chi_{1}(n)\chi_{2}(m)G(un,\chi_{1};m)G(um,\chi_{2};n).$
###### Lemma 9.
Let $p\geq 3$ be a prime, $\alpha\geq 2,m\geq 2$ be two integers . Then for
any integer $n$ with $(n,p)=1$, we have the identity
$\sum_{\chi\mod
p^{\alpha}}{|G(n,\chi;p^{\alpha})|^{2m}}=4^{(m-1)}\cdot\phi^{2}{(p^{\alpha})}\cdot
p^{(m-1)\alpha}.$
###### Proof..
By the definition of $G(n,\chi;p^{\alpha}),$ we have
$\displaystyle|G(n,\chi;p^{\alpha})|^{2}$ $\displaystyle=$
$\displaystyle\underset{a=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{b=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\chi(a)\overline{\chi(b)}e\left(\frac{n(a^{2}-b^{2})}{p^{\alpha}}\right)$
$\displaystyle=$
$\displaystyle\underset{a=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\chi(a)\underset{b=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{nb^{2}(a^{2}-1)}{p^{\alpha}}\right)~{}.$
Hence, by this formula we have
$\displaystyle\sum_{\chi\mod p^{\alpha}}|G(n,\chi;p^{\alpha})|^{2m}$
$\displaystyle=$ $\displaystyle\sum_{\chi\mod
p^{\alpha}}\underset{x_{1}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{x_{2}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{x_{m}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\chi(x_{1}\cdots
x_{m})\prod_{i=1}^{m}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right)$
$\displaystyle=$ $\displaystyle\phi(p^{\alpha})\underset{x_{1}x_{2}\cdots
x_{m}\equiv 1\mod
p^{\alpha}}{\underset{x_{1}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{x_{2}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{x_{m}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}}\prod_{i=1}^{m}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right).$
Then by Lemma 5 we have
$None$ $\sum_{\chi\mod
p^{\alpha}}|G(n,\chi;p^{\alpha})|^{2m}=\phi(p^{\alpha})\sum_{k=0}^{m}{{m\choose
k}A(m,k)},$
where
$A(m,k)=\underset{x_{1}x_{2}\cdots x_{m}\equiv 1\mod
p^{\alpha}}{\underset{\begin{subarray}{c}x_{1}=1\\\ p^{\alpha-1}\parallel
x_{1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{k}=1\\\
p^{\alpha-1}\parallel
x_{k}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{\begin{subarray}{c}x_{k+1}=1\\\
p^{\alpha}\mid
x_{k+1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{m}=1\\\
p^{\alpha}\mid
x_{m}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}}\prod_{i=1}^{m}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right).$
Now, in order to prove Lemma 9, we need to calculate $A(m,k)$.
$\displaystyle A(m,k)$ $\displaystyle=$
$\displaystyle\underset{x_{1}x_{2}\cdots x_{m}\equiv 1\mod
p^{\alpha}}{\underset{\begin{subarray}{c}x_{1}=1\\\ p^{\alpha-1}\parallel
x_{1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{k}=1\\\
p^{\alpha-1}\parallel
x_{k}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{\begin{subarray}{c}x_{k+1}=1\\\
p^{\alpha}\mid
x_{k+1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{m}=1\\\
p^{\alpha}\mid
x_{m}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}}\prod_{i=1}^{m}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right)$
$\displaystyle=$ $\displaystyle 2\phi(p^{\alpha})\underset{x_{1}x_{2}\cdots
x_{m-1}\equiv 1\mod p^{\alpha}}{\underset{\begin{subarray}{c}x_{1}=1\\\
p^{\alpha-1}\parallel
x_{1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{k}=1\\\
p^{\alpha-1}\parallel
x_{k}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\underset{\begin{subarray}{c}x_{k+1}=1\\\
p^{\alpha}\mid
x_{k+1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{m-1}=1\\\
p^{\alpha}\mid
x_{m-1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}}\prod_{i=1}^{m-1}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right)$
$\displaystyle=$ $\displaystyle 2\phi(p^{\alpha})A(m-1,k).$
Hence, by induction on $m$, we have
$None$ $A(m,k)=2^{m-k}\phi^{m-k}(p^{\alpha})A(k,k).$
Next, we shall calculate $A(k,k)$. By the definition, we have
$A(k,k)=\underset{x_{1}x_{2}\cdots x_{k}\equiv 1\mod
p^{\alpha}}{\underset{\begin{subarray}{c}x_{1}=1\\\ p^{\alpha-1}\parallel
x_{1}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}\cdots\underset{\begin{subarray}{c}x_{k}=1\\\
p^{\alpha-1}\parallel
x_{k}^{2}-1\end{subarray}}{\overset{p^{\alpha}}{{\sum}^{\prime}}}}\prod_{i=1}^{k}\left(\underset{y_{i}=1}{\overset{p^{\alpha}}{{\sum}^{\prime}}}e\left(\frac{ny_{i}^{2}(x_{i}^{2}-1)}{p^{\alpha}}\right)\right).$
Write $x_{i}=r_{i}p^{\alpha-1}+\varepsilon_{i}(1\leq r_{i}\leq
p-1,\varepsilon_{i}=\pm 1)$ for $i=1,2,\cdots,k$. Then by Lemma 5, we have
$\displaystyle A(k,k)$ $\displaystyle=$ $\displaystyle
p^{k(\alpha-1)}\underset{\begin{subarray}{c}\varepsilon_{1}r_{1}+\varepsilon_{2}r_{2}+\cdots+\varepsilon_{k}r_{k}\equiv
0\mod p\\\
\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{k}=1\end{subarray}}{\underset{r_{1}=1}{\overset{p-1}{{\sum}}}\underset{r_{2}=1}{\overset{p-1}{{\sum}}}\cdots\underset{r_{k}=1}{\overset{p-1}{{\sum}}}}\prod_{i=1}^{k}\left(\left(\frac{2n\varepsilon_{i}r_{i}}{p}\right)G(1;p)-1\right)$
$\displaystyle=$ $\displaystyle
p^{k(\alpha-1)}\underset{\begin{subarray}{c}r_{1}+r_{2}+\cdots+r_{k}\equiv
0\mod p\\\
\varepsilon_{1}\varepsilon_{2}\cdots\varepsilon_{k}=1\end{subarray}}{\underset{r_{1}=1}{\overset{p-1}{{\sum}}}\underset{r_{2}=1}{\overset{p-1}{{\sum}}}\cdots\underset{r_{k}=1}{\overset{p-1}{{\sum}}}}\prod_{i=1}^{k}\left(\left(\frac{2nr_{i}}{p}\right)G(1;p)-1\right)$
$\displaystyle=$ $\displaystyle
2^{k-1}p^{k(\alpha-1)}\underset{r_{1}+r_{2}+\cdots+r_{k}\equiv 0\mod
p}{\underset{r_{1}=1}{\overset{p-1}{{\sum}}}\underset{r_{2}=1}{\overset{p-1}{{\sum}}}\cdots\underset{r_{k}=1}{\overset{p-1}{{\sum}}}}\prod_{i=1}^{k}\left(\left(\frac{2nr_{i}}{p}\right)G(1;p)-1\right)$
$\displaystyle=$ $\displaystyle
2^{k-1}p^{k(\alpha-1)}\cdot\underset{r_{1}+r_{2}+\cdots+r_{k}\equiv 0\mod
p}{\underset{r_{1}=1}{\overset{p-1}{{\sum}}}\underset{r_{2}=1}{\overset{p-1}{{\sum}}}\cdots\underset{r_{k}=1}{\overset{p-1}{{\sum}}}}\Bigg{(}(-1)^{k}$
$\displaystyle+\left.\sum^{k}_{j=1}(-1)^{k-j}{k\choose
j}\left(\frac{2n}{p}\right)^{j}G^{j}(1;p)\left(\frac{r_{1}r_{2}\cdots
r_{j}}{p}\right)\right).$
By Lemma 4 and Lemma 6, the above equality becomes
$\displaystyle A(k,k)$ $\displaystyle=$ $\displaystyle
2^{k-1}p^{k(\alpha-1)}(-1)^{k}\left(\frac{1}{p}\left((p-1)^{k}+(p-1)(-1)^{k}\right)\right.$
$\displaystyle+\sum^{\lfloor k/2\rfloor}_{j=1}(-1)^{2j}{k\choose
2j}\left(\frac{2n}{p}\right)^{2j}G^{2j}(1;p)(-1)^{k-2j}\left(\frac{-1}{p}\right)^{j}p^{j-1}(p-1)\bigg{)}.$
By Lemma 7, we have
$\displaystyle A(k,k)$ $\displaystyle=$ $\displaystyle
2^{k-1}p^{k(\alpha-1)-1}\Big{(}(-1)^{k}(p-1)^{k}+(p-1)+\sum^{\lfloor
k/2\rfloor}_{j=1}{k\choose 2j}p^{2j}(p-1)\Big{)}$ $\displaystyle=$
$\displaystyle
2^{k-1}p^{k(\alpha-1)-1}\Big{(}(-1)^{k}(p-1)^{k}+(p-1)\big{(}(p+1)^{k}+(1-p)^{k}\big{)}\big{/}2\Big{)}$
$\displaystyle=$ $\displaystyle
2^{k-2}p^{k(\alpha-1)-1}\Big{(}(p+1)(1-p)^{k}+(p-1)(p+1)^{k}\Big{)}.$
Hence, by (3) we have
$A(m,k)=2^{m-2}p^{m(\alpha-1)-1}\Big{(}(-1)^{k}(p+1)(p-1)^{m}+(p-1)^{m-k+1}(p+1)^{k}\Big{)}.$
Finally, by (2) we have
$\displaystyle\sum_{\chi\mod p^{\alpha}}|G(n,\chi;p^{\alpha})|^{2m}$
$\displaystyle=$ $\displaystyle\phi(p^{\alpha})\sum_{k=0}^{m}{{m\choose
k}2^{m-2}p^{m(\alpha-1)-1}\Big{(}(-1)^{k}(p+1)(p-1)^{m}+(p-1)^{m-k+1}(p+1)^{k}\Big{)}}$
$\displaystyle=$
$\displaystyle\phi(p^{\alpha})2^{m-2}p^{m(\alpha-1)-1}(p+1)(p-1)^{m}\sum^{m}_{k=0}{m\choose
k}(-1)^{k}$
$\displaystyle~{}+\phi(p^{\alpha})2^{m-2}p^{m(\alpha-1)-1}\sum^{m}_{k=0}{m\choose
k}(p-1)^{m-k+1}(p+1)^{k}$ $\displaystyle=$ $\displaystyle
0+\phi(p^{\alpha})2^{m-2}p^{m(\alpha-1)-1}(p-1)(2p)^{m}$ $\displaystyle=$
$\displaystyle 4^{m-1}\phi^{2}(p^{\alpha})p^{\alpha(m-1)}.$
This completes the proof of Lemma 9. ∎
###### Proof of Theorem 1..
Since $q$ is an odd square-full number, let
$q=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots
p_{\omega(q)}^{\alpha_{\omega(q)}}$, we have $\alpha_{i}\geq
2,i=1,\cdots\omega(q)$. For any integer $n$ with $(n,q)=1$, by Lemma 8 and
Lemma 9, we obtain
$\displaystyle\sum\limits_{\chi\mod q}|G(n,\chi;q)|^{2m}$ $\displaystyle=$
$\displaystyle\prod^{\omega(q)}_{\begin{subarray}{c}i=1\\\
p_{i}^{\alpha_{i}}\parallel q\end{subarray}}\sum_{\chi\mod
p_{i}^{\alpha_{i}}}|G(nq/p_{i}^{\alpha_{i}},\chi;p_{i}^{\alpha_{i}})|^{2m}$
$\displaystyle=$ $\displaystyle\prod^{\omega(q)}_{\begin{subarray}{c}i=1\\\
p_{i}^{\alpha_{i}}\parallel
q\end{subarray}}\big{(}4^{m-1}p_{i}^{\alpha_{i}(m-1)}\phi^{2}(p_{i}^{\alpha_{i}})\big{)}$
$\displaystyle=$ $\displaystyle 4^{(m-1)\omega(q)}\cdot
q^{m-1}\cdot\phi^{2}(q)~{}.$
This completes the proof of theorem 1. ∎
## References
* [1] Tom M. Apostol, Introduction to Analytic Number Theory, Spring-Verlag, New York, 1976.
* [2] T. Cochrane and Z. Y. Zheng, Pure and mixed exponential sums, Acta Arith, 91 (1999), no.3, 249-278.
* [3] Y. He and W. P. Zhang, On the $2k$-th power mean value of the generalized quadratic Gauss sums, Bull. Korean Math. Soc. 48 (2011), No.1, 9-15.
* [4] L. K. Hua, Introduction to Number Theory, Science Press, Beijing, 1979.
* [5] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207.
* [6] W. P. Zhang, Moments of generalized quadratic Gauss sums weighted by L-functions, J. Number Theory 92 (2002), no.2, 304-314.
* [7] W. P. Zhang and H. Liu, On the general Gauss sums and their fourth power mean, Osaka J. Math. 42 (2005), no.1 189-199
|
arxiv-papers
| 2011-07-13T06:38:06 |
2024-09-04T02:49:20.463832
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng Liu and Quan-Hui Yang",
"submitter": "Quan-Hui Yang",
"url": "https://arxiv.org/abs/1107.2470"
}
|
1107.2482
|
# Maximum Matchings via Glauber Dynamics
Anant Jindal Laxmi Niwas Mittal Institute of Information Technology, India.
anantjindal1@gmail.com Gazal Kochar Laxmi Niwas Mittal Institute of
Information Technology, India. gkochar@gmail.com Manjish Pal Indian Institute
of Technology Gandhinagar, India. manjish_pal@iitgn.ac.in
###### Abstract
In this paper we study the classic problem of computing a maximum cardinality
matching in general graphs $G=(V,E)$. This problem has been studied
extensively more than four decades. The best known algorithm for this problem
till date runs in $O(m\sqrt{n})$ time due to Micali and Vazirani [24]. Even
for general bipartite graphs this is the best known running time (the
algorithm of Karp and Hopcroft [16] also achieves this bound). For regular
bipartite graphs one can achieve an $O(m)$ time algorithm which, following a
series of papers, has been recently improved to $O(n\log n)$ by Goel, Kapralov
and Khanna (STOC 2010) [15]. In this paper we present a randomized algorithm
based on the Markov Chain Monte Carlo paradigm which runs in $O(m\log^{2}n)$
time, thereby obtaining a significant improvement over [24].
We use a Markov chain similar to the _hard-core model_ for Glauber Dynamics
with _fugacity_ parameter $\lambda$, which is used to sample independent sets
in a graph from the Gibbs Distribution [31], to design a faster algorithm for
finding maximum matchings in general graphs. Motivated by results which show
that in the hard-core model one can prove fast mixing times (for e.g. it is
known that for $\lambda$ less than a critical threshold the mixing time of the
hard-core model is $O(n\log n)$ [27], we define an analogous Markov chain
(depending upon a parameter $\lambda$) on the space of all possible partial
matchings of a given graph $G$, for which the probability of a particular
matching $M$ in the stationary follows the Gibbs distribution which is:
$\displaystyle\pi(M)=\frac{\lambda^{|M|}}{\sum_{x\in\Omega}\lambda^{|x|}}$
where $\Omega$ is the set of all possible matchings in $G$.
We prove upper and lower bounds on the mixing time of this Markov chain.
Although our Markov chain is essentially a simple modification of the one used
for sampling independent sets from the Gibbs distribution, their properties
are quite different. Our result crucially relies on the fact that the mixing
time of our Markov Chain is independent of $\lambda$, a significant deviation
from the recent series of works [11, 26, 28, 29, 30] which achieve
computational transition (for estimating the partition function) on a
threshold value of $\lambda$. As a result we are able to design a randomized
algorithm which runs in $O(m\log^{2}n)$ time that provides a major improvement
over the running time of the algorithm due to Micali and Vazirani. Using the
conductance bound, we also prove that mixing takes $\Omega(\frac{m}{k})$ time
where $k$ is the size of the maximum matching.
## 1 Introduction
Given an unweighted undirected graph $G=(V,E)$ with $|E|=m$ and $|V|=n$, a
matching $M$ is a set of edges belonging to $E$ such that no two edges in $M$
are incident on a vertex. If there is a matching of size $n/2$ (for $n$ even),
then it is called a _perfect matching_. The Maximum Matching problem is to
find the maximum sized matching in a given graph. The computational complexity
of this problem has been studied extensively for more than four decades
starting with an algorithm of Edmonds.
### 1.1 General Graphs
Edmonds’s celebrated paper ‘Paths, Trees and Flowers’ [9] was the first to
give an efficient algorithm (also called the _blossom shrinking algorithm_)
for finding maximum matching in general graphs. This algorithm can be
implemented in $O(n^{4})$ time. The running time was subsequently improved in
a number of papers [10, 20, 22]. All these papers were variants of Edmonds
algorithm. Even and Kariv [12] obtained an improvement to $O(n^{2.5})$ which
was improved by Micali and Vazirani [24] who gave an $O(m\sqrt{n})$ time
algorithm for the problem by a careful handling of blossoms. This is the best
known algorithm for finding maximum matchings in general bipartite graphs.
### 1.2 Bipartite Graphs
For bipartite graphs, the problem can easily be solved using the max-flow
algorithm by Ford and Fulkerson, an algorithm usually taught in an
undergraduate algorithms course [21], which has a running time of $O(mn)$. The
first algorithm for this problem was given by Konig [19]. Hopcroft and Karp
[16] gave an algorithm that runs in $O(m\sqrt{n})$ time. This algorithm is an
exact and deterministic algorithm. The problem becomes significantly simpler
for regular bipartite graphs. In a _d-regular_ bipartite graph every vertex
has degree $d$. When $d$ is a power of 2, Gabor and Kariv were able to achieve
an $O(m)$ algorithm. After significant efforts, the ideas used there were used
by Cole, Ost and Schirra [4] to obtain a get an $O(m)$ algorithm for general
$d$.
In a recent line of attack by Goel, Kapralov and Khanna [13, 14], the authors
were able to use sampling based methods to get improved running time. In the
most recent paper they were able to achieve a running time of $O(n\log n)$ for
$d$-regular graphs [15]. Their algorithm performs an appropriately truncated
random-walk on a modified graph to successively find augmenting path.
## 2 Our Results
In this paper we give a Markov Chain Monte Carlo algorithm for finding a
maximum matching in general bipartite graphs. Our algorithm is in the spirit
similar to [15] which also is a ‘truncated random walk’ based algorithm,
however the stationary distribution of the underlying Markov Chains in their
case is different from ours. Inspired from the hard-core model with fugacity
parameter $\lambda$ of sampling independent sets from graphs we define a
similar Markov Chain over the space of all possible partial matching such that
its stationary distribution $\pi(\cdot)$ is the Gibbs distribution, ie. given
a matching $M$ its probability $\pi$ is
$\displaystyle\pi(M)=\frac{\lambda^{|M|}}{\sum_{\sigma\in\Omega}\lambda^{|\sigma|}}$
where $\Omega$ is the set of all possible matchings in $G$. Notice that
$\pi(M)$ is maximum for maximum matchings and if $\lambda$ is a significantly
large number, $\pi(M)$ tends to 1 for maximum matchings. Our algorithm is
extremely simple (being a standard in the MCMC paradigm). Starting from a
fixed matching we start a random walk in $\Omega$ according to the underlying
graph $\tilde{G}$ of the Markov chain. After $T_{mix}$ (mixing time) steps the
distribution reached by the algorithm is roughly the same as the Gibbs
distribution. More formally, the variation distance of ${\cal D}^{t}$ (the
distribution after $t$ steps) from the Gibbs distribution is less than
$\frac{1}{2e}$. This just leaves the task of proving an upper-bound on the
mixing time of the Markov Chain, for which we resort to the Coupling Method
introduced by Bubley and Dyer [2]. We define a metric $\Phi(\cdot,\cdot)$ and
apply the bound from [2], to show that the Markov Chain mixes in $O(m\log n)$
time. We also use the conductance method to show that the mixing time will be
at least $\Omega(\frac{m}{k})$ where $k$ is the size of the maximum matching.
Thus upto logarithmic factors the bounds are same when $k$ is small (at most
polylogarithmic). The main result of our paper can be concisely written as
follows:
###### Theorem 1 (Main).
There exists a randomized algorithm which given a graph $G=(V,E)$ with $|V|=n$
and $|E|=m$ finds a maximum matching in $O(m\log^{2}n)$ time with high
probability.
Important remark: It has been pointed to us independently by Yuval Peres,
Jonah Sherman, Piyush Srivastava and other anonymous reviewers that the
coupling used in this paper doesn’t have the right marginals because of which
the mixing time bound doesn’t hold, and also the main result presented in the
paper. We thank them for reading the paper with interest and promptly pointing
out this mistake.
### 2.1 Organization
The paper is organized as follows: in Section 3 we give a brief description of
the basic idea and technique that are underlying our algorithm. Subsequently
in Section 4 we give an overview of the MCMC paradigm which includes basic
preliminaries and definitions regarding Markov Chain and Mixing. Section 7 is
devoted to the details of the chain being used by us and proving that indeed
it has the desired properties. We then prove upper and lower bounds on its
mixing time using the coupling method and conductance argument in Section 8
and Section 9 respectively. We end the paper with a conclusion and some open
problems.
## 3 The Idea: Maximum Matchings via Glauber Dynamics
Our ideas are inspired mainly from the results in the Glauber dynamics of the
hard-core model of fugacity parameter $\lambda$, for sampling independent sets
from the Gibbs distribution. According to the Gibbs distribution the
probability of an independent set $I$ is given by
$\displaystyle{\cal
G}(I)=\frac{\lambda^{|I|}}{\sum_{\rho\in\Omega}\lambda^{|\rho|}}$
where $\Omega$ is the set of all possible independent sets in $G$ and ${\cal
Z}=\sum_{\rho\in\Omega}\lambda^{|\rho|}$ is also called the _partition
function_. Clearly for $\lambda=1$ the partition function value is same as the
number of independent sets in the graph (computing which is a #P-Hard
problem).
In the hard-core model given a particular configuration $\sigma$ of an
independent set ($\sigma$ can be thought of an $n$-dimensional 0/1 vector
which is 1 for all the vertices which are in the independent set and 0
otherwise), we choose a vertex randomly uniformly, if this vertex is already
present in the independent set we keep it with probability
$\frac{\lambda}{1+\lambda}$ and discard it with probability
$\frac{1}{1+\lambda}$ otherwise the vertex is not in the independent set and
if this vertex can be added to the independent set (i.e. none of its neighbors
are already present in the independent set) then again it is added with with
probability $\frac{\lambda}{1+\lambda}$ and rejected with probability
$\frac{1}{1+\lambda}$. The beauty of this Markov Chain is that the stationary
distribution is the Gibbs distribution. The details of this chain can be found
in [31].
The key difference between the mixing time of Glauber Dynamics for independent
sets and our case is that one can achieve fast mixing time in the former case
only for small values of $\lambda$. In fact intuitively one should not be able
to obtain fast mixing times for large values of $\lambda$ because such a
result would imply that we can design a randomized polynomial time algorithm
for finding maximum independent set in a graph, which is an NP-Hard problem.
This intuition has led to a series of papers [11, 26, 28, 29, 30] which
ultimately has been successful in proving that there exists a threshold value
of $\lambda=\lambda_{c}$ such that if $\lambda>\lambda_{c}$ then estimating
the partition function is hard (the exact technical condition is that unless
$NP=RP$ no FPRAS exists for estimating ${\cal Z}$) and for
$\lambda<\lambda_{c}$ one can obtain an FPTAS for the same problem. Previous
to this result, the computational complexity of estimating the partition
function (and counting the number of independent sets) was only understood for
special graphs [5, 32]. Apart from these results substantial attention has
been given to obtain good bounds on the mixing time of this chain for trees
[28].
We define a Markov Chain which is tuned to our need. In our case $\sigma$ is a
set of edges which form a partial matching. We make a simple modification to
the above chain, wherein instead of picking a random vertex we pick a random
edge $e_{r}\in E$ and perform the same experiment with the parameter $\lambda$
as in the case of independent sets (notice that the chosen edge won’t be added
if in the present matching there is an edge sharing an end point with
$e_{r}$). We then show that this chain is aperiodic and irreducible with
stationary as the Gibbs distribution over the space of all possible partial
matchings with parameter $\lambda$. We then use the techniques of bounding the
mixing time to achieve a $\lambda$ independent upper bound. This remarkable
property allows us to exploit the nature of Gibbs distribution (which we
obtain for very large values of $\lambda$) without getting an overhead on the
mixing time.
### 3.1 Markov Chain Monte Carlo
Markov Chain Monte Carlo algorithms have played a significant role in
statistics, econometrics, physics and computing science over the last two
decades. For some high-dimensional problems in geometry, such as computing the
volume of a convex body in $d$ dimensions, MCMC simulation is the only known
general approach for providing a solution within time polynomial in $d$ [6].
For a number of other hard problems like approximating the permanent [18],
approximate counting [17], the only known FPRASs (_Fully Polynomial time
Randomized Approximation Schemes_) rely on the MCMC paradigm. In this paper,
we use this method to obtain a faster algorithm for the classical problem of
finding maximum matchings in general graphs, a problem which is known to be
solvable in polynomial time.
The Markov Chain Monte Carlo (MCMC) method is a simple and frequently used
approach for sampling from the Gibbs distribution of a statistical mechanical
system. The idea goes like this, we design a Markov chain whose state space is
$\Omega$ whose stationary distribution is the desired Gibbs distribution.
Starting at an arbitrary state, we simulate the Markov chain on $\Omega$ until
it is sufficiently close to its stationary distribution. We then output the
final state which is a sample from (close to) the desired distribution. The
required length of the simulation, in order to get close to the stationary
distribution, is traditionally referred to as the mixing time $\tau$ or
$T_{mix}$ and the aim is to bound the mixing time to ensure that the
simulation is efficient. For a detailed understanding of the theory of Markov
Chains we would recommend the recent excellent book by Levin, Peres and Wilmer
[23].
## 4 Preliminaries
### 4.1 Markov chains
Consider a stochastic process $(X_{t})_{t=0}^{\infty}$ on a finite state space
$\Omega$. Let P denote a non-negative matrix of size $|\Omega|\times|\Omega|$
which satisfies
$\sum_{j\epsilon\Omega}P_{ij}=1\mbox{ }\forall i\in\Omega$
The process is called a Markov chain if for all times $t$ and $i,j\in\Omega$
probability of going from $i^{th}$ state to $j^{th}$ state is independent of
the path by which $i^{th}$ state is reached i.e. if $X_{t}$ is the state of
the process at time $t$ then
$\mbox{P}[X_{t+1}|X_{t}=x_{t},X_{t-1}=x_{t-1}\dots
X_{0}=x_{0}]=\mbox{P}[X_{t+1}|X_{t}=x_{t}]$
A distribution $\pi$ is called a _stationary distribution_ if it satisfies
$\pi P=\pi$. A necessary and sufficient condition for a chain to have a unique
stationary distribution is that the chain is
* 1.
_Irreducible_ : for all $i,j\in\Omega$ there exists a time $t$ such that
$P^{t}_{ij}>$ 0; and
* 2.
_Aperiodic_ : for all $i\in\Omega$, GCD {$t:P^{t}_{ii}>$ 0} = 1.
A Markov Chain which has both of the above properties is called _ergodic_. For
an ergodic Markov chain, if a distribution $\pi$ satisfies the detailed
balance equations
$\pi_{i}P_{ij}=\pi_{j}P_{ji}$
for all $i,j\in\Omega$ then $\pi$ is the (unique) stationary distribution and
such a chain is called _reversible_.
### 4.2 Mixing Time
The notion of mixing time is defined as a way to measure the closeness of the
distribution after $t$ steps w.r.t. the stationary distribution. The total
variation distance between two discrete probability distributions over a
finite space $\Omega$ is defined as the half of the $l_{1}$ norm of the
corresponding probability vectors.
$d_{TV}(\mu,\nu)=\frac{1}{2}\sum_{\omega\in\Omega}|\mu(\omega)-\nu(\omega)|$
If $P_{t}$ is the probability distribution after $t$ steps in the random walk
then $T_{mix}$ is the minimum $t$ for which,
$d_{TV}(P_{t},\pi)\leq\frac{1}{2e}$
where $\pi$ is the stationary distribution. Therefore, if we intend to get
close to a stationary distribution we just have to truncate the random walk on
the state space after $\tau$ steps.
### 4.3 Conductance
The conductance of a Markov chain is defined as the following quantity,
$\phi(G)=\displaystyle\min_{S\subset V}\frac{\sum_{i\in
S,j\in\bar{S}}\pi_{i}p_{ij}}{(\sum_{i\in
S}\pi_{i})(\sum_{i\in\bar{S}}\pi_{i})}$
Another quantity of our interest here is $T_{relax}$, the _relaxation time_ of
the Markov chain. $T_{relax}$ is defined as the inverse of conductance i.e.
$T_{relax}=\frac{1}{\phi(G)}$. It is known that $T_{mix}$ and $T_{relax}$ obey
the following inequality [1].
$T_{relax}+1\leq T_{mix}.$
Also it is known [28] that
$\displaystyle T_{relax}=\Omega\left(\frac{1}{\phi}\right),$
a bound usually used to prove lower bounds on the Mixing time of Markov
Chains.
## 5 The Chain
In this section, we describe the chain considered by our algorithm which is
essentially a modification of the hard-core model of Glauber Dynamics. Recall
that our objective is to come up with a chain whose stationary distribution
ensures that the probability of being at a maximum matching is the largest.
Recall that first we need to ensure that our chain is aperiodic and
irreducible.
We use the following natural modification of the Markov chain for the hard-
core model of Glauber dynamics.
* •
Choose an edge $e_{r}$ uniformly at random from $E$.
* •
Let
$\sigma^{\prime}=\begin{cases}\sigma\bigcup\\{e_{r}\\},&\text{with
probability}\frac{\lambda}{1+\lambda}\\\
\sigma\backslash\\{e_{r}\\},&\text{with
probability}\frac{1}{1+\lambda}\end{cases}$
* •
If $\sigma^{\prime}$ is a valid matching, move to state $\sigma^{\prime}$
otherwise remain at state $\sigma$.
We are now prepared to prove that this is a valid Markov Chain with stationary
as the Gibbs distribution, where we define Gibbs distribution as the following
distribution over the set of all possible matchings $\Omega$ as
$\displaystyle{\cal
G}(M)=\frac{\lambda^{|M|}}{\sum_{x\in\Omega}\lambda^{|x|}}$
In the rest, we will call $Z=\sum_{x\in\Omega}\lambda^{|x|}$.
###### Lemma 1.
$X_{t}$ is ergodic with stationary distribution as $\cal G$.
###### Proof.
Since for every state there is some probability by which the walk can remain
in the same state, the chain is aperiodic. Also the underlying graph is
connected because given any matching there is a at least one path to reach any
other matching (consider the path that first drops all the edges of the
initial matching and adds the edges of the new matching one by one).
To show that the stationary of this distribution is $\cal G$, we would show
that the chain is reversible w.r.t. $\cal G$. Consider two distinct states $i$
and $j$ in the Markov Chain. Assume w.l.o.g that there is an edge from $i$ to
$j$ (if there is no edge from $i$ to $j$ then the balance equations
corresponding to $i$ and $j$ are trivially satisfied). Let the configuration
$i$ has $t$ edges in it then according to our construction, $j$ will have
either $t+1$ or $t-1$ states (it won’t be $t$ because we have assumed that the
two states are distinct). We will just look at the case when $j$ has $t+1$
edges, the other case is analogous. Since we are interested in showing
reversibility w.r.t. $\cal G$, $\pi_{i}=\frac{\lambda^{t}}{Z}$ and
$\pi_{j}=\frac{\lambda^{t+1}}{Z}$. Therefore,
$\pi_{i}P_{ij}=\frac{\lambda^{t}}{Z}\cdot\frac{\lambda}{m(1+\lambda)}$, and
$\pi_{j}P_{ji}=\frac{\lambda^{t+1}}{Z}\cdot\frac{1}{m(1+\lambda)}$. Thus
$\pi_{i}P_{ij}=\pi_{j}P_{ji}=\frac{\lambda^{t+1}}{mZ(1+\lambda)}$
∎
## 6 Upper Bound using Coupling
In this section we prove an upper bound on $T_{mix}$ of the chain defined in
the previous section. Our bound is based on the coupling argument introduced
by Bubley and Dyer [2]. We first give a description of the coupling method.
### 6.1 Coupling Method
A coupling of a Markov chain on state space is a stochastic process
$(\sigma_{t},\eta_{t})$ on $|\Omega|\times|\Omega|$ such that:
* •
$\sigma_{t}$ and $\eta_{t}$ are copies of the original Markov chain and
* •
if $\sigma_{t}=\eta_{t}$, then $\sigma_{t+1}=\eta_{t+1}$
Thus the chains follow each other after the first instant when they hit each
other. In order to measure the distance between the two copies of the chain,
one introduces a distance function $\Phi$ on the product state space
$\Omega\times\Omega$ so that
$\Phi=\Phi(\sigma_{t},\eta_{t})=0\Longleftrightarrow\sigma_{t}=\eta_{t}$. For
two states $\sigma$ and $\eta$ let $\rho(\sigma,\eta)$ be the set of all paths
from $\sigma$ to $\eta$ in the Markov Chain. The following theorem due to to
Bubley and Dyer is used to prove mixing time on Markov chain.
###### Theorem 2.
Let $\Phi$ be an integer-valued metric defined on $\Omega\times\Omega$ which
takes values in {0,1…D} such that, for all $\sigma,\eta\in\Omega$ there exists
a path $\xi\in\rho(\sigma,\eta)$ with
$\displaystyle\Phi(\sigma,\eta)=\sum_{i}\Phi(\xi^{i},\xi^{i+1})$
Suppose there exists a constant $\beta<$1 and a coupling
$(\sigma_{t},\eta_{t})$ of the Markov chain such that, for all
$\sigma_{t},\eta_{t}$,
$\displaystyle
E[\Phi(\sigma_{t+1},\eta_{t+1})]\leq\beta\Phi(\sigma_{t},\eta_{t})$
Then the mixing time is bounded by
$\displaystyle\tau\leq\frac{\log(2eD)}{1-\beta}$
###### Proof.
Can be found in [2, 31]. ∎
In order to bound the mixing time, we will define a coupling so as to minimize
the time until both copies of the Markov chain reach the same state, and we
will do that by defining the coupling in such a way that on every step both
markov chains reach towards same state. The aim is to prove a good upper bound
on $E[\Phi(\sigma_{t+1},\eta_{t+1})]$ in terms of $\Phi(\sigma_{t},\eta_{t})$.
In the following subsection we define the coupling:
#### 6.1.1 Coupling
Consider the following process $(\sigma_{t},\eta_{t})$ on
$|\Omega|\times|\Omega|$ where $\Omega$ is the space of all possible
matchings.
###### Definition 1.
Choose an edge uniformly at random,
* 1.
If insertion is possible in both $\sigma$ and $\eta$ then add it probability
$\frac{\lambda}{1+\lambda}$ and remove it with probability
$\frac{1}{1+\lambda}$.
* 2.
If insertion is possible in one and not possible in the other then remove that
edge if it is already present in one matching.
Notice that here we are relying on the fact that one can insert an edge if it
is already present in it. It is easy to verify that this indeed is a coupling
for the Markov chain defined in the previous section.
We use the following distance function for the aforementioned coupling. Let
$\sigma\oplus\eta=\left\\{e\in
E|e\in\left((\sigma\setminus\eta)\bigcup(\eta\setminus\sigma)\right)\right\\}\mbox{
where }\sigma\mbox{ and }\eta\in\Omega$
define the distance function $\Phi(\sigma_{t},\eta_{t})=|\sigma\oplus\eta|$
which is the number of edges present in one but not in the other. Our
objective is to upper-bound $E[\Phi_{(}\sigma_{t+1},\eta_{t+1})]$ in terms of
$d_{t}=\Phi(\sigma_{t},\eta_{t})$. Based on the definition of our coupling the
following cases may arise once we pick an edge $e_{r}$ uniformly randomly:
* 1.
$e_{r}$ can be added to both $\sigma_{t}$ and $\eta_{t}$: The subevents are
(a) $e_{r}$ was present in both of them, in this case
$\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}$, (b) $e_{r}$ is not present in both of
them, in which case again $\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}$ and (c)
$e_{r}$ is present in one but not in other, in which case
$\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}-1$.
* 2.
$e_{r}$ can be added in exactly one of $\sigma_{t}$ and $\eta_{t}$: The sub
events for this case are (a) $e_{r}$ is not present in both, which gives
$\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}$ (b)$e_{r}$ is present in one matching
and not in other, in which case $\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}-1$
* 3.
$e_{r}$ can’t be added to any one of $\sigma_{t}$ and $\eta_{t}$: In this case
$\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}$.
Using the above events we can prove the following:
###### Lemma 2.
$\displaystyle
E[\Phi(\sigma_{t+1},\eta_{t+1})]=\Phi(\sigma_{t},\eta_{t})\left(1-\frac{1}{m}\right)$.
###### Proof.
Since $\Phi(\sigma_{t+1},\eta_{t+1})$ can only take two values either $d$ or
$d-1$ we only need to calculate the the probability of the happening of one of
these cases. This happens when either the event 1(c) or the event 2(b) takes
place (as mentioned above). Thus,
$\mbox{{Pr}}[\Phi(\sigma_{t+1},\eta_{t+1})=d_{t}-1]=\mbox{{Pr}}[1(c)\cup
2(b)]$
clearly the distance becomes $d_{t}-1$ when the chosen edge is one of the
edges in $\sigma\oplus\eta$. We can divide $\sigma\oplus\eta$ in to two sets
$U$ and $\bar{U}$. $U$ is the set of edges which can be added to the matching
in which it is not present, and $\bar{U}$ is the set of edge which can’t be
added to the matching in which it is not present. Using this notation we can
write the desired probability as
$\displaystyle\displaystyle\mbox{{Pr}}[1(c)\cup 2(b)]$ $\displaystyle=$
$\displaystyle\sum_{e\in
U}\frac{1}{m}\left(\frac{\lambda}{1+\lambda}+\frac{1}{1+\lambda}\right)+\sum_{e\in\bar{U}}\frac{1}{m}$
$\displaystyle=$
$\displaystyle\frac{|\sigma\oplus\eta|}{m}=\frac{\Phi(\sigma_{t},\eta_{t})}{m}$
Therefore,
$\displaystyle\displaystyle E[\Phi(\sigma_{t+1},\eta_{t+1})]$ $\displaystyle=$
$\displaystyle(d_{t}-1)\frac{\Phi(\sigma_{t},\eta_{t})}{m}+d_{t}\left(1-\frac{\Phi(\sigma_{t},\eta_{t})}{m}\right)$
$\displaystyle=$
$\displaystyle\Phi(\sigma_{t},\eta_{t})\left(1-\frac{1}{m}\right)(\mbox{ since
}d_{t}=\Phi(\sigma_{t},\eta_{t}))$
∎
We can now prove the following:
###### Lemma 3.
$T_{mix}=O(m\log n)$.
###### Proof.
Given any $\sigma$ and $\eta$ we define a $d=\Phi(\sigma,\eta)$ length path as
$\sigma=\xi^{1},\xi^{2}\dots,\xi^{d}=\eta$ such that $\xi^{i+1}$ is the state
obtained by removing exactly one edge from
$\xi^{i}\in\sigma\bigcap(\sigma\oplus\eta)$ for $i=1,2\dots j$ where $\xi^{j}$
consists only of edges which do not belong to $\sigma\oplus\eta$ and for all
$k\geq j$, $\xi^{k+1}$ is obtained by adding one edge to $\xi^{k}$ which
belongs to $\eta\bigcap(\sigma\oplus\eta)$. Since $\xi^{i},\xi^{i+1}=1$ for
all $i=1,2\dots d-1$, we have
$\displaystyle\Phi(\sigma,\eta)=\sum_{i}\Phi(\xi^{i},\xi^{i+1})$
Also we can write
$E[\Phi(\sigma_{t+1},\eta_{t+1})]=\beta\Phi(\sigma_{t},\eta_{t})(\mbox{ with
}\beta=\left(1-\frac{1}{m}\right))$
this allows us to apply the result from Theorem 2 which gives the following
result
$\displaystyle T_{mix}\leq\frac{\log 2eD}{\displaystyle
1-\left(1-\frac{1}{m}\right)}\leq m\log(4en)=O(m\log n)$
where we have used $\beta=(1-1/m)$ and $D=2n$. ∎
Our algorithm is concisely presented as follows:
Input: A Graph $G=(V,E)$ with $|V|=n$ and $|E|=m$
$\sigma_{0}\leftarrow anymatching$;
$\lambda=2^{m}$;
for _$t=0$ _to_ $10m\log n$_ do
choose an edge $e_{r}$ uniformly randomly ;
$\sigma^{\prime}=\begin{cases}\sigma_{t}\bigcup\\{e_{r}\\},&\text{with
probability}\frac{\lambda}{1+\lambda}\\\
\sigma_{t}\backslash\\{e_{r}\\},&\text{with
probability}\frac{1}{1+\lambda}\end{cases}$;
if $\sigma^{\prime}$ is a valid matching;
$\sigma_{t+1}=\sigma^{\prime}$;
else;
$\sigma_{t+1}=\sigma_{t}$;
end for
return _$\sigma_{t}$_
Algorithm 1 RandMatching
###### Theorem 3.
The probability that the random walk in Algorithm 1 ends on a maximum matching
is at least $\frac{20}{189}$.
###### Proof.
Let ${\cal M}_{i}$ be the set all of matchings of size $i$ (and $S_{i}$ be its
cardinality) in the given graph. Also, let $k$ be the size of the maximum
matching. We need to find the probability that the random walk lands up on a
maximum matching after $T=10m\log n\geq T_{mix}$ steps. Let $X_{T}$ be the
state after $T$ steps, and $\pi_{T}$ be the probability distribution after $T$
steps then by definition of mixing time and triangle inequality,
$\displaystyle\left|\sum_{\omega\in{\cal
M}_{k}}\pi_{T}(\omega)-\sum_{\omega\in{\cal M}_{k}}{\cal
G}(\omega)\right|\leq\sum_{\omega\in{\cal M}_{k}}|\pi_{T}(\omega)-{\cal
G}(\omega)|\leq\sum_{\omega\in\Omega}|\pi_{T}(\omega)-{\cal
G}(\omega)|\leq\frac{1}{e}$
where $\sum_{\omega\in{\cal M}_{k}}\pi_{T}(\omega):=\mbox{Pr}_{k}(\pi_{T})$ is
the probability of reaching a maximum matching after $T$ steps (the success
probability of the algorithm) and $\sum_{\omega\in{\cal M}_{k}}{\cal
G}(\omega):=\mbox{Pr}_{k}({\cal G})$ is the probability of finding a maximum
matching according to the Gibbs distribution. We need to find the probability
that $X_{T}$ is a maximum matching. Thus we have,
$\mbox{Pr}_{k}(\pi_{T})\in\left[\mbox{Pr}_{k}({\cal
G})-\frac{1}{e},\mbox{Pr}_{k}({\cal G})+\frac{1}{e}\right]$
Also,
$\displaystyle\mbox{Pr}_{k}({\cal G})$ $\displaystyle=$
$\displaystyle\frac{\lambda^{k}S_{k}}{\displaystyle\sum_{i=0}^{k}\lambda^{i}S_{i}}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{k}S_{k}}{\displaystyle\lambda^{k}S_{k}+\lambda^{k-1}S_{k-1}+\lambda^{k-2}S_{k-2}+\dots+S_{0}}$
dividing by $\lambda^{k}$ both numerator and denominator
$\displaystyle=$ $\displaystyle\frac{S_{k}}{S_{k}+\frac{\displaystyle
S_{k-1}}{\lambda}+\frac{S_{k-2}}{\lambda^{2}}+...+\frac{S_{0}}{\lambda^{k}}}$
We put $\lambda=S_{m}$ where $S_{m}$=$\displaystyle\max_{i}^{k}S_{i}$
$\displaystyle=$ $\displaystyle\frac{S_{k}}{\displaystyle
S_{k}+\frac{S_{k-1}}{S_{m}}+\frac{S_{k-2}}{S_{m}^{2}}+\dots+\frac{S_{0}}{S_{m}^{k}}}=\frac{1}{\displaystyle
1+\frac{S_{k-1}}{S_{k}S_{m}}+\frac{S_{k-2}}{S_{k}S_{m}^{2}}+\dots+\frac{S_{0}}{S_{k}S_{m}^{k}}}$
By definition of $S_{m}$, $\frac{S_{i}}{S_{m}}$ is always $\leq 1$, hence we
have
$\displaystyle\frac{1}{\displaystyle
1+\frac{S_{k-1}}{S_{k}S_{m}}+\frac{S_{k-2}}{S_{k}S_{m}^{2}}+\dots+\frac{S_{0}}{S_{k}S_{m}^{k}}}$
$\displaystyle\geq$ $\displaystyle\frac{1}{\displaystyle
1+\frac{1}{S_{k}}+\frac{1}{S_{k}S_{m}}+\frac{1}{S_{k}S_{m}^{2}}...+\frac{1}{S_{k}S_{m}^{k-1}}}$
$\displaystyle\geq$ $\displaystyle\frac{1}{\displaystyle
1+\frac{1}{S_{k}}\left(1+\frac{1}{S_{m}}+\frac{1}{S_{m}^{2}}\dots+\frac{1}{S_{m}^{k-1}}\right)}$
$\displaystyle\geq$ $\displaystyle\frac{1}{\displaystyle
1+\frac{1}{S_{k}}\left(\frac{1-\left(\frac{1}{S_{m}}\right)^{k}}{1-\frac{1}{S_{m}}}\right)}$
$\displaystyle\geq$ $\displaystyle\frac{1}{\displaystyle
1+\frac{1}{S_{k}}\left(\frac{S_{m}^{k}-1}{S_{m}^{k-1}(S_{m}-1)}\right)}$
Since
$\displaystyle\displaystyle\left(\frac{S_{m}^{k}-1}{S_{m}^{k-1}(S_{m}-1)}\right)$
$\displaystyle=$
$\displaystyle\left(1+\frac{S_{m}^{k-1}-1}{S_{m}^{k-1}(S_{m}-1)}\right)=1+\Theta\displaystyle\left(\frac{1}{S_{m}}\right)\leq\frac{11}{10}$
$\displaystyle\mbox{Pr}_{k}({\cal G})$ $\displaystyle\geq$
$\displaystyle\frac{1}{1+\frac{1}{\displaystyle\frac{11S_{k}}{10}}}\geq\frac{10}{21}$
This gives us,
$\mbox{Pr}_{k}(\pi_{T})\geq\frac{10}{21}-\frac{1}{e}\geq\frac{20}{189}$
∎
Notice that our proof still goes through even if we choose a $\lambda$ that is
larger than $S_{m}$ (we can take $\lambda=2^{m}$). Therefore $\lambda$ can be
represented using $m$ bits. We can now prove the main theorem,
###### Theorem 4.
Given a graph $G=(V,E)$ with $|V|=n$ and $|E|=m$, there exists a randomized
algorithm that runs in $O(m\log^{2}n)$ time and finds a maximum matching with
high probability.
###### Proof.
Each step of the algorithm runs in $O(1)$ time (we just need to maintain an
array which indicates whether the $i^{th}$ vertex is occupied in the matching
or not), thus one call of Algorithm 1 runs in $O(m\log n)$ time which by
Theorem 3 lands on a maximum matching with probability $\frac{20}{169}$. Thus
calling it $10\log n$ times independently, ensures that we land on a maximum
matching in one call is at least $1-\left(\frac{169}{189}\right)^{10\log
n}=1-\frac{1}{n^{\Omega(1)}}$. ∎
## 7 Lower Bound via Conductance
The conductance method as defined in the Section is used to obtain lower
bounds on the mixing time of the a Markov chain. To get a lower bound on
$T_{relax}$ we need an upper-bound on $\phi$, and by definition of $\phi$, for
any cut $(S,\bar{S})$.
$\phi\leq\displaystyle\left(\frac{\sum_{i\in
S,j\in\bar{S}}\pi_{i}P_{ij}}{(\sum_{i\in
S}\pi_{i})(\sum_{i\in\bar{S}}\pi_{i})}\right)$
This allows us to observe:
###### Lemma 4.
For any graph $G$, the conductance of our Markov Chain satisfies
$\phi\leq O\left(\frac{k}{m}\right)$
where $k$ is the size of maximum matching.
###### Proof.
To give an upper bound on conductance we need to construct a cut $(S,\bar{S})$
for which we can estimate the above quantity. Let $S$ be consisting of exactly
one matching which is the maximum matching $m_{k}$ where $k$ is the size of
the maximum matching. Thus the number of edges going out of $S$ is $k$.
$\displaystyle\frac{\sum_{i\in S,j\in\bar{S}}\pi_{i}P_{ij}}{(\sum_{i\in
S}\pi_{i})(\sum_{i\in\bar{S}}\pi_{i})}=\frac{\frac{\lambda^{k}}{Z}\cdot
k\cdot\frac{1}{m(\lambda+1)}}{\frac{\lambda^{k}}{Z}\left(1-\frac{\lambda^{k}}{Z}\right)}=\frac{k}{(1+\lambda)m}\cdot\frac{1}{\left(1-\frac{\lambda^{k}}{Z}\right)}$
Also using the terminology of Theorem 3
$\displaystyle\displaystyle\frac{\lambda^{k}}{Z}=\frac{\lambda^{k}}{\sum_{i=0}^{k}S_{i}\lambda^{i}}=\frac{1}{\sum_{i=0}^{k}S_{k-i}\frac{1}{\lambda^{i}}}\leq\frac{1}{\sum_{i=0}^{k}\frac{1}{\lambda^{i}}}=\frac{\lambda^{k}(\lambda-1)}{\lambda^{k+1}-1}=1-\Theta\left(\frac{1}{\lambda}\right)$
Therefore,
$\displaystyle\frac{\sum_{i\in S,j\in\bar{S}}\pi_{i}P_{ij}}{(\sum_{i\in
S}\pi_{i})(\sum_{i\in\bar{S}}\pi_{i})}=\frac{k}{(1+\lambda)m}\cdot\frac{1}{\Theta\left(\frac{1}{\lambda}\right)}=\Theta\left(\frac{k}{m}\right)$
Thus the result follows. ∎
As a result of the previous lemma we have the following result.
###### Lemma 5.
For any graph $G$, the mixing time of our Markov chain satisfies,
$T_{mix}\geq\Omega\left(\frac{m}{k}\right)$
###### Proof.
Follows from results in Section 4.3. ∎
Thus the lower bound is sharp (upto logarithmic factors) if the size of the
matching in the given graph is small (say at most $O(\mbox{poly}\log n)$).
A Note on Other methods to prove Lower Bound: There are other methods, apart
from conductance, which can be used to prove lower bounds on mixing time.
Although powerful and useful in many contexts, it is not clear whether such
methods could be applied to our chain. For eg. the Wilson’s method [1] expects
the knowledge of one eigenvector of the transition vector that is different
from the all 1’s vector and the corresponding eigenvalue lies in the range
$\left(0,\frac{1}{2}\right)$. Since the matrix $P$ for our case is an
exponential sized matrix with apparently no useful pattern in the entries, it
is not clear how to come up with such an eigenvector. In fact, we made several
educated guesses for coming up with such a vector all of which failed to serve
our purpose.
## 8 Conclusion
In this paper, we gave a new randomized algorithm for finding maximum
matchings in general bipartite graphs that runs in $O(m\log^{2}n)$ time that
improves upon the running time of Micali and Vazirani. Our algorithm was based
on the MCMC paradigm which performs a truncated random walk on the Markov
Chain defined by the Glauber Dynamics with parameter $\lambda$. Apart from the
benefit of being very simple (both in analysis and implementation) our
algorithm is the first near linear time complexity algorithm for the maximum
matching problem for general graphs. Moreover, unlike [15] the running time of
our algorithm is not a random variable.
To our knowledge this is for the first time Glauber dynamics and the nature of
Gibbs distribution has been exploited to design an faster algorithm for a
problem for which efficient solutions are already known, and we hope this idea
can be of use in other problems as well. The obvious open problem will be to
improve both the upper-bounds and lower bounds on the mixing time. Is it
possible to improve upon the present bound to get an $O(m)$ time algorithm?
Also one can explore the possibility of proving a tighter lower bound of
$\Omega(m)$ by using more refined techniques. More specifically, it would be
interesting to see if one can obtain an explicit eigenvector of the transition
matrix $P$ (that is different from all 1’s vector) and apply Wilson’s to get
an improved lower bound on the mixing time.
## 9 Acknowledgements
Manjish Pal would like to thank Eric Vigoda for answering his mails about the
Gibbs distribution. Thanks to Purushottam Kar for his feedback on the
presentation of the paper.
## References
* [1] N. Berestycki, Eight lectures on Mixing Times, 2009.
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* [3] R. Cole and J.E. Hopcroft, On edge coloring bipartite graphs, SIAM J. Comput., 11(3), 1982, pp. 540-546.
* [4] R. Cole, K. Ost, and S. Schirra, Edge-coloring bipartite multigraphs in $O(E\log D)$ time, Combinatorica, 21(1), 2001, pp. 5-12.
* [5] M. Dyer, A. Frieze and M. Jerrum, On Counting Independent Sets In Sparse Graphs, SIAM Journal on Computing 31(5), 2001, pp. 1527-1541.
* [6] M. Dyer, A. Frieze and R. Kannan, A random polynomial time algorithm for approximating the volume of convex sets, Journal of the Association for Computing Machinery, 38,1991, pp. 1-17.
* [7] M. Dyer and C. Greenhill, On Markov chains for independent sets, Journal of Algorithms 35(1) 2000, pp. 17-49.
* [8] M. Dyer, L. A. Goldberg and M. Jerrum, Markov chain comparison, Probability Surveys Vol. 3, 2006, pp. 89 111.
* [9] J. Edmonds, Paths, Trees and Flowers, Canadian J. 1965, pp. 449-467
* [10] H. Gabow, Implementation of Algorithms for Maximum Matching of Graphs, PhD Thesis, 1973
* [11] A. Galanis, Q. Ge, D. Stefankovic, E. Vigoda and L. Yang., Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model, arXiv:1105.5131v1 , 2011.
* [12] H.N. Gabow and O. Kariv, Algorithms for edge coloring bipartite graphs and multigraphs, SIAM J. Comput., 11(1), 1982, pp. 117 129.
* [13] A. Goel, M. Kapralov, and S. Khanna, Perfect matchings via uniform sampling in regular bipartite graphs, Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms, 2009, pp. 11-17.
* [14] A. Goel, M. Kapralov, and S. Khanna, Perfect matchings in $O(n^{1.5})$ time in regular bipartite graphs., arXiv:0902.1617v2, 2009.
* [15] A. Goel, M. Kapralov, and S. Khanna, Perfect Matchings in $O(n\log n)$ Time in Regular Bipartite Graphs, STOC, 2010.
* [16] J.E. Hopcroft and R.M. Karp, An $n^{\frac{5}{2}}$ algorithm for maximum matchings in bipartite graphs, SIAM J.Comput., 2(4),1973, pp. 225-231.
* [17] Mark Jerrum and Alistair Sinclair, The Markov chain Monte Carlo method: an approach to approximate counting and integration , ”Approximation Algorithms for NP-hard Problems,” D.S.Hochbaum ed., PWS Publishing, Boston, 1996.
* [18] M.R. Jerrum and A.J. Sinclair, Approximating the permanent, SIAM Journal on Computing, 1989 vol. 18, pp.1149 1178.
* [19] D. K$\ddot{o}$nig, Uber graphen und ihre anwendung auf determinententheorie und mengenlehre, Math. Annalen, 77,1916, pp. 453-465.
* [20] T. Kameda and I. Munro, A $O(|V|\cdot|E|)$ Algorithm for Maximum Matchings of Graphs, Computing, vol.12, pp. 91-98, 1974.
* [21] R. Kleinberg and Eva Tardos, Algorithm Design, Addison Wesley, 2005.
* [22] E.G. Lawler, Combinatorial Optimization Theory, Holt, Rinehart and Winston, Chapter 6, pp. 217-239, 1976.
* [23] D. A. Levin, Y. Peres, E. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009.
* [24] S. Micali and V. V. Vazirani, An $O(|E|\sqrt{|V|})$ time Algorithm for Maximum Matchings, FOCS, 1980, pp.17-27.
* [25] R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.
* [26] E. Mossel, D. Weitz and N. Wormald, On the hardness of sampling independent sets beyond the tree threshold, Probability Theory and Related Fields, 2009 vol. 143, no. 3-4, pp. 401-439.
* [27] F. Martinelli, A. Sinclair and D. Weitz, Fast mixing for independent sets, colorings and other models on trees, SODA, 2004, pp. 456-465
* [28] R. Restrepo, D. Stefankovic, J.C. Vera, E. Vigoda and L. Yang, Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees, arXiv:1007.2255v1, 2010.
* [29] A. Sly, Computational Transition at the Uniqueness Threshold, FOCS, 2010, pp. 287-296.
* [30] D. Weitz, Counting independent sets up to the tree threshold, STOC, 2006, pp. 140-149.
* [31] E. J.Vigoda, Sampling from Gibbs distributions, 1999, Phd Thesis.
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|
arxiv-papers
| 2011-07-13T08:03:38 |
2024-09-04T02:49:20.470012
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anant Jindal, Gazal Kochar, Manjish Pal",
"submitter": "Manjish Pal",
"url": "https://arxiv.org/abs/1107.2482"
}
|
1107.2512
|
# Deformation of algebras associated to group cocycles
Makoto Yamashita makotoy@ms.u-tokyo.ac.jp
###### Abstract.
We define a deformation of algebras endowed with coaction of the reduced group
algebras. The deformation parameter is given by a $2$-cocycle over the group.
We prove $K$-theory isomorphisms for the cocycles which can be perturbed to
the trivial one.
###### Key words and phrases:
deformation, Fell bundle, K-theory
###### 2010 Mathematics Subject Classification:
Primary 46L80; Secondary 46L65, 58B34
## 1\. Introduction
Deformation of algebras has been an important principle in the study of
operator algebras and noncommutative geometry. The noncommutative torus, whose
‘function algebra’ is generated by two unitaries $u,v$ satisfying
$uv=e^{i\theta}vu$ for $\theta\in\mathbb{R}$, is one of the most famous
examples which lead to many interesting ideas in the early studies of
noncommutative geometry by Connes [MR823176] and others. The relation between
the generators of the algebra of noncommutative torus indicates that it can be
thought as a deformation of the algebra of functions on the $2$-torus.
It turned out that the noncommutative torus is an example of a more general
deformation procedure called the $\theta$-deformation due to Rieffel
[MR1002830]. It takes any $C^{*}$-algebra admitting an action of a torus
$\mathbb{T}^{n}$ as the original algebra, and the deformation parameter is
given by a skewsymmetric form on $\mathbb{R}^{n}$. He showed that the
$\theta$-deformations have the same $K$-groups as the original algebras
[MR1237992], extending the case of the noncommutative torus by
Pimsner–Voiculescu [MR587369].
The noncommutative torus can be also thought as the twisted group algebra for
the case of the discrete group $\mathbb{Z}^{2}$. Recently, a $K$-theory
isomorphism result of the reduced twisted group algebras
$C^{*}_{r,\omega}(\Gamma)$ was proved for any discrete group $\Gamma$
satisfying the Baum–Connes conjecture with the compact operator algebra
coefficient by Echterhoff et al. [MR2608195]. They showed that the $K$-groups
of the twisted algebras $C^{*}_{r,\omega}(\Gamma)$ do not change if $\omega$
is given by a real $2$-cocycle on $\Gamma$, which can be thought as a
continuous perturbation of the trivial cocycle. We note that Mathai
[MR2218025] also proved the $K$-theory invariance under twisting by such
cocycles for a slightly different class of groups, building on Lafforgue’s
Banach algebraic approach [MR1914617] to the Baum–Connes conjecture.
In this paper we unify the two frameworks mentioned above. We thus define a
way to deform the $C^{*}$-algebras admitting coactions of the compact quantum
group $C^{*}_{r}(\Gamma)$ (in other words, cross sectional algebras of Fell
bundles over $\Gamma$ [MR1488064]), and the deformation parameter is given by
a $\mathrm{U}(1)$-valued $2$-cocycle on $\Gamma$. Our main result (Theorem 1)
is that, when $\Gamma$ satisfies the Baum–Connes conjecture with coefficients
and the cocycle comes from an $\mathbb{R}$-valued $2$-cocycle, the $K$-groups
of the deformed algebra are isomorphic to those of the original algebra.
We also remark that there are several similar formalisms of deformation of
operator algebras which do not fall into our approach. The deformation of Fell
bundles due to Abadie–Exel [MR1856986] seems to be closest to ours. The
deformation of function algebras of compact groups which appeared in the study
of ergodic actions with full multiplicity by Wassermann [MR990110] is a close
analogue of the twisted group algebra. Finally, there is a similar
$K$-theoretic invariance result by Neshveyev–Tuset [arXiv:1103.4346] for
certain $C^{*}$-algebraic compact quantum groups and its homogeneous spaces,
for the $q$-deformations of simply connected simple compact Lie groups.
#### Acknowledgements
This paper was written during the author’s stay at Department of Mathematical
Sciences, Copenhagen University. He would like to thank Copenhagen University
for their support and hospitality. He is also grateful to Ryszard Nest,
Takeshi Katsura, and Reiji Tomatsu for stimulating discussions and fruitful
comments.
## 2\. Preliminaries
When $A$ and $B$ are $C^{*}$-algebras, $A\otimes B$ denotes their minimal
tensor product unless otherwise specified. Likewise when $H$ and $K$ are
Hilbert spaces, $H\otimes K$ denotes their tensor product as a Hilbert space.
When $H$ is a Hilbert space and $X$ is a Hilbert $C^{*}$-module over $A$, we
let $H\otimes X$ denote the tensor product Hilbert $C^{*}$-module over $A$. We
let $\mathcal{L}\left(X\right)$ denote the algebra of the endomorphisms of a
Hilbert $C^{*}$-module $X$.
When $A$ is a $C^{*}$-algebra, we let $M(A)$ denote the multiplier algebra.
The coactions of locally compact quantum groups on $C^{*}$-algebras are
assumed to be the continuous ones. The crossed products with respect to
(co)actions of locally compact quantum groups on $C^{*}$-algebras are
understood to be the reduced ones unless otherwise specified. Our convention
is that, when $\alpha$ is an action of a discrete group $\Gamma$ on a
$C^{*}$-algebra $A\subset B(H)$, the reduced crossed product is the
$C^{*}$-algebra generated by the operators
$\lambda_{g}\otimes\operatorname{Id}_{H}$ for $g\in\Gamma$ and the ones
$\tilde{\alpha}(a)\colon\delta_{g}\otimes\xi\mapsto\delta_{g}\otimes\alpha_{g}(a)\xi\quad(g\in\Gamma,\xi\in
H,a\in A)$
on $\ell^{2}(\Gamma)\otimes H$.
### 2.1. Group cocycles
Let $\Gamma$ be a discrete group. When $(G,+)$ is a commutative group, a
$G$-valued $2$-cocycle $\omega$ on $\Gamma$ is a map
$\omega\colon\Gamma\times\Gamma\rightarrow G$ satisfying the cocycle identity
(1)
$\omega(g_{0},g_{1})+\omega(g_{0}g_{1},g_{2})=\omega(g_{1},g_{2})+\omega(g_{0},g_{1}g_{2}).$
We always assume that $\omega$ satisfies the normalization condition
$\omega(g,e)=\omega(e,g)=1\quad(g\in\Gamma).$
In this paper we consider the cases $G=\mathbb{R}$ and $G=\mathrm{U}(1)$ as
the target group of cocycles. Note that when $\omega$ is an
$\mathbb{R}$-valued $2$-cocycle, we obtain a $\mathrm{U}(1)$-valued cocycle
$e^{i\omega}$ by putting $e^{i\omega}(g,h)=e^{i\omega(g,h)}$.
When $\omega$ is a $\mathrm{U}(1)$-valued $2$-cocycle on $\Gamma$, the twisted
reduced group $C^{*}$-algebra $C^{*}_{r,\omega}(\Gamma)$ is defined to be the
$C^{*}$-algebraic span of the operators $\lambda^{(\omega)}_{g}\in
B(\ell^{2}\Gamma)$ for $g\in\Gamma$ defined by
$\lambda^{(\omega)}_{g}\delta_{h}=\omega(g,h)\delta_{gh}.$
Given $\Gamma$ and $\omega$, we can consider the fundamental unitary
$W=\sum_{g}\delta_{g}\otimes\lambda_{g}$ and another unitary operator
$\sum_{g,h}\omega(g,h)\delta_{g}\otimes\delta_{h}$ representing $\omega$, both
on $\ell^{2}(\Gamma)^{\otimes 2}$. Then the unitary operator
(2)
$W^{(\omega)}=W\omega\colon\delta_{h}\otimes\delta_{k}\mapsto\beta(h,k)\delta_{h}\otimes\delta_{hk}$
in the von Neumann algebra $\ell^{\infty}(\Gamma)\otimes B(\ell^{2}(\Gamma))$
is called the regular $\omega$-representation unitary. The algebra
$C^{*}_{r,\omega}(\Gamma)$ can be also defined as the $C^{*}$-algebraic span
of the operators $\phi\otimes\iota(W^{(\omega)})$ for
$\phi\in\ell^{1}(\Gamma)=\ell^{\infty}(\Gamma)_{*}$.
The generators $(\lambda^{(\omega)}_{g})_{g\in\Gamma}$ satisfy the relations
$\displaystyle\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}$
$\displaystyle=\omega(g,h)\lambda^{(\omega)}_{gh},$
$\displaystyle(\lambda^{(\omega)}_{g})^{*}$
$\displaystyle=\overline{\omega(g,g^{-1})}\lambda^{(\omega)}_{g^{-1}}.$
From this we see that the vector state for $\delta_{e}$ is tracial. This trace
is called the standard trace $\tau$ on $C^{*}_{r,\omega}(\Gamma)$.
Two cocycles $\omega$ and $\omega^{\prime}$ are said to be cohomologous when
there exists a map $\psi\colon\Gamma\rightarrow\mathrm{U}(1)$ satisfying
$\psi(g)\psi(h)\omega(g,h)\overline{\psi(gh)}=\omega^{\prime}(g,h).$
If this is the case, the algebras $C^{*}_{r,\omega}(\Gamma)$ and
$C^{*}_{r,\omega^{\prime}}(\Gamma)$ are isomorphic via the map
$\lambda^{(\omega)}_{g}\mapsto\overline{\psi(g)}\lambda^{(\omega^{\prime})}_{g}$.
We let $\overline{\omega}$ denote the complex conjugate cocycle
$\overline{\omega}(g,h)=\overline{\omega(g,h)}$. Then the twisted algebra
$C^{*}_{r,\overline{\omega}}(\Gamma)$ is antiisomorphic to
$C^{*}_{r,\omega}(\Gamma)$ as follows. The formula
(3) $\tilde{\omega}(g,h)=\omega(h^{-1},g^{-1})$
defines another cocycle on $\Gamma$. On one hand, the map
$\lambda^{(\omega)}_{g}\mapsto\lambda^{(\tilde{\omega})}_{g^{-1}}$ defines an
antiisomorphism from $C^{*}_{r,\omega}(\Gamma)$ to
$C^{*}_{r,\tilde{\omega}}(\Gamma)$. On the other hand, the equality
$\begin{split}\omega(h^{-1},g^{-1})\omega(g,h)\omega(gh,h^{-1}g^{-1})&=\omega(h^{-1},g^{-1})\omega(h,h^{-1}g^{-1})\omega(g,g^{-1})\\\
&=\omega(h,h^{-1})\omega(g,g^{-1})\end{split}$
shows that $\overline{\omega}$ and $\tilde{\omega}$ are cohomologous with
respect to the map $g\mapsto\omega(g,g^{-1})$.
### 2.2. Crossed product presentation of twisted group algebras
The reduced group algebra $C^{*}_{r}(\Gamma)$ admits the structure of (the
function algebra of) a compact quantum group by the coproduct map
$\delta(\lambda_{g})=\lambda_{g}\otimes\lambda_{g}$.
Suppose that $\alpha$ and $\beta$ are $\mathrm{U}(1)$-valued $2$-cocycles on
$\Gamma$. Then, with the unitary regular $\beta$-representation unitary (2),
we have
$W^{(\beta)}(\lambda^{(\alpha\cdot\beta)}_{g}\otimes\operatorname{Id}_{\ell^{2}(\Gamma)})(W^{(\beta)})^{*}=\lambda^{(\alpha)}_{g}\otimes\lambda^{(\beta)}_{g}$
for any $g\in\Gamma$. This way we obtain a $C^{*}$-algebra homomorphism
$C^{*}_{r,\alpha\cdot\beta}(\Gamma)\rightarrow C^{*}_{r,\alpha}(\Gamma)\otimes
C^{*}_{r,\beta}(\Gamma),\quad\lambda^{(\alpha\cdot\beta)}_{g}\mapsto\lambda^{(\alpha)}_{g}\otimes\lambda^{(\beta)}_{g}.$
When either of $\alpha$ or $\beta$ is trivial, we obtain the coactions
$\displaystyle\delta^{(\omega)}_{l}$ $\displaystyle\colon
C^{*}_{r,\omega}(\Gamma)\rightarrow C^{*}_{r}(\Gamma)\otimes
C^{*}_{r,\omega}(\Gamma),$ $\displaystyle\delta^{(\omega)}_{r}$
$\displaystyle\colon C^{*}_{r,\omega}(\Gamma)\rightarrow
C^{*}_{r,\omega}(\Gamma)\otimes C^{*}_{r}(\Gamma)$
of $C^{*}_{r}(\Gamma)$ on the twisted group algebra
$C^{*}_{r,\omega}(\Gamma)$. Note that these two carry the same data because
$C^{*}_{r}(\Gamma)$ is cocommutative.
The crossed product algebra
$C^{*}_{r,\omega}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma)$ with respect to
the coaction $\delta^{(\omega)}_{r}$ is the $C^{*}$-algebra generated by
$\delta^{(\omega)}_{r}(C^{*}_{r,\omega}(\Gamma))$ and $1\otimes C_{0}(\Gamma)$
in $B(\ell^{2}(\Gamma)\otimes\ell^{2}(\Gamma))$.
This crossed product is actually isomorphic to the compact operator algebra
$\mathcal{K}(\ell^{2}(\Gamma))\simeq\Gamma\ltimes_{\lambda}C_{0}(\Gamma)\simeq
C^{*}_{r}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma),$
where $\lambda$ in the middle denotes the left translation action of $\Gamma$
on $C_{0}(\Gamma)$. This isomorphism is given by the map
(4)
$C^{*}_{r,\omega}(\Gamma)\rtimes_{\delta^{(\omega)}_{r}}C_{0}(\Gamma)\rightarrow
C^{*}_{r}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma),\quad\lambda^{(\omega)}_{g}\delta_{h}\mapsto\omega(g,h)\lambda_{g}\delta_{h}.$
The crossed product
$C^{*}_{r,\omega}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma)$ admits the dual
action $\hat{\delta}^{(\omega)}_{r}$ of $\Gamma$ defined by
$(\hat{\delta}^{(\omega)}_{r})_{k}(\lambda^{(\omega)}_{g}\delta_{h})=\lambda^{(\omega)}_{g}\delta_{hk^{-1}}.$
If we regard $\hat{\delta}^{(\omega)}_{r}$ as an action of $\Gamma$ on
$C^{*}_{r}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma)$ via the isomorphism (4),
the dual coaction can be expressed as
(5)
$(\hat{\delta}^{(\omega)}_{r})_{k}(\lambda_{g}\delta_{h})=\overline{\omega(g,h)}\omega(g,hk^{-1})\lambda_{g}\delta_{hk^{-1}}.$
By the Takesaki–Takai duality, the crossed product
$\mathcal{K}(\ell^{2}(\Gamma))\rtimes_{\hat{\delta}^{(\omega)}_{r}}\Gamma$ is
strongly Morita equivalent to $C^{*}_{r,\omega}(\Gamma)$.
### 2.3. Coaction of quantum groups and braided tensor products
Suppose that $A$ is a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra. Thus, $A$ admits a
coaction $\alpha$ of $C^{*}_{r}(\Gamma)$ given by a homomorphism
$\alpha\colon A\rightarrow C^{*}_{r}(\Gamma)\otimes A$
which satisfies the multiplicativity
$\iota\otimes\alpha\circ\alpha=\delta\otimes\iota\circ\alpha$ and the
condition that $C^{*}_{r}(\Gamma)_{1}\alpha(A)$ is dense
$C^{*}_{r}(\Gamma)\otimes A$, called the cancellation property or continuity
of $\alpha$. We write the coaction as
$\alpha(x)=\sum_{g}\lambda_{g}\otimes\alpha^{(g)}(x)$. Then
$x=\alpha^{(g)}(x)$ is equivalent to $\alpha(x)=\lambda_{g}\otimes x$. Note
that linear span $A_{\text{fin}}$ of such elements, the elements of finite
spectrum, are dense in $A$. This fact will be frequently utilized later to
verify the images of various homomorphisms.
Suppose that $A$ is represented on a Hilbert space $H$. Then a unitary $X\in
M(C^{*}_{r}(\Gamma)\otimes\mathcal{K}(H))$ is said to be a covariant
representation for $\alpha$ if it satisfies
$\delta\otimes\iota(X)=X_{13}X_{23}$ and $X^{*}(1\otimes a)X=\alpha(a)$.
By analogy with the case of $\Gamma=\mathbb{Z}^{2}$ [MR2738561]*Section 3, we
would like to consider ‘the diagonal coaction’
$\alpha\otimes\delta^{(\omega)}_{l}$ of $C^{*}_{r}(\Gamma)$ on $A\otimes
C^{*}_{r,\omega}(\Gamma)$. Nonetheless, a naive attempt
$A\otimes C^{*}_{r,\omega}(\Gamma)\rightarrow C^{*}_{r}(\Gamma)\otimes
A\otimes C^{*}_{r,\omega}(\Gamma),\quad a\otimes
x\mapsto\alpha(a)_{12}\delta^{(\omega)}_{l}(x)_{13}$
does not define an algebra homomorphism unless $\Gamma$ is commutative. To
remedy this we use the notion of braided tensor product instead.
We consider an action $\mathrm{Ad}^{(\omega)}$ of $\Gamma$ on
$C^{*}_{r,\omega}(\Gamma)$ given by
$\mathrm{Ad}^{(\omega)}_{g}(\lambda^{(\omega)}_{h})=\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}(\lambda^{(\omega)}_{g})^{*}=\omega(g,h)\omega(gh,g^{-1})\overline{\omega(g,g^{-1})}\lambda^{(\omega)}_{ghg^{-1}}.$
Let $\widetilde{\mathrm{Ad}}^{(\omega)}$ denote the algebra homomorphism
$C^{*}_{r,\omega}(\Gamma)\rightarrow M(C_{0}(\Gamma)\otimes
C^{*}_{r,\omega}(\Gamma)),\quad
x\mapsto\sum_{h}\delta_{h}\otimes\mathrm{Ad}^{(\omega)}_{h^{-1}}(x).$
This is implemented as the adjoint by the $\omega$-representation unitary
$W^{(\omega)}$, and satisfies
$\iota\otimes\widetilde{\mathrm{Ad}}^{(\omega)}\circ\widetilde{\mathrm{Ad}}^{(\omega)}=\hat{\delta}\otimes\iota\circ\widetilde{\mathrm{Ad}}^{(\omega)}$.
Hence it defines a coaction of the dual quantum group
$(C_{0}(\Gamma),\hat{\delta})$.
Combined with the coaction $\delta^{(\omega)}_{l}$ of $C^{*}_{r}(\Gamma)$, the
algebra $C^{*}_{r,\omega}(\Gamma)$ becomes a
$\Gamma$-Yetter–Drinfeld-$C^{*}$-algebra [MR2566309]. Indeed, it amounts to
verifying the commutativity of the diagram [MR2566309]*Definition 3.1
(6)
$\begin{CD}C^{*}_{r,\omega}(\Gamma)@>{\delta^{(\omega)}_{l}}>{}>\hat{S}\otimes
C^{*}_{r,\omega}(\Gamma)@>{\iota\otimes\widetilde{\mathrm{Ad}}^{(\omega)}}>{}>M(\hat{S}\otimes
S\otimes C^{*}_{r,\omega}(\Gamma))\\\
@V{}V{\widetilde{\mathrm{Ad}}^{(\omega)}}V@V{}V{\Sigma_{12}}V\\\ M(S\otimes
C^{*}_{r,\omega}(\Gamma))@>{\iota\otimes\delta^{(\omega)}_{l}}>{}>M(S\otimes\hat{S}\otimes
C^{*}_{r,\omega}(\Gamma))@>{\mathrm{Ad}_{W}}>{}>M(S\otimes\hat{S}\otimes
C^{*}_{r,\omega}(\Gamma))\end{CD},$
where $\hat{S}=C^{*}_{r}(\Gamma)$, $S=C_{0}(\Gamma)$, $W$ is the fundamental
unitary $\sum_{h}\delta_{h}\otimes\lambda_{h}$ in $M(C_{0}(\Gamma)\otimes
C^{*}_{r}(\Gamma))$, and $\Sigma$ is the transposition of tensors. If we track
the image of $\lambda^{(\omega)}_{g}\in C^{*}_{r,\omega}(\Gamma)$ along the
top-right arrows, we obtain
$\lambda^{(\omega)}_{g}\mapsto\lambda_{g}\otimes\lambda^{(\omega)}_{g}\mapsto\sum_{h}\lambda_{g}\otimes\delta_{h}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}\mapsto\sum_{h}\delta_{h}\otimes\lambda_{g}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}$
Similarly, if we go along the left-bottom arrows, we obtain
$\lambda^{(\omega)}_{g}\mapsto\sum_{h}\delta_{h}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}\mapsto\sum_{h}\delta_{h}\otimes\lambda_{hgh^{-1}}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}\\\
\mapsto\sum_{h}\delta_{h}\otimes\lambda_{g}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h},$
where we used
$\begin{split}\delta^{(\omega)}_{l}((\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h})&=\overline{\omega(h,h^{-1})}\omega(h^{-1},g)\omega(h^{-1}g,h)\lambda_{h^{-1}gh}\otimes\lambda^{(\omega)}_{h^{-1}gh}\\\
&=\lambda_{h^{-1}gh}\otimes(\lambda^{(\omega)}_{h})^{*}\lambda^{(\omega)}_{g}\lambda^{(\omega)}_{h}.\end{split}$
Combining these, we conclude that the diagram (6) is commutative.
As proved in [MR2566309]*Theorem 3.2, a Yetter–Drinfeld algebra is the same
thing as an algebra endowed with a coaction of the Drinfeld dual. In our
setting the Drinfeld dual $\mathsf{D}(\Gamma)$ of $\Gamma$ is represented by
the algebra $C_{0}(\mathsf{D}(\Gamma))=C_{0}(\Gamma)\otimes C^{*}_{r}(\Gamma)$
endowed with the coproduct
$\Delta=(\Sigma\circ\mathrm{Ad}_{W})_{23}\circ\hat{\delta}\otimes\delta\colon\delta_{h}\otimes\lambda_{g}\mapsto\sum_{h^{\prime}h^{\prime\prime}=h}(\delta_{h^{\prime}}\otimes\lambda_{h^{\prime\prime}gh^{\prime\prime-1}})\otimes(\delta_{h^{\prime\prime}}\otimes\lambda_{g}).$
The above Yetter–Drinfeld algebra structure on $C^{*}_{r,\omega}(\Gamma)$
corresponds to the coaction
$C^{*}_{r,\omega}(\Gamma)\rightarrow M(C_{0}(\mathsf{D}(\Gamma))\otimes
C^{*}_{r,\omega}(\Gamma)),\quad\lambda^{(\omega)}_{g}\mapsto\sum_{h}\delta_{h}\otimes\lambda_{h^{-1}gh}\otimes\mathrm{Ad}^{(\omega)}_{h^{-1}}(\lambda^{(\omega)}_{g}).$
Let $A$ be a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra and $\omega$ be a
$\mathrm{U}(1)$-valued $2$-cocycle on $\Gamma$. The braided tensor product
$A\boxtimes C^{*}_{r,\omega}(\Gamma)$ of $A$ and $C^{*}_{r,\omega}(\Gamma)$
[MR2566309]*Definition 3.3 is the $C^{*}$-algebra of operators on the Hilbert
$C^{*}$-module $\ell^{2}(\Gamma)\otimes A\otimes C^{*}_{r,\omega}(\Gamma)$
generated by the operators of the form
$\alpha(a)_{12}\widetilde{\mathrm{Ad}}^{(\omega)}(x)_{13}$ for $a\in A$ and
$x\in C^{*}_{r,\omega}(\Gamma)$. By means of the conditional expectation
$\iota\otimes\tau$ from $A\otimes C^{*}_{r,\omega}(\Gamma)$ onto $A$, we may
regard $A\boxtimes C^{*}_{r,\omega}(\Gamma)$ as a subalgebra of
$\mathcal{L}\left(\ell^{2}(\Gamma)\otimes A\otimes\ell^{2}(\Gamma)\right)$.
Note that our convention (the Yetter–Drinfeld algebra being the second
component in the braided tensor product) is different from that of
[MR2566309]*Definition 3.3.
By [MR2182592]*Proposition 8.3, we have
$A\boxtimes
C^{*}_{r,\omega}(\Gamma)=\overline{\alpha(A)_{12}\widetilde{\mathrm{Ad}}^{(\omega)}(C^{*}_{r,\omega}(\Gamma))_{13}}$
as linear spaces of $\mathcal{L}\left(\ell^{2}(\Gamma)\otimes
A\otimes\ell^{2}(\Gamma)\right)$.
By [MR2566309]*the remark after Definition 3.3, the braided tensor product
$A\boxtimes C^{*}_{r,\omega}(\Gamma)$ admits a coaction
$\alpha\otimes\delta^{(\omega)}_{l}$ of $C^{*}_{r}(\Gamma)$ which we shall
call the diagonal coaction. It is given by
$\alpha\otimes\delta^{(\omega)}_{l}(\alpha(a)_{12}\widetilde{\mathrm{Ad}}^{(\omega)}(x)_{13})=\delta\otimes\iota(\alpha(a))_{123}\iota\otimes\widetilde{\mathrm{Ad}}^{(\omega)}(\delta^{(\omega)}_{l}(x))_{124}.$
### 2.4. Exterior equivalence of actions
Let us briefly recall the notion of exterior equivalence between the
(co)actions on $C^{*}$-algebras by the locally compact quantum groups of our
interest.
Let $\alpha$ and $\beta$ be actions of $\Gamma$ on a $C^{*}$-algebra $A$.
These two actions are said to be exterior equivalent when there exists a
family $(u_{g})_{g\in\Gamma}$ of unitaries in $M(A)$ satisfying
$u_{g}\alpha_{g}(u_{h})=u_{gh}$ and
$\beta_{g}=\mathrm{Ad}_{u_{g}}\circ\alpha_{g}$ for any $g,h\in\Gamma$. Two
actions $\alpha$ and $\beta$ of $\Gamma$ on different algebras $A$ and $B$ are
said to be outer conjugate if there is an isomorphism $\phi\colon B\rightarrow
A$ such that the action $(\phi\beta_{g}\phi^{-1})_{g}$ on $A$ is exterior
equivalent to $\alpha$.
Outer conjugate actions define isomorphic crossed products with conjugate dual
(co)actions. An action is exterior equivalent to the trivial one if and only
if it is the conjugation action with respect to a group homomorphism from
$\Gamma$ into the unitary group of $M(A)$.
Similarly, two coactions $\alpha$ and $\beta$ of $C^{*}_{r}(\Gamma)$ on a
$C^{*}$-algebra $A$ is said to be exterior equivalent when there is a unitary
element $X$ in $C^{*}_{r}(\Gamma)\otimes A$ satisfying
$X_{23}\iota\otimes\alpha(X)=\delta\otimes\iota(X)$ and
$X\alpha(x)X^{*}=\beta(x)$ for $x\in A$. Such $X$ is called an
$\alpha$-cocycle.
## 3\. Deformation of algebras
###### Definition 1.
Let $A$ be a $C^{*}$-algebra with a coaction $\alpha$ of $C^{*}_{r}(\Gamma)$,
and $\omega$ be a $\mathrm{U}(1)$-valued $2$-cocycle on $\Gamma$. We define
the deformation $A_{\alpha,\omega}$ of $A$ with respect to $\alpha$ and
$\omega$ (the $\omega$-deformation of $A$) to be the fixed point algebra
$(A\boxtimes C^{*}_{r,\overline{\omega}}(\Gamma))^{C^{*}_{r}(\Gamma)}$ under
the diagonal coaction $\alpha\otimes\delta^{(\overline{\omega})}_{l}$. When
there is no source of confusion for $\alpha$ we write $A_{\omega}$ instead of
$A_{\alpha,\omega}$.
###### Proposition 2.
Let $\Gamma$, $\omega$, and $A$ be as above. Then the deformed algebra
$A_{\omega}$ is isomorphic to the subalgebra $A^{\prime}_{\omega}$ of
$C^{*}_{r,\omega}(\Gamma)\otimes A$ consisting of the elements $x$ satisfying
$\iota\otimes\alpha(x)_{213}=\delta^{(\omega)}_{l}\otimes\iota(x)$.
###### Proof.
Note that the $C^{*}$-algebras $A\boxtimes
C^{*}_{r,\overline{\omega}}(\Gamma)$ and $C^{*}_{r,\omega}(\Gamma)\otimes
A\otimes C^{*}_{r,\overline{\omega}}(\Gamma)$ are represented on
$\ell^{2}(\Gamma)\otimes A\otimes C^{*}_{r,\overline{\omega}}(\Gamma)$. We
have a homomorphism $\Phi$ from the former to the latter by $x\mapsto
W^{(\overline{\omega})}_{13}x(W^{(\overline{\omega})})^{*}_{13}$. The effect
of $\Phi$ on the generators of $A\boxtimes
C^{*}_{r,\overline{\omega}}(\Gamma)$ is described by
$\displaystyle\alpha(x)_{12}$
$\displaystyle\mapsto\sum_{g}\lambda^{(\omega)}_{g}\otimes\alpha^{(g)}(x)\otimes\lambda^{(\overline{\omega})}_{g},$
$\displaystyle\widetilde{\mathrm{Ad}}^{(\overline{\omega})}(y)_{13}$
$\displaystyle\mapsto y_{3}.$
Thus the image of $\Phi$ is $A^{\prime}_{\omega}\otimes
C^{*}_{r,\omega}(\Gamma)$, and the corresponding coaction of
$C^{*}_{r}(\Gamma)$ is simply given by
$(\iota\otimes\delta^{(\overline{\omega})}_{l})_{213}$. Hence the fixed point
algebra is given by $A^{\prime}_{\omega}$. ∎
###### Corollary 3.
When the $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra $(A,\alpha)$ is given by the pair
$(C^{*}_{r}(\Gamma),\delta)$, the deformed algebra $A_{\omega}$ is isomorphic
to $C^{*}_{r,\omega}(\Gamma)$.
###### Proof.
By Proposition 2, we may identify the braided tensor product with the
subalgebra of $C^{*}_{r,\omega}(\Gamma)\otimes C^{*}_{r}(\Gamma)$ spanned by
$\lambda^{(\omega)}_{g}\otimes\lambda_{g}$ for $g\in\Gamma$. As this is equal
to the image of $\delta^{(\omega)}_{l}$, we obtain the assertion. ∎
###### Corollary 4.
Let $A$ be a $C^{*}$-algebra with a coaction $\alpha$ of $C^{*}_{r}(\Gamma)$.
When the cocycle $\omega$ is trivial, the deformed algebra $A_{\omega}$ is
isomorphic to $A$.
###### Proof.
In this case the algebra $A^{\prime}_{\omega}$ in Proposition 2 is the image
of $\alpha$. Hence we obtain $A_{\omega}\simeq A$. ∎
###### Corollary 5.
When the coaction $\alpha$ is trivial, $A_{\omega}$ is isomorphic to $A$ for
any $2$-cocycle $\omega$.
###### Remark 6.
When $a\in A_{\text{fin}}$, we can consider an element
$\sum_{g}\lambda^{(\omega)}_{g}\otimes\alpha^{(g)}(a)$ in
$A^{\prime}_{\omega}$. We let $a^{(\omega)}$ denote the corresponding element
in $A_{\omega}$. The $\omega$-deformation $A_{\omega}$ can be regarded as a
certain $C^{*}$-algebraic completion of the vector space
$\left\\{a^{(\omega)}\mid a\in A_{\text{fin}}\right\\}\simeq A_{\text{fin}}$
endowed with the twisted $*$-algebra structure
$\displaystyle a^{(\omega)}b^{(\omega)}$
$\displaystyle=\sum_{g,h}\omega(g,h)(\alpha^{(g)}(a)\alpha^{(h)}(b))^{(\omega)},$
$\displaystyle(a^{(\omega)})^{*}$
$\displaystyle=\sum_{g}\overline{\omega(g,g^{-1})}(\alpha^{(g)}(a)^{*})^{(\omega)}.$
###### Example 7.
Let $A$ be a $\mathbb{T}^{n}$-$C^{*}$-algebra for some $n$, and
$(\theta_{jk})_{jk}$ be a skewsymmetric real matrix of size $n$. Then the
$\theta$-deformation $A_{\theta}$ of $A$ is given by $(A\otimes
C(\mathbb{T}^{n})_{\theta})^{\mathbb{T}^{n}}$, where
$C(\mathbb{T}^{n})_{\theta}$ is the universal $C^{*}$-algebra generated by $n$
unitaries $u_{1},\ldots,u_{n}$ satisfying
$u_{j}u_{k}=e^{i\theta_{jk}}u_{k}u_{j}$, and $\mathbb{T}^{n}$ acts on
$A\otimes C(\mathbb{T}^{n})_{\theta}$ by the diagonal action.
The algebra $C(\mathbb{T}^{n})_{\theta}$ can be regarded as the twisted group
algebra of $\mathbb{Z}^{n}$ with the $2$-cocycle $\omega(x,y)=e^{i(\theta
x,y)}$. By Proposition 2, $A_{\theta}$ can be identified with $A_{\omega}$.
###### Example 8.
Let $B$ be a $\Gamma$-$C^{*}$-algebra. Then the reduced crossed product
$\Gamma\ltimes B$ is a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra by the dual
coaction. If $\omega$ is a $2$-cocycle on $\Gamma$, the deformed algebra
$(\Gamma\ltimes B)_{\omega}$ can be identified with the twisted reduced
crossed product $\Gamma\ltimes_{\alpha,\omega}B$ [MR0241994].
There is another coaction of $C^{*}_{r}(\Gamma)$ on $A\boxtimes
C^{*}_{r,\omega}(\Gamma)$, given by
$\alpha(x)_{12}\widetilde{\mathrm{Ad}}^{(\overline{\omega})}(y)_{13}\mapsto\iota\otimes\alpha\circ\alpha(x)_{123}\widetilde{\mathrm{Ad}}^{(\overline{\omega})}(y)_{24}.$
We denote this coaction by $\alpha_{\omega}$. It is implemented as the adjoint
with the dual fundamental unitary
$\hat{W}=\sum_{g}\lambda_{g}\otimes\delta_{g}$. It can be easily seen from the
definitions that the two coactions $\alpha_{\omega}$ and
$\alpha\otimes\delta^{(\overline{\omega})}_{l}$ of $C^{*}_{r}(\Gamma)$ commute
with each other. Hence $\alpha_{\omega}$ restricts to the fixed point
subalgebra $A_{\omega}$ of $\alpha\otimes\delta^{(\overline{\omega})}_{l}$.
###### Remark 9.
When $\omega$ and $\eta$ are $\mathrm{U}(1)$-valued $2$-cocycles on $\Gamma$,
we have $(A_{\omega})_{\eta}=A_{\omega\cdot\eta}$ for any
$C^{*}_{r}(\Gamma)$-$C^{*}$-algebra $A$.
We have the following generalization of the isomorphism (4).
###### Proposition 10.
The crossed product algebra $C_{0}(\Gamma)\ltimes_{\alpha}A_{\omega}$ is
isomorphic to the corresponding algebra $C_{0}(\Gamma)\ltimes_{\alpha}A$ of
the untwisted case.
###### Proof.
We identify $A_{\omega}$ with the algebra $A^{\prime}_{\omega}$ of Proposition
2. Thus, the crossed product
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega}}A_{\omega}$ is represented by the
$C^{*}$-algebra of operators generated by $(\delta_{h})_{1}$ and
$\sum_{g}\lambda_{g}\otimes\lambda^{(\omega)}_{g}\otimes\alpha^{(g)}(x)$ on
$\ell^{2}(\Gamma)^{\otimes 2}\otimes A$.
Let $V$ be the unitary operator
$\delta_{k}\otimes\delta_{k^{\prime}}\mapsto\overline{\omega}(k^{-1},k)\omega(k^{-1},k^{\prime})\delta_{k}\otimes\delta_{k^{\prime}}$.
The assertion follows once we prove that the image of
$\Phi=\mathrm{Ad}_{V_{12}}\colon A^{\prime}_{\omega}\rightarrow
B(\ell^{2}(\Gamma)^{\otimes 2}\otimes A)$ is equal to
$C_{0}(\Gamma)\ltimes
A^{\prime}=\vee\left\\{(\delta_{h})_{1},\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x)\mid
h\in\Gamma,x\in A\right\\}.$
If $h\in\Gamma$ and $x\in A^{\prime}_{\omega}$ has finite spectrum, the action
of $\Phi(\alpha_{\omega}(x)(\delta_{h})_{1})$ on the vector
$\delta_{k}\otimes\delta_{k^{\prime}}\otimes b$ is given by
$\sum_{g}\delta_{h,k}\omega(k^{-1},k)\overline{\omega(k^{-1},k^{\prime})}\overline{\omega(k^{-1}g^{-1},gk)}\omega(k^{-1}g^{-1},gk^{\prime})\delta_{gh}\otimes\delta_{gk^{\prime}}\otimes\alpha^{(g)}(x)b.$
Using the cocycle identity for $\omega$, we see that this is equal to
$\sum_{g}\omega(g,h)\delta_{h,k}\delta_{gh}\otimes\delta_{gk^{\prime}}\otimes\alpha^{(g)}(x)b,$
which is equal to the action of
$\sum_{g}\omega(g,h)\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x)(\delta_{h})_{1}$.
This operator is indeed in $C_{0}(\Gamma)\ltimes A^{\prime}$. ∎
We have the following expression of $\hat{\alpha}_{\omega}$
(7)
$(\hat{\alpha}_{\omega})_{k}((\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x))(\delta_{h})_{1})\\\
=\omega(g,hk^{-1})\overline{\omega(g,h)}(\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x))(\delta_{hk^{-1}})_{1}),$
regarded as an action on $C_{0}(\Gamma)\ltimes_{\alpha}A$ via the isomorphism
$\Phi$ in the proof of Proposition 10. By the Takesaki–Takai duality,
$A_{\omega}$ is strongly Morita equivalent to the crossed product
$\Gamma\ltimes_{\hat{\alpha}_{\omega}}C_{0}(\Gamma)\ltimes_{\alpha_{\omega}}A_{\omega}$
with respect to the dual action $\hat{\alpha}_{\omega}$ of $\alpha_{\omega}$
by $\Gamma$.
For each $g\in\Gamma$, let $A_{g}$ denote the corresponding spectral subspace
consisting of the elements $x\in A$ satisfying $\alpha^{(g)}(x)=x$. Recall
that the Fell bundle $(A_{g})_{g\in\Gamma}$ associated to $A$ has the
approximation property [MR1488064]*Definition 4.4 when there is a sequence
$a_{i}$ of functions from $\Gamma$ into $A_{e}$ satisfying
(8) $\sup_{i}\left\|\sum_{g}a_{i}(g)^{*}a_{i}(g)\right\|<\infty$
and
(9) $\lim_{i}\sum_{h}a_{i}(gh)^{*}ba_{i}(h)=b\quad(g\in\Gamma,b\in A_{g}).$
If $\Gamma$ is amenable, any $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra has the
approximation property. This property also holds when $A$ is given as
$\Gamma\ltimes_{\beta}B$ for some amenable action $\beta$ of a discrete group
$\Gamma$ on a unital $C^{*}$-algebra $B$.
###### Lemma 11.
Let $\Gamma$, $\omega$, and $A$ be as above. The Fell bundle associated to $A$
has the approximation property if and only if the one associated to
$A_{\omega}$ has the approximation property.
###### Proof.
The algebra $(A_{\omega})_{e}$ is naturally isomorphic to $A_{e}$. Hence we
may regard $a_{i}$ as a sequence of functions with values in $A_{\omega}$.
Then the condition (8) is automatic. The other one (9) follows from the
equalities
$\displaystyle\left\|b^{(\omega)}\right\|$ $\displaystyle=\left\|b\right\|,$
$\displaystyle a_{i}(gh)^{*}b^{(\omega)}a_{i}(h)$
$\displaystyle=(a_{i}(gh)^{*}ba_{i}(h))^{(\omega)}$
for any $g\in\Gamma$ and $b\in A_{g}$. ∎
We have the following adaptation of [MR1237992]*Theorem 4.1 in our context.
###### Proposition 12.
Let $\Gamma$, $\omega$, and $A$ be as above, and suppose that the Fell bundle
associated to $A$ has the approximation property. Then $A_{\omega}$ is nuclear
if and only if $A$ is nuclear.
###### Proof.
The Fell bundle associated to $A_{\omega}$ also has the approximation property
by Lemma 11. By Remark 9, it is enough to prove that $A_{\omega}$ is nuclear
when $A$ is nuclear.
By the amenability of the Fell bundle associated to $A_{\omega}$, the maximal
and the reduced crossed product coincide for the dual action of $\Gamma$ on
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega}}A_{\omega}$ [MR1895615]*Corollary 3.6.
Since $C_{0}(\Gamma)\ltimes_{\alpha_{\omega}}A_{\omega}$ is nuclear by
Proposition 10, we conclude that its crossed product by $\Gamma$ is also
nuclear, c.f. [MR1926869]*the proof of Theorem 5.3, (2) $\Rightarrow$ (3). ∎
###### Proposition 13.
Let $\Gamma$, $\omega$ be as above, and $A$ be a
$C^{*}_{r}(\Gamma)$-$C^{*}$-algebra represented on a Hilbert space $H$.
Suppose that there is a covariant representation $X\in
M(C^{*}_{r}(\Gamma)\otimes\mathcal{K}(H))$ of $C^{*}_{r}(\Gamma)$. Then, the
action of $C^{*}_{r,\omega}(\Gamma)\otimes A$ on $\ell^{2}(\Gamma)\otimes H$
restricts to the one of the algebra $A^{\prime}_{\omega}$ of Proposition 2 on
$X^{*}(\delta_{e}\otimes H)$.
###### Proof.
Recall that the dual fundamental unitary
$\hat{W}=\sum_{g}\lambda_{g}^{*}\otimes\delta_{g}$ satisfies
$\delta(x)=\hat{W}^{*}(1\otimes x)\hat{W}$. Hence
$\delta\otimes\iota(X^{*})=\delta\otimes\iota(X^{*})_{213}=X^{*}_{13}X^{*}_{23}$
implies
$X^{*}_{13}X^{*}_{23}(\delta_{e}\otimes\delta_{e}\otimes\xi)=\mathrm{Ad}_{\hat{W}^{*}_{12}}(X^{*}_{23})(\delta_{e}\otimes\delta_{e}\otimes\xi)=\hat{W}^{*}_{12}X^{*}_{23}(\delta_{e}\otimes\delta_{e}\otimes\xi).$
for $\xi\in H$. Thus, any $\eta\in X^{*}(\delta_{e}\otimes H)$ satisfies
$X^{*}_{13}(\delta_{e}\otimes\eta)=\hat{W}^{*}_{12}(\delta_{e}\otimes\eta)$.
Conversely, if we had
$X^{*}_{13}(\delta_{e}\otimes\xi)=\hat{W}^{*}_{12}(\delta_{e}\otimes\xi)$ for
some $\xi\in\ell^{2}(\Gamma)\otimes H$, we can write $\xi$ as
$\sum_{g}\delta_{g}\otimes\xi_{g}$ and conclude that
$X^{*}(\delta_{e}\otimes\xi_{g})=\delta_{g}\otimes\xi_{g}$ for any $g$, i.e.,
$\xi=X^{*}(\delta_{e}\otimes\sum_{g}\xi_{g})$. Hence we can identify
$X^{*}(\delta_{e}\otimes H)$ with the subspace $\left\\{\xi\mid
X^{*}_{13}\xi=\hat{W}^{*}_{12}\xi\right\\}$ of
$\delta_{e}\otimes\ell^{2}(\Gamma)\otimes H$ via the embedding
$\xi\mapsto\delta_{e}\otimes\xi$.
By the covariance of $X$, we can characterize $A^{\prime}_{\omega}$ as the
subalgebra of $C^{*}_{r,\omega}(\Gamma)\otimes A$ satisfying
$\hat{W}^{*}_{12}(1\otimes a)\hat{W}_{12}=X^{*}_{13}(1\otimes a)X_{13}.$
If $\xi\in X^{*}(\delta_{e}\otimes H)$ and $a\in A^{\prime}_{\omega}$, one has
$X^{*}_{13}(1\otimes a)(\delta_{e}\otimes\xi)=\hat{W}^{*}_{12}(1\otimes
a)\hat{W}_{12}X^{*}_{13}(\delta_{e}\otimes\xi)=\hat{W}^{*}_{12}(1\otimes
a)(\delta_{e}\otimes\xi),$
which proves the assertion. ∎
###### Proposition 14.
Let $\alpha$ and $\beta$ be exterior equivalent coactions of
$C^{*}_{r}(\Gamma)$ on $A$, and $\omega$ be a $2$-cocycle on $\Gamma$. Then
the corresponding deformed algebras $A_{\alpha,\omega}$ and $A_{\beta,\omega}$
are strongly Morita equivalent.
###### Proof.
Let $U$ be an $\alpha$-cocycle satisfying $U\alpha(x)U^{*}=\beta(x)$. As in
the standard argument, the rank $1$ Hilbert $A$-module $A$ admits a coaction
of $C^{*}_{r}(\Gamma)$ defined by
$X_{U}\colon\xi\otimes
x\mapsto\alpha(x)U^{*}\xi_{1}\quad(\xi\in\ell^{2}(\Gamma),x\in A).$
This coaction is covariant with respect to the coaction $\alpha$ on $A$ for
the left $A$-module structure and $\beta$ for the right. Then, as in
Proposition 13, we can take the closed subspace $X_{U}(\delta_{e}\otimes A)$
in the Hilbert $C^{*}$-module $C^{*}_{r,\omega}(\Gamma)\otimes A$ which is
closed under the left action of $A^{\prime}_{\alpha,\omega}$ and the right
action of $A^{\prime}_{\beta,\omega}$. This bimodule is the imprimitivity
bimodule between the two algebras. ∎
###### Corollary 15.
Let $A$ be a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra and $\omega$ be a $2$-cocycle
on $\Gamma$. Then the deformed algebra $A_{\omega}$ is strongly Morita
equivalent to the twisted crossed product
$\Gamma\ltimes_{\hat{\alpha},\omega}C_{0}(\Gamma)\ltimes_{\alpha}A$.
###### Proof.
The double dual coaction of $C^{*}_{r}(\Gamma)$ on the iterated crossed
product
$C^{*}_{r}(\Gamma)\ltimes_{\hat{\alpha}}C_{0}(\Gamma)\ltimes_{\alpha}A$ and
the amplification of $\alpha$ on $\mathcal{K}(\ell^{2}(\Gamma))\otimes A$ are
outer conjugate by the Takesaki–Takai duality. The assertion follows from
Proposition 14 and the natural identification $(\mathcal{K}\otimes
A)_{\omega}\simeq\mathcal{K}\otimes A_{\omega}$. ∎
This corollary shows that the twisted crossed product (Example 8) is the
universal example up to the strong Morita equivalence. We can also see that
Proposition 10, and the resulting strong Morita equivalence between
$A_{\omega}$ and
$\Gamma\ltimes_{\hat{\alpha}_{\omega}}C_{0}(\Gamma)\ltimes_{\alpha}A$ is an
adaptation of the ‘untwisting’ of twisted crossed products by Packer–Raeburn
[MR1002543]*Corollary 3.7.
### 3.1. $K$-theory isomorphism of deformed algebras
Let $\omega$ be a normalized $\mathrm{U}(1)$-valued $2$-cocycle on $\gamma$.
For each $k\in\Gamma$, consider the unitary element
(10)
$v_{k}=\left(\sum_{g}\omega(gk,k^{-1})\delta_{g}\right)\left(\sum_{h}\lambda_{hk^{-1}h^{-1}}\delta_{h}\right)$
in
$M(C^{*}_{r}(\Gamma)\rtimes_{\delta_{r}}C_{0}(\Gamma))=B(\ell^{2}(\Gamma))$.
The second sum is actually the unitary $\rho_{k}$ which implements the right
translation $\delta_{h}\mapsto\delta_{hk^{-1}}$. From the relation
$\lambda_{g^{\prime-1}}v_{k}\lambda_{g^{\prime}}=\sum_{g}\omega(gk,k^{-1})\delta_{g^{\prime-1}g}\rho_{k}=\sum_{g}\omega(g^{\prime}gk,k^{-1})\delta_{g}\rho_{k},$
we conclude
$v_{k}\lambda_{g^{\prime}}v_{k}^{-1}=\lambda_{g^{\prime}}\sum_{g}\omega(g^{\prime}gk,k^{-1})\overline{\omega(gk,k^{-1})}\delta_{g}.$
Combining this and
$v_{k}\delta_{h^{\prime}}v_{k}^{-1}=\rho_{k}\delta_{h^{\prime}}\rho_{k}^{-1}=\delta_{h^{\prime}k^{-1}},$
we obtain
$v_{k}\lambda_{g^{\prime}}\delta_{h^{\prime}}v_{k}^{-1}=\omega(g^{\prime}h^{\prime},k^{-1})\overline{\omega(h^{\prime},k^{-1})}\lambda_{g^{\prime}}\delta_{h^{\prime}k^{-1}}.$
By the cocycle condition for $\omega$ and (5), we see that the right hand side
above is equal to
$(\hat{\delta}^{(\omega)}_{r})_{k}(\lambda_{g^{\prime}}\delta_{h^{\prime}})$.
The failure of the multiplicativity of $(v_{k})_{k}$ is controlled by the
cocycle $\omega$. By
$\begin{split}v_{k}v_{k^{\prime}}&=\sum_{g}\omega(gk,k^{-1})\delta_{g}\rho_{k}\sum_{h}\omega(hk^{\prime},k^{\prime-1})\delta_{h}\rho_{k^{\prime}}\\\
&=\sum_{g=hk^{-1}}\omega(gk,k^{-1})\omega(hk^{\prime},k^{\prime-1})\delta_{g}\rho_{kk^{\prime}}\end{split}$
and the cocycle relation (1) for $g_{0}=gkk^{\prime}$, $g_{1}=k^{\prime-1}$,
and $g_{2}=k^{-1}$, we obtain
(11) $v_{k}v_{k^{\prime}}=\omega(k^{\prime-1},k^{-1})v_{kk^{\prime}}.$
###### Remark 16.
Suppose that the cocycle $\omega$ above is of the form $e^{i\omega_{0}}$ for
some $\mathbb{R}$-valued $2$-cocycle $\omega_{0}$. Then we obtain its opposite
cocycle $\tilde{\omega}_{0}$ as in (3). When $H$ is a finite subgroup of
$\Gamma$, the $2$-cocycle $\tilde{\omega}_{0}|_{H}$ is a coboundary because of
$H^{2}(H,\mathbb{R})$ is trivial. Hence there exists a map $\phi$ from $H$
into $\mathbb{R}$ satisfying
(12)
$\tilde{\omega}_{0}(h_{0},h_{1})=\phi(h_{0})-\phi(h_{0}h_{1})+\phi(h_{1})\quad(h_{0},h_{1}\in
H).$
The normalization condition on $\omega$ implies the one $\phi(e)=0$ for
$\phi$. The unitaries $(e^{-i\phi(h)}v_{h})_{h\in H}$ are multiplicative by
(11), and they implement the action $\hat{\delta}^{(\omega)}_{r}|_{H}$ on
$C^{*}_{r}\rtimes C_{0}(\Gamma)$ modulo the isomorphism (4) by (5).
Now, assume that $\omega$ is induced by an $\mathbb{R}$-valued $2$-cocycle
$\omega_{0}$ as above. Our goal is to show that the $K$-groups of $A_{\omega}$
are isomorphic to those of $A$.
Let $I$ denote the closed unit interval $[0,1]$. Generalizing the method of
[MR2608195]*Section 1, we put $\omega_{\theta}=e^{i\theta\omega_{0}}$ for
$\theta\in I$ and consider the following $C^{*}$-$C(I)$-algebra
$A_{\omega_{\star}}$ over $I$, whose fiber at $\theta$ is given by
$A_{\omega_{\theta}}$.
First, we consider the Hilbert space $L^{2}(I;\ell^{2}(\Gamma))\simeq
L^{2}(I)\otimes\ell^{2}(\Gamma)$, and the operators
$\lambda^{(\omega_{\star})}_{g}$ for $g\in\Gamma$ defined by
$(\lambda^{(\omega_{\star})}_{g}\xi)_{\theta}=\lambda^{(\omega_{\theta})}_{g}\xi_{\theta}\quad(\xi\in
L^{2}(I;\ell^{2}(\Gamma)),\theta\in I).$
Thus we obtain a $C^{*}$-$C(I)$-algebra $C^{*}_{r,\omega_{\star}}(\Gamma)$,
given as the $C^{*}$-algebra generated by these operators and the natural
action of $C(I)$ on $L^{2}(I;\ell^{2}(\Gamma))$.
Next, we see that $C^{*}_{r,\omega_{\star}}(\Gamma)$ is a
$\Gamma$-Yetter–Drinfeld algebra by the coaction
$\lambda^{(\omega_{\star})}_{g}\mapsto\lambda_{g}\otimes\lambda^{(\omega_{\star})}_{g},f\mapsto
1\otimes f\quad(g\in\Gamma,f\in C(I))$
of $C^{*}_{r}(\Gamma)$ and the one
$\lambda^{(\omega_{\star})}_{g}\mapsto\delta_{h}\otimes(\lambda^{(\omega_{\star})}_{h})^{*}\lambda^{(\omega_{\star})}_{g}\lambda^{(\omega_{\star})}_{h},f\mapsto
1\otimes f\quad(g,h\in\Gamma,f\in C(I))$
of $C_{0}(\Gamma)$. This $C(I)$-algebra and its $\Gamma$-Yetter–Drinfeld
algebra structure is induced by the twisted fundamental unitary
$W^{(\omega_{\star})}=W\omega_{\star}\colon\delta_{g}\otimes\delta_{h}\mapsto(e^{\theta
i\omega_{0}(g,h)}\delta_{g}\otimes\delta_{g,h})_{\theta}$
on $L^{2}(I;\ell^{2}(\Gamma)^{\otimes 2})$ which commutes with $C(I)$.
Thus we can take the braided tensor product $A\boxtimes
C^{*}_{r,\omega_{\star}}(\Gamma)$ which is again a $C(I)$-$C^{*}$-algebra with
a compatible coaction of $C^{*}_{r}(\Gamma)$. The algebra $A_{\omega_{\star}}$
is defined to be the fixed point algebra for this coaction. By an argument
analogous to Proposition 2, this is isomorphic to the subalgebra
$A^{\prime}_{\omega_{\star}}$ of $A\otimes C^{*}_{r,\omega_{\star}}(\Gamma)$
consisting of the elements $a$ satisfying
$\alpha_{13}(a)=\delta^{(\omega_{\star})}_{12}(a)$.
The algebra $A_{\omega_{\star}}$ admits a coaction $\alpha_{\omega_{\star}}$
of $C^{*}_{r}(\Gamma)$ defined in the obvious way. The crossed product
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A_{\omega_{\star}}$ is a
$\Gamma$-$C^{*}$-$C(I)$-algebra, and the evaluation at each fiber is a
$\Gamma$-homomorphism.
###### Lemma 17.
The $C^{*}$-$C(I)$-algebra
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A_{\omega_{\star}}$ is
isomorphic to the constant field with fiber $C_{0}(\Gamma)\ltimes_{\alpha}A$.
###### Proof.
The proof is essentially the same as that of Proposition 10. The formula
$(V\delta_{k}\otimes\delta_{k^{\prime}})_{\theta}=\overline{\omega_{\theta}}(k^{-1},k)\omega_{\theta}(k^{-1},k^{\prime})\delta_{k}\otimes\delta_{k^{\prime}}.$
defines unitary operator $V$ on $L^{2}(I;\ell^{2}(\Gamma)^{\otimes 2})$ which
commutes with $C(I)$. When $x\in A$ and $h\in\Gamma$, the constant section
$(\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x))(\delta_{h})_{1}$
of $C(I)\otimes C_{0}(\Gamma)\ltimes_{\alpha}A^{\prime}$ is mapped to the
element
$\left(\left(\sum_{g}\overline{\omega_{\theta}(g,h)}\lambda_{g}\otimes\lambda^{(\omega_{\theta})}_{g}\otimes\alpha^{(g)}(x)\right)(\delta_{h})_{1}\right)_{\theta}$
of
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A^{\prime}_{\omega_{\star}}$. ∎
Thus, $A_{\omega_{\star}}$ is
$\operatorname{\mathcal{R}\mathit{KK}}(I,-,-)$-equivalent to the crossed
product of $C(I)\otimes C_{0}(\Gamma)\ltimes_{\alpha}A$ by an action of
$\Gamma$ corresponding to $\hat{\alpha}_{\star}$ via the isomorphism of Lemma
17. Using (7), we can express this action as
(13)
$\left((\hat{\alpha}_{\star})_{k}\left(\left(\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x)\right)(\delta_{h})_{1}\right)\right)_{\theta}\\\
=\omega_{\theta}(g,hk^{-1})\overline{\omega_{\theta}(g,h)}\left(\sum_{g}\lambda_{g}\otimes\lambda_{g}\otimes\alpha^{(g)}(x)\right)(\delta_{hk^{-1}})_{1}$
for $x\in A$ and $h,k\in\Gamma$.
###### Remark 18.
The $C^{*}$-$C(I)$-algebra $A_{\omega_{\star}}$ becomes a continuous field of
$C^{*}$-algebras when the Fell bundle associated to $A$ has the approximation
property [MR990592]*Corollary 2.7, see also the proof of Proposition 12.
###### Proposition 19.
Let $H$ be any finite subgroup of $\Gamma$ and $\theta\in I$. Then the
restriction of $\hat{\alpha}_{\star}$ to $H$ is outer conjugate to the
restriction of the constant field of the action
$\hat{\alpha}_{\omega_{\theta}}$.
###### Proof.
We first prove the assertion for the case $\theta=0$. As in Remark 16, we can
take a map $\phi$ from $H$ to $\mathbb{R}$ satisfying (12). Now, consider the
unitaries
$(w_{k})_{\theta^{\prime}}=e^{-i\theta^{\prime}\phi(k)}\sum_{g}\omega_{\theta^{\prime}}(gk,k^{-1})\delta_{g}\quad(\theta^{\prime}\in
I)$
in $M(C(I)\otimes C_{0}(\Gamma))$ for $k\in H$. This is a
$\hat{\delta}_{r}$-cocycle. Indeed, we have
(14)
$\left(w_{k}(\hat{\delta}_{r})_{k}(w_{k^{\prime}})\right)_{\theta^{\prime}}\\\
=e^{-i\theta^{\prime}\phi(k)}\left(\sum_{g}\omega_{\theta^{\prime}}(gk,k^{-1})\delta_{g}\right)e^{-i\theta^{\prime}\phi(k^{\prime})}\left(\sum_{g^{\prime}}\omega_{\theta^{\prime}}(g^{\prime}k^{\prime},k^{\prime-1})\delta_{g^{\prime}k^{-1}}\right)\\\
=e^{-i\theta^{\prime}(\phi(k)+\phi(k^{\prime}))}\sum_{g}\omega_{\theta^{\prime}}(gk,k^{-1})\omega_{\theta^{\prime}}(gkk^{\prime},k^{\prime-1})\delta_{g}.$
Using (12), one sees that $e^{-i\theta^{\prime}(\phi(k)+\phi(k^{\prime}))}$ is
equal to
$e^{-\theta^{\prime}\phi(kk^{\prime})}\overline{\omega_{\theta^{\prime}}(k^{\prime-1},k^{-1})}$.
Applying (1) for $g_{0}=gk$, $g_{1}=k^{\prime-1}$, and $g_{2}=k^{-1}$, we see
that the right hand side of (14) is equal to $w_{kk^{\prime}}$.
If we regard $(w_{k})_{k}$ as elements of $M(C_{0}(\Gamma)\ltimes_{\alpha}A)$,
they are $\hat{\alpha}_{\star}$-cocycle by definition of the dual (co)action.
We next see that they implement the conjugation between $\hat{\alpha}$ and
$\hat{\alpha}_{\omega_{\star}}$. Indeed, recalling that $\hat{\alpha}$ is the
conjugation by $(\rho_{g})_{g}$, we can compute
$\left(\mathrm{Ad}_{(w_{k})_{1}}\circ\hat{\alpha}_{k}\left(\left(\sum_{g}\lambda_{g}\alpha^{(g)}(x)\right)(\delta_{h})_{1}\right)\right)_{\theta^{\prime}}\\\
=\mathrm{Ad}_{(v_{k}^{(\theta^{\prime})})_{1}}\left(\left(\sum_{g}\lambda_{g}\alpha^{(g)}(x)\right)(\delta_{h})_{1}\right)\\\
=\sum_{g}\omega_{\theta^{\prime}}(gh,k^{-1})\overline{\omega_{\theta^{\prime}}(h,k^{-1})}\lambda_{g}\delta_{hk^{-1}}\lambda_{g}\alpha^{(g)}(x)(\delta_{h})_{1}$
using the unitaries $(v_{k}^{(\theta^{\prime})})_{k}$ defined in the same way
as (10) but $\omega$ being replaced by $\omega_{\theta^{\prime}}$. By (13) and
the cocycle identity for $\omega_{\theta^{\prime}}$, the right hand side of
the above formula is indeed equal to $\hat{\alpha}_{\star}$. Thus we obtain
the outer conjugacy of the actions of $H$ for $\theta=0$.
For the general value of $\theta$, we can argue in the same way as above that
the actions $\hat{\alpha}$ and $\hat{\alpha}_{\omega_{\theta}}$ are outer
conjugate. Thus we can compose the above outer conjugacy with the constant
field of conjugacy between $\hat{\alpha}$ and
$\hat{\alpha}_{\omega_{\theta}}$, which implies the assertion for the
arbitrary value of $\theta$. ∎
We recall that the ‘left hand side’ of the Baum–Connes conjecture with
coefficients can be computed in the following way.
###### Proposition 20 ([MR2608195]*Proposition 1.6).
Let $G$ be a second countable locally compact group, and $A$ and $B$ be
$G$-$C^{*}$-algebras. If $z\in\operatorname{\mathit{KK}}^{G}(A,B)$ induces an
isomorphism $K^{H}_{*}(A)\rightarrow K^{H}_{*}(B)$ for any compact subgroup
$H$ of $G$, the Kasparov product with $z$ induces an isomorphism from
$K^{\text{top}}_{*}(G;A)$ to $K^{\text{top}}_{*}(G;B)$.
###### Theorem 1.
Let $\Gamma$ be a discrete group satisfying the Baum–Connes conjecture with
coefficients, $A$ be a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra, and $\omega_{0}$
be an $\mathbb{R}$-valued $2$-cocycle on $\Gamma$. Then the $K$-groups
$K_{i}(A_{\omega})$ ($i=0,1$) of the deformed algebra $A_{\omega}$ are
isomorphic to $K_{i}(A)$ for the cocycle $\omega=e^{i\omega_{0}}$.
###### Proof.
It is enough to show that the evaluation map $\operatorname{ev}_{\theta}$ at
$\theta\in I$ for the $C^{*}$-$C(I)$-algebra
$\Gamma\ltimes_{\hat{\alpha}_{\omega_{\star}}}C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A_{\omega_{\star}}$
induces an isomorphism in the $K$-theory for any $\theta$.
Proposition 19 implies that for any finite group $H$ of $\Gamma$, the
$H$-homomorphism $\operatorname{ev}_{\theta}$ induces an isomorphism of the
crossed products by $H$. By the Green–Julg isomorphism $K^{H}_{*}(X)\simeq
K_{*}(H\ltimes X)$ which holds for any $H$-$C^{*}$-algebra $X$, we obtain that
$\operatorname{ev}_{\theta}$ induces an isomorphism on the $K^{H}$-groups. By
Proposition 20, $\operatorname{ev}_{\theta}$ induces an isomorphism
$K^{\text{top}}_{*}(\Gamma;C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A_{\omega_{\star}})\simeq
K^{\text{top}}_{*}(\Gamma;C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\theta}}}A_{\omega_{\theta}}).$
By the assumption on $\Gamma$, the both hand sides are isomorphic to the
$K$-groups of the crossed products by $\Gamma$. ∎
We have a slight variation of the above theorem for the groups satisfying the
strong Baum–Connes conjecture.
###### Theorem 2.
Let $\Gamma$ be a discrete group satisfying the strong Baum–Connes conjecture,
$A$ be a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra, and $\omega_{0}$ be an
$\mathbb{R}$-valued $2$-cocycle on $\Gamma$. Then the deformed algebra
$A_{\omega}$ is $\operatorname{\mathit{KK}}$-equivalent to $A$ for the cocycle
$\omega=e^{i\omega_{0}}$.
###### Proof.
Recall the following formulation of the strong Baum–Connes conjecture due to
Meyer–Nest [MR2193334]. The group $\Gamma$ satisfies the conjecture with
arbitrary coefficients [MR2193334]*Definition 9.1 if and only if the descent
functor
$\operatorname{\mathit{KK}}^{\Gamma}\rightarrow\operatorname{\mathit{KK}},A\mapsto\Gamma\ltimes_{r}A$
maps weak equivalences to isomorphisms [MR2193334]*p. 213.
The evaluation maps for the $C^{*}$-$C(I)$-algebra
$C_{0}(\Gamma)\ltimes_{\alpha_{\omega_{\star}}}A_{\omega_{\star}}$ are weak
equivalences by Proposition 19. Thus, the reduced crossed products by $\Gamma$
are $\operatorname{\mathit{KK}}$-equivalent if $\Gamma$ satisfies the strong
Baum–Connes conjecture. ∎
###### Remark 21.
Suppose that $A$ is nuclear, the Fell bundle associated to $A$ has the
approximation property, and that $\Gamma$ satisfies the strong Baum–Connes
conjecture. Then the continuous field $A_{\omega_{\star}}$ is an
$\operatorname{\mathcal{R}\mathit{KK}}$-fibration in the sense of [MR2511635]
by Proposition 12, Theorem 2, and [MR2511635]*Corollary 1.6.
### 3.2. Deformation of equivariant spectral triples
We see that the ‘equivariant Dirac operators’ for a given coaction of
$C^{*}_{r}(\Gamma)$ give isospectral deformations on the
$\omega$-deformations, which induce the same index map modulo the $K$-theory
isomorphism of Theorem 1.
As in Proposition 13, let $(A,H,X)$ be a covariant representation of a
$C^{*}_{r}(\Gamma)$-$C^{*}$-algebra $A$ on $H$. Suppose that $D$ is a
(possibly unbounded) self-adjoint operator on $H$, and $\mathcal{A}$ is a
subalgebra of $A$ such that $a(1+D^{2})^{-1}$ is compact and $[D,a]$ is
bounded for any $a\in\mathcal{A}$. Thus, $(\mathcal{A},D,H)$ is an odd
spectral triple. By abuse of notation, we let
$\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes D$ the closure of the operator
$\xi\otimes\eta\mapsto\xi\otimes D\eta$ for $\xi\in\ell^{2}(\Gamma)$ and
$\eta\in\operatorname{dom}(D)$.
We assume that $\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes D$ commutes $X$
(in particular, $X$ preserves the domain of
$(\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes D)$) and one has
$\alpha^{g}(a)\in\mathcal{A}$ for any $a\in\mathcal{A}$ and $g\in\Gamma$.
These conditions respectively correspond to the equivariance of the Dirac
operator and the smoothness of the action. We shall call such a spectral
triple as a $C^{*}_{r}(\Gamma)$-equivariant spectral triple. By the
equivariance of $D$, the operator $\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes
D$ restricts to $X^{*}(\delta_{e}\otimes H)$. This restriction is unitarily
equivalent to $D$.
Let $\mathcal{A}_{\text{fin}}$ denote the subalgebra of $\mathcal{A}$
consisting of the elements with finite $\alpha$-spectrum. Then the commutators
of $\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes D$ and
$\sum_{g}\lambda^{(\omega)}_{g}\otimes\alpha^{(g)}(a)\in A^{\prime}_{\omega}$
for $a\in\mathcal{A}_{\text{fin}}$ are bounded. Thus, if we let
$\mathcal{A}_{\omega,\text{fin}}$ denote the algebra generated by the
$a^{(\omega)}$ for $a\in\mathcal{A}_{\text{fin}}$, we obtain a new spectral
triple
$(\mathcal{A}_{\omega,\text{fin}},\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes
D|_{X^{*}(\delta_{e}\otimes H)},X^{*}(\delta_{e}\otimes H)),$
which is an isospectral deformation of the original triple. By means of the
unitary operator $X$ between $H$ and $X^{*}(\delta_{e}\otimes H)$, we consider
this as a spectral triple represented on $H$, denoted by
$(\mathcal{A}_{\omega,\text{fin}},D,H).$
If the original spectral triple $(A,D,H)$ is even, the above construction
gives an even spectral triple over $\mathcal{A}_{\omega,\text{fin}}$ provided
$X$ is compatible with the grading on $H$, that is $X\in
M(C^{*}_{r}(\Gamma)\otimes\mathcal{K}(H)^{\text{even}})$.
Assume that $(A,D,H)$ is an even triple, and let $F=D\left|D\right|^{-1}$ be
the phase of $D$. The above construction of the deformed spectral triple give
a Fredholm module $(F,H)$ over $A_{\omega}$, which is in
$\operatorname{\mathit{KK}}_{0}(A_{\omega},\mathbb{C})$. The next theorem
shows that this element induce the essentially same map on the $K$-group if
$\omega$ is a real $2$-cocycle.
###### Theorem 3.
Suppose that $\Gamma$ satisfies the Baum–Connes conjecture with coefficients
and $\omega_{0}$ be an $\mathbb{R}$-valued $2$-cocycle on $\Gamma$. Let $A$ be
a $C^{*}_{r}(\Gamma)$-$C^{*}$-algebra admitting an equivariant even spectral
triple $(H,D)$. Then the even Fredholm module $(F,H)$ for
$F=D\left|D\right|^{-1}$ induce the same map modulo the isomorphism given in
Theorem 1.
###### Proof.
The isomorphisms of the $K$-groups are induced by the evaluation maps of the
$C^{*}$-$C(I)$-algebra $A^{\prime}_{\omega_{\star}}$.
The algebra $A^{\prime}_{\omega_{\star}}$ acts on the field of Hilbert space
$X(\delta_{e}\otimes H)\otimes L^{2}(I)$ over $I$, and its elements have the
bounded commutator with the self-adjoint operator
$\left(\operatorname{Id}_{\ell^{2}(\Gamma)}\otimes F|_{X(\delta_{e}\otimes
H)}\right)\otimes\operatorname{Id}_{L^{2}(I)}.$
This operator defines an element of
$\operatorname{\mathcal{R}\mathit{KK}}(I;A^{\prime}_{\omega_{\star}},C(I))$.
It is clear from the construction that, if we specialize this element to a
point $\theta\in I$, we obtain the Fredholm module $(F,H)$ on
$A_{\omega_{\theta}}$. ∎
###### Remark 22.
There is a corresponding statement for the odd equivariant spectral triples.
It can be proved in the same way, or can be reduced to the even case by taking
the graded tensor product with the standard odd spectral triple over
$C^{\infty}(S^{1})$.
## 4\. Concluding remarks
###### Remark 23.
Suppose that $G$ is a compact group, $\omega$ is a $2$-cocycle on the dual
$\hat{G}$ of $G$. Wassermann [MR990110] defined a deformation $C(G)_{\omega}$
of $C(G)$ as in [MR990110], endowed with the action $\lambda^{\omega}$ of $G$.
When $G$ is commutative, this construction can be identified with ours. More
generally, we can generalize this construction to arbitrary $2$-cocycles over
discrete quantum groups.
When $A$ is a $G$-$C^{*}$-algebra, we can define its deformation by
$A_{\omega}=(A\otimes C(G)_{\omega})^{G}$. We may expect similar phenomenons
in this context too, but we lack nontrivial examples in this context. For
example, the $\mathrm{U}(1)$-valued $2$-cocycles on the duals of semisimple
compact Lie groups which can be perturbed to the trivial one are always
induced from the dual of the maximal torus [MR996457][arXiv:1011.4569]. In
general, suppose that $H$ is a subgroup of $G$ and $\omega$ is a cocycle in
$L(H)\otimes L(H)\subset L(G)\otimes L(G)$. Then we have the natural
identification $C(G)_{\omega}=\operatorname{Ind}^{G}_{H}C(H)_{\omega}$ which
leads to $A_{\omega}\simeq(\operatorname{Res}^{G}_{H}A)_{\omega}$ for any
$G$-$C^{*}$-algebra $A$. Hence we can reduce the computation to $\hat{H}$
which is an ordinary discrete group if $H$ is commutative. We note that a
recent work of Kasprzak [MR2736961] handles this situation.
###### Remark 24.
The compact quantum groups $C^{*}_{r}(\Gamma)$ can be characterized as the
commutative ones among the general compact quantum groups. The arguments in
Section 3.1 depend on this commutativity in the following way. If $G$ is a
compact group as above and $A$ is a $C^{*}$-algebra endowed with an action
$\alpha$ of $G$, we can define the deformation of $A$ by taking the fixed
point algebra $(A\otimes C(G)_{\omega})^{\alpha\otimes\lambda^{\omega}}$. When
$G$ is commutative, this algebra is invariant under $\alpha$ (or
$\lambda_{\omega}$) by
$\alpha_{g}\otimes\iota\circ\alpha_{h}\otimes\lambda^{\omega}_{h}=\alpha_{gh}\otimes\lambda^{\omega}_{h}=\alpha_{hg}\otimes\lambda^{\omega}_{h}=\alpha_{h}\otimes\lambda^{\omega}_{h}\circ\alpha_{g}\otimes\iota.$
Then we can take the crossed product $G\ltimes_{\alpha}(A\otimes
C(G)_{\omega})^{\alpha\otimes\lambda^{\omega}}$, which is isomorphic to the
corresponding algebra for the case $\omega=1$. This way we can reduce the
problem of $(A\otimes C(G)_{\omega})^{\alpha\otimes\lambda^{\omega}}$ to the
corresponding one for the actions of $\hat{G}$ on $G\ltimes_{\alpha}(A\otimes
C(G))^{\alpha\otimes\lambda}$.
###### Remark 25.
For a noncommutative compact quantum group $G$, one may consider another form
of deformation of the function algebra with respect to a $2$-cocycle on the
dual discrete quantum group. Namely, if $\hat{\delta}$ is the coproduct of
$C^{*}G$ and $\omega$ is a $2$-cocycle,
$\delta_{\omega}=\omega\hat{\delta}\omega^{-1}$ defines another coproduct on
$C^{*}G$. Thus the dual Hopf algebra $H_{\omega}$ of
$(C^{*}G,\delta_{\omega})$ can be regarded as a deformation of $C(G)$.
Moreover, the cocycle condition for $\omega$ can be relaxed to the twist
condition for some associator $\Phi$. A result of Neshveyev–Tuset
[arXiv:1102.0248] for $q$-deformations of simply connected simple compact Lie
groups suggests that the $K$-theory of $H_{\omega}$ do not change if $\omega$
and the associator $\Phi_{\omega}$ vary continuously in an appropriate sense.
## References
|
arxiv-papers
| 2011-07-13T10:16:33 |
2024-09-04T02:49:20.478398
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Makoto Yamashita",
"submitter": "Makoto Yamashita",
"url": "https://arxiv.org/abs/1107.2512"
}
|
1107.2612
|
# Commuting time geometry of ergodic Markov chains
Peter G. Doyle Jean Steiner
(Version 1A9 dated 5 March 2008
No Copyright††thanks: The authors hereby waive all copyright and related or
neighboring rights to this work, and dedicate it to the public domain. This
applies worldwide. )
###### Abstract
We show how to map the states of an ergodic Markov chain to Euclidean space so
that the squared distance between states is the expected commuting time. We
find a minimax characterization of commuting times, and from this we get
monotonicity of commuting times with respect to equilibrium transition rates.
All of these results are familiar in the case of time-reversible chains, where
techniques of classical electrical theory apply. In presenting these results,
we take the opportunity to develop Markov chain theory in a ‘conformally
correct’ way.
## 1 Overview
In an eye-opening paper, Chandra, Raghavan, Ruzzo, Smolensky, and Tiwari [1]
revealed the central importance of expected commuting times for the theory of
time-reversible Markov chains. Here we extend the discussion to general, non-
time-reversible chains.
We begin by showing how to embed the states in a Euclidean space so that the
squared distance between states is the commuting time. In the time-reversible
case, Leibon et al. have used Euclidean embeddings to great effect as a way to
visualize a chain, and reveal natural clustering of states. Our embedding
theorem shows that non-time-reversible chains should be amenable to the same
treatment.
Looking beyond the Euclidean embedding, we find a natural minimax
characterization of commuting times. From this we get the monotonicity law for
commuting times: If all equilibrium interstate transition rates are increased,
then all commuting times are diminished. For time-reversible chains, this
monotonicity law is an ancient and powerful tool. It is questionable how
useful it will prove to be in the general case.
In presenting these results, we will be taking a ‘conformally correct’
approach to Markov chains. Briefly, a conformal change to a Markov chain
changes its equilibrium measure, but not its equilibrium transition rates. The
opportunity to develop this conformally correct approach is at least as
important to us as the particular results we’ll be discussing here.
## 2 The problem
The _commuting time_ $T_{ab}$ between two states $a,b$ of an ergodic Markov
chain is the expected time, starting from $a$, to go to $b$ and then back to
$a$. Evidently $T_{ab}=T_{ba}$ and
$T_{ac}\leq T_{ab}+T_{bc}.$
Thus it might seem natural to think of $T_{ab}$ as a measure of the distance
between $a$ and $b$. But in fact it is most natural to think of $T_{ab}$ as
the _squared distance_ between $a$ and $b$. The reason is that, as we will
see, there is a natural way to identify the states of the chain with points in
a Euclidean space having quadratic form $||x||^{2}$ such that for any states
$a,b$ we have
$T_{ab}=||a-b||^{2}.$
Now that we are interpreting $T_{ab}$ as a squared distance, the inequality
$T_{ac}\leq T_{ab}+T_{bc}$ tells us that
$||a-c||^{2}\leq||a-b||^{2}+||b-c||^{2}.$
This means that all angles $\angle abc$ are acute (at least weakly: some might
be right angles).
Realizing commuting times as squared distances is straight-forward for time-
reversible chains. Here’s a sketch, meant only for orientation: We won’t rely
on any of this below. Time-reversible chains correspond exactly to resistor
networks, with $T_{ab}$ corresponding to the effective resistance between $a$
and $b$. This effective resistance is the energy of a unit current flow from
$a$ to $b$. The energy of a flow is its squared distance with respect to the
energy norm on flows. If we associate to state $i$ the unit current flow from
$i$ to some arbitrary reference vertex (the ‘ground’), then the difference
between the flows associated to $a$ and $b$ will be the unit current flow from
$a$ to $b$, having square norm $T_{ab}$.
The trick will be to extend this result to non-time-reversible chains. Now, it
may in fact be the case that to any chain there corresponds a time-reversible
chain having the same $T$, up to multiplication by a positive constant. This
would immediately take care of the extension beyond the time-reversible case.
It is easy enough to compute what the transition rates of this time-reversible
chain would have to be, but we don’t know that they are always positive. We
leave this question for another day.
Before proceeding, we should observe that the triangle inequality for squared
lengths is not in itself a sufficient condition for realizability of a
Euclidean simplex. It _is_ sufficient for tetrahedra (four vertices in
3-space), but for five vertices we have the following counterexample. Take
$T=\left(\begin{array}[]{ccccc}0&7&7&7&13\\\ 7&0&12&12&7\\\ 7&12&0&12&7\\\
7&12&12&0&7\\\ 13&7&7&7&0\end{array}\right)$
This matrix is not realizable because the associated quadratic form with
matrix
$\frac{1}{2}\left(\begin{array}[]{llll}14&2&2&13\\\ 2&14&2&13\\\ 2&2&14&13\\\
13&13&13&26\end{array}\right)$
is not positive definite: It has the eigenvalue
$\frac{1}{2}(22-\sqrt{523})\approx-0.434597$. Since we’re going to see that
commuting time matrices are always realizable, this means in particular that
this matrix $T$ cannot arise as the matrix of commuting times of a Markov
chain.
## 3 The short answer
Below we will give the honest solution to this problem, developing in a
thoroughgoing way what we will call the ‘conformally correct’ approach to
Markov chains. Here we just extract the answer to our embedding question, and
present it in a way that should be immediately accessible to those familiar
with the standard theory of Markov chains, as developed for example in
Grinstead and Snell [6]. The only caveat is that we will be using tensor
notation, i.e. writing some indices up rather than down. You can look at
section 5 below for remarks about this, but if you prefer you can just view
this as an idiosyncracy, as long as you bear in mind that
$\tensor{Z}{{}_{i}^{j}}$ represents a different array of numbers from
$Z_{ij}$.
Consider a discrete-time Markov chain with transition probabilities
$\tensor{P}{{}_{i}^{j}}=\mathrm{Prob}(\mbox{next at $j$}|\mbox{start at
$i$}).$
Assume the chain is ergodic so there is a unique equilibrium measure $w^{i}$
with
$\sum_{i}w^{i}\tensor{P}{{}_{i}^{j}}=w^{j}$
and
$\sum_{i}w^{i}=1.$
Define
$\Delta^{ij}=w^{i}(\tensor{I}{{}_{i}^{j}}-\tensor{P}{{}_{i}^{j}}),$
and note that
$\sum_{i}\Delta^{ij}=\sum_{j}\Delta^{ij}=0.$
Now define
$\tensor{Z}{{}_{i}^{j}}=(\tensor{I}{{}_{i}^{j}}-w^{j})+(\tensor{P}{{}_{i}^{j}}-w^{j})+(\tensor{{P^{(2)}}}{{}_{i}^{j}}-w^{j})+\ldots,$
where
$\tensor{{P^{(2)}}}{{}_{i}^{j}}=\sum_{k}\tensor{P}{{}_{i}^{k}}\tensor{P}{{}_{k}^{j}}$
represents the matrix square of $\tensor{P}{{}_{i}^{j}}$, and the elided terms
involve higher matrix powers. Using conventional matrix notation if we define
$\tensor{{P^{(\infty)}}}{{}_{i}^{j}}=w^{j}$ we can write
$\displaystyle Z$ $\displaystyle=$
$\displaystyle(I-{P^{(\infty)}})+(P-{P^{(\infty)}})+(P^{(2)}-{P^{(\infty)}})+\ldots$
$\displaystyle=$ $\displaystyle(I-P+{P^{(\infty)}})^{-1}-{P^{(\infty)}}.$
(Note that Grinstead and Snell [6] use the alternate definition
$Z=(I-P+{P^{(\infty)}})^{-1}$, which is less congenial but works just as well
in this context.)
Set
$Z_{ij}=\frac{1}{w^{j}}\tensor{Z}{{}_{i}^{j}}.$
$Z_{ij}$ acts like an inverse to $\Delta^{ij}$ in the sense that for any
$u^{i}$ with $\sum_{i}u^{i}=0$, we have
$\sum_{jk}u^{j}Z_{jk}\Delta^{kl}=u^{l}$
and
$\sum_{jk}\Delta^{ij}Z_{jk}u^{k}=u^{i}.$
Standard Markov chain theory tells us that the expected time $M_{ab}$ to hit
state $b$ starting from state $a$ is
$M_{ab}=Z_{bb}-Z_{ab}.$
So for the commuting time we have
$T_{ab}=M_{ab}+M_{ba}=Z_{aa}-Z_{ab}-Z_{ba}-Z_{bb}.$
For a vector $x=(x_{i})_{i=1,\ldots,n}$ define
$||x||^{2}=\sum_{ij}x_{i}\Delta^{ij}x_{j}.$
Please note that this does not make $\Delta^{ij}$ the matrix of the quadratic
form in the usual sense, because in general $\Delta^{ij}\neq\Delta^{ji}$. The
matrix of the form in the usual sense is the symmetrized version
$\frac{1}{2}(\Delta^{ij}+\Delta^{ji})$.
Because
$\sum_{i}\Delta^{ij}=\sum_{j}\Delta^{ij}=0$
we have the key identity
$||x||^{2}=-\frac{1}{2}\sum_{ij}\Delta^{ij}(x_{i}-x_{j})^{2}.$
Recalling the definition of $\Delta^{ij}$ gives
$||x||^{2}=\frac{1}{2}\sum_{ij}w^{i}\tensor{P}{{}_{i}^{j}}(x_{i}-x_{j})^{2}.$
Thus the quadratic form $||x||^{2}$ is weakly positive definite, but not
strictly so, because it vanishes for constant vectors:
$||(c,\ldots,c)||^{2}=0.$
It becomes strictly positive definite if we identify vectors differing by a
constant vector:
$(x_{i})_{i=1,\ldots,n}\equiv(z_{i}+c)_{i=1,\ldots,n}.$
This Euclidean space (vectors mod constant vectors, with the pushed-down
quadratic form) is where we will embed our chain.
To get the embedding, map state $a$ to the vector
$f(a)=(Z_{ai})_{i=1,\ldots,n}.$
For the difference between the images of $a$ and $b$ we have
$(f(a)-f(b))_{i}=Z_{ai}-Z_{bi}=\sum_{k}(\tensor{\delta}{{}_{a}^{k}}-\tensor{\delta}{{}_{b}^{k}})Z_{ki},$
with $\tensor{\delta}{{}_{i}^{j}}$ the Kronecker delta. We want to see that
$f(a)-f(b)$ has square norm $T_{ab}$.
From the generalized inverse relationship between $Z_{ij}$ and $\Delta^{ij}$
and the fact that
$\sum_{k}\tensor{\delta}{{}_{a}^{k}}-\tensor{\delta}{{}_{b}^{k}}=0$
we have
$\sum_{i}(Z_{ai}-Z_{bi})\Delta^{ij}=\sum_{ki}(\tensor{\delta}{{}_{a}^{k}}-\tensor{\delta}{{}_{b}^{k}})Z_{ki}\Delta^{ij}=\tensor{\delta}{{}_{a}^{j}}-\tensor{\delta}{{}_{b}^{j}}.$
So
$\displaystyle||f(a)-f(b)||^{2}$ $\displaystyle=$
$\displaystyle\sum_{ij}(Z_{ai}-Z_{bi})\Delta^{ij}(Z_{aj}-Z_{bj})$
$\displaystyle=$
$\displaystyle\sum_{j}(\tensor{\delta}{{}_{a}^{j}}-\tensor{\delta}{{}_{b}^{j}})(Z_{aj}-Z_{bj})$
$\displaystyle=$ $\displaystyle Z_{aa}-Z_{ab}-Z_{ba}+Z_{bb}$ $\displaystyle=$
$\displaystyle T_{ab}.$
There you have it.
## 4 What just happened
We want to explain the proof we have just given in more conceptual terms.
Let $V$ be a finite-dimensional real vector space, and ${V^{\star}}$ the dual
space, consisting of linear functionals $\phi:V\to{\bf R}$. For
$u\in{V^{\star}}$, $x\in V$ write
$\langle u,x\rangle_{V}=u(x)$
for the natural pairing between $V$ and ${V^{\star}}$. Identify $V$ with
${V^{\star\star}}$ as usual:
$\langle x,u\rangle_{V^{\star}}=u(x)=\langle u,x\rangle_{V}.$
To a map $f:V\to W$ we associate the adjoint map $f^{\star}:W^{\star}\to
V^{\star}$, such that for $u\in W^{\star}$, $x\in V$
$\langle f^{\star}(u),x\rangle_{V}=u(f(x)).$
A bilinear form on $V$ arises from a linear map
$\phi:V\to{V^{\star}}$
via
$L_{\phi}(x,y)=\langle\phi(x),y\rangle_{V}.$
The adjoint map
${\phi^{\star}}:{V^{\star}}\to V$
yields the transposed bilinear form
$L_{\phi^{\star}}(x,y)=\langle{\phi^{\star}}(x),y\rangle_{V}=\langle
x,\phi(y)\rangle_{V^{\star}}=\langle\phi(y),x\rangle_{V}=L_{\phi}(y,x).$
If $\phi$ is invertible the inverse
${\phi^{-1}}:{V^{\star}}\to V$
yields the form $L_{\phi^{-1}}$ on ${V^{\star}}$:
$L_{\phi^{-1}}(u,v)=\langle\phi^{-1}(u),v\rangle_{V^{\star}}=\langle
v,{\phi^{-1}}(u)\rangle_{V}.$
The forms $L_{\phi^{\star}}$ and $L_{\phi^{-1}}$ are conjugate, because
$L_{\phi^{-1}}(u,v)=\langle
v,{\phi^{-1}}(u)\rangle_{V}=L_{\phi}({\phi^{-1}}(v),{\phi^{-1}}(u))=L_{\phi^{\star}}({\phi^{-1}}(u),{\phi^{-1}}(v)).$
Going back the other way,
$L_{\phi^{\star}}(x,y)=L_{\phi^{-1}}(\phi(x),\phi(y)).$
From these two equations, we get two distinct ways to conjugate $L_{\phi}$ to
$L_{\phi^{-1\star}}$. Plugging $\phi=({\phi^{-1}})^{-1}$ into the first and
putting $(x,y)$ for $(u,v)$, we get
$L_{\phi}(x,y)=L_{\phi^{-1\star}}(\phi(x),\phi(y)).$
Plugging $\phi=({\phi^{\star}})^{\star}$ into the second we get
$L_{\phi}(x,y)=L_{\phi^{-1\star}}({\phi^{\star}}(x),{\phi^{\star}}(y)).$
Now putting ${\phi^{\star}}$ for $\phi$ we see that in fact there were two
ways to conjugate $L_{\phi^{-1}}$ to $L_{\phi^{\star}}$:
$L_{\phi^{\star}}(x,y)=L_{\phi^{-1}}(\phi(x),\phi(y))=L_{\phi^{-1}}({\phi^{\star}}(x),{\phi^{\star}}(y)).$
Having two ways to conjugate $L_{\phi}$ to $L_{\phi^{-1\star}}$ gives us an
automorphism ${\phi^{-1}}\circ{\phi^{\star}}$ of $L_{\phi}$:
$L_{\phi}(x,y)=L_{\phi}({\phi^{-1}}({\phi^{\star}}(x)),{\phi^{-1}}({\phi^{\star}}(y))).$
Along with ${\phi^{-1}}\circ{\phi^{\star}}$ we also have the inverse
automorphism ${\phi^{-1\star}}\circ\phi$:
$L_{\phi}(x,y)=L_{\phi}({\phi^{-1\star}}(\phi(x)),{\phi^{-1\star}}(\phi(y))).$
We could also consider powers other than $-1$ of our automorphism, but we
don’t need to, because the conjugacy between $L_{\phi}$ and $L_{\phi^{\star}}$
is canonical (in the sense of being equivariant with respect to taking duals
and inverses) up to this factor of two. The difference between them, as
measured by the automorphism ${\phi^{-1}}\circ{\phi^{\star}}$, measures the
antisymmetry of $L_{\phi}$. It is destined to play an important role in our
future.
Looking now at the level of quadratic forms $Q_{\phi}(x)=L_{\phi}(x,x)$,
everything in sight is conjugate:
$Q_{\phi}(x)=Q_{\phi^{\star}}(x);$
$Q_{\phi^{-1}}(u)=Q_{\phi^{-1\star}}(u)=Q_{\phi}({\phi^{-1}}(u))=Q_{\phi}({\phi^{-1\star}}(u)).$
All this nonsense can be made much more concrete using matrices. Let $V={\bf
R}^{n}$ and represent $x\in V$, $u\in{V^{\star}}$ as column and row vectors
respectively, so that the pairing is just multplying a row vector by a column
vector:
$\langle u,x\rangle_{V}=ux.$
Denote transposition of matrices by $\star$. Write
$L_{\phi}(x,y)=x^{\star}Ay,$
so that
$\phi(x)=x^{\star}A=(A^{\star}x)^{\star}.$
Now
${\phi^{-1}}(u)=(uA^{-1})^{\star}=A^{-1\star}u^{\star},$
so
$L_{\phi^{-1}}(u,v)=\langle
v,{\phi^{-1}}(u)\rangle_{V}=vA^{-1\star}u^{\star}=uA^{-1}v^{\star}.$
Good!
Now to see the two conjugacies of $L_{\phi^{\star}}$ with $L_{\phi^{-1}}$:
$A^{\star}A^{-1}A=A^{\star};$ $AA^{-1}A^{\star}=A^{\star}.$
These combine to give two automorphisms of $L_{\phi}$:
$(A^{-1}A^{\star})^{\star}A(A^{-1}A^{\star})=AA^{-1\star}AA^{-1}A^{\star}=A;$
$(A^{-1\star}A)^{\star}A(A^{-1\star}A)=A^{\star}A^{-1}A^{-1\star}A=A.$
Hmm. Why didn’t we do it this way in the first place?
So, here’s what happened with our Markov chain. We started with the space
$V={\bf R}^{n}/\mathrm{1}$ with quadratic form
$L_{\phi}(x,y)=\sum_{ij}x_{i}\Delta^{ij}y_{j}$, embedded the states in
${V^{\star}}={\bf R}^{n}\perp\mathrm{1}$ with quadratic form
$L_{\phi^{-1}}(u,v)=\sum_{ij}u^{i}Z_{ij}v^{j}$, and proved that
$L_{\phi^{-1}}$ is positive definite by showing that it is conjugate to
$L_{\phi}$.
## 5 Tensor notation for Markov chains
As you will already have noticed, we are using tensor notation, rather than
trying to work within the confines of matrix notation, as is usual in the
theory of Markov chains. For our purposes, a tensor may be viewed as an array
where some of the indices are written as superscripts rather than subscripts.
Thus, for example, we write the transition rates for a Markov chain as
$\tensor{P}{{}_{i}^{j}}$, and the equilbrium measure as $w^{i}$.
Where the indices of a tensor are placed makes a difference: Thus
$\tensor{Z}{{}_{i}^{j}}$ represents a different array from $Z_{ij}$. We may
‘raise’ and ‘lower’ these indices as is usual with tensors, though in this
case the procedure is simpler than usual, because to raise or lower an index
$i$ we just multiply or divide by the entries of $w^{i}$. Thus we get $Z_{ij}$
from $\tensor{Z}{{}_{i}^{j}}$ by lowering the index $j$:
$Z_{ij}=\frac{1}{w^{i}}\tensor{Z}{{}_{i}^{j}}.$
We get back to $\tensor{Z}{{}_{i}^{j}}$ from $Z_{ij}$ by raising the index
$j$:
$\tensor{Z}{{}_{i}^{j}}=w^{j}Z_{ij}.$
We will still be able to use matrix notation to multiply matrices (two-index
tensors) and vectors (one-index tensors). The beautiful thing is that when we
do this, the indices take care of themselves, as long as the indices that get
summed over when multiplying matrices are paired high with low. To show by
example what this means, if we write $C=AB$, it will entail (among other
things) that
$\tensor{C}{{}_{i}^{j}}=\tensor{(AB)}{{}_{i}^{j}}=\sum_{k}\tensor{A}{{}_{i}^{k}}\tensor{B}{{}_{k}^{j}}=\sum_{k}\tensor{A}{{}_{i}{}_{k}}\tensor{B}{{}^{k}{}^{j}},$
and
$\tensor{C}{{}_{i}{}_{j}}=\tensor{(AB)}{{}_{i}{}_{j}}=\sum_{k}\tensor{A}{{}_{i}^{k}}\tensor{B}{{}_{k}{}_{j}}=\sum_{k}\tensor{A}{{}_{i}{}_{k}}\tensor{B}{{}^{k}_{j}}=\sum_{k}\tensor{A}{{}_{i}{}_{k}}w^{k}\tensor{B}{{}_{k}{}_{j}}=\sum_{k}\tensor{A}{{}_{i}{}_{k}}\tensor{B}{{}^{k}{}^{j}}\frac{1}{w^{j}}.$
Note. If you’re familiiar with the Einstein summation convention, be aware
that we don’t use it here. It wouldn’t work well in this context, because we
want to write $w^{i}Z_{ij}$ without automatically summing over $i$.
Fortunately, for our purposes, using the notation of matrix multiplication
turns out to be even more convenient than the summation convention.
## 6 What it means to be conformally correct
We have said that we want our approach to be ‘conformally correct’. Before we
go further, a word about what this means. (Skip this if you don’t care.)
Conformal equivalence of Markov chains is most natural for continuous time
chains. In that context two chains with transition rates
$\tensor{A}{{}_{i}^{j}}$ and $\tensor{B}{{}_{i}^{j}}$ are conformally
equivalent if
$\tensor{B}{{}_{i}^{j}}=\frac{1}{a_{i}}\tensor{A}{{}_{i}^{j}}$
where all $a_{i}>0$. Generally we will also want the additional condition that
$\sum_{i}w^{i}a_{i}=1$ where $w^{i}$ is the equilibrium probability of being
at $i$ for the $A$ chain. With this ‘volume condition’ the equilibrium
probability of being at $i$ for the $B$ chain will be $w^{i}a_{i}$ and
$B^{ij}=w^{i}a_{i}\tensor{B}{{}_{i}^{j}}=w^{i}a_{i}\frac{1}{a_{i}}\tensor{A}{{}_{i}^{j}}=A^{ij}.$
Thus while the raw transition rates $\tensor{A}{{}_{i}^{j}}$ are not conformal
invariants, when we raise the index $i$ we get a new array
$A^{ij}=w^{i}\tensor{A}{{}_{i}^{j}}$ whose entries are conformal invariants:
They tell the rate at which transitions are made from $i$ to $j$ when the
chain is in equilibrium.
It is possible to talk about conformal equivalence of discrete time chains,
but it is not as pleasant as for continuous-time chains. This is true so often
in the theory of Markov chains! And yet, for simplicity, we want to talk about
discrete-time chains. So our approach will be to do everything in such a way
that the discussion would be conformally invariant when translated from
discrete to continuous time.
So that’s what it means for chains to be conformally equivalent. As for
‘conformal correctness’, we mean an approach that seeks to identify and
emphasize quantities that are conformally invariant. And why should we do
this? Because it will pay.
## 7 Visualizing commuting times
One way to determine the expected commuting time $T_{ab}$ between $a$ and $b$
is to run the chain for a long time $T$ (beware of confusion!), paying
attention to when the chain is at $a$ or $b$ and ignoring other states. If $R$
is the number of runs of $a$’s (which is within $1$ of the number of runs of
$b$’s), then
$T_{ab}\approx T/R.$
To keep track of $R$ we imagine painting our Markovian particle green when it
reaches $a$ and red when it reaches $b$. Let $r_{ab}$ be the equilibrium rate
at which red particles are being painted green. Ignoring end effects, over our
long time interval $T$, $R$ above is the number of times a red particle gets
painted green, thus roughly $Tr_{ab}$, and it follows that
$T_{ab}=\frac{1}{r_{ab}}.$
This is an instance of the general principle from renewal theory that when
events happen at rate $r$, the expected time between events is $1/r$.
Note. This painting business is very close to a model developed by Kingman [8]
and Kelly [7]. (See exercise 1 in section 3.3 of Doyle and Snell [4].)
However, I don’t know that Kingman and Kelley ever made the connection to
commuting times, and it is possible that their discussion concerned only time-
reversible chains. Somebody should check this.
It is high time to observe that if $\hat{T}_{ab}$ is the commuting time for
the time-reversed chain (according to the general convention that time-
reversed quantities wear hats), we have
$T_{ab}=T_{ba}=\hat{T}_{ab}=\hat{T}_{ba}.$
We claim to be able to see this from our way of approximating $T_{ab}$ by
observing the chain over a long time. If we reverse a record of the chain
moving forward for a long time, we see roughly a record of the time-reversed
chain starting in equlibrium. In fact if we started the original chain in
equilibrium we’re golden. If we started the chain not in equilibirum (e.g. by
starting at $a$, as we might well be tempted to do), there will be problems
toward the end of the time-reversed record, as the time-reversed chain gets
drawn to end where the forward chain began. But this effect is negligible when
$T$ is large.
## 8 The Laplacian and the cross-potential
Consider a discrete-time Markov chain with transition probabilities
$\tensor{P}{{}_{i}^{j}}=\mathrm{Prob}(\mbox{next at $j$}|\mbox{start at
$i$}).$
Assume the chain is ergodic, so that there is a unique equilibrium measure
$w^{i}$ with
$\sum_{i}w^{i}\tensor{P}{{}_{i}^{j}}=w^{j},$ $\sum_{i}w^{i}=1.$
Define the _Laplacian_
$\Delta^{ij}=w^{i}(\tensor{I}{{}_{i}^{j}}-\tensor{P}{{}_{i}^{j}}).$
For $i\neq j$, $-\Delta^{ij}$ tells the equilibrium rate of transitions from
$i$ to $j$; $\Delta^{ii}$ tells the total rate of transitions to and from
states other than $i$. The time-reversed Markov chain has Laplacian
$\hat{\Delta}^{ij}=\Delta^{ji}$. A time-reversible chain has
$\Delta^{ij}=\Delta^{ji}$.
We have
$\sum_{i}\Delta^{ij}=\sum_{j}\Delta^{ij}=0.$
So considered as a matrix, $\Delta^{ij}$ is not invertible. However, it has a
generalized inverse $Z_{ij}$ with the property that for any measure of total
mass 0, which is to say for any $u^{i}$ with $\sum_{i}u^{i}=0$, we have
$\sum_{jk}u^{j}Z_{jk}\Delta^{kl}=u^{l}$
and
$\sum_{jk}\Delta^{ij}Z_{jk}u^{k}=u^{i}.$
An equivalent way to write this is
$\sum_{jk}\Delta^{ij}Z_{jk}\Delta^{kl}=\Delta^{il},$
because if we think of $\Delta^{ij}$ as a matrix, its rows and columns both
span the space of measures with total mass 0.
A sensible choice for the generalized inverse $Z_{ij}$ is
$Z_{ij}=\frac{1}{w^{j}}\tensor{Z}{{}_{i}^{j}}$
where
$\tensor{Z}{{}_{i}^{j}}=(\tensor{I}{{}_{i}^{j}}-w^{j})+(\tensor{P}{{}_{i}^{j}}-w^{j})+(\tensor{{P^{(2)}}}{{}_{i}^{j}}-w^{j})+\ldots,$
where
$\tensor{{P^{(2)}}}{{}_{i}^{j}}=\sum_{k}\tensor{P}{{}_{i}^{k}}\tensor{P}{{}_{k}^{j}}$
represents the matrix square of $\tensor{P}{{}_{i}^{j}}$, and the elided terms
involve higher matrix powers. Define
$\tensor{{P^{(\infty)}}}{{}_{i}^{j}}=w^{j}$, to suggest that the ‘infinitieth
power’ of $\tensor{P}{{}_{i}^{j}}$ has all rows equal to the vector $w^{i}$.
We can write
$\displaystyle Z$ $\displaystyle=$
$\displaystyle(I-{P^{(\infty)}})+(P-{P^{(\infty)}})+(P^{(2)}-{P^{(\infty)}})+\ldots$
$\displaystyle=$ $\displaystyle(I-P+{P^{(\infty)}})^{-1}-{P^{(\infty)}}.$
This naturally translates into the formula we’ve given for
$\tensor{Z}{{}_{i}^{j}}$, and from there, by ‘lowering the index j’, we get
$Z_{ij}$.
For this choice of $Z$ we have the natural interpretation that
$\tensor{Z}{{}_{i}^{j}}$ is the expected excess number of visits to $j$ for a
chain starting at $i$ compared to a chain starting in equilibrium. For the
time-reversed chain we get
$\tensor{\hat{Z}}{{}_{ij}}=\tensor{Z}{{}_{ji}},$
and so in particular if the chain is time-reversible we have $Z_{ij}=Z_{ji}$.
This is all very well, but we still do not want to prescribe this particular
choice of $Z$ because it is not conformally invariant: It depends on the
equilibrium measure $w^{i}$, and not just on the Laplacian ‘matrix’
$\Delta^{ij}$. This makes it insufficiently canonical for us.
What _is_ canonical is the bilinear form
$B(u,v)=\sum_{ij}u^{i}Z_{ij}v^{j}$
when $u$ and $v$ are restricted to the subspace $S$ of measures of total mass
$0$:
$S=\\{u^{i}:\sum_{i}u^{i}=0\\}$
Fixing $a,b,c,d$ and setting
$u=\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}};\;v=\tensor{\delta}{{}_{c}^{i}}-\tensor{\delta}{{}_{d}^{i}}$
gives us the _cross-potential_
$N_{abcd}=B(\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}},\tensor{\delta}{{}_{c}^{i}}-\tensor{\delta}{{}_{d}^{i}})=Z_{ac}-Z_{ad}-Z_{bc}+Z_{bd}.$
$N$ satisfies
$N_{bacd}=N_{abdc}=-N_{abcd}.$
For the time-reversed process
$\hat{N}_{abcd}=N_{cdab}.$
Clearly, knowing $N$ is the same as knowing $B$, or $\Delta$. If we know $w$
as well as $N$ we can recover our sensible-but-not-canonical $Z$:
$Z_{ij}=\sum_{kl}N_{ikjl}w^{k}w^{l}.$
Different choices of $w$ in this formula lead to different $Z$’s, but they all
determine the same bilinear form $B$. From $Z$ and $w$ we can recover $P$.
In general, it is useful to think of an ergodic Markov chain as specified by
the cross-potential $N$, which determines its conformally invariant
properties, together with the equilibrium measure $w$. Expressing formulas in
these terms allows us to see the extent to which quantities are conformally
invariant (like $N$, $B$, and $\Delta$) or not (like $w$, $Z$, $P$).
Complaint. $N$ and $w$ together don’t quite determine the original transition
rates for a continuous-time Markov chain, or rather, they wouldn’t do so if we
had some way to distinguish between remaining at $i$ and moving from $i$ to
$i$. Such a distinction is not possible for discrete-time chains represented
by matrices, but we could handle it in the continuous case by allowing for
non-zero transition rates on the diagonal. Better yet, we could reformulate
Markov chain theory in the context of queuing networks based on $1$-complexes
(graphs where loops and multiple edges are allowed). This would give us a way
to distinguish different ways of stepping from $i$ to $j$. A further step
would be to allow a general distribution for the time it takes to make a
transition for $i$ to $j$. This would be very helpful when watching the chain
only when it is in a subset of its states, as in the case above where we
contemplated watching the chain only when it is at $a$ or $b$. We didn’t say
just what we meant by this, because it doesn’t conveniently fit into the usual
formulation of Markov chain theory.
## 9 Probabilistic and electrical interpretation
We may interpret $N_{abcd}$ probabilistically as the equilibrium concentration
difference between $c$ and $d$ due to a unit flow of particles entering at $a$
and leaving at $b$. Here’s what this means. Introduce Markovian particles at
$a$ at a unit rate, and remove them when they reach $b$. Write the ‘dynamic
equilibrium’ measure of particles at $i$ as $w^{i}\phi_{i}$, so that
$\phi_{i}$ tells the concentration of particles relative to the ‘static
equilibrium’ measure $w^{i}$. Conservation of particles implies that
$w^{i}\phi_{i}\sum_{j}\tensor{P}{{}_{i}^{j}}-\sum_{j}w^{j}\phi_{j}\tensor{P}{{}_{j}^{i}}=\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}}.$
We hasten to rewrite this in the conformally correct form
$\sum_{j}\phi_{j}\Delta^{ji}=\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}}.$
Since also
$\sum_{j}(Z_{aj}-Z_{bj})\Delta^{ji}=\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}}$
and since the Laplacian $\Delta$ kills only constants, if follows that
$\phi_{j}=Z_{aj}-Z_{bj}+C,$
and thus
$\phi_{c}-\phi_{d}=Z_{ac}-Z_{bc}-Z_{ad}+Z_{bd}=N_{abcd}.$
From this probabilistic interpretation of $N$ we can see that
$N_{abab}=C_{ab}$, the commuting time between $a$ and $b$. Indeed, in the
particle-painting scenario introduced earlier, $C_{ab}$ is the reciprocal of
the rate at which red particles are turning green at $a$. Paying attention
only to green particles, we see green particles appearing at $a$ at rate
$1/C_{ab}$, and disappearing at $b$. The equilibrium concentration of green
particles at $i$ is the probability $p_{i}$ of hitting $a$ before $b$ for the
time-reversed chain, and in particular $p_{a}=1$ and $p_{b}=0$, so the
concentration difference between $a$ and $b$ is $1$. Multiplying the green
flow by $C_{ab}$ normalizes it to a unit flow with concentration difference
$C_{ab}$ between $a$ and $b$. So
$C_{ab}=N_{abab}.$
If we embellish this probabilistic scenario by imagining that our particles
carry a positive charge, we may identify the net flow of particles with
electrical current; the concentration of particles (relative to the
equilibrium measure) with electrical potential; and differences of
concentration with voltage drop. With this terminology, $N_{abcd}$ tells the
voltage drop between $c$ and $d$ due to a unit current from $a$ to $b$.
Traditionally this way of talking is reserved for time-reversible Markov
chains, which are precisely those for which we have the ‘reciprocity law’
$N_{abcd}=N_{cdab}$. For such chains, if we build a resistor network where
nodes $i\neq j$ are joined by a resistor of conductance (i.e., reciprocal
resistance) $-\Delta^{ij}$, then $N_{abcd}$ will indeed be the voltage drop
between $c$ and $d$ due to a unit current from $a$ to $b$. We propose to
extend this way of talking to non-time-reversible chains.
In electrical terms, the voltage drop $N_{abab}$ between $a$ and $b$ due to a
unit current between $a$ and $b$ is the _effective resistance_. This is the
same as the reciprocal of the current that flows when a $1$-volt battery is
connected up between $a$ and $b$—which is what we get in effect when we
measure commuting times using green and red paint. So the commuting time
$C_{ab}=N_{abab}$ is the same as the effective resistance between $a$ and $b$.
The connection of commuting time to effective resistance, and the general
recognition that commuting times play a key role in understanding Markov
chains, is due to Chandra et al. [1].
Note. Now we are in a position to understand the significance of the name
‘cross-potential’. This name is meant to indicate the connection of $N_{abcd}$
to the cross-ratio of complex function theory. If we extend our notions about
Markov chains to cover Brownian motion on the Riemann sphere, we get
$\displaystyle N_{abcd}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}(\log|a-c|-\log|a-d|-\log|b-c|+\log|b-d|)$
$\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\log\left|\frac{a-c}{a-d}\frac{b-d}{b-c}\right|)$
$\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\Re\log\frac{a-c}{a-d}\frac{b-d}{b-c}.$
We don’t have to specify a metric on the sphere here, because the Laplacian is
a conformal invariant in two dimensions. Thinking of the sphere as being an
electrical conductor with constant conductivity (say, 1 mho ‘per square’), the
electrical interpretation becomes exact. The advantage of having $N$ to take
four ‘arguments’ now becomes apparent, because $N_{abcb}=\infty$. That’s why
engineers using look for cracks in nuclear reactor cooling pipes with a
emph4-point probe. To get a sensible generalization of $C_{ab}$ we will need
to do some kind of renormalization, which will introduce a dependence on the
metric. We should not be sorry about this, because it brings curvature into
the picture—and you know that can’t be bad.
## 10 Realization
Now, finally, to realize commuting times as squared distances. From the
bilinear form $B$ we get the quadratic form
$Q(u)=||u||^{2}=B(u,u)=\sum_{ij}u^{i}Z_{ij}u^{j}.$
$C_{ab}=N_{abab}=Q(\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}})=||\tensor{\delta}{{}_{a}}-\tensor{\delta}{{}_{b}}||^{2}.$
So if we map $i$ to $\tensor{\delta}{{}_{i}}$ then the commuting time $C_{ab}$
becomes the squared distance between the images in the $Q$-norm.
That is, if what we’re calling the $Q$-norm is indeed a norm. Is $Q$ really
positive definite?
To understand better what is going on here, it is useful to look at the
bilinear form
$L(\phi,\psi)=\sum_{ij}\phi_{i}\Delta^{ij}\psi_{j},$
where we think of $\phi$ and $\psi$ as being defined only modulo additive
constants. If we think of $\phi_{i}$ as the potential of the measure
$\sum_{i}\phi_{i}\Delta^{ik},$
then this is the same bilinear form as before, except that now instead of
measures of total mass $0$ it takes as its arguments the corresponding
potentials, the first with respect to the original chain, and the second with
respect to the time-reversed chain:
$L(\phi,\psi)=B(\sum_{i}\phi_{i}\Delta^{ik},\sum_{i}\psi_{i}\Delta^{ki})=B(\sum_{i}\phi_{i}\Delta^{ik},\sum_{i}\hat{\Delta}^{ik}\psi_{i}).$
This follows from the formula $\Delta Z\Delta=\Delta$ above.
Now to get the equivalent of $Q$ in this context we restrict to the subspace
$V=\\{(\phi,\psi):\sum_{i}\phi_{i}\Delta^{ik}=\sum_{j}\Delta^{kj}\psi_{j}\\}$
and take as our quadratic form
$R((\phi,\psi))=L(\phi,\psi).$
In the case of a time-reversible chain, $V$ is just the diagonal $\phi=\psi$,
and
$Q(\phi\Delta)=R((\phi,\phi))=L(\phi,\phi)=\sum_{ij}\phi_{i}\Delta^{ij}\phi_{j}=\frac{1}{2}\sum_{ij}(-\Delta^{ij})(\phi_{i}-\phi_{j})^{2}.$
This is evidently positive-definite. Indeed, if we associate to $(\phi,\phi)$
the vector with $n\choose 2$ coordinates
$\sqrt{-\Delta^{ij}}(\phi_{i}-\phi_{j})$, $i<j$, then we will have embedded
the normed space $(V,R)$, and along with it our Markov chain, in Euclidean
$n\choose 2$-space.
Electrically, what we have done here is to account for the energy being
dissipated in the network by adding up the energy dissipated by individual
resistors. And there should be some kind of probabilistic interpretation as
well.
That’s how it works for time-reversible chains, for which
$\Delta^{ij}=\Delta^{ji}$. However, the argument extends to the general case
by what amounts to a trick. The key is the observation that for
$(\phi,\psi)\in V$ we have
$L(\phi,\psi)=L(\phi,\phi)=L(\psi,\psi).$
(But please note that in general $L(\phi,\psi)\neq L(\psi,\phi)$!) So
$Q(\phi\Delta)=R((\phi,\psi))=L(\phi,\psi)=L(\phi,\phi)=\sum_{ij}\phi_{i}\Delta^{ij}\phi_{j}=\frac{1}{2}\sum_{ij}(-\Delta^{ij})(\phi_{i}-\phi_{j})^{2}.$
So there is the positive-definiteness we need.
Now, though, we don’t see any natural way to interpret the terms of the sum
electrically or probabilistically. (Which is not to say that there isn’t one!)
In putting $\phi$ in both slots of $L$ we leave the subspace $V$, and thereby
commit what appears to be an unnatural act. But it seems to have paid off.
## 11 Minimax characterization of commuting times and hitting probabilities
Fix states $a\neq b$, and let
$S_{a,b}=\\{\phi|\phi_{a}=1,\phi_{b}=0\\}$
Here we really should be thinking of $\phi$ as being defined only up to an
additive constant, which means we should write $\phi_{a}-\phi_{b}=1$, but
we’re going to be sloppy about this, because we want to focus attention on two
distinguished elements of $S_{a,b}$ which are naturally $1$ and $a$ and $0$ at
$b$. These are
$\bar{\phi}_{i}=\mathrm{Prob}(\mbox{hit $a$ before $b$ starting at $i$ going
backward in time})$
and
$\bar{\psi}_{i}=\mathrm{Prob}(\mbox{hit $a$ before $b$ starting at $i$ going
forward in time}).$
We’ve met $\bar{\phi}$ before: It’s proportional to the equilibrium
concentration of green particles in our painting scenario. $\bar{\psi}$ is the
analogous quantity for the reversed chain. The pair $(\bar{\phi},\bar{\psi})$
belongs to our subset $V$, because
$(\bar{\phi}\Delta)^{i}=(\Delta\bar{\psi})^{i}=r_{ab}(\tensor{\delta}{{}_{a}^{i}}-\tensor{\delta}{{}_{b}^{i}}).$
Here we once again are writing $r_{ab}=\frac{1}{T_{ab}}$ for the equilibrium
rate of commuting between $a$ and $b$. Observe that any $f$ we have
$L(\bar{\phi},f)=L(f,\bar{\psi})=r_{ab}(f_{a}-f_{b}).$
So whenever $f$ is in $S_{a,b}$ we have
$L(\bar{\phi},f)=L(f,\bar{\psi})=r_{ab},$
and in particular
$L(\bar{\phi},\bar{\psi})=r_{ab}.$
Theorem.
$r_{ab}=\frac{1}{T_{ab}}\min_{\alpha}\max_{\phi+\psi=2\alpha}L(\phi,\psi).$
Here and below, $\alpha$, $\phi$, and $\psi$ are restricted to lie in
$S_{a,b}$, i.e. to take value $1$ at $a$ and $0$ at $b$.
Proof. Whatever $\alpha$ is, we may take $\phi=\bar{\phi}$ (and thus
$\psi=2\alpha-\bar{\phi}$), and have
$L(\phi,\psi)=L(\bar{\phi},\psi)=r_{ab}$
as above. So
$\min_{\alpha}\max_{\phi+\psi=2\alpha}L(\phi,\psi)\geq r_{ab}.$
To prove the inequality in the other direction, and in the process identify
where the minimax is achieved, take
$\alpha=\frac{1}{2}(\bar{\phi}+\bar{\psi}).$
If $\phi+\psi=2\alpha$ then we can write
$\phi=\bar{\phi}+f$
and
$\psi=\bar{\psi}-f,$
where $f_{a}=f_{b}=0$.
Now
$L(\bar{\phi},f)=L(f,\bar{\psi})=r_{ab}(f_{a}-f_{b})=0,$
so
$L(\phi,\psi)=L(\bar{\phi}+f,\bar{\psi}-f)=L(\bar{\phi},\bar{\psi})-L(f,f)=r_{ab}-L(f,f).$
And even though we claim it is a travesty to put the same $f$ into both slots
of $L$, we still have
$L(f,f)\geq 0:$
That was the upshot of our embedding investigation. So
$L(\phi,\psi)\leq r_{ab},$
still assuming $\alpha=\frac{1}{2}(\bar{\phi}+\bar{\psi})$ and
$\phi+\psi=2\alpha$. Hence
$\min_{\alpha}\max_{\phi+\psi=2\alpha}L(\phi,\psi)\geq
r_{ab}.\quad\rule{5.69054pt}{7.11317pt}$
In the time-reversible case, where $\Delta^{ij}=\Delta^{ji}$, this minimax can
be reduced to a straight minimum. That’s because in this case for any $g,f$ we
have $L(f,g)=L(g,f)$, and hence
$L(g+f,g-f)=L(g,g)-L(f,f).$
So to maximize $L(\phi,\psi)$ while fixing the sum $\phi+\psi=2\alpha$ we take
$\phi=\psi=\alpha$.
Corollary. When $\Delta^{ij}$ is symmetric
$r_{ab}=\min_{\phi(a)=1,\phi(b)=0}L(\phi,\phi).\quad\rule{5.69054pt}{7.11317pt}$
This minimum principle for resistances was known already to 19th century
physicists, specifically Thomson (a.k.a. Kelvin), Maxwell, and Rayleigh: For
more about this, see Doyle and Snell [4].
Having a straight minimum is a lot better than having a minimax, because now
we can plug in any $\phi$ with $\phi(a)=1,\phi(b)=0$ and get an upper bound
for $r_{ab}$, corresponding to a lower bound for $T_{ab}$. This method is a
staple of electrical theory—the part of electrical theory that doesn’t extend
to non-time-reversible chains because it depends on the relation
$L(f,g)=L(g,f)$.
For time-reversible chains there are also complementary methods for finding
lower bounds for $r_{ab}$, and thus upper bounds for $T_{ab}$. These emerge
from the minimum principle through the mystery of convex duality. In practice,
though, it is generally conceptually simpler to work instead with the
monotonicity law described in the next section. This monotonicity law extends
to all chains, but sadly, for all we can tell thus far, its usefulness appears
to get left behind.
## 12 Monotonicity
From the minimax characterization of commuting times we immediately get the
following:
Monotonicity Law Commuting times decrease monotonically when equilibrium
interstate transition increase: Using barred and unbarred quantities to refer
to two different Markov chains, if $\Delta^{ij}\leq\bar{\Delta}^{ij}$ for all
$i\neq j$ then $\bar{T_{ij}}\leq T_{ij}$ for all $i,j$.
Actually it would be better to think of $\Delta$ and $\bar{\Delta}$ here as
referring to conformal classes of chains, rather than individual chains,
because as we know $\Delta^{ij}$ and $T_{ij}$ are conformal invariants.
This law holds for all chains, time-reversible or not. As we said above, for
time-reversible chains this law can be used to get upper and lower bounds for
commuting times, and hence for hitting probabilities: This is discussed in
great detail by Doyle and Snell [4].
Sadly, even though the law extends to the non-time-reversible case, its
usefulness does not extend, at least not in any obvious way. How can this be?
There seem to be a number of reasons.
First, for time-reversible chains, if we block transitions back and forth
between states $c,d$, requiring the particle to remain where it is when it
attempts to make such a transition, we get a new $\bar{\Delta}$ dominated by
the original $\Delta$ in the sense that $\bar{\Delta}^{ij}\leq\Delta^{ij}$ for
$i\neq j$. Electrically speaking, blocking transitions between $c$ and $d$
amounts to cutting the wire between them. In the non-time-reversible case,
this will change the equilibrium measure $w^{i}$ and thereby destroy the
relation $\bar{\Delta}^{ij}\leq\Delta^{ij}$ that we need for monotonicity.
Second, for time-reversible chains, it is simple and natural to introduce
intermediate states. Electrically speaking, introducing a state between $c$
and $d$ amounts to dividing the ‘wire’ connecting $c$ and $d$ into two pieces,
if only in our mind’s eye. By combining this with the putting or taking of
wires, we can produce chains to bound $T_{ab}$ above or below as closely as we
please. And we can do this in such a way that our approximating chains are
easy to analyze. Here lies the third apparent shortcoming of the non-time-
reversible case: A seeming paucity of chains whose commuting times are easy to
compute.
So, of what use is this monotonicity law in the non-time-reversible case? That
remains to be seem.
## 13 The obstruction to time-reversibility
Let $M_{ij}$ be the expected time to reach $j$ starting from $i$. Coppersmith,
Tetali, and Winkler showed that a Markov chain is time-reversible just if for
all $a,b,c$
$M_{ab}+M_{bc}+M_{ca}=M_{ac}+M_{cb}+M_{ba}.$
And in this case the expected time to traverse a cycle of any length will be
the same in either direction. Note that the $M_{ij}$s themselves are not
conformally invariant, these cycle sums are. For a cycle of length $2$, the
cycle sum is our best friend the commuting time.
We always have
$M_{ab}+M_{bc}+M_{ca}=\hat{M}_{ac}+\hat{M}_{cb}+\hat{M}_{ba}$
(look at a long record of the chain backwards), so an equivalent condition is
that for all $a,b,c$
$M_{ab}+M_{bc}+M_{ca}=\hat{M}_{ab}+\hat{M}_{bc}+\hat{M}_{ca}.$
This is true despite the fact that in general
$\hat{M}{ab}\neq M_{ba}.$
So, why is this true? It comes down to the fact that a conformal class of
chains is reversible just if our bilinear for $L(\phi,\psi)$ on
$V=\\{x^{i}|\sum_{i}x_{i}=0\\}$ is symmetric. To any bilinear form
$\sum_{ij}u^{i}Z_{ij}v^{j}$ on $V$ their corresponds a natural cohomology
class
$Z_{ij}-Z_{ji},$
which is to say, an antisymmetric matrix defined up to addition of a matrix of
the form $B_{ij}=a_{i}-a_{j}$. This class represents the obstruction to
symmetrizing the matrix of the form within its $ab$-equivalence class. This
class vanishes just if it integrates to $0$ around any cycle, and cycles of
length $3$ span the space of cycles. Indeed, they span it in a very redundant
way. To verify reversibility, it would suffice to check any basis for the
space of cycles, e.g. only cycles of length $3$ involving the fixed state $n$
(the ‘ground’).
## 14 More to be said
The next step would be discuss how to use the knee-jerk mapping to make a
chain time-reversible without changing its commuting times. The knee-jerk
method will produce the desired time-reversible chain whenever such a thing
exists, but we still don’t know if this is always the case. What we do know is
that if it turns out that no suitable time-reversible chain exists, the knee-
jerk method will delivers a time-reversible chain whose commuting times agree
as well as possible with those of the original chain. (See Coppersmith et al.
[2], Doyle [3] .)
Then we should discuss uniformization of Markov chains, whereby we prescribe a
canonical representative chain within each conformal class (or in other words,
we prescribe a canonical $w$ to accompany a given $N$). This canonical chain
extremizes the Kemeny constant $K$, which is the expected time to hit a point
chosen according to the equilibrium distribution. (As Kemeny observed, $K$
doesn’t depend on where you start.) The extremal chain is characterized by
constancy of the expected time $K_{i}$ to hit $i$ starting from equilibrium
(the so-called ‘preKemeny non-constant’). It’s easy to write down the
transition probabilities for this extremal chain. But, are they necessarily
positive?
Beyond this lies the extension of this whole business to diffusion on
surfaces, where we must renormalize hitting times because Brownian motion in
dimension $2$ never hits a given point. (Cf. Doyle and Steiner [5].) Now to
uniformize we extremize not Kemeny’s constant, but a variant with a correction
term involving the Gaussian curvature. Again, it is easy to write down the
extremizing metric, or rather the extremizing area measure, which is not a
priori positive everywhere. For spheres, all the round metrics tie for the
extremum. For tori, the flat metrics win. For higher genus surfaces, the
winners are not hyperbolic surfaces, nor should they be, because having
constant curvature is a local condition that doesn’t know thick from thin. The
canonical measure is sensitive to thickness in a conformally correct way. But
is it a positive measure? If it isn’t, could it still be good for something?
## References
* [1] A. K. Chandra, P. Raghavan, W.L. Ruzzo, R. Smolensky, and P. Tiwari. The electrical resistance of a graph captures its commute and cover times. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pages 574–586, Seattle, May 1989.
* [2] D. Coppersmith, P. Doyle, P. Raghavan, and M. Snir. Random walks on weighted graphs, and applications to on-line algorithms. Journal of the ACM, 40:454–476, 1993.
* [3] Peter G. Doyle and Jim Reeds. The knee-jerk mapping, arXiv:math/0606068v1 [math.PR].
* [4] Peter G. Doyle and J. Laurie Snell. Random Walks and Electric Networks. The Mathematical Association of America, 1984, arXiv:math/0001057v1 [math.PR].
* [5] Peter G. Doyle and Jean Steiner. Hide and seek on surfaces and Markov chains.
* [6] Charles M. Grinstead and J. Laurie Snell. Introduction to Probability. The American Mathematical Society, 1997.
* [7] F. Kelly. Reversibility and Stochastic Networks. Wiley, 1979.
* [8] J. F. C. Kingman. Markov population processes. j. Appl. Prob., 6:1–18, 1969.
|
arxiv-papers
| 2011-07-13T17:36:17 |
2024-09-04T02:49:20.491986
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Peter G. Doyle and Jean Steiner",
"submitter": "Peter G. Doyle",
"url": "https://arxiv.org/abs/1107.2612"
}
|
1107.2621
|
# Depth and minimal number of generators of square free monomial ideals
Dorin Popescu Dorin Popescu, Institute of Mathematics ”Simion Stoilow”,
Research unit 5, University of Bucharest, P.O.Box 1-764, Bucharest 014700,
Romania dorin.popescu@imar.ro
###### Abstract.
Let $I$ be an ideal of a polynomial algebra over a field generated by square
free monomials of degree $\geq d$. If $I$ contains more monomials of degree
$d$ than $(n-d)/(n-d+1)$ multiplied with the number of square free monomials
of $S$ of degree $d$ then $\operatorname{depth}_{S}I\leq d$, in particular the
Stanley’s Conjecture holds in this case.
Key words : Monomial Ideals, Depth, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from the CNCSIS grant PN II-542/2009 of Romanian Ministry of
Education, Research and Inovation is gratefully acknowledged.
Let $S=K[x_{1},\ldots,x_{n}]$ be the polynomial algebra in $n$-variables over
a field $K$ and $I\subset S$ a square free monomial ideal. Let $d$ be a
positive integer and $\rho_{d}(I)$ be the number of all square free monomials
of degree $d$ of $I$.
The proposition below was repaired using an idea of Y. Shen to whom we owe
thanks.
###### Proposition 1.
If $I$ is generated by square free monomials of degree $\geq d$ and
$\rho_{d}(I)>((n-d)/(n-d+1)){n\choose d}$ then $\operatorname{depth}_{S}I\leq
d$.
###### Proof.
Apply induction on $n$. If $n=d$ then there exists nothing to show. Suppose
that $n>d$. Let $\nu_{i}$ be the number of the square free monomials of degree
$d$ from $I\cap(x_{i})$. We may consider two cases renumbering the variables
if necessary.
Case 1 $\nu_{1}>((n-d)/(n-d+1)){n-1\choose d-1}$.
Let $S^{\prime}:=K[x_{2},\ldots,x_{n}]$ and
$x_{1}c_{1},\ldots,x_{1}c_{\nu_{1}}$, $c_{i}\in S^{\prime}$ be the square free
monomials of degree $d$ from $I\cap(x_{1})$. Then $J=(I:x_{1})\cap S^{\prime}$
contains $(c_{1},\ldots,c_{\nu_{1}})$ and so
$\rho_{d-1}(J)\geq\nu_{1}>((n-d)/(n-d+1)){n-1\choose d-1}.$ By induction
hypothesis, we get $\operatorname{depth}_{S^{\prime}}J\leq d-1$. It follows
$\operatorname{depth}_{S}JS\leq d$ and so $\operatorname{depth}_{S}I\leq d$ by
[7, Proposition 1.2].
Case 2 $\nu_{i}\leq((n-d)/(n-d+1)){n-1\choose d-1}$ for all $i\in[n]$.
We get $\sum_{i=1}^{n}\nu_{i}\leq n((n-d)/(n-d+1)){n-1\choose d-1}$. Let
$A_{i}$ be the set of the square free monomials of degree $d$ from
$I\cap(x_{i})$. A square free monomial from $I$ of degree $d$ will be present
in $d$-sets $A_{i}$ and it follows
$\rho_{d}(I)=|\cup_{i=1}^{n}A_{i}|\leq(n/d)((n-d)/(n-d+1)){n-1\choose
d-1}=((n-d)/(n-d+1)){n\choose d}$
if $n\geq d+1$. Contradiction!
###### Remark 2.
If $I$ is generated by square free monomials of degree $\geq d$, then
$\operatorname{depth}_{S}I\geq d$. Indeed, since $I$ has a square free
resolution the last shift in the resolution of $I$ is at most $n$. Thus if $I$
is generated in degree $\geq d$, then the resolution can have length at most
$n-d$, which means that the depth of $I$ is greater than or equal to $d$ (this
argument belongs to J. Herzog). Hence in the setting of the above proposition
we get $\operatorname{depth}_{S}I=d$.
###### Corollary 3.
Let $I$ be an ideal generated by $\mu(I)$ square free monomials of degree $d$.
If $\mu(I)>((n-d)/(n-d+1)){n\choose d}$ then $\operatorname{depth}_{S}I=d$.
###### Example 4.
Let $I=(x_{1}x_{2},x_{2}x_{3})\subset S:=K[x_{1},x_{2},x_{3}].$ Then $d=2$ and
$\mu(I)=2>(1/2){3\choose 2}$. It follows that $\operatorname{depth}_{S}I=2$ by
the above corollary.
###### Example 5.
Let
$I=(x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{2}x_{3},x_{2}x_{5},x_{3}x_{4},x_{3}x_{5},x_{4}x_{5})\subset\\\
S:=K[x_{1},\ldots,x_{5}]$. Then $d=2$ and $\mu(I)=8>(3/4){5\choose 2}$ and so
$\operatorname{depth}_{S}I=2$.
Next lemma presents a nice class of square free monomial ideals $I$ with
$\mu(I)={n\choose d+1}\leq((n-d)/(n-d+1)){n\choose d}$ but
$\operatorname{depth}_{S}I=d$. We suppose that $n\geq 3$. Let $w$ be the only
square free monomial of degree $n$ of $S$, that is $w=\Pi_{j=1}^{n}x_{i}$. Set
$f_{i}=w/(x_{i}x_{i+1})$ for $1\leq i<n$, $f_{n}=w/(x_{1}x_{n})$ and let
$L_{n}:=(f_{1},\ldots,f_{n-1})$, $I_{n}:=(L,f_{n})$ be ideals of $S$ generated
in degree $d=n-2$. We will see that $\operatorname{depth}_{S}I_{n}=n-2$ even
$\mu(I_{n})=n={n\choose d+1}$.
###### Lemma 6.
Then $\operatorname{depth}_{S}L_{n}=n-1$ and
$\operatorname{depth}_{S}I_{n}=n-2$.
###### Proof.
Apply induction on $n\geq 3$. If $n=3$ then $L_{3}=(x_{3},x_{1})$,
$I_{3}=(x_{1},x_{2},x_{3})$ and the result is trivial. Assume that $n>3$. Note
that $(L_{n}:x_{n})=L_{n-1}S=(I_{n}:x_{n})$ because
$f_{n},f_{n-1}\in(L_{n}:x_{n})$. We have
$L_{n}=(L_{n}:x_{n})\cap(x_{n},L_{n})=(L_{n-1}S)\cap(x_{n},f_{n-1}),$
$I_{n}=(I_{n}:x_{n})\cap(x_{n},I_{n})=(L_{n-1}S)\cap(x_{n},f_{n-1},f_{n})=(L_{n-1}S)\cap(x_{n},u)\cap(x_{1},x_{n-1},x_{n}),$
where $u=w/(x_{1}x_{n-1}x_{n})$. But $(x_{1},x_{n-1})$ is a minimal prime
ideal of $L_{n-1}S$ and so we may remove $(x_{1},x_{n-1},x_{n})$ above, that
is $I_{n}=(L_{n-1}S)\cap(x_{n},u)$. On the other hand,
$(L_{n-1}S)+(x_{n},u)=(x_{n},I_{n-1})$ and
$(L_{n-1}S)+(x_{n},f_{n-1})=(x_{n},L_{n-1})S$ because $f_{n-1}\in L_{n-1}S$.
We have the following exact sequences
$0\rightarrow S/L_{n}\rightarrow S/L_{n-1}S\oplus S/(x_{n},f_{n-1})\rightarrow
S/(x_{n},L_{n-1}S)\rightarrow 0,$ $0\rightarrow S/I_{n}\rightarrow
S/L_{n-1}S\oplus S/(x_{n},u)\rightarrow S/(x_{n},I_{n-1}S)\rightarrow 0.$
By induction hypothesis $\operatorname{depth}L_{n-1}=n-2$ and
$\operatorname{depth}I_{n-1}=n-3$ and so
$\operatorname{depth}_{S}S/(x_{n},L_{n-1}S)=n-3$,
$\operatorname{depth}_{S}S/(x_{n},I_{n-1}S)=n-4$. As
$\operatorname{depth}_{S}S/(x_{n},f_{n-1})=\operatorname{depth}_{S}S/(x_{n},u)=n-2$,
it follows $\operatorname{depth}_{S}S/L_{n}=n-2$,
$\operatorname{depth}_{S}S/I_{n}=n-3$ by the Depth Lemma applied to the above
exact sequences.
Now, let $I$ be an arbitrary square free monomial ideal and $P_{I}$ the poset
given by all square free monomials of $I$ (a finite set) with the order given
by the divisibility. Let ${\mathcal{P}}$ be a partition of $P_{I}$ in
intervals $[u,v]=\\{w\in P_{I}:u|w,w|v\\}$, let us say
$P_{I}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define
$\operatorname{sdepth}{\mathcal{P}}=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and
$\operatorname{sdepth}_{S}I=\operatorname{max}_{\mathcal{P}}\operatorname{sdepth}{\mathcal{P}}$,
where ${\mathcal{P}}$ runs in the set of all partitions of $P_{I}$. This is
the so called the Stanley depth of $I$, in fact this is an equivalent
definition given in a general form by [1].
For instance, in Example 4, we have
$P_{I}=\\{x_{1}x_{2},x_{2}x_{3},x_{1}x_{2}x_{3}\\}$ and we may take
${\mathcal{P}}:\ \
P_{I}=[x_{1}x_{2},x_{1}x_{2}x_{3}]\cup[x_{2}x_{3},x_{2}x_{3}]$ with
$\operatorname{sdepth}_{S}{\mathcal{P}}=2$. Moreover, it is clear that
$\operatorname{sdepth}_{S}I=2$.
###### Remark 7.
If $I$ is generated by $\mu(I)>{n\choose d+1}$ square free monomials of degree
$d$ then $\operatorname{sdepth}_{S}I=d$. Since $((n-d)/(n-d+1)){n\choose
d}\geq{n\choose d+1}$, the Proposition 1 says that in a weaker case case
$\operatorname{depth}_{S}I\leq\operatorname{sdepth}_{S}I$, which was in
general conjectured by Stanley [8]. Stanley’s Conjecture holds for
intersections of four monomial prime ideals of $S$ by [2] and [4] and for
square free monomial ideals of $K[x_{1},\ldots,x_{5}]$ by [3] (a short
exposition on this subject is given in [5]). It is worth to mention that
Proposition 1 holds in the stronger case when $\mu(I)>{n\choose d+1}$ (see
[6]), but the proof is much more complicated and the easy proof given in the
present case has its importance.
In the Example 5 we have
$P_{I}=[x_{1}x_{2},x_{1}x_{2}x_{4}]\cup[x_{1}x_{3},x_{1}x_{3}x_{5}]\cup[x_{1}x_{4},x_{1}x_{4}x_{5}]\cup[x_{2}x_{3},x_{1}x_{2}x_{3}]\cup[x_{3}x_{4},x_{1}x_{3}x_{4}]\cup[x_{3}x_{5},x_{3}x_{4}x_{5}]\cup[x_{4}x_{5},x_{2}x_{4}x_{5}]\cup[x_{2}x_{3}x_{4},x_{2}x_{3}x_{4}]\cap[x_{2}x_{3}x_{5},x_{2}x_{3}x_{5}]\cup(\cup_{\alpha}[\alpha,\alpha])$,
where $\alpha$ runs in the set of square free monomials of $I$ of degree
$4,5$. It follows that $\operatorname{sdepth}_{S}I=3$. But as we know
$\operatorname{depth}_{S}I=2$.
## References
* [1] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [2] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
* [3] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
* [4] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, arXiv.AC/1009.5646.
* [5] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187-194.
* [6] D. Popescu, Depth of factors of square free monomial ideals, Preprint, 2011.
* [7] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784.
* [8] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
|
arxiv-papers
| 2011-07-13T18:18:00 |
2024-09-04T02:49:20.501322
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1107.2621"
}
|
1107.2629
|
# Construction of wedge-local nets of observables through Longo-Witten
endomorphisms
Yoh Tanimoto111Supported in part by the ERC Advanced Grant 227458 OACFT
“Operator Algebras and Conformal Field Theory”.
Dipartimento di Matematica, Università di Roma “Tor Vergata”
Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy.
e-mail: tanimoto@mat.uniroma2.it
###### Abstract
A convenient framework to treat massless two-dimensional scattering theories
has been established by Buchholz. In this framework, we show that the
asymptotic algebra and the scattering matrix completely characterize the given
theory under asymptotic completeness and standard assumptions.
Then we obtain several families of interacting wedge-local nets by a purely
von Neumann algebraic procedure. One particular case of them coincides with
the deformation of chiral CFT by Buchholz-Lechner-Summers. In another case, we
manage to determine completely the strictly local elements. Finally, using
Longo-Witten endomorphisms on the $U(1)$-current net and the free fermion net,
a large family of wedge-local nets is constructed.
## 1 Introduction
Construction of interacting models of quantum field theory in physical four-
dimensional spacetime has been a long-standing open problem since the birth of
quantum theory. Recently, operator-algebraic methods have been applied to
construct models with weaker localization property [18, 19, 10, 7, 22]. It is
still possible to calculate the two-particle scattering matrix for these
weakly localized theories and they have been shown to be nontrivial. However,
the strict locality still remains difficult. Indeed, of these deformed
theories, strictly localized contents have been shown to be trivial in higher
dimensions [7]. In contrast, in two-dimensional spacetime, a family of
strictly local theories has been constructed and nontrivial scattering
matrices have been calculated [23]. The construction of local nets of
observables is split up into two procedures: construction of wedge-local nets
and determination of strictly local elements. In this paper we present a
purely von Neumann algebraic procedure to construct wedge-local nets based on
chiral CFT and completely determine strictly local elements for some of these
wedge-local nets. Furthermore, we show that the pair of the S-matrix and the
asymptotic algebra forms a complete invariant of the given net and give a
simple formula to recover the original net from these data.
In algebraic approach to quantum field theory, or algebraic QFT, theories are
realized as local nets of operator algebras. Principal examples are
constructed from local quantum fields, or in mathematical terms, from
operator-valued distributions which commute in spacelike regions. However,
recent years purely operator-algebraic constructions of such nets have been
found. A remarkable feature of these new constructions is that they first
consider a single von Neumann algebra (instead of a family of von Neumann
algebras) which is acted on by the spacetime symmetry group in an appropriate
way. The construction procedure relying on a single von Neumann algebra has
been proposed in [4] and resulted in some intermediate constructions [18, 19,
7, 22] and even in a complete construction of local nets [23]. This von
Neumann algebra is interpreted as the algebra of observables localized in a
wedge-shaped region. There is a prescription to recover the strictly localized
observables [4]. However, the algebras of strictly localized observables are
not necessarily large enough and they can be even trivial [7]. When it turned
out to be sufficiently large, one had to rely on the modular nuclearity
condition, a sophisticated analytic tool [8, 23].
Among above constructions, the deformation by Buchholz, Lechner and Summers
starts with an arbitrary wedge-local net. When one applies the BLS deformation
to chiral conformal theories in two dimensions, things get considerably
simplified. We have seen that the theory remains to be asymptotically complete
in the sense of waves [6] even after the deformation and the full S-matrix has
been computed [15]. In this paper we carry out a further construction of
wedge-local nets based on chiral conformal nets. It turns out that all these
construction are related with endomorphisms of the half-line algebra in the
chiral components recently studied by Longo and Witten [26]. Among such
endomorphisms, the simplest ones are translations and inner symmetries. We
show that the construction related to translations coincides with the BLS
deformation of chiral CFT. The construction related to inner symmetries is new
and we completely determine the strictly localized observables under some
technical conditions. Furthermore, by using the family of endomorphisms on the
$U(1)$-current net considered in [26], we construct a large family of wedge-
local nets parametrized by inner symmetric functions. All these wedge-local
nets have nontrivial S-matrix, but the strictly local part of the wedge-local
nets constructed through inner symmetries has trivial S-matrix. The strict
locality of the other constructions remains open. Hence, to our opinion, the
true difficulty lies in strict locality.
Another important question is how large the class of theories is obtained by
this procedure. The class of S-matrices so far obtained is considered rather
small, since any of such S-matrices is contained in the tensor product of
abelian algebras in chiral components, which corresponds to the notion of
local diagonalizability in quantum information. In this paper, however, we
show that a massless asymptotically complete theory is completely
characterized by its asymptotic behaviour and the S-matrix, and the whole
theory can be recovered with a simple formula. Hence we can say that this
formula is sufficiently general.
In Section 2 we recall standard notions of algebraic QFT and scattering
theory. In Section 3 we show that the pair of S-matrix and the asymptotic
algebra is a complete invariant of a massless asymptotically complete net. In
Section 4 we construct wedge-local nets using one-parameter endomorphisms of
Longo-Witten. It is shown that the case of translations coincides with the BLS
deformation of chiral CFT and the strictly local elements are completely
determined for the case of inner symmetries. A common argument is summarized
in Section 4.1. Section 5 is devoted to the construction of wedge-local nets
based on a specific example, the $U(1)$-current net. A similar construction is
obtained also for the free fermionic net. Section 6 summarizes our
perspectives.
## 2 Preliminaries
### 2.1 Poincaré covariant net
We recall the algebraic treatment of quantum field theory [20]. A (local)
Poincaré covariant net ${\mathcal{A}}$ on ${\mathbb{R}}^{2}$ assigns to each
open bounded region $O$ a von Neumann algebra ${\mathcal{A}}(O)$ on a
(separable) Hilbert space ${\mathcal{H}}$ satisfying the following conditions:
1. (1)
Isotony. If $O_{1}\subset O_{2}$, then
${\mathcal{A}}(O_{1})\subset{\mathcal{A}}(O_{2})$.
2. (2)
Locality. If $O_{1}$ and $O_{2}$ are causally disjoint, then
$[{\mathcal{A}}(O_{1}),{\mathcal{A}}(O_{2})]=0$.
3. (3)
Poincaré covariance. There exists a strongly continuous unitary representation
$U$ of the (proper orthochronous) Poincaré group
${{\mathcal{P}}^{\uparrow}_{+}}$ such that for any open region $O$ it holds
that
$U(g){\mathcal{A}}(O)U(g)^{*}={\mathcal{A}}(gO),\mbox{ for
}g\in{{\mathcal{P}}^{\uparrow}_{+}}.$
4. (4)
Positivity of energy. The joint spectrum of the translation subgroup
${\mathbb{R}}^{2}$ of ${{\mathcal{P}}^{\uparrow}_{+}}$ of $U$ is contained in
the forward lightcone
$V_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}^{2}:p_{0}\geq|p_{1}|\\}$.
5. (5)
Existence of the vacuum. There is a unique (up to a phase) unit vector
$\Omega$ in ${\mathcal{H}}$ which is invariant under the action of $U$, and
cyclic for $\bigvee_{O\Subset{\mathbb{R}}^{2}}{\mathcal{A}}(O)$.
6. (6)
Additivity. If $O=\bigcup_{i}O_{i}$, then
${\mathcal{A}}(O)=\bigvee_{i}{\mathcal{A}}(O_{i})$.
From these axioms, the following property automatically follows (see [2])
1. (7)
Reeh-Schlieder property. The vector $\Omega$ is cyclic and separating for each
${\mathcal{A}}(O)$.
It is convenient to extend the definition of net also to a class of unbounded
regions called wedges. By definition, the standard left and right wedges are
as follows:
$\displaystyle W_{\mathrm{L}}$ $\displaystyle:=$
$\displaystyle\\{(t_{0},t_{1}):t_{0}>t_{1},t_{0}<-t_{1}\\}$ $\displaystyle
W_{\mathrm{R}}$ $\displaystyle:=$
$\displaystyle\\{(t_{0},t_{1}):t_{0}<t_{1},t_{0}>-t_{1}\\}$
The wedges $W_{\mathrm{L}}$, $W_{\mathrm{R}}$ are invariant under Lorentz
boosts. They are causal complements of each other. All the regions obtained by
translations of standard wedges are still called left- and right-wedges,
respectively. Moreover, a bounded region obtained as the intersection of a
left wedge and a right wedge is called a double cone. Let $O^{\prime}$ denote
the causal complement of $O$. It holds that
$W_{\mathrm{L}}^{\prime}=W_{\mathrm{R}}$, and if
$D=(W_{\mathrm{L}}+a)\cap(W_{\mathrm{R}}+b)$ is a double cone,
$a,b\in{\mathbb{R}}^{2}$, then
$D^{\prime}=(W_{\mathrm{R}}+a)\cup(W_{\mathrm{L}}+b)$. It is easy to see that
$\Omega$ is still cyclic and separating for ${\mathcal{A}}(W_{\mathrm{L}})$
and ${\mathcal{A}}(W_{\mathrm{R}})$.
We assume the following properties as natural conditions.
* •
Bisognano-Wichmann property. The modular group $\Delta^{it}$ of
${\mathcal{A}}(W_{\mathrm{R}})$ with respect to $\Omega$ is equal to
$U(\Lambda(-2\pi t))$, where $\Lambda(t)=\left(\begin{matrix}\cosh t&\sinh
t\\\ \sinh t&\cosh t\end{matrix}\right)$ denotes the Lorentz boost.
* •
Haag duality. If $O$ is a wedge or a double cone, then it holds that
${\mathcal{A}}(O)^{\prime}={\mathcal{A}}(O^{\prime})$.
If ${\mathcal{A}}$ is Möbius covariant (conformal), then the Bisognano-
Wichmann property is automatic [5], and Haag duality is equivalent to strong
additivity ([29], see also Section 2.2). These properties are valid even in
massive interacting models [23]. Duality for wedge regions (namely
${\mathcal{A}}(W_{\mathrm{L}})^{\prime}={\mathcal{A}}(W_{\mathrm{R}})$)
follows from Bisognano-Wichmann property [31], and it implies that the dual
net indeed satisfies the Haag duality [2].
### 2.2 Chiral conformal nets
In this Section we introduce a fundamental class of examples of Poincaré
covariant nets. For this purpose, first we explain nets on the one-dimensional
circle $S^{1}$. An open nonempty connected nondense subset $I$ of the circle
$S^{1}$ is called an interval. A (local) Möbius covariant net
${\mathcal{A}}_{0}$ on $S^{1}$ assigns to each interval a von Neumann algebra
${\mathcal{A}}_{0}(I)$ on a (separable) Hilbert space ${\mathcal{H}}_{0}$
satisfying the following conditions:
1. (1)
Isotony. If $I_{1}\subset I_{2}$, then
${\mathcal{A}}_{0}(I_{1})\subset{\mathcal{A}}_{0}(I_{2})$.
2. (2)
Locality. If $I_{1}\cap I_{2}=\emptyset$, then
$[{\mathcal{A}}_{0}(I_{1}),{\mathcal{A}}_{0}(I_{2})]=0$.
3. (3)
Möbius covariance. There exists a strongly continuous unitary representation
$U_{0}$ of the Möbius group ${\rm PSL}(2,{\mathbb{R}})$ such that for any
interval $I$ it holds that
$U_{0}(g){\mathcal{A}}_{0}(I)U_{0}(g)^{*}={\mathcal{A}}_{0}(gI),\mbox{ for
}g\in{\rm PSL}(2,{\mathbb{R}}).$
4. (4)
Positivity of energy. The generator of the one-parameter subgroup of rotations
in the representation $U_{0}$ is positive.
5. (5)
Existence of the vacuum. There is a unique (up to a phase) unit vector
$\Omega_{0}$ in ${\mathcal{H}}_{0}$ which is invariant under the action of
$U_{0}$, and cyclic for $\bigvee_{I\Subset S^{1}}{\mathcal{A}}_{0}(I)$.
We identify the circle $S^{1}$ as the one-point compactification of the real
line ${\mathbb{R}}$ by the Cayley transform:
$t=i\frac{z-1}{z+1}\Longleftrightarrow
z=-\frac{t-i}{t+i},\phantom{...}t\in{\mathbb{R}},\phantom{..}z\in
S^{1}\subset{\mathbb{C}}.$
Under this identification, we refer to translations $\tau$ and dilations
$\delta$ of ${\mathbb{R}}$ and these are contained in ${\rm
PSL}(2,{\mathbb{R}})$. It is known that the positivity of energy is equivalent
to the positivity of the generator of translations [25].
From the axioms above, the following properties automatically follow (see
[17])
1. (6)
Reeh-Schlieder property. The vector $\Omega_{0}$ is cyclic and separating for
each ${\mathcal{A}}_{0}(I)$.
2. (7)
Additivity. If $I=\bigcup_{i}I_{i}$, then
${\mathcal{A}}_{0}(I)=\bigvee_{i}{\mathcal{A}}_{0}(I_{i})$.
3. (8)
Haag duality on $S^{1}$. For an interval $I$ it holds that
${\mathcal{A}}_{0}(I)^{\prime}={\mathcal{A}}_{0}(I^{\prime})$, where
$I^{\prime}$ is the interior of the complement of $I$ in $S^{1}$.
4. (9)
Bisognano-Wichmann property. The modular group $\Delta_{0}^{it}$ of
${\mathcal{A}}_{0}({\mathbb{R}}_{+})$ with respect to $\Omega_{0}$ is equal to
$U_{0}(\delta(-2\pi t))$, where $\delta$ is the one-parameter group of
dilations.
###### Example 2.1.
At this level, we have a plenty of examples: The simplest one is the
$U(1)$-current net which will be explained in detail in Section 5.1. Among
others, the most important family is the loop group nets [17, 32]. Even a
classification result has been obtained for a class of nets on $S^{1}$ [21].
A net ${\mathcal{A}}_{0}$ on $S^{1}$ is said to be strongly additive if it
holds that
${\mathcal{A}}_{0}(I)={\mathcal{A}}_{0}(I_{1})\vee{\mathcal{A}}_{0}(I_{2})$,
where $I_{1}$ and $I_{2}$ are intervals obtained by removing an interior point
of $I$.
Let us denote by ${\rm Diff}(S^{1})$ the group of orientation-preserving
diffeomorphisms of the circle $S^{1}$. This group naturally includes ${\rm
PSL}(2,{\mathbb{R}})$. A Möbius covariant net ${\mathcal{A}}_{0}$ on $S^{1}$
is said to be conformal or diffeomorphism covariant if the representation
$U_{0}$ of ${\rm PSL}(2,{\mathbb{R}})$ associated to ${\mathcal{A}}_{0}$
extends to a projective unitary representation of ${\rm Diff}(S^{1})$ such
that for any interval $I$ and $x\in{\mathcal{A}}_{0}(I)$ it holds that
$\displaystyle
U_{0}(g){\mathcal{A}}_{0}(I)U_{0}(g)^{*}={\mathcal{A}}_{0}(gI),\mbox{ for
}g\in{\rm Diff}(S^{1}),$ $\displaystyle U_{0}(g)xU_{0}(g)^{*}=x,\mbox{ if
}{\rm supp}(g)\subset I^{\prime},$
where ${\rm supp}(g)\subset I^{\prime}$ means that $g$ acts identically on
$I$.
Let ${\mathcal{A}}_{0}$ be a Möbius covariant net on $S^{1}$. If a unitary
operator $V_{0}$ commutes with the translation unitaries
$T_{0}(t)=U_{0}(\tau(t))$ and it holds that
$V_{0}{\mathcal{A}}_{0}({\mathbb{R}}_{+})V_{0}^{*}\subset{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
then we say that $V_{0}$ implements a Longo-Witten endomorphism of
${\mathcal{A}}_{0}$. In particular, $V_{0}$ preserves $\Omega_{0}$ up to a
scalar since $\Omega_{0}$ is the unique invariant vector under $T_{0}(t)$.
Such endomorphisms have been studied first in [26] and they found a large
family of endomorphisms for the $U(1)$-current net, its extensions and the
free fermion net.
Let us denote two lightlines by
${\mathrm{L}}_{\pm}:=\\{(t_{0},t_{1})\in{\mathbb{R}}^{2}:t_{0}\pm t_{1}=0\\}$.
Note that any double cone $D$ can be written as a direct product of intervals
$D=I_{+}\times I_{-}$ where $I_{+}\subset{\mathrm{L}}_{+}$ and
$I_{-}\subset{\mathrm{L}}_{-}$. Let ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ be
two Möbius covariant nets on $S^{1}$ defined on the Hilbert spaces
${\mathcal{H}}_{1},{\mathcal{H}}_{2}$ with the vacuum vectors
$\Omega_{1},\Omega_{2}$ and the representations $U_{1},U_{2}$ of ${\rm
PSL}(2,{\mathbb{R}})$. From this pair, we can construct a two-dimensional net
${\mathcal{A}}$ as follows: For a double cone $D=I_{+}\times I_{-}$, we set
${\mathcal{A}}(D)={\mathcal{A}}_{1}(I_{+})\otimes{\mathcal{A}}_{2}(I_{-})$.
For a general open region $O\subset{\mathbb{R}}$, we set
${\mathcal{A}}(O):=\bigvee_{D\subset O}{\mathcal{A}}(D)$. We set
$\Omega:=\Omega_{1}\otimes\Omega_{2}$ and define the representation $U$ of
${\rm PSL}(2,{\mathbb{R}})\times{\rm PSL}(2,{\mathbb{R}})$ by $U(g_{1}\times
g_{2}):=U_{1}(g_{1})\otimes U_{2}(g_{2})$. By recalling that ${\rm
PSL}(2,{\mathbb{R}})\times{\rm PSL}(2,{\mathbb{R}})$ contains the Poincaré
group ${{\mathcal{P}}^{\uparrow}_{+}}$, it is easy to see that ${\mathcal{A}}$
together with $U$ and $\Omega$ is a Poincaré covariant net. We say that such
${\mathcal{A}}$ is chiral and ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$ are
referred to as the chiral components. If ${\mathcal{A}}_{1},{\mathcal{A}}_{2}$
are conformal, then the representation $U$ naturally extends to a projective
representation of ${\rm Diff}(S^{1})\times{\rm Diff}(S^{1})$.
### 2.3 Scattering theory for Borchers triples
A Borchers triple on a Hilbert space ${\mathcal{H}}$ is a triple
$({\mathcal{M}},T,\Omega)$ of a von Neumann algebra ${\mathcal{M}}\subset
B({\mathcal{H}})$, a unitary representation $T$ of ${\mathbb{R}}^{2}$ on
${\mathcal{H}}$ and a vector $\Omega\in{\mathcal{H}}$ such that
* •
${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathcal{M}})\subset{\mathcal{M}}$ for
$(t_{0},t_{1})\in W_{\mathrm{R}}$, the standard right wedge.
* •
The joint spectrum ${\rm sp}\,T$ is contained in the forward lightcone
$V_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}_{2}:p_{0}\geq|p_{1}|\\}$.
* •
$\Omega$ is a unique (up to scalar) invariant vector under $T$, and cyclic and
separating for ${\mathcal{M}}$.
By the theorem of Borchers [4, 16], the representation $T$ extends to the
Poincaré group ${{\mathcal{P}}^{\uparrow}_{+}}$, with Lorentz boosts
represented by the modular group of ${\mathcal{M}}$ with respect to $\Omega$.
With this extension $U$, ${\mathcal{M}}$ is Poincaré covariant in the sense
that if $gW_{\mathrm{R}}\subset W_{\mathrm{R}}$ for
$g\in{{\mathcal{P}}^{\uparrow}_{+}}$, then
$U(g){\mathcal{M}}U(g)^{*}\subset{\mathcal{M}}$.
The relevance of Borchers triples comes from the fact that we can construct
wedge-local nets from them: Let ${\mathcal{W}}$ be the set of wedges, i.e. the
the set of all $W=gW_{\mathrm{R}}$ or $W=gW_{\mathrm{L}}$ where $g$ is a
Poincaré transformation. A wedge-local net ${\mathcal{W}}\ni
W\mapsto{\mathcal{A}}(W)$ is a map from ${\mathcal{W}}$ to the set of von
Neumann algebras which satisfy isotony, locality, Poincaré covariance,
positivity of energy, and existence of vacuum, restricted to ${\mathcal{W}}$.
A wedge-local net associated with the Borchers triple
$({\mathcal{M}},T,\Omega)$ is the map defined by
${\mathcal{A}}(W_{R}+a)=T(a){\mathcal{M}}T(a)^{\ast}$ and
${\mathcal{A}}(W_{R}^{\prime}+a)=T(a){\mathcal{M}}^{\prime}T(a)^{\ast}$. This
can be considered as a notion of nets with a weaker localization property. It
is clear that there is a one-to-one correspondence between Borchers triples
and wedge-local nets. A further relation with local nets will be explained at
the end of this section. For simplicity, we study always Borchers triples,
which involve only a single von Neumann algebra.
We denote by ${\mathcal{H}}_{+}$ (respectively by ${\mathcal{H}}_{-}$) the
space of the single excitations with positive momentum, (respectively with
negative momentum) i.e.,
${\mathcal{H}}_{+}=\\{\xi\in{\mathcal{H}}:T(t,t)\xi=\xi\mbox{ for
}t\in{\mathbb{R}}\\}$ (respectively
${\mathcal{H}}_{-}=\\{\xi\in{\mathcal{H}}:T(t,-t)\xi=\xi\mbox{ for
}t\in{\mathbb{R}}\\}$).
Our fundamental examples come from Poincaré covariant nets. For a Poincaré
covariant net ${\mathcal{A}}$, we can construct a Borchers triple as follows:
* •
${\mathcal{M}}={\mathcal{A}}(W_{\mathrm{R}})$
* •
$T:=U|_{{\mathbb{R}}^{2}}$, the restriction of $U$ to the translation
subgroup.
* •
$\Omega$: the vacuum vector.
Indeed, the first condition follows from the Poincaré (in particular,
translation) covariance of the nets and the other conditions are assumed
properties of $U$ and $\Omega$ of the net. If $({\mathcal{M}},T,\Omega)$ comes
from a chiral conformal net
${\mathcal{A}}={\mathcal{A}}_{1}\otimes{\mathcal{A}}_{2}$, then we say this
triple is chiral, as well. This simple construction by tensor product of
chiral nets is considered to be the “undeformed net”. We will exhibit later
different constructions.
Given a Borchers triple $({\mathcal{M}},T,\Omega)$, we can consider the
scattering theory with respect to massless particles [15], which is an
extension of [6]: For a bounded operator $x\in B({\mathcal{H}})$ we write
$x(a)={\hbox{\rm Ad\,}}T(a)(x)$ for $a\in{\mathbb{R}}^{2}$. Furthermore, we
define a family of operators parametrized by ${\mathcal{T}}$:
$x_{\pm}(h_{\mathcal{T}}):=\int dt\,h_{\mathcal{T}}(t)x(t,\pm t),$
where
$h_{\mathcal{T}}(t)=|{\mathcal{T}}|^{-\varepsilon}h(|{\mathcal{T}}|^{-\varepsilon}(t-{\mathcal{T}}))$,
$0<\varepsilon<1$ is a constant, ${\mathcal{T}}\in{\mathbb{R}}$ and $h$ is a
nonnegative symmetric smooth function on ${\mathbb{R}}$ such that $\int
dt\,h(t)=1$.
###### Lemma 2.2 ([6] Lemma 2(b), [15] Lemma 2.1).
Let $x\in{\mathcal{M}}$, then the limits
$\Phi^{\mathrm{out}}_{+}(x):=\underset{{\mathcal{T}}\to+\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{+}(h_{\mathcal{T}})$
and
$\Phi^{\mathrm{in}}_{-}(x):=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,x_{-}(h_{\mathcal{T}})$
exist and it holds that
* •
$\Phi^{\mathrm{out}}_{+}(x)\Omega=P_{+}x\Omega$ and
$\Phi^{\mathrm{in}}_{-}(x)\Omega=P_{-}x\Omega$
* •
$\Phi^{\mathrm{out}}_{+}(x){\mathcal{H}}_{+}\subset{\mathcal{H}}_{+}$ and
$\Phi^{\mathrm{in}}_{-}(x){\mathcal{H}}_{-}\subset{\mathcal{H}}_{-}$.
* •
${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{out}}_{+}(x))=\Phi^{\mathrm{out}}_{+}({\hbox{\rm
Ad\,}}U(g)(x))$ and ${\hbox{\rm
Ad\,}}U(g)(\Phi^{\mathrm{in}}_{-}(x))=\Phi^{\mathrm{in}}_{-}({\hbox{\rm
Ad\,}}U(g)(x))$ for $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that
$gW_{\mathrm{R}}\subset W_{\mathrm{R}}$.
Furthermore, the limits $\Phi^{\mathrm{out}}_{+}(x)$ (respectively
$\Phi^{\mathrm{in}}_{-}(x)$) depends only on $P_{+}x\Omega$ (respectively on
$P_{-}x\Omega$).
Similarly we define asymptotic objects for the left wedge $W_{\mathrm{L}}$.
Since $J{\mathcal{M}}^{\prime}J={\mathcal{M}}$, where $J$ is the modular
conjugation for ${\mathcal{M}}$ with respect to $\Omega$, we can define for
any $y\in{\mathcal{M}}^{\prime}$
$\Phi_{+}^{\mathrm{in}}(y):=J\Phi_{+}^{\mathrm{out}}(JyJ)J,\,\,\Phi_{-}^{\mathrm{out}}(y):=J\Phi_{-}^{\mathrm{in}}(JyJ)J.$
Then we have the following.
###### Lemma 2.3 ([15], Lemma 2.2).
Let $y\in{\mathcal{M}}^{\prime}$. Then
$\Phi_{+}^{\mathrm{in}}(y)=\underset{{\mathcal{T}}\to-\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y_{+}(h_{\mathcal{T}}),\,\,\Phi_{-}^{\mathrm{out}}(y)=\underset{{\mathcal{T}}\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,y_{-}(h_{\mathcal{T}}).$
These operators depend only on the respective vectors
$\Phi_{+}^{\mathrm{in}}(y)\Omega=P_{+}y\Omega$,
$\Phi_{-}^{\mathrm{out}}(y)\Omega=P_{-}y\Omega$ and we have
1. (a)
$\Phi_{+}^{\mathrm{in}}(y){\mathcal{H}}_{+}\subset{\mathcal{H}}_{+},\,\,\Phi_{-}^{\mathrm{out}}(y){\mathcal{H}}_{-}\subset{\mathcal{H}}_{-}$,
2. (b)
${\hbox{\rm
Ad\,}}U(g)(\Phi_{+}^{\mathrm{in}}(y))=\Phi_{+}^{\mathrm{in}}({\hbox{\rm
Ad\,}}U(g)(y))$ and ${\hbox{\rm
Ad\,}}U(g)(\Phi_{-}^{\mathrm{out}}(y))=\Phi_{-}^{\mathrm{out}}({\hbox{\rm
Ad\,}}U(g)(y))$ for $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that
$gW_{\mathrm{L}}\subset W_{\mathrm{L}}$.
For $\xi_{+}\in{\mathcal{H}}_{+}$, $\xi_{-}\in{\mathcal{H}}_{-}$, there are
sequences of local operators $\\{x_{n}\\}\subset{\mathcal{M}}$ and
$\\{y_{n}\\}\subset{\mathcal{M}}^{\prime}$ such that
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{+}x_{n}\Omega=\xi_{+}$
and
$\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,P_{-}y_{n}\Omega=\xi_{-}$.
With these sequences we define collision states as in [15]:
$\displaystyle\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{in}}_{+}(x_{n})\Phi^{\mathrm{in}}_{-}(y_{n})\Omega$
$\displaystyle\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{{{\mathrm{s}\textrm{-}\lim}}}\,\Phi^{\mathrm{out}}_{+}(x_{n})\Phi^{\mathrm{out}}_{-}(y_{n})\Omega.$
We interpret $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ (respectively
$\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$) as the incoming
(respectively outgoing) state which describes two non-interacting waves
$\xi_{+}$ and $\xi_{-}$. These asymptotic states have the following natural
properties.
###### Lemma 2.4 ([15], Lemma 2.3).
For the collision states $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$ and
$\eta_{+}{\overset{{\mathrm{in}}}{\times}}\eta_{-}$ it holds that
1. 1.
$\langle\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-},\eta_{+}{\overset{{\mathrm{in}}}{\times}}\eta_{-}\rangle=\langle\xi_{+},\eta_{+}\rangle\cdot\langle\xi_{-},\eta_{-}\rangle$.
2. 2.
$U(g)(\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-})=(U(g)\xi_{+}){\overset{{\mathrm{in}}}{\times}}(U(g)\xi_{-})$
for all $g\in{{\mathcal{P}}^{\uparrow}_{+}}$ such that $gW_{\mathrm{R}}\subset
W_{\mathrm{R}}$.
And analogous formulae hold for outgoing collision states.
Furthermore, we set the spaces of collision states: Namely, we let
${\mathcal{H}}^{\mathrm{in}}$ (respectively ${\mathcal{H}}^{\mathrm{out}}$) be
the subspace generated by $\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$
(respectively $\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}$). From Lemma
2.4, we see that the following map
$S:\xi_{+}{\overset{{\mathrm{out}}}{\times}}\xi_{-}\longmapsto\xi_{+}{\overset{{\mathrm{in}}}{\times}}\xi_{-}$
is an isometry. The operator
$S:{\mathcal{H}}^{\mathrm{out}}\to{\mathcal{H}}^{\mathrm{in}}$ is called the
scattering operator or the S-matrix of the Borchers triple
$({\mathcal{M}},U,\Omega)$. We say the waves in the triple are interacting if
$S$ is not a constant multiple of the identity operator on
${\mathcal{H}}^{\mathrm{out}}$. We say that the Borchers triple is
asymptotically complete (and massless) if it holds that
${\mathcal{H}}^{\mathrm{in}}={\mathcal{H}}^{\mathrm{out}}={\mathcal{H}}$. We
have seen that a chiral net and its BLS deformations (see Section 4.2.2) are
asymptotically complete [15]. If the Borchers triple
$({\mathcal{M}},T,\Omega)$ is constructed from a Poincaré covariant net
${\mathcal{A}}$, then we refer to these objects and notions as
$S,{\mathcal{H}}_{\pm}$ and asymptotic completeness of ${\mathcal{A}}$, etc.
This notion of asymptotic completeness concerns only massless excitations.
Indeed, if one considers the massive free model for example, then it is easy
to see that all the asymptotic fields are just the vacuum expectation (mapping
to ${\mathbb{C}}{\mathbbm{1}}$).
To conclude this section, we put a remark on the term “wedge-local net”. If a
Borchers triple $({\mathcal{M}},T,\Omega)$ comes from a Haag dual Poincaré
covariant net ${\mathcal{A}}$, then the local algebras are recovered by the
formula ${\mathcal{A}}(D)=T(a){\mathcal{M}}T(a)^{*}\cap
T(b){\mathcal{M}}^{\prime}T(b)^{*}={\mathcal{A}}(W_{\mathrm{R}}+a)\cap{\mathcal{A}}(W_{\mathrm{L}}+b)$,
where $D=(W_{\mathrm{R}}+a)\cap(W_{\mathrm{L}}+b)$ is a double cone.
Furthermore, if ${\mathcal{A}}$ satisfies Bisognano-Wichmann property, then
the Lorentz boost is obtained from the modular group, hence all the components
of the net are regained from the triple. Conversely, for a given Borchers
triple, one can define a “local net” by the same formula above. In general,
this “net” satisfies isotony, locality, Poincaré covariance and positivity of
energy, but not necessarily satisfies additivity and cyclicity of vacuum [4].
Addivity is usually used only in the proof of Reeh-Schlieder property, thus we
do not consider it here. If the “local net” constructed from a Borchers triple
satisfies cyclicity of vacuum, we say that the original Borchers triple is
strictly local. In this respect, a Borchers triple or a wedge-local net is
considered to have a weaker localization property. Hence the search for
Poincaré covariant nets reduces to the search for strictly local nets. Indeed,
by this approach a family of (massive) interacting Poincaré nets has been
obtained [23].
## 3 Asymptotic chiral algebra and S-matrix
### 3.1 Complete invariant of nets
Here we observe that asymptotically complete (massless) net ${\mathcal{A}}$ is
completely determined by its behaviour at asymptotic times. This is
particularly nice, since the search for Poincaré covariant nets is reduced to
the search for appropriate S-matrices. Having seen the classification of a
class of chiral components [21], one would hope even for a similar
classification result for massless asymptotically complete nets.
Specifically, we construct a complete invariant of a net with Bisognano-
Wichmann property consisting of two elements. We already know the first
element, the S-matrix. Let us construct the second element, the asymptotic
algebra. An essential tool is half-sided modular inclusion (see [33, 1] for
the original references). Indeed, we use an analogous argument as in [31,
Lemma 5.5]. Let ${\mathcal{N}}\subset{\mathcal{M}}$ be an inclusion of von
Neumann algebras. If there is a cyclic and separating vector $\Omega$ for
${\mathcal{N}},{\mathcal{M}}$ and ${\mathcal{M}}\cap{\mathcal{N}}^{\prime}$,
then the inclusion ${\mathcal{N}}\subset{\mathcal{M}}$ is said to be standard
in the sense of [13]. If
$\sigma^{\mathcal{M}}_{t}({\mathcal{N}})\subset{\mathcal{N}}$ for
$t\in{\mathbb{R}}_{\pm}$ where $\sigma^{\mathcal{M}}_{t}$ is the modular
automorphism of ${\mathcal{M}}$ with respect to $\Omega$, then it is called a
$\pm$half-sided modular inclusion.
###### Theorem 3.1 ([33, 1]).
If $({\mathcal{N}}\subset{\mathcal{M}},\Omega)$ is a standard $+$(respectively
$-$)half-sided modular inclusion, then there is a Möbius covariant net
${\mathcal{A}}_{0}$ on $S^{1}$ such that
${\mathcal{A}}_{0}({\mathbb{R}}_{-})={\mathcal{M}}$ and
${\mathcal{A}}_{0}({\mathbb{R}}_{-}-1)={\mathcal{N}}$ (respectively
${\mathcal{A}}_{0}({\mathbb{R}}_{+})={\mathcal{M}}$ and
${\mathcal{A}}_{0}({\mathbb{R}}_{+}+1)={\mathcal{N}}$).
If a unitary representation $T_{0}$ of ${\mathbb{R}}$ with positive spectrum
satisfies $T_{0}(t)\Omega=\Omega$ for $t\in{\mathbb{R}}$, ${\hbox{\rm
Ad\,}}T_{0}(t)({\mathcal{M}})\subset{\mathcal{M}}$ for $t\leq 0$ (respectively
$t\geq 0$) and ${\hbox{\rm Ad\,}}T_{0}(-1)({\mathcal{M}})={\mathcal{N}}$
(respectively ${\hbox{\rm Ad\,}}T_{0}(1)({\mathcal{M}})={\mathcal{N}}$), then
$T_{0}$ is the representation of the translation group of the Möbius covariant
net constructed above.
We put
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(O):=\\{\Phi^{\mathrm{out}}_{+}(x),x\in{\mathcal{A}}(O)\\}$.
We will show that
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))\subset\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})$
is a standard $+$half-sided modular inclusion when restricted to
${\mathcal{H}}_{+}$. Indeed, $\Phi^{\mathrm{out}}_{+}$ commutes with
${\hbox{\rm Ad\,}}U(g_{t})$ where $g_{t}=\Lambda(-2\pi t)$ is a Lorentz boost
(Lemma 2.2), and $\tilde{{\mathcal{A}}}(W_{\mathrm{R}}+(-1,1))$ is sent into
itself under ${\hbox{\rm Ad\,}}U(g_{t})$ for $t\geq 0$. Hence by Bisognano-
Wichmann property,
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))\subset\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})$
is a $+$half-sided modular inclusion. In addition, when restricted to
${\mathcal{H}}_{+}$, this inclusion is standard. To see this, note that
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1)+(1,1))=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(0,2))$
because $\Phi^{\mathrm{out}}_{+}$ is invariant under $T(1,1)$, and hence
$\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(D)\subset\left(\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}}+(-1,1))^{\prime}\cap\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})\right)$,
where $D=W_{\mathrm{R}}\cap(W_{\mathrm{L}}+(0,2))$. It follows that
$\overline{\left(\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}\left(W_{\mathrm{R}}+(-1,1)\right)^{\prime}\cap\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(W_{\mathrm{R}})\right)\Omega}\supset\overline{\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}(D)\Omega}=\overline{P_{+}{\mathcal{A}}(D)\Omega}={\mathcal{H}}_{+},$
which is the standardness on ${\mathcal{H}}_{+}$. Then we obtain a Möbius
covariant net on $S^{1}$ acting on ${\mathcal{H}}_{+}$, which we denote by
${\mathcal{A}}^{\mathrm{out}}_{+}$. Similarly we get a Möbius covariant net
${\mathcal{A}}^{\mathrm{out}}_{-}$ on ${\mathcal{H}}_{-}$. Two nets
${\mathcal{A}}^{\mathrm{out}}_{+}$ and ${\mathcal{A}}^{\mathrm{out}}_{-}$ act
like tensor product by Lemma 2.4, and span the whole space ${\mathcal{H}}$
from the vacuum $\Omega$ by asymptotic completeness. In other words,
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ is a
chiral Möbius covariant net on ${\mathbb{R}}^{2}$ acting on ${\mathcal{H}}$.
We call this chiral net
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ the
(out-)asymptotic algebra of the given net ${\mathcal{A}}$. Similarly one
defines ${\mathcal{A}}^{\mathrm{in}}_{+}$ and
${\mathcal{A}}^{\mathrm{in}}_{-}$.
Let $({\mathcal{M}},T,\Omega)$, where
${\mathcal{M}}:={\mathcal{A}}(W_{\mathrm{R}})$, be the Borchers triple
associated to an asymptotically complete Poincare covariant net
${\mathcal{A}}$ which satisfies Bisognano-Wichmann property and Haag duality.
Our next observation is that ${\mathcal{M}}$ can be recovered from asymptotic
fields.
###### Proposition 3.2.
It holds that
${\mathcal{M}}=\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}=\tilde{{\mathcal{A}}}^{\mathrm{out}}_{+}({\mathbb{R}}_{-})\vee\tilde{{\mathcal{A}}}^{\mathrm{in}}_{-}({\mathbb{R}}_{+})$.
###### Proof.
The inclusion
${\mathcal{M}}\supset\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}$
is obvious from the definition of asymptotic fields. The converse inclusion is
established by the modular theory: From the assumption of Bisognano-Wichmann
property, the modular automorphism of ${\mathcal{M}}$ with respect to $\Omega$
is the Lorentz boosts $U(\Lambda(-2\pi t))$. Furthermore, it holds that
${\hbox{\rm Ad\,}}U(\Lambda(-2\pi
t))(\Phi^{\mathrm{out}}_{+}(x))=\Phi^{\mathrm{out}}_{+}({\hbox{\rm
Ad\,}}U(\Lambda(-2\pi t))(x))$ by Lemma 2.2. An analogous formula holds for
$\Phi^{\mathrm{in}}$. Namely, the algebra in the middle term of the statement
is invariant under the modular group.
By the assumed asymptotic completeness, the algebra in the middle term spans
the whole space ${\mathcal{H}}$ from the vacuum $\Omega$ as well. Hence by a
simple consequence of Takesaki’s theorem [30, Theorem IX.4.2] [31, Theorem
A.1], these two algebras coincide.
The last equation follows by the definition of asymptotic algebra and their
invariance under translations in respective directions. ∎
###### Proposition 3.3.
It holds that $S\cdot\Phi^{\mathrm{out}}_{\pm}(x)\cdot
S^{*}=\Phi^{\mathrm{in}}_{\pm}(x)$ and
$S\cdot\tilde{{\mathcal{A}}}^{\mathrm{out}}_{\pm}({\mathbb{R}}_{\mp})\cdot
S^{*}=\tilde{{\mathcal{A}}}^{\mathrm{in}}_{\pm}({\mathbb{R}}_{\mp})$.
###### Proof.
This follows from the calculation, using Lemmata 2.2, 2.3 and 2.4,
$\displaystyle\Phi^{\mathrm{in}}_{+}(x)(\xi{\overset{{\mathrm{in}}}{\times}}\eta)$
$\displaystyle=$
$\displaystyle(P_{+}x\xi){\overset{{\mathrm{in}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle
S\left((P_{+}x\xi){\overset{{\mathrm{out}}}{\times}}\eta\right)$
$\displaystyle=$ $\displaystyle
S\cdot\Phi^{\mathrm{out}}_{+}(x)(\xi{\overset{{\mathrm{out}}}{\times}}\eta)$
$\displaystyle=$ $\displaystyle S\cdot\Phi^{\mathrm{out}}_{+}(x)\cdot
S^{*}(\xi{\overset{{\mathrm{in}}}{\times}}\eta),$
and asymptotic completeness. The equation for “$-$” fields is proved
analogously. The last equalities are simple consequences of the formulae for
asymptotic fields. ∎
###### Theorem 3.4.
The out-asymptotic net
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$ and
the S-matrix $S$ completely characterizes the original net ${\mathcal{A}}$ if
it satisfies Bisognano-Wichmann property, Haag duality and asymptotic
completeness.
###### Proof.
The wedge algebra is recovered by
${\mathcal{M}}={\mathcal{A}}(W_{\mathrm{R}})=\\{\Phi^{\mathrm{out}}_{+}(x),\Phi^{\mathrm{in}}_{-}(y):x,y\in{\mathcal{M}}\\}^{\prime\prime}$
by Proposition 3.2. In the right-hand side, $\Phi^{\mathrm{in}}_{-}$ is
recovered from $\Phi^{\mathrm{out}}_{-}$ and $S$ by Proposition 3.3. Hence the
wedge algebra is completely recovered from the data
$\Phi^{\mathrm{out}}_{\pm}$ and $S$, or ${\mathcal{A}}^{\mathrm{in}}_{\pm}$
and $S$ by Proposition 3.2. By Haag duality, the data of wedge algebras are
enough to recover the local algebras. By Bisognano-Wichmann property, the
representation $U$ of the whole Poincaré group is recovered from the modular
data. ∎
###### Remark 3.5.
Among the conditions on ${\mathcal{A}}$, Bisognano-Wichmann property is
satisfied in almost all known examples. Haag duality can be satisfied by
extending the net [2] without changing the S-matrix. Hence we consider them as
standard assumptions. On the other hand, asymptotic completeness is in fact a
very strong condition. For example, a conformal net is asymptotically complete
if and only if it is chiral [31]. Hence the class of asymptotically complete
nets could be very small even among Poincaré covariant nets. But a clear-cut
scattering theory is available only for asymptotically complete cases. The
general case is under investigation [14].
### 3.2 Chiral nets as asymptotic nets
We can express the modular objects of the interacting net in terms of the ones
of the asymptotic chiral net.
###### Proposition 3.6.
Let $\Delta^{\mathrm{out}}$ and $J^{\mathrm{out}}$ be the modular operator and
the modular conjugation of
${\mathcal{A}}_{+}^{\mathrm{out}}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{-}^{\mathrm{out}}({\mathbb{R}}_{+})$
with respect to $\Omega$. Then it holds that $\Delta=\Delta^{\mathrm{out}}$
and $J=SJ^{\mathrm{out}}$.
###### Proof.
First we note that the modular objects of ${\mathcal{A}}(W_{\mathrm{R}})$
restrict to ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$ by Takesaki’s theorem
[30, Theorem IX.4.2]. Indeed,
${\mathcal{A}}^{\mathrm{out}}_{+}({\mathbb{R}}_{+})$ and
${\mathcal{A}}^{\mathrm{out}}_{-}({\mathbb{R}}_{-})$ are subalgebras of
${\mathcal{A}}(W_{\mathrm{R}})$ and invariant under ${\hbox{\rm
Ad\,}}\Delta^{it}$, or equivalently under the Lorentz boosts ${\hbox{\rm
Ad\,}}U(\Lambda(-2\pi t))$ by Bisognano-Wichmann property, as we saw in the
proof of Proposition 3.2, then the projections onto the respective subspaces
commute with the modular objects. Let us denote these restrictions by
$\Delta_{+}^{it},J_{+},\Delta_{-}^{it}$ and $J_{-}$, respectively.
We identify ${\mathcal{H}}_{+}\otimes{\mathcal{H}}_{-}$ and the full Hilbert
space ${\mathcal{H}}$ by the action of
${\mathcal{A}}^{\mathrm{out}}_{+}\otimes{\mathcal{A}}^{\mathrm{out}}_{-}$. By
Bisognano-Wichmann property and Lemma 2.4, we have
$\displaystyle\Delta^{it}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle(U(\Lambda(-2\pi
t))\xi){\overset{{\mathrm{out}}}{\times}}(U(\Lambda(-2\pi t))\eta)$
$\displaystyle=$ $\displaystyle\Delta_{+}^{it}\xi\otimes\Delta_{-}^{it}\eta$
$\displaystyle=$
$\displaystyle(\Delta_{+}\otimes\Delta_{-})^{it}\cdot\xi\otimes\eta,$
which implies that $\Delta=\Delta_{+}\otimes\Delta_{-}=\Delta^{\mathrm{out}}$.
As for modular conjugations, we take $x\in{\mathcal{A}}(W_{\mathrm{R}})$ and
$y\in{\mathcal{A}}(W_{\mathrm{R}})^{\prime}={\mathcal{A}}(W_{\mathrm{L}})$ and
set $\xi=\Phi^{\mathrm{out}}_{+}(x)\Omega$ and
$\eta=\Phi^{\mathrm{out}}_{-}(y)\Omega$. Then we use Lemma 2.3 to see
$\displaystyle J\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle
J\cdot\Phi^{\mathrm{out}}_{+}(x)\Phi^{\mathrm{out}}_{-}(y)\Omega$
$\displaystyle=$
$\displaystyle\Phi^{\mathrm{in}}_{+}(JxJ)\Phi^{\mathrm{in}}_{-}(JyJ)\Omega$
$\displaystyle=$ $\displaystyle(J\xi){\overset{{\mathrm{in}}}{\times}}(J\eta)$
$\displaystyle=$ $\displaystyle
S\cdot(J_{+}\xi){\overset{{\mathrm{out}}}{\times}}(J_{-}\eta)$
$\displaystyle=$ $\displaystyle S\cdot(J_{+}\otimes J_{-})\cdot\xi\otimes\eta$
from which one infers that $J=S\cdot(J_{+}\otimes J_{-})=S\cdot
J^{\mathrm{out}}$. ∎
Theorem 3.4 tells us that chiral conformal nets can be viewed as free field
nets for massless two-dimensional theory (cf. [31]). Let us formulate the
situation the other way around. Let
${\mathcal{A}}_{+}\otimes{\mathcal{A}}_{-}$ be a chiral CFT, then it is an
interesting open problem to characterize unitary operators which can be
interpreted as a S-matrix of a net whose asymptotic net is the given
${\mathcal{A}}_{+}\otimes{\mathcal{A}}_{-}$. We restrict ourselves to point
out that there are several immediate necessary conditions: For example, $S$
must commute with the Poincaré symmetry of the chiral net since it coincides
with the one of the interacting net. Analogously it must hold that
$(J_{+}\otimes J_{-})S(J_{+}\otimes J_{-})=S^{*}$. Furthermore, the algebra of
the form as in Proposition 3.2 must be strictly local.
If one has an appropriate operator $S$, an interacting Borchers triple can be
constructed by (cf. Propositions 3.2, 3.3)
* •
${\mathcal{M}}_{S}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})\\}^{\prime\prime}$,
* •
$U:=U_{+}\otimes U_{-}$,
* •
$\Omega:=\Omega_{+}\otimes\Omega_{-}$.
By the formula for the modular conjugation in Proposition 3.6, it is immediate
to see that
${\mathcal{M}}_{S}^{\prime}:=\\{{\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\mathcal{A}}_{+}({\mathbb{R}}_{+}),y^{\prime}\in{\mathcal{A}}_{-}({\mathbb{R}}_{-})\\}^{\prime\prime}.$
Then for
$x\in{\mathcal{A}}_{+}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{-}({\mathbb{R}}_{+})$
it holds that
$\Phi^{\mathrm{out}}_{+}(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$ and
$\Phi^{\mathrm{in}}_{-}({\hbox{\rm Ad\,}}S({\mathbbm{1}}\otimes y))={\hbox{\rm
Ad\,}}S({\mathbbm{1}}\otimes y)$. Similarly, we have
$\Phi^{\mathrm{out}}_{-}({\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}}))={\hbox{\rm
Ad\,}}S(x^{\prime}\otimes{\mathbbm{1}})$ and
$\Phi^{\mathrm{in}}_{+}({\mathbbm{1}}\otimes y^{\prime})={\mathbbm{1}}\otimes
y^{\prime}$ for $x^{\prime}\in{\mathcal{A}}_{+}({\mathbb{R}}_{+})$ and
$y^{\prime}\in{\mathcal{A}}_{-}({\mathbb{R}}_{-})$. From this it is easy to
see that $S$ is indeed the S-matrix of the constructed Borchers triple.
In the following Sections we will construct unitary operators which comply
with these conditions except strict locality. To my opinion, however, the true
difficulty is the strict locality, which has been so far established only for
“regular” massive models [23]. But it is also true that the class of
S-matrices constructed in the present paper can be seen rather small (see the
discussion in Section 6).
## 4 Construction through one-parameter semigroup of endomorphisms
In this Section, we construct families of Borchers triples using one-parameter
semigroup of endomorphisms of Longo-Witten type. The formula to define the von
Neumann algebra is very simple and the proofs use a common argument based on
spectral decomposition.
Our construction is based on chiral conformal nets on $S^{1}$, and indeed one
family can be identified as the BLS deformation of chiral nets (see Section
4.2). But in our construction, the meaning of the term “deformation” is not
clear and we refrain from using it. From now on, we consider only chiral net
with the identical components
${\mathcal{A}}_{1}={\mathcal{A}}_{2}={\mathcal{A}}_{0}$ for simplicity. It is
not difficult to generalize it to “heterotic case” where
${\mathcal{A}}_{1}\neq{\mathcal{A}}_{2}$.
### 4.1 The commutativity lemma
The following Lemma is the key of all the arguments and will be used later in
this Section concerning one-parameter endomorphisms. Typical examples of the
operator $Q_{0}$ in Lemma will be the generator of one-dimensional
translations $P_{0}$ (Section 4.2), or of one-parameter inner symmetries of
the chiral component (Section 4.4).
As a preliminary, we give a remark on tensor product. See [12] for a general
account on spectral measure and measurable family. Let $E_{0}$ be a
projection-valued measure on $Z$ (typically, the spectral measure of some
self-adjoint operator) and $V(\lambda)$ be a measurable family of operators
(bounded or not). Then one can define an operator
$\int_{Z}V(\lambda)\otimes
dE_{0}(\lambda)(\xi\otimes\eta):=\int_{Z}V(\lambda)\xi\otimes
dE_{0}(\lambda)\eta.$
If $V(\lambda)$ is unbounded, the vector $\xi$ should be in a common domain of
$\\{V(\lambda)\\}$. As we will see, this will not matter in our cases. For two
bounded measurable families $V,V^{\prime}$, it is easy to see that
$\int_{Z}V(\lambda)\otimes
dE_{0}(\lambda)\cdot\int_{Z}V^{\prime}(\lambda)\otimes
dE_{0}(\lambda)=\int_{Z}V(\lambda)V^{\prime}(\lambda)\otimes dE_{0}(\lambda).$
###### Lemma 4.1.
We fix a parameter $\kappa\in{\mathbb{R}}$. Let $Q_{0}$ be a self-adjoint
operator on ${\mathcal{H}}_{0}$ and Let $Z\subset{\mathbb{R}}$ be the spectral
supports of $Q_{0}$. If it holds that $[x,{\hbox{\rm Ad\,}}e^{is\kappa
Q_{0}}(x^{\prime})]=0$ for $x,x^{\prime}\in B({\mathcal{H}}_{0})$ and $s\in
Z$, then we have that
$\displaystyle[x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})]$ $\displaystyle=0,$
$\displaystyle[{\mathbbm{1}}\otimes x,{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}({\mathbbm{1}}\otimes x^{\prime})]$ $\displaystyle=0.$
###### Proof.
We prove only the first commutation relation, since the other is analogous.
Let $Q_{0}=\int_{Z}s\cdot dE_{0}(s)$ be the spectral decomposition of $Q_{0}$.
According to this spectral decomposition, we can decompose only the second
component:
$Q_{0}\otimes Q_{0}=Q_{0}\otimes\int_{Z}s\cdot dE_{0}(x)=\int_{Z}sQ_{0}\otimes
dE_{0}(s).$
Hence we can describe the adjoint action of $e^{i\kappa Q_{0}\otimes Q_{0}}$
explicitly:
$\displaystyle{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})$ $\displaystyle=$
$\displaystyle\int_{Z}e^{is\kappa Q_{0}}\otimes
dE(s)\cdot(x^{\prime}\otimes{\mathbbm{1}})\cdot\int_{Z}e^{-is\kappa
Q_{0}}\otimes dE_{0}(s)$ $\displaystyle=$
$\displaystyle\int_{Z}\left({\hbox{\rm Ad\,}}e^{is\kappa
Q_{0}}(x^{\prime})\right)\otimes dE_{0}(s)$
Then it is easy to see that this commutes with $x\otimes{\mathbbm{1}}$ by the
assumed commutativity. ∎
### 4.2 Construction of Borchers triples with respect to translation
The objective here is to apply the commutativity lemma in Section 4.1 to the
endomorphism of translation. Then it turns out that the Borchers triples
obtained by the BLS deformation of a chiral triple coincide with this
construction. A new feature is that our construction involves only von Neumann
algebras.
#### 4.2.1 Construction of Borchers triples
Let $({\mathcal{M}},T,\Omega)$ be a chiral Borchers triple with chiral
component ${\mathcal{A}}_{0}$ and $T_{0}(t)=e^{itP_{0}}$ the chiral
translation: Namely,
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
$T(t_{0},t_{1})=T_{0}\left(\frac{t_{0}-t_{1}}{\sqrt{2}}\right)\otimes
T_{0}\left(\frac{t_{0}+t_{1}}{\sqrt{2}}\right)$ and
$\Omega=\Omega_{0}\otimes\Omega_{0}$.
Note that $T_{0}(t)$ implements a Longo-Witten endomorphism of
${\mathcal{A}}_{0}$ for $t\geq 0$. In this sense, the construction of this
Section is considered to be based on the endomorphisms $\\{{\hbox{\rm
Ad\,}}T_{0}(t)\\}$. A nontrivial family of endomorphisms will be featured in
Section 5.
We construct a new Borchers triple on the same Hilbert space
${\mathcal{H}}={\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$ as follows. Let us
fix $\kappa\in{\mathbb{R}}_{+}$.
* •
${\mathcal{M}}_{P_{0},\kappa}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$,
* •
the same $T$ from the chiral triple,
* •
the same $\Omega$ from the chiral triple.
###### Theorem 4.2.
Let $\kappa\geq 0$. Then the triple $({\mathcal{M}}_{P_{0},\kappa},T,\Omega)$
is a Borchers triple with the S-matrix $S_{P_{0},\kappa}=e^{i\kappa
P_{0}\otimes P_{0}}$.
###### Proof.
The vector $\Omega_{0}\otimes\Omega_{0}$ is obviously invariant under $T$ and
$T$ has the spectrum contained in $V_{+}$. The generator $P_{0}$ of one-
dimensional translations obviously commutes with one-dimensional translation
$T_{0}$, hence $P_{0}\otimes P_{0}$ commutes with $T=T_{0}\otimes T_{0}$, so
does $e^{i\kappa P_{0}\otimes P_{0}}$. We claim that
${\mathcal{M}}_{P_{0},\kappa}$ is preserved under translations in the right
wedge. Indeed, if $(t_{0},t_{1})\in W_{\mathrm{R}}$, then we have
${\hbox{\rm Ad\,}}T(t_{0},t_{1})\left({\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}({\mathbbm{1}}\otimes y)\right)={\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}\left({\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathbbm{1}}\otimes y)\right).$
and ${\hbox{\rm Ad\,}}T(t_{0},t_{1})({\mathbbm{1}}\otimes
y)\in{\mathbbm{1}}\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$ and it is
obvious that ${\hbox{\rm
Ad\,}}T(t_{0},t_{1})(x\otimes{\mathbbm{1}})\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathbbm{1}}$,
hence the generators of the von Neumann algebra ${\mathcal{M}}_{P_{0},\kappa}$
are preserved.
We have to show that $\Omega$ is cyclic and separating for
${\mathcal{M}}_{P_{0},\kappa}$. Note that it holds that $e^{i\kappa
P_{0}\otimes P_{0}}\cdot\xi\otimes\Omega_{0}=\xi\otimes\Omega_{0}$ for any
$\kappa\in{\mathbb{R}},\xi\in{\mathcal{H}}_{0}$, by the spectral calculus. Now
cyclicity is seen by noting that
$\displaystyle(x\otimes{\mathbbm{1}})\cdot{\hbox{\rm Ad\,}}e^{i\kappa
P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes y)\cdot\Omega$ $\displaystyle=$
$\displaystyle(x\otimes{\mathbbm{1}})\cdot e^{i\kappa P_{0}\otimes
P_{0}}\cdot(x\Omega_{0})\otimes\Omega_{0}$ $\displaystyle=$
$\displaystyle(x\Omega_{0})\otimes(y\Omega_{0})$
and by the cyclicity of $\Omega$ for the original algebra
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
Finally we show separating property as follows: we set
${\mathcal{M}}^{1}_{P_{0},\kappa}=\\{{\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}(x^{\prime}\otimes{\mathbbm{1}}),{\mathbbm{1}}\otimes
y^{\prime}:x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}),y^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\\}^{\prime\prime}.$
Note that $\Omega$ is cyclic for ${\mathcal{M}}^{1}_{P_{0},\kappa}$ by an
analogous proof for ${\mathcal{M}}_{P_{0},\kappa}$, thus for the separating
property, it suffices to show that ${\mathcal{M}}_{P_{0},\kappa}$ and
${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute. Let
$x,y^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
First, $x\otimes{\mathbbm{1}}$ and ${\mathbbm{1}}\otimes y^{\prime}$ obviously
commute. Next, we apply Lemma 4.1 to $x,x^{\prime}$ and $Q_{0}=P_{0}$ to see
that $x\otimes{\mathbbm{1}}$ and ${\hbox{\rm Ad\,}}e^{i\kappa P_{0}\otimes
P_{0}}(x^{\prime}\otimes{\mathbbm{1}})$ commute: Indeed, the spectral support
of $P_{0}$ is ${\mathbb{R}}_{+}$, and for $s\in{\mathbb{R}}_{+}$, $x$ and
${\hbox{\rm Ad\,}}e^{is\kappa P_{0}}(x^{\prime})$ commute since $P_{0}$ is the
generator of one-dimensional translations and since
$x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
Similarly, for $y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$, ${\hbox{\rm
Ad\,}}e^{i\kappa P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes y)$ and
${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute. This implies that
${\mathcal{M}}_{P_{0},\kappa}$ and ${\mathcal{M}}^{1}_{P_{0},\kappa}$ commute.
The S-matrix corresponds to the unitary used to twist the chiral net as we saw
in the discussion at the end of Section 3.2. ∎
Now that we have constructed a Borchers triple, it is possible to express its
modular objects in terms of the ones of the chiral triple by an analogous
argument as Proposition 3.6. Then one sees that
${\mathcal{M}}^{1}_{P_{0},\kappa}$ is indeed the commutant
${\mathcal{M}}_{P_{0},\kappa}^{\prime}$.
#### 4.2.2 BLS deformation
We briefly review the BLS deformation [7]. Let $({\mathcal{M}},T,\Omega)$ be a
Borchers triple. We denote by ${\mathcal{M}}^{\infty}$ the subset of elements
of ${\mathcal{M}}$ which are smooth under the action of $\alpha={\hbox{\rm
Ad\,}}T$ in the norm topology. Then one can define for any
$x\in{\mathcal{M}}^{\infty}$, and a matrix
$\Theta_{\kappa}=\left(\begin{matrix}0&\kappa\\\ \kappa&0\end{matrix}\right)$,
the warped convolution
$x_{\kappa}=\int
dE(a)\,\alpha_{\Theta_{\kappa}a}(x):=\lim_{\varepsilon\searrow
0}(2\pi)^{-2}\int d^{2}a\,d^{2}b\,f(\varepsilon a,\varepsilon b)e^{-ia\cdot
b}\alpha_{\Theta_{\kappa}a}(x)T(b)$
on a suitable domain, where $dE$ is the spectral measure of $T$ and
$f\in\mathscr{S}({\mathbb{R}}^{2}\times{\mathbb{R}}^{2})$ satisfies
$f(0,0)=1$. We set
${\mathcal{M}}_{\kappa}:=\\{x_{\kappa}:x\in{\mathcal{M}}^{\infty}\\}^{\prime\prime}.$
For $\kappa>0$, the following holds.
###### Theorem 4.3 ([7]).
If $({\mathcal{M}},T,\Omega)$ is a Borchers triple, then
$({\mathcal{M}}_{\kappa},T,\Omega)$ is also a Borchers triple.
We call the latter the BLS deformation of the original triple
$({\mathcal{M}},T,\Omega)$. One of the main results of this paper is to obtain
the BLS deformation by a simple procedure.
We have determined the property of deformed scattering theory in [15]. In our
notation $M^{2}=P_{0}\otimes P_{0}$ we have the following.
###### Theorem 4.4.
For any $\xi\in{\mathcal{H}}_{+}$ and $\eta\in{\mathcal{H}}_{-}$, the
following relations hold:
$\displaystyle\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta=e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(\xi\otimes\eta),$
$\displaystyle\xi{\overset{{\mathrm{in}}}{\times}}_{\kappa}\eta=e^{\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(\xi\otimes\eta),$
where on the left-hand sides there appear the collision states of the deformed
theory.
#### 4.2.3 Reproduction of BLS deformation
Let $({\mathcal{M}},T,\Omega)$ be a chiral Borchers triple. In this Section we
show that the Borchers triple $({\mathcal{M}}_{P_{0},\kappa},T,\Omega)$
obtained above is unitarily equivalent to the BLS deformation
$({\mathcal{M}}_{\kappa},T,\Omega)$. Then we can calculate the asymptotic
fields very simply. We use symbols ${\overset{{\mathrm{out}}}{\times}}$ and
${\overset{{\mathrm{out}}}{\times}}_{\kappa}$ to denote collision states with
respect to the corresponding Borchers triples with ${\mathcal{M}}$
(undeformed) and ${\mathcal{M}}_{\kappa}$, respectively. Recall that for the
undeformed chiral triple, all these products
${\overset{{\mathrm{out}}}{\times}}$, ${\overset{{\mathrm{in}}}{\times}}$ and
$\otimes$ coincide [15].
###### Theorem 4.5.
Let us put ${\mathcal{N}}_{P_{0},\kappa}:={\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}{\mathcal{M}}_{P_{0},\kappa}$.
Then it holds that ${\mathcal{N}}_{P_{0},\kappa}={\mathcal{M}}_{\kappa}$,
hence we have the coincidence of two Borchers triples
$({\mathcal{N}}_{P_{0},\kappa},T,\Omega)=({\mathcal{M}}_{\kappa},T,\Omega)$.
###### Proof.
In [15], we have seen that the deformed BLS triple is asymptotically complete.
Furthermore, we have
$\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta=e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\xi{\overset{{\mathrm{out}}}{\times}}\eta.$
As for observables, let $x\in A_{0}({\mathbb{R}}_{-})$ and we use the notation
$x_{\Theta_{\kappa}}$ from [7] 222The reader is suggested to look at the
notation $F_{Q}$ in [7], where $F$ is an observable in ${\mathcal{M}}$ and $Q$
is a $2\times 2$ matrix. We keep the symbol $Q$ for a generator of one-
parameter automorphisms, hence we changed the notation to avoid confusions..
For the asymptotic field $\Phi^{\mathrm{out}}_{\kappa,+}$ of BLS deformation,
we have
$\displaystyle\Phi_{\kappa,+}^{\mathrm{out}}((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}})\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$
$\displaystyle((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}\xi){\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$
$\displaystyle(x\xi){\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta$
$\displaystyle=$ $\displaystyle e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\cdot(x\xi){\overset{{\mathrm{out}}}{\times}}\eta$ $\displaystyle=$
$\displaystyle e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}\cdot
x\otimes{\mathbbm{1}}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$
$\displaystyle=$ $\displaystyle{\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}(x\otimes{\mathbbm{1}})\cdot
e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}\cdot\xi{\overset{{\mathrm{out}}}{\times}}\eta$ $\displaystyle=$
$\displaystyle{\hbox{\rm Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(x\otimes{\mathbbm{1}})\cdot\xi{\overset{{\mathrm{out}}}{\times}}_{\kappa}\eta,$
(see Appendix A for the second equality) hence, we have
$\Phi_{\kappa,+}^{\mathrm{out}}((x\otimes{\mathbbm{1}})_{\Theta_{\kappa}})={\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}(x\otimes{\mathbbm{1}})$.
Analogously we have $\Phi_{\kappa,-}^{\mathrm{in}}(({\mathbbm{1}}\otimes
y)_{\Theta_{\kappa}})={\hbox{\rm Ad\,}}e^{\frac{i\kappa}{2}P_{0}\otimes
P_{0}}({\mathbbm{1}}\otimes y)$ for $y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$.
Note that by definition we have
${\mathcal{N}}_{P_{0},\kappa}=\\{{\hbox{\rm
Ad\,}}e^{-\frac{i\kappa}{2}P_{0}\otimes
P_{0}}(x\otimes{\mathbbm{1}}),{\hbox{\rm
Ad\,}}e^{\frac{i\kappa}{2}P_{0}\otimes P_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}.$
Since the image of the right-wedge algebra by $\Phi^{\mathrm{out}}_{+}$ and
$\Phi^{\mathrm{in}}_{-}$ remains in the right-wedge algebra, from the above
observation, we see that
${\mathcal{N}}_{P_{0},\kappa}\subset{\mathcal{M}}_{\kappa}$ [15]. To see the
converse inclusion, recall that it has been proved that the modular group
$\Delta^{it}$ of the right-wedge algebra with respect to $\Omega$ remains
unchanged under the BLS deformation. We have that ${\hbox{\rm
Ad\,}}\Delta^{it}(e^{i\kappa P_{0}\otimes P_{0}})=e^{i\kappa P_{0}\otimes
P_{0}}$, hence it is easy to see that ${\mathcal{N}}_{P_{0},\kappa}$ is
invariant under ${\hbox{\rm Ad\,}}\Delta^{it}$. By the theorem of Takesaki
[30, Theorem IX.4.2], there is a conditional expectation from
${\mathcal{M}}_{\kappa}$ onto ${\mathcal{N}}_{P_{0},\kappa}$ which preserves
the state $\langle\Omega,\cdot\Omega\rangle$ and in particular,
${\mathcal{M}}_{\kappa}={\mathcal{N}}_{P_{0},\kappa}$ if and only if $\Omega$
is cyclic for ${\mathcal{N}}_{P_{0},\kappa}$. We have already seen the
cyclicity in Theorem 4.2, thus we obtain the thesis.
The translation $T$ and $\Omega$ remain unchanged under
$e^{-\frac{i\kappa}{2}P_{0}\otimes P_{0}}$, which established the unitary
equivalence between two Borchers triples. ∎
###### Remark 4.6.
It is also possible to formulate Theorem 3.4 for Borchers triple, although the
asymptotic algebra will be neither local nor conformal in general. From this
point of view, Theorem 4.5 is just a corollary of the coincidence of S-matrix.
Here we preferred a direct proof, instead of formulating non local net on
${\mathbb{R}}$.
### 4.3 Endomorphisms with asymmetric spectrum
Here we briefly describe a generalization of the construction in previous
Sections. Let ${\mathcal{A}}_{0}$ be a local net on $S^{1}$, $T_{0}$ be the
representation of the translation. We assume that there is a one-parameter
family $V_{0}(t)=e^{iQ_{0}t}$ of unitary operators with a positive or negative
generator $Q_{0}$ such that $V_{0}(t)$ and $T_{0}(s)$ commute and ${\hbox{\rm
Ad\,}}V_{0}(t)({\mathcal{A}}_{0}({\mathbb{R}}_{+}))\subset{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
for $t\geq 0$. With these ingredients, we have the following:
###### Theorem 4.7.
The triple
* •
${\mathcal{M}}_{Q_{0},\kappa}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}e^{\pm i\kappa Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$,
* •
$T:=T_{0}\otimes T_{0}$,
* •
$\Omega:=\Omega_{0}\otimes\Omega_{0}$,
where $\pm$ corresponds to ${\rm sp}\,Q_{0}\subset{\mathbb{R}}_{\pm}$, is a
Borchers triple with the S-matrix $e^{\pm i\kappa Q_{0}\otimes Q_{0}}$ for
$\kappa\geq 0$.
The proof is analogous to Theorem 4.2 and we refrain from repeating it here.
The construction looks very simple, but to our knowledge, there are only few
examples. The one-parameter group of translation itself has been studied in
the previous Sections. Another one-parameter family of unitaries with a
negative generator $\\{\Gamma(e^{-{\frac{\kappa}{P_{1}}}})\\}$ has been found
for the $U(1)$-current [26], where $P_{1}$ is the generator on the one-
particle space, $\kappa\geq 0$ and $\Gamma$ denotes the second quantization.
Indeed, by Borchers’ theorem [4, 16], such one-parameter group together with
the modular group forms a representation of the “$ax+b$” group, thus it is
related somehow with translation.
### 4.4 Construction of Borchers triples through inner symmetry in chiral CFT
#### 4.4.1 Inner symmetry
Let ${\mathcal{A}}_{0}$ be a conformal (Möbius) net on $S^{1}$. An
automorphism of ${\mathcal{A}}_{0}$ is a family of automorphisms
$\\{\alpha_{0,I}\\}$ of local algebras $\\{{\mathcal{A}}_{0}(I)\\}$ with the
consistency condition $\alpha_{0,J}|_{{\mathcal{A}}_{0}(I)}=\alpha_{0,I}$ for
$I\subset J$. If each $\alpha_{0,I}$ preserves the vacuum state $\omega$, then
$\alpha_{0}$ is said to be an inner symmetry. An inner symmetry $\alpha_{0}$
is implemented by a unitary $V_{\alpha_{0}}$ defined by
$V_{\alpha_{0}}x\Omega=\alpha_{0,I}(x)\Omega$, where
$x\in{\mathcal{A}}_{0}(I)$. This definition does not depend on the choice of
$I$ by the consistency condition. If $\alpha_{0,t}$ is a one-parameter family
of weakly continuous automorphisms, then the implementing unitaries satisfy
$V_{\alpha_{0}}(t)V_{\alpha_{0}}(s)=V_{\alpha_{0}}(t+s)$ and
$V_{\alpha_{0}}(0)={\mathbbm{1}}$, hence there is a self-adjoint operator
$Q_{0}$ such that $V_{\alpha_{0}}(t)=e^{itQ_{0}}$ and $Q_{0}\Omega=0$.
Furthermore, $e^{itQ_{0}}$ commutes with modular objects [30]:
$J_{0}e^{itQ_{0}}J_{0}=e^{itQ_{0}}$, or $J_{0}Q_{0}J_{0}=-Q_{0}$ (note that
$J_{0}$ is an anti-unitary involution). If $\alpha_{t}$ is periodic with
period $2\pi$, namely $a_{0,t}=a_{0,t+2\pi}$ then it holds that
$V_{\alpha_{0}}(t)=V_{\alpha_{0}}(t+2\pi)$ and the generator $Q_{0}$ has a
discrete spectrum ${\rm sp}\,Q_{0}\subset{\mathbb{Z}}$. For the technical
simplicity, we restrict ourselves to the study of periodic inner symmetries.
We may assume that the period is $2\pi$ by a rescaling of the parameter.
###### Example 4.8.
We consider the loop group net ${\mathcal{A}}_{G,k}$ of a (simple, simply
connected) compact Lie group $G$ at level $k$ [17, 32], the net generated by
vacuum representations of loop groups $LG$ [28]. On this net, the original
group $G$ acts as a group of inner symmetries. We fix a maximal torus in $G$
and choose a one-parameter group in the maximal torus with a rational
direction, then it is periodic. Any one-parameter group is contained in a
maximal torus, so there are a good proportion of periodic one-parameter groups
in $G$ (although generic one-parameter groups have irrational direction, hence
not periodic). In particular, in the $SU(2)$-loop group net
${\mathcal{A}}_{SU(2),k}$, any one-parameter group in $SU(2)$ is periodic
since $SU(2)$ has rank $1$.
An inner automorphism $\alpha_{0}$ commutes with Möbius symmetry because of
Bisognano-Wichmann property. Hence it holds that
$U_{0}(g)Q_{0}U_{0}(g)^{*}=Q_{0}$. Furthermore, if the net ${\mathcal{A}}_{0}$
is conformal, then $\alpha_{0}$ commutes also with the diffeomorphism symmetry
[11]. Let $G$ be a group of inner symmetries and ${\mathcal{A}}_{0}^{G}$ be
the assignment: $I\mapsto{\mathcal{A}}_{0}(I)^{G}|_{{\mathcal{H}}_{0}^{G}}$,
where ${\mathcal{A}}_{0}(I)^{G}$ denotes the fixed point algebra of
${\mathcal{A}}_{0}(I)$ with respect to $G$ and
${\mathcal{H}}_{0}^{G}:=\overline{\\{x\Omega_{0}:x\in{\mathcal{A}}_{0}^{G}(I),I\subset
S^{1}\\}}$. Then it is easy to see that ${\mathcal{A}}_{0}^{G}$ is a Möbius
covariant net and it is referred to as the fixed point subnet of
${\mathcal{A}}_{0}$ with respect to $G$.
We can describe the action $\alpha_{0}$ of a periodic one-parameter group of
inner symmetries in a very explicit way, which can be considered as the
“spectral decomposition” of $\alpha_{0}$. Although it is well-known, we
summarize it here with a proof for the later use. This will be the basis of
the subsequent analysis.
###### Proposition 4.9.
Any element $x\in{\mathcal{A}}_{0}(I)$ can be written as $x=\sum_{n}x_{n}$,
where $x_{n}\in{\mathcal{A}}_{0}(I)$ and $\alpha_{0,t}(x_{n})=e^{int}x_{n}$.
We denote
${\mathcal{A}}_{0}(I)_{n}=\\{x\in{\mathcal{A}}_{0}(I):\alpha_{0,t}(x)=e^{int}x\\}$.
It holds that
${\mathcal{A}}_{0}(I)_{m}{\mathcal{A}}_{0}(I)_{n}\subset{\mathcal{A}}_{0}(I)_{m+n}$
and ${\mathcal{A}}_{0}(I)_{m}E_{0}(n){\mathcal{H}}_{0}\subset
E_{0}(m+n){\mathcal{H}}_{0}$, where $E_{0}(n)$ denotes the spectral projection
of $Q_{0}$ corresponding to the eigenvalue $n\in{\mathbb{Z}}$.
###### Proof.
Let us fix an element $x\in{\mathcal{A}}_{0}(I)$. The Fourier transform
$x_{n}:=\int_{0}^{2\pi}\alpha_{s}(x)e^{-ins}\,ds$
(here we consider the weak integral using the local normality of
$\alpha_{0,t}$) is again an element of ${\mathcal{A}}_{0}(I)$, since
${\mathcal{A}}_{0}(I)$ is invariant under $\alpha_{0,t}$. Furthermore it is
easy to see that
$\displaystyle\alpha_{0,t}(x_{n})$ $\displaystyle=$
$\displaystyle\alpha_{0,t}\left(\int_{0}^{2\pi}\alpha_{0,s}(x)e^{-ins}\,ds\right)$
$\displaystyle=$ $\displaystyle\int_{0}^{2\pi}\alpha_{0,s+t}(x)e^{-ins}\,ds$
$\displaystyle=$ $\displaystyle
e^{int}\int_{0}^{2\pi}\alpha_{0,s}(x)e^{-ins}\,ds=e^{int}x_{n},$
hence we have $x_{n}\in{\mathcal{A}}_{0}(I)_{n}$.
By assumption, $\alpha_{0,t}(x)={\hbox{\rm Ad\,}}e^{itQ_{0}}(x)$ and ${\rm
sp}\,Q_{0}\subset{\mathbb{Z}}$. If we define $x_{l,m}=E_{0}(l)xE_{0}(m)$, it
holds that ${\hbox{\rm Ad\,}}e^{itQ_{0}}x_{l,m}=e^{i(l-m)t}x_{l,m}$. The
integral and this decomposition into matrix elements are compatible, hence for
$x\in{\mathcal{A}}_{0}(I)$ we have
$x_{n}=\sum_{l-m=n}x_{l,m}.$
Now it is clear that $x=\sum_{n}x_{n}$ where each summand is a different
matrix element, hence the sum is strongly convergent. Furthermore from this
decomposition we see that
${\mathcal{A}}_{0}(I)_{m}{\mathcal{A}}_{0}(I)_{n}\subset{\mathcal{A}}_{0}(I)_{m+n}$
and ${\mathcal{A}}_{0}(I)_{m}E_{0}(n){\mathcal{H}}_{0}\subset
E_{0}(m+n){\mathcal{H}}_{0}$. ∎
At the end of this Section, we exhibit a simple formula for the adjoint action
${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}$ on the tensor product
Hilbert space ${\mathcal{H}}:={\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$.
###### Lemma 4.10.
For $x_{m}\in{\mathcal{A}}_{0}(I)_{m},y_{n}\in{\mathcal{A}}_{0}(I)_{n}$, it
holds that ${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x_{m}\otimes{\mathbbm{1}})=x_{m}\otimes e^{im\kappa Q_{0}}$ and
${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y_{n})=e^{in\kappa Q_{0}}\otimes y_{n}$.
###### Proof.
Recall that ${\rm sp}\,Q_{0}\in{\mathbb{Z}}$. Let $Q_{0}=\sum_{l}l\cdot
E_{0}(l)$ be the spectral decomposition of $Q_{0}$. As in the proof of Lemma
4.1, we decompose only the second component of $Q_{0}\otimes Q_{0}$ to see
that
$\displaystyle Q_{0}\otimes Q_{0}$ $\displaystyle=$ $\displaystyle
Q_{0}\otimes\left(\sum_{l}l\cdot E_{0}(l)\right)=\sum_{l}lQ_{0}\otimes
E_{0}(l)$ $\displaystyle e^{i\kappa Q_{0}\otimes Q_{0}}$ $\displaystyle=$
$\displaystyle\sum_{l}e^{il\kappa Q_{0}}\otimes E_{0}(l)$
$\displaystyle{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x_{m}\otimes{\mathbbm{1}})$ $\displaystyle=$
$\displaystyle\sum_{l}{\hbox{\rm Ad\,}}e^{il\kappa Q_{0}}(x_{m})\otimes
E_{0}(l)$ $\displaystyle=$ $\displaystyle\sum_{l}e^{iml\kappa}x_{m}\otimes
E_{0}(l)$ $\displaystyle=$ $\displaystyle x_{m}\otimes e^{im\kappa Q_{0}}.$
∎
###### Proposition 4.11.
For each $l\in{\mathbb{Z}}$ there is a cyclic and separating vector $v\in
E_{0}(l){\mathcal{H}}_{0}$ for a local algebra ${\mathcal{A}}_{0}(I)$.
###### Proof.
It is enough to note that the decomposition ${\mathbbm{1}}=\sum_{l}E_{0}(l)$
is compatible with the decomposition of the whole space with respect to
rotations, since inner symmetries commute with any Möbius transformation.
Hence each space $E_{0}(l){\mathcal{H}}_{0}$ is a direct sum of eigenspace of
rotation. It is a standard fact that a eigenvector of rotation which has
positive spectrum is cyclic and separating for each local algebra (see the
standard proof of Reeh-Schlieder property, e.g. [2]). ∎
We put $E(l,l^{\prime}):=E_{0}(l)\otimes E_{0}(l^{\prime})$.
###### Corollary 4.12.
Each space $E(l,l^{\prime}){\mathcal{H}}$ contains a cyclic and separating
vector $v$ for ${\mathcal{A}}_{0}(I)\otimes{\mathcal{A}}_{0}(J)$ for any pair
of intervals $I,J$.
#### 4.4.2 Construction of Borchers triples and their intersection property
Let ${\mathcal{A}}_{0}$ be a Möbius covariant net and $\alpha_{0,t}$ be a
periodic one-parameter group of inner symmetries. The automorphisms can be
implemented as $\alpha_{0,t}={\hbox{\rm Ad\,}}e^{itQ_{0}}$ as explained in
Section 4.4.1. The self-adjoint operator $Q_{0}$ is referred to as the
generator of the inner symmetry.
We construct a Borchers triple as in Section 4.2.1. Let
$\kappa\in{\mathbb{R}}$ be a real parameter (this time $\kappa$ can be
positive or negative) and we put
$\displaystyle{\mathcal{M}}_{Q_{0},\kappa}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa
Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})\\}^{\prime\prime}$
$\displaystyle T(t_{0},t_{1})$ $\displaystyle:=$ $\displaystyle
T_{0}\left(\frac{t_{0}-t_{1}}{\sqrt{2}}\right)\otimes
T_{0}\left(\frac{t_{0}+t_{1}}{\sqrt{2}}\right)$ $\displaystyle\Omega$
$\displaystyle:=$ $\displaystyle\Omega_{0}\otimes\Omega_{0}$
###### Theorem 4.13.
The triple $({\mathcal{M}}_{Q_{0},\kappa},T,\Omega)$ above is a Borchers
triple with a nontrivial scattering operator $S_{Q_{0},\kappa}=e^{i\kappa
Q_{0}\otimes Q_{0}}$.
###### Proof.
As remarked in Section 4.4.1, $Q_{0}$ commutes with Möbius symmetry $U_{0}$,
hence $Q_{0}\otimes Q_{0}$ and the translation $T=T_{0}\otimes T_{0}$ commute.
Since
$({\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+}),T,\Omega)$
is a Borchers triple (see Section 2), it holds that ${\hbox{\rm
Ad\,}}T(t_{0},t_{1}){\mathcal{M}}\subset{\mathcal{M}}$ for $(t_{0},t_{1})\in
W_{\mathrm{R}}$ and $T(t_{0},t_{1})\Omega=\Omega$ and $T$ has the joint
spectrum contained in $V_{+}$.
Since $\alpha_{0,t}$ is a one-parameter group of inner symmetries, it holds
that
$\alpha_{0,s}({\mathcal{A}}_{0}({\mathbb{R}}_{-}))={\mathcal{A}}_{0}({\mathbb{R}}_{-})$
and
$\alpha_{0,t}({\mathcal{A}}_{0}({\mathbb{R}}_{+}))={\mathcal{A}}_{0}({\mathbb{R}}_{+})$
for $s,t\in{\mathbb{R}}$. By Lemma 4.1, for
$x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})$ it holds that
$[x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\kappa Q_{0}\otimes
Q_{0}}(x^{\prime}\otimes{\mathbbm{1}})]=0.$
Then one can show that $({\mathcal{M}}_{Q_{0},\kappa},T,\Omega)$ is a Borchers
triple as in the proof of Theorem 4.2. The formula for the S-matrix can be
proved analogously as in Section 3.2. ∎
We now proceed to completely determine the intersection property of
${\mathcal{M}}_{Q_{0},\kappa}$. As a preliminary, we describe the elements in
${\mathcal{M}}_{Q_{0},\kappa}$ in terms of the original algebra
${\mathcal{M}}$ componentwise. On
${\mathcal{M}}={\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
there acts the group $S^{1}\otimes S^{1}$ by the tensor product action:
$(s,t)\mapsto\alpha_{s,t}:=\alpha_{0,s}\otimes\alpha_{0,t}={\hbox{\rm
Ad\,}}(e^{isQ_{0}}\otimes e^{itQ_{0}})$. According to this action, we have a
decomposition of an element $z\in{\mathcal{M}}$ into Fourier components as in
Section 4.4.1:
$z_{m,n}:=\int_{S^{1}\times S^{1}}\alpha_{s,t}(z)e^{-i(ms+nt)}\,ds\,dt,$
which is still an element of ${\mathcal{M}}$, and with
$E(l,l^{\prime}):=E_{0}(l)\otimes E_{0}(l^{\prime})$, these components can be
obtained by
$z_{m,n}=\underset{l^{\prime}-k^{\prime}=n}{\sum_{l-k=m}}E(l,l^{\prime})zE(k,k^{\prime}).$
One sees that ${\hbox{\rm Ad\,}}(e^{isQ_{0}}\otimes e^{itQ_{0}})$ acts also on
${\mathcal{M}}_{Q_{0},\kappa}$ since it commutes with ${\hbox{\rm
Ad\,}}e^{i\kappa Q_{0}\otimes Q_{0}}$. We still write this action by $\alpha$.
We can take their Fourier components by the same formula and the formula with
spectral projections still holds.
###### Lemma 4.14.
An element $z_{\kappa}\in{\mathcal{M}}_{Q_{0},\kappa}$ has the components of
the form
$(z_{\kappa})_{m,n}=z_{m,n}(e^{in\kappa Q_{0}}\otimes{\mathbbm{1}}),$
where $z=(z_{m,n})$ is some element in ${\mathcal{M}}$. Similarly, an element
$z^{\prime}_{\kappa}\in{\mathcal{M}}_{Q,\kappa}^{\prime}$ has the components
of the form
$(z^{\prime}_{\kappa})_{m,n}=z_{m,n}({\mathbbm{1}}\otimes e^{im\kappa
Q_{0}}),$
where $z^{\prime}=(z_{m,n})$ is some element in ${\mathcal{M}}^{\prime}$.
###### Proof.
We will show only the former statement since the latter is analogous. First we
consider an element of a simple form
$(x_{m}\otimes{\mathbbm{1}})S({\mathbbm{1}}\otimes y_{n})S^{*}$, where
$x_{m}\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})_{m}$ and
$y_{n}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+})_{n}$. We saw in Proposition 4.10
that this is equal to $(x_{m}\otimes y_{n})(e^{i\kappa
nQ_{0}}\otimes{\mathbbm{1}})$, thus this has the asserted form. Note that the
linear space spanned by these elements for different $m,n$ is closed even
under product. For a finite product and sum, the thesis is linear with respect
to $x$ and $y$, hence we obtain the desired decomposition. The von Neumann
algebra ${\mathcal{M}}_{Q_{0},\kappa}$ is linearly generated by these
elements. Recalling that $z_{m,n}$ is a matrix element with respect to the
decomposition ${\mathbbm{1}}=\sum_{l,l^{\prime}}E(l,l^{\prime})$, we obtain
the Lemma. ∎
Now we are going to determine the intersection of wedge algebras. At this
point, we need to use unexpectedly strong additivity and conformal covariance
(see Section 2). The fixed point subnet ${\mathcal{A}}_{0}^{\alpha_{0}}$ of a
strongly additive net ${\mathcal{A}}_{0}$ on $S^{1}$ with respect to the
action $\alpha_{0}$ of a compact group $S^{1}$ of inner symmetry is again
strongly additive [35].
###### Example 4.15.
The loop group nets ${\mathcal{A}}_{SU(N),k}$ are completely rational [17,
34], hence in particular they are strongly additive. Moreover, they are
conformal [28].
If ${\mathcal{A}}_{0}$ is diffeomorphism covariant, the strong additivity
follows from the split property and the finiteness of $\mu$-index [27]. We
have plenty of examples of nets which satisfy strong additivity and conformal
covariance since it is known that complete rationality passes to finite index
extensions and finite index subnets [24].
###### Theorem 4.16.
Let ${\mathcal{A}}_{0}$ be strongly additive and conformal and $e^{isQ_{0}}$
implement a periodic family of inner symmetries with the generator $Q_{0}$. We
write, with a little abuse of notation, $T(t_{+},t_{-}):=T_{0}(t_{+})\otimes
T_{0}(t_{-})$. For $t_{+}<0$ and $t_{-}>0$ we have
${\mathcal{M}}_{Q_{0},\kappa}\cap\left({\hbox{\rm
Ad\,}}T(t_{+},t_{-})({\mathcal{M}}_{Q_{0},\kappa}^{\prime})\right)={\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-})),$
where $G$ is the group of automorphisms generated by ${\hbox{\rm
Ad\,}}e^{i\kappa Q_{0}}$.
###### Proof.
Let us consider an element from the intersection. From Lemma 4.14, we have two
descriptions of such an element, namely,
$\displaystyle(z_{\kappa})_{m,n}$ $\displaystyle=$ $\displaystyle
z_{m,n}(e^{in\kappa
Q_{0}}\otimes{\mathbbm{1}}),\,\,\,z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+}),$
$\displaystyle(z^{\prime}_{\kappa})_{m,n}$ $\displaystyle=$ $\displaystyle
z^{\prime}_{m,n}({\mathbbm{1}}\otimes e^{im\kappa
Q_{0}}),\,\,\,z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{+})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-}).$
If these elements have to coincide, each $(m,n)$ component has to coincide. Or
equivalently, it should happen that $z_{m,n}(e^{in\kappa Q_{0}}\otimes
e^{-im\kappa Q_{0}})=z^{\prime}_{m,n}$.
Recall that an inner symmetry commutes with diffeomorphisms [11]. This implies
that the fixed point subalgebra contains the representatives of
diffeomorphisms. Furthermore, the fixed point subalgebra by a compact group is
again strongly additive [35]. This means that
$\displaystyle{\mathcal{A}}_{0}^{\alpha_{0}}((-\infty,t_{+}))\vee{\mathcal{A}}_{0}^{\alpha_{0}}((0,\infty))={\mathcal{A}}_{0}^{\alpha_{0}}((t_{+},0)^{\prime}),$
$\displaystyle{\mathcal{A}}_{0}^{\alpha_{0}}((-\infty,0))\vee{\mathcal{A}}_{0}^{\alpha_{0}}((t_{-},\infty))={\mathcal{A}}_{0}^{\alpha_{0}}((0,t_{-})^{\prime}),$
where $I^{\prime}$ means the complementary interval in $S^{1}$.
We claim that if for
$z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$
and
$z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{+})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-})$
there holds $z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}$, then
$z=z^{\prime}\in\left({\mathcal{A}}_{0}({\mathbb{R}}_{-})\cap{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{0})\right)\otimes\left({\mathcal{A}}_{0}({\mathbb{R}}_{+})\cap{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{-})\right)$.
Indeed, since
$z\in{\mathcal{A}}_{0}({\mathbb{R}}_{-})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{+})$,
it commutes with $U(g_{+}\times g_{-})$ with ${\rm
supp}(g_{+})\subset{\mathbb{R}}_{+}$ and ${\rm
supp}(g_{-})\subset{\mathbb{R}}_{-}$. Similarly,
$z^{\prime}\in{\mathcal{A}}_{0}({\mathbb{R}}_{+}+t_{0})\otimes{\mathcal{A}}_{0}({\mathbb{R}}_{-}+t_{1})$
commutes with $U(g_{+}\times g_{-})$ with ${\rm
supp}(g_{+})\subset{\mathbb{R}}_{-}+t_{+}$ and ${\rm
supp}(g_{-})\subset{\mathbb{R}}_{+}+t_{-}$. Furthermore, the unitary
$e^{im\kappa Q}\otimes e^{-in\kappa Q}$ which implements an inner symmetry
commutes with any action of diffeomorphism [11]. Recall that the fixed point
subalgebra is strongly additive, hence by the assumed equality
$z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}$, this element
commutes with
${\mathcal{A}}_{0}^{\alpha_{0}}((t_{+},0)^{\prime})\otimes{\mathcal{A}}_{0}^{\alpha_{0}}((0,t_{-})^{\prime})$.
In particular, it commutes with any diffeomorphism of $S^{1}\times S^{1}$
supported in $(t_{+},0)^{\prime}\times(0,t_{-})^{\prime}$. There is a sequence
of diffeomorphisms $g_{i}$ which take ${\mathbb{R}}_{-}\times{\mathbb{R}}_{+}$
to $(t_{+}-\varepsilon_{i},0)\times(0,t_{-}+\varepsilon_{i})$ with support
disjoint from $(t_{+},0)\times(0,t_{-})$ for arbitrary small
$\varepsilon_{i}>0$. This fact and the diffeomorphism covariance imply that
$z$ is indeed contained in
${\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$. By a
similar reasoning, one sees that
$z^{\prime}\in{\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$
as well. Now by Reeh-Schlieder property for
${\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$ we have
$z=z^{\prime}$ since $z\Omega=z\cdot(e^{im\kappa Q}\otimes e^{-in\kappa
Q})\Omega=z^{\prime}\Omega$.
Thus, if $z_{m,n}(e^{im\kappa Q}\otimes e^{-in\kappa Q})=z^{\prime}_{m,n}$,
then
$z_{m,n}=z^{\prime}_{m,n}\in{\mathcal{A}}_{0}((t_{+},0))\otimes{\mathcal{A}}_{0}((0,t_{-}))$.
Furthermore, by Corollary 4.12, there is a separating vector $v\in
E(l,l^{\prime}){\mathcal{H}}$. Now it holds that $e^{inl\kappa-
iml^{\prime}\kappa}z_{m,n}v=z^{\prime}_{m,n}v$, hence from the separating
property of $v$ it follows that $e^{inl\kappa-
iml^{\prime}\kappa}z_{m,n}=z^{\prime}_{m,n}$ for each pair
$(l,l^{\prime})\in{\mathbb{Z}}\times{\mathbb{Z}}$. This is possible only if
both $n\kappa$ and $m\kappa$ are $2\pi$ multiple of an integer or
$z_{m,n}=z^{\prime}_{m,n}=0$. This is equivalent to that ${\hbox{\rm
Ad\,}}e^{i\kappa mQ_{0}}\otimes e^{i\kappa nQ_{0}}(z)=z$, namely, $z$ is an
element of the fixed point algebra
${\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-}))$ by
the action ${\hbox{\rm Ad\,}}e^{i\kappa mQ_{0}}\otimes e^{i\kappa nQ_{0}}$ of
$G\times G$. ∎
Note that the size of the intersection is very sensitive to the parameter
$\kappa$: If $\kappa$ is $2\pi$-multiple of a rational number, then the
inclusion $[{\mathcal{A}}_{0},{\mathcal{A}}_{0}^{G}]$ has finite index.
Otherwise, it has infinite index.
Finally, we comment on the net generated by the intersection. The intersection
takes a form of chiral net ${\mathcal{A}}_{0}^{G}\otimes{\mathcal{A}}_{0}^{G}$
where $G$ is generated by ${\hbox{\rm Ad\,}}e^{i\kappa Q_{0}}$, hence the
S-matrix is trivial [15]. This result is expected also from [31], where Möbius
covariant net has always trivial S-matrix. Our construction is based on inner
symmetries which commute with Möbius symmetry, hence the net of strictly local
elements is necessarily Möbius covariant, then it should have trivial
S-matrix. But from this simple argument one cannot infer that the intersection
should be asymptotically complete, or equivalently chiral. This exact form of
the intersection can be found only by the present argument.
#### 4.4.3 Construction through cyclic group actions
Here we briefly comment on the actions by the cyclic group ${\mathbb{Z}}_{k}$.
In previous Sections, we have constructed Borchers triples for the action of
$S^{1}$. It is not difficult to replace $S^{1}$ by a finite group
${\mathbb{Z}}_{k}$. Indeed, the main ingredient was the existence of the
Fourier components. For ${\mathbb{Z}}_{k}$-actions, the discrete Fourier
transform is available and all the arguments work parallelly (or even more
simply). For the later use, we state only the result without repeating the
obvious modification of definitions and proofs.
###### Theorem 4.17.
Let ${\mathcal{A}}_{0}$ be a strongly additive conformal net on $S^{1}$ and
$\alpha_{0,n}={\hbox{\rm Ad\,}}e^{i\frac{2\pi n}{k}Q_{0}}$ be an action of
${\mathbb{Z}}_{k}$ as inner symmetries. Then, for $n\in{\mathbb{Z}}_{k}$, the
triple
$\displaystyle{\mathcal{M}}_{Q_{0},n}$ $\displaystyle:=$
$\displaystyle\\{x\otimes{\mathbbm{1}},{\hbox{\rm Ad\,}}e^{i\frac{2\pi
n}{k}Q_{0}\otimes Q_{0}}({\mathbbm{1}}\otimes
y):x\in{\mathcal{A}}_{0}({\mathbb{R}}_{-}),y\in{\mathcal{A}}({\mathbb{R}}_{+})\\}^{\prime\prime}$
$\displaystyle T$ $\displaystyle:=$ $\displaystyle T_{0}\otimes T_{0}$
$\displaystyle\Omega$ $\displaystyle:=$
$\displaystyle\Omega_{0}\otimes\Omega_{0}$
is an asymptotically complete Borchers triple with S-matrix $e^{i\frac{2\pi
n}{k}Q_{0}\otimes Q_{0}}$. As for strictly local elements, we have
${\mathcal{M}}_{Q_{0},n}\cap\left({\hbox{\rm
Ad\,}}T(t_{+},t_{-})({\mathcal{M}}_{Q,n}^{\prime})\right)={\mathcal{A}}_{0}^{G}((t_{+},0))\otimes{\mathcal{A}}_{0}^{G}((0,t_{-})),$
where $G$ is the group of automorphisms of ${\mathcal{A}}_{0}$ generated by
${\hbox{\rm Ad\,}}e^{i\frac{2\pi n}{k}Q_{0}}$.
Note that, although the generator $Q_{0}$ of inner symmetries of the cyclic
group $Z_{k}$ is not unique, we used it always in the form $e^{i\kappa Q_{0}}$
or $e^{i\kappa Q_{0}\otimes Q_{0}}$ and these operators are determined by the
automorphisms. Spectral measures can be defined in terms of these
exponentiated operators uniquely on (the dual of) the cyclic group
${\mathbb{Z}}_{k}$. In this way, the choice of $Q_{0}$ does not appear in the
results and proofs.
## 5 Construction through endomorphisms on the $U(1)$-current net
### 5.1 The $U(1)$-current net and Longo-Witten endomorphisms
In this Section we will construct a family of Borchers triples for a specific
net on $S^{1}$. Since we need explicit formulae for the relevant operators, we
briefly summarize here some facts about the net called the $U(1)$-current net,
or the (chiral part of) free massless bosonic field. On this model, there has
been found a family of Longo-Witten endomorphisms [26]. We will construct a
Borchers triple for each of these endomorphisms. This model has been studied
with the algebraic approach since the fundamental paper [9]. We refer to [25]
for the notations and the facts in the following.
A fundamental ingredient is the irreducible unitary representation of the
Möbius group with the lowest weight $1$: Namely, we take the irreducible
representation of ${\rm PSL}(2,{\mathbb{R}})$ of which the smallest eigenvalue
of the rotation subgroup is $1$. We call the Hilbert space
${\mathcal{H}}^{1}$. We take a specific realization of this representation.
Namely, let $C^{\infty}(S^{1},{\mathbb{R}})$ be the space of real-valued
smooth functions on $S^{1}$. This space admits a seminorm
$\|f\|:=\sum_{k\geq 0}2k|\hat{f}_{k}|^{2},$
where $\hat{f}_{k}$ is the $k$-th Fourier component of $f$, and a complex
structure
$(\widehat{{\mathcal{I}}f})_{k}=-i\mathrm{sign}(k)\hat{f}_{k}.$
Then, by taking the quotient space by the null space with respect to the
seminorm, we obtain the complex Hilbert space ${\mathcal{H}}^{1}$. We say
$C^{\infty}(S^{1},{\mathbb{R}})\subset{\mathcal{H}}^{1}$. On this space, there
acts ${\rm PSL}(2,{\mathbb{R}})$ by naturally extending the action on
$C^{\infty}(S^{1},{\mathbb{R}})$.
Let us denote ${\mathcal{H}}^{n}:={\mathcal{H}}^{\otimes n}$ for a nonnegative
integer $n$. On this space, there acts the symmetric group $\mathrm{Sym}(n)$.
Let $Q_{n}$ be the projection onto the invariant subspace with respect to this
action. We put ${\mathcal{H}}^{n}_{s}:=Q_{n}{\mathcal{H}}^{n}$ and the
symmetric Fock space
${\mathcal{H}}^{\Sigma}_{s}:=\bigoplus_{n}{\mathcal{H}}^{n}_{s},$
and this will be the Hilbert space of the $U(1)$-current net on $S^{1}$. For
$\xi\in{\mathcal{H}}^{1}$, we denote by $e^{\xi}$ a vector of the form
$\sum_{n}\frac{1}{n!}\xi^{\otimes
n}=1\oplus\xi\oplus\left(\frac{1}{2}\xi\otimes\xi\right)\oplus\cdots$. Such
vectors form a total set in ${\mathcal{H}}^{\Sigma}_{s}$. The Weyl operator of
$\xi$ is defined by
$W(\xi)e^{\eta}=e^{-\frac{1}{2}\langle\xi,\xi\rangle-\langle\xi,\eta\rangle}e^{\xi+\eta}$.
The Hilbert space ${\mathcal{H}}^{\Sigma}_{s}$ is naturally included in the
unsymmetrized Fock space:
${\mathcal{H}}^{\Sigma}:=\bigoplus_{n}({\mathcal{H}}^{1})^{\otimes
n}={\mathbb{C}}\oplus{\mathcal{H}}^{1}\oplus\left({\mathcal{H}}^{1}\otimes{\mathcal{H}}^{1}\right)\oplus\cdots$
We denote by $Q_{\Sigma}$ the projection onto ${\mathcal{H}}^{\Sigma}_{s}$.
For an operator $X_{1}$ on the one particle space ${\mathcal{H}}^{1}$, we
define the second quantization of $X_{1}$ on ${\mathcal{H}}^{\Sigma}_{s}$ by
$\Gamma(X_{1}):=\bigoplus_{n}(X_{1})^{\otimes n}=1\oplus
X_{1}\oplus\left(X_{1}\otimes X_{1}\right)\oplus\cdots$
Obviously, $\Gamma(X_{1})$ restricts to the symmetric Fock space
${\mathcal{H}}^{\Sigma}_{s}$. We still write this restriction by
$\Gamma(X_{1})$ if no confusion arises. For a unitary operator $V_{1}\in
B({\mathcal{H}}^{1})$ and $\xi\in{\mathcal{H}}^{1}$, it holds that
$\Gamma(V_{1})e^{\xi}=e^{V_{1}\xi}$ and ${\hbox{\rm
Ad\,}}\Gamma(V_{1})(W(\xi))=W(V_{1}\xi)$. On the one particle space
${\mathcal{H}}^{1}$, there acts the Möbius group ${\rm PSL}(2,{\mathbb{R}})$
irreducibly by $U_{1}$. Then ${\rm PSL}(2,{\mathbb{R}})$ acts on
${\mathcal{H}}^{\Sigma}$ and on ${\mathcal{H}}^{\Sigma}_{s}$ and by
$\Gamma(U_{1}(g))$, $g\in{\rm PSL}(2,{\mathbb{R}})$. The representation of the
translation subgroup in ${\mathcal{H}}^{1}$ is denoted by
$T_{1}(t)=e^{itP_{1}}$ with the generator $P_{1}$.
The $U(1)$-current net ${{\mathcal{A}}^{(0)}}$ is defined as follows:
${{\mathcal{A}}^{(0)}}(I):=\\{W(f):f\in
C^{\infty}(S^{1},{\mathbb{R}})\subset{\mathcal{H}}^{1},{\rm supp}(f)\subset
I\\}^{\prime\prime}.$
The vector
$1\in{\mathbb{C}}={\mathcal{H}}^{0}\subset{\mathcal{H}}^{\Sigma}_{s}$ serves
as the vacuum vector $\Omega_{0}$ and $\Gamma(U_{1}(\cdot))$ implements the
Möbius symmetry. We denote by $T^{\Sigma}_{s}$ the representation of one-
dimensional translation of ${{\mathcal{A}}^{(0)}}$.
For this model, a large family of endomorphisms has been found by Longo and
Witten.
###### Theorem 5.1 ([26], Theorem 3.6).
Let $\varphi$ be an inner symmetric function on the upper-half plane
$\SS_{\infty}\subset{\mathbb{C}}$: Namely, $\varphi$ is a bounded analytic
function of $\SS_{\infty}$ with the boundary value $|\varphi(p)|=1$ and
$\varphi(-p)=\overline{\varphi(p)}$ for $p\in{\mathbb{R}}$. Then
$\Gamma(\varphi(P_{1}))$ commutes with $T^{\Sigma}_{s}$ (in particular
$\Gamma(\varphi(P_{1}))\Omega_{0}=\Omega_{0}$) and ${\hbox{\rm
Ad\,}}\Gamma(\varphi(P_{1}))$ preserves
${{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$. In other words,
$\Gamma(\varphi(P_{1}))$ implements a Longo-Witten endomorphism of
${{\mathcal{A}}^{(0)}}$.
### 5.2 Construction of Borchers triples
In this Section, we construct a Borchers triple for a fixed $\varphi$, the
boundary value of an inner symmetric function (see Section 5.1). Many
operators are naturally defined on the unsymmetrized Fock space, hence we
always keep in mind the inclusion
${\mathcal{H}}^{\Sigma}_{s}\subset{\mathcal{H}}^{\Sigma}$. The full Hilbert
space for the two-dimensional Borchers triples will be
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$.
On ${\mathcal{H}}^{m}$, there act $m$ commuting operators
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
i\leq m\\}.$
We construct a unitary operator by the functional calculus on the
corresponding spectral measure. We set
* •
$P_{i,j}^{m,n}:=({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})\otimes({\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}})$,
which acts on ${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$, $1\leq i\leq m$ and
$1\leq j\leq n$.
* •
$S^{m,n}_{\varphi}:=\prod_{i,j}\varphi(P_{i,j}^{m,n})$, where
$\varphi(P_{i,j}^{m,n})$ is the functional calculus on
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$.
* •
$S_{\varphi}:=\bigoplus_{m,n}S_{\varphi}^{m,n}=\bigoplus_{m,n}\prod_{i,j}\varphi(P^{m,n}_{i,j})$
By construction, the operator $S_{\varphi}$ acts on
${\mathcal{H}}^{\Sigma}\otimes{\mathcal{H}}^{\Sigma}$. Furthermore, it is easy
to see that $S_{\varphi}$ commutes with both $Q_{\Sigma}\otimes{\mathbbm{1}}$
and ${\mathbbm{1}}\otimes Q_{\Sigma}$: In other words, $S_{\varphi}$ naturally
restricts to partially symmetrized subspaces
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ and
${\mathcal{H}}^{\Sigma}\otimes{\mathcal{H}}^{\Sigma}_{s}$ and to the totally
symmetrized space
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$. Note that
$S^{m,n}_{\varphi}$ is a unitary operator on the Hilbert spaces
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{n}$ and $S_{\varphi}$ is the direct
sum of them.
Let $E_{1}\otimes E_{1}\otimes\cdots\otimes E_{1}$ be the joint spectral
measure of operators
$\\{{\mathbbm{1}}\otimes\cdots\otimes\underset{j\mbox{-th}}{P_{1}}\otimes\cdots\otimes{\mathbbm{1}}:1\leq
j\leq n\\}$. The operators $\\{\varphi(P_{i,j}^{m,n}):1\leq i\leq m,1\leq
j\leq n\\}$ and $S_{\varphi}^{m,n}$ are compatible with the spectral measure
$\left(\overset{m\mbox{-}\mathrm{times}}{\overbrace{E_{1}\otimes
E_{1}\otimes\cdots\otimes
E_{1}}}\right)\otimes\left(\overset{n\mbox{-}\mathrm{times}}{\overbrace{E_{1}\otimes
E_{1}\otimes\cdots\otimes E_{1}}}\right)$ and one has
$\varphi(P_{i,j}^{m,n})=\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes\left({\mathbbm{1}}\otimes\cdots\underset{j\mbox{-th}}{dE_{1}(p_{j})}\otimes\cdots{\mathbbm{1}}\right).$
For $m=0$ or $n=0$ we set $\varphi_{i,j}^{m,n}={\mathbbm{1}}$ as a convention.
According to this spectral decomposition, we decompose $S_{\varphi}$ with
respect only to the right component as in the commutativity Lemma 4.1:
$\displaystyle S_{\varphi}$ $\displaystyle=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}\varphi(P_{i,j}^{m,n})$
$\displaystyle=$
$\displaystyle\bigoplus_{m,n}\prod_{i,j}\int\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes
dE_{0}(p_{1})\otimes\cdots\otimes dE(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{m,n}\int\prod_{i,j}\left({\mathbbm{1}}\otimes\cdots\otimes\underset{i\mbox{-th}}{\varphi(p_{j}P_{1})}\otimes\cdots{\mathbbm{1}}\right)\otimes
dE_{0}(p_{1})\otimes\cdots\otimes dE(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\bigoplus_{m}\prod_{j}(\varphi(p_{j}P_{1}))^{\otimes
m}\otimes dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\bigoplus_{m}(\varphi(p_{j}P_{1}))^{\otimes
m}\otimes dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n})$ $\displaystyle=$
$\displaystyle\bigoplus_{n}\int\prod_{j}\Gamma(\varphi(p_{j}P_{1}))\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n}),$
where the integral and the product commute in the third equality since the
spectral measure is disjoint for different values of $p$’s, and the sum and
the product commute in the fifth equality since the operators in the integrand
act on mutually disjoint spaces, namely on
${\mathcal{H}}^{m}\otimes{\mathcal{H}}^{\Sigma}$ for different $m$. Since all
operators appearing in the integrand in the last expression are the second
quantization operators, this formula naturally restricts to the partially
symmetrized space ${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$.
###### Lemma 5.2.
It holds for $x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$ that
$[x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0,$
on the Hilbert space
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}_{s}$.
###### Proof.
The operator $S_{\varphi}$ is disintegrated into second quantization operators
as we saw above. If $\varphi$ is an inner symmetric function, then so is
$\varphi(p_{j}\cdot)$, $p_{j}\geq 0$, thus each $\Gamma(\varphi(p_{j}P_{1}))$
implements a Longo-Witten endomorphism.
Note that $S_{\varphi}$ restricts naturally to
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ by construction and
$x\otimes{\mathbbm{1}}$ and $x^{\prime}\otimes{\mathbbm{1}}$ extend naturally
to ${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$ since the right-
components of them are just the identity operator ${\mathbbm{1}}$. Then we
calculate the commutation relation on
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$. This is done in the
same way as Lemma 4.1: Namely, we have
${\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})=\bigoplus_{n}\int{\hbox{\rm
Ad\,}}\left(\prod_{j}\Gamma(\varphi(p_{j}P_{1}))\right)(x^{\prime})\otimes
dE_{1}(p_{1})\otimes\cdots\otimes dE_{1}(p_{n}).$
And this commutes with $x\otimes{\mathbbm{1}}$. Indeed, since
$x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-})$ and
$x^{\prime}\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$, hence ${\hbox{\rm
Ad\,}}\Gamma(\varphi(p_{j}))(x^{\prime})\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})$
for any $p_{j}\geq 0$ by Theorem 5.1 of Longo-Witten, and by the fact that the
spectral support of $E_{1}$ is positive. Precisely, we have
$[x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}(x^{\prime}\otimes{\mathbbm{1}})]=0$ on
${\mathcal{H}}^{\Sigma}_{s}\otimes{\mathcal{H}}^{\Sigma}$.
Now all operators $S_{\varphi}$, $x\otimes{\mathbbm{1}}$ and
$x^{\prime}\otimes{\mathbbm{1}}$ commute with ${\mathbbm{1}}\otimes
Q_{\Sigma}$, we obtain the thesis just by restriction. ∎
Finally we construct a Borchers triple by following the prescription at the
end of Section 3.1.
###### Theorem 5.3.
The triple
* •
${\mathcal{M}}_{\varphi}:=\\{x\otimes{\mathbbm{1}},{\hbox{\rm
Ad\,}}S_{\varphi}({\mathbbm{1}}\otimes
y):x\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{-}),y\in{{\mathcal{A}}^{(0)}}({\mathbb{R}}_{+})\\}^{\prime\prime}$
* •
$T=T_{0}\otimes T_{0}$ of ${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$
* •
$\Omega=\Omega_{0}\otimes\Omega_{0}$ of
${{\mathcal{A}}^{(0)}}\otimes{{\mathcal{A}}^{(0)}}$
is an asymptotically complete Borchers triple with S-matrix $S_{\varphi}$.
###### Proof.
This is almost a repetition of the proof of Theorem 4.2. Namely, the
conditions on $T$ and $\Omega$ are readily satisfied since they are same as
the chiral triple. The operators $S_{\varphi}$ and $T$ commute since both are
the functional calculus of the same spectral measure, hence $T(t_{0},t_{1})$
sends ${\mathcal{M}}_{\varphi}$ into itself for $(t_{0},t_{1})\in
W_{\mathrm{R}}$. The vector $\Omega$ is cyclic for ${\mathcal{M}}_{\varphi}$
since ${\mathcal{M}}_{\varphi}\Omega\supset\\{x\otimes{\mathbbm{1}}\cdot
S_{\varphi}\cdot{\mathbbm{1}}\otimes
y\cdot\Omega\\}=\\{x\otimes{\mathbbm{1}}\cdot{\mathbbm{1}}\otimes
y\cdot\Omega\\}$ and the latter is dense by the Reeh-Schlieder property of the
chiral net. The separating property of $\Omega$ is shown through Lemma 5.2. ∎
###### Remark 5.4.
In this approach, the function $\varphi$ itself appears in two-particle
scattering, not the square as in [22]. Thus, although the formulae look
similar, the present construction contains much more examples.
#### Intersection property for constant functions $\varphi$
For the simplest cases $\varphi(p)=1$ or $\varphi(p)=-1$, we can easily
determine the strictly local elements. Indeed, for $\varphi(p)=1$,
$S_{\varphi}={\mathbbm{1}}$ and the Borchers triple coincides with the one
from the original chiral net. For $\varphi(p)=-1$,
$S_{\varphi}^{m,n}=(-1)^{mn}\cdot{\mathbbm{1}}$ and it is not difficult to see
that if one defines an operator $Q_{0}:=2P_{e}-{\mathbbm{1}}$, where $P_{e}$
is a projection onto the “even” subspace $\bigoplus_{n}{\mathcal{H}}^{2n}_{s}$
of ${\mathcal{H}}^{\Sigma}_{s}$, then $e^{i\pi Q_{0}}$ implements a
${\mathbb{Z}}_{2}$-action of inner symmetries on ${{\mathcal{A}}^{(0)}}$ and
$S_{\varphi}=e^{i\pi Q_{0}\otimes Q_{0}}$. Then Theorem 4.17 applies to find
that the strictly local elements are of the form
${{\mathcal{A}}^{(0)}}^{{\mathbb{Z}}_{2}}\otimes{{\mathcal{A}}^{(0)}}^{{\mathbb{Z}}_{2}}$
where the action of ${\mathbb{Z}}_{2}$ is realized by ${\hbox{\rm
Ad\,}}e^{i\pi nQ_{0}}$.
### 5.3 Free fermionic case
As explained in [26], one can construct a family of endomorphisms on the
Virasoro net ${\rm Vir}_{c}$ with the central charge $c=\frac{1}{2}$ by
considering the free fermionic field. With a similar construction using the
one-particle space on which the Möbius group acts irreducibly and projectively
with the lowest weight $\frac{1}{2}$, one considers the free fermionic
(nonlocal) net on $S^{1}$, which contains ${\rm Vir}_{\frac{1}{2}}$ with index
$2$.
The endomorphisms are implemented again by the second quantization operators.
By “knitting up” such operators as is done for bosonic $U(1)$-current case,
then by restricting to the observable part ${\rm Vir}_{\frac{1}{2}}$, we
obtain a family of Borchers triples with the asymptotic algebra ${\rm
Vir}_{\frac{1}{2}}\otimes{\rm Vir}_{\frac{1}{2}}$ with nontrivial S-matrix. In
the present article we omit the detail, and hope to return to this subject
with further investigations.
## 6 Conclusion and outlook
We showed that any two-dimensional massless asymptotically complete model is
characterized by its asymptotic algebra which is automatically a chiral Möbius
net, and the S-matrix. Then we reinterpreted the Buchholz-Lechner-Summers
deformation applied to chiral conformal net in this framework: It corresponds
to the S-matrix $e^{i\kappa P_{0}\otimes P_{0}}$. Furthermore we obtained
wedge-local nets through periodic inner symmetries which have S-matrix
$e^{i\kappa Q_{0}\otimes Q_{0}}$. We completely determined the strictly local
contents in terms of the fixed point algebra when the chiral component is
strongly additive and conformal. Unfortunately, the S-matrix restricted to the
strictly local part is trivial. For the $U(1)$-current net and the Virasoro
net ${\rm Vir}_{c}$ with $c=\frac{1}{2}$, we obtained families of wedge-local
nets parametrized by inner symmetric functions $\varphi$.
One important lesson is that construction of wedge-local nets should be
considered as an intermediate step to construct strictly local nets: Indeed,
any Möbius covariant net has trivial S-matrix [31], hence the triviality of
S-matrix in the construction through inner symmetries is interpreted as a
natural consequence. Although the S-matrix as a Borchers triple is nontrivial,
this should be treated as a false-positive. The true nontriviality should be
inferred by examining the strictly local part. On the other hand, we believe
that the techniques developed in this paper will be of importance in the
further explorations in strictly local nets. The sensitivity of the strictly
local part to the parameter $\kappa$ in the case of the construction with
respect to inner symmetries gives another insight.
Apart from the problem of strict locality, a more systematic study of the
necessary or sufficient conditions for S-matrix is desired. Such a
consideration could lead to a classification result of certain classes of
massless asymptotically complete models. For the moment, a more realistic
problem would be to construct S-matrix with the asymptotic algebra
${\mathcal{A}}_{N}\otimes{\mathcal{A}}_{N}$, where ${\mathcal{A}}_{N}$ is a
local extension of the $U(1)$-current net [9, 26]. A family of Longo-Witten
endomorphisms has been constructed also for ${\mathcal{A}}_{N}$, hence a
corresponding family of wedge-local net is expected and recently a similar
kind of endomorphisms has been found for a more general family of nets on
$S^{1}$ [3]. Or a general scheme of deforming a given Wightman-field theoretic
net has been established [22]. The family of S-matrices constructed in the
present paper seems rather small, since there is always a pair of spectral
measures and their tensor product diagonalizes the S-matrix. This could mean
in physical terms that the interaction between two waves is not very strong.
We hope to address these issues in future publications.
#### Acknowledgment.
I am grateful to Roberto Longo for his constant support. I wish to thank
Marcel Bischoff, Kenny De Commer, Wojciech Dybalski and Daniele Guido for
useful discussions.
## Appendix Appendix A A remark on BLS deformation
In the proof of Theorem 4.5 we used the fact that
$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}\xi\otimes\Omega_{0}=x\xi\otimes\Omega_{0}$.
The equation (2.2) from [7] translates in our notation to
$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}=\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B}{\hbox{\rm
Ad\,}}U(\kappa t_{1},\kappa t_{0})(x\otimes{\mathbbm{1}})FdE(t_{0},t_{1}),$
where $B$ is a bounded subset in ${\mathbb{R}}^{2}$ and $F$ is a finite
dimensional subspace in ${\mathcal{H}}$. Now it is easy to see that
$(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}(\xi\otimes\Omega_{0})=x\xi\otimes\Omega_{0}$.
Indeed, we have $\xi\otimes\Omega_{0}\in E({\mathrm{L}}_{+})$, where
${\mathrm{L}}_{+}=\\{(p_{0},p_{1})\in{\mathbb{R}}^{2}:p_{0}+p_{1}=0\\}$, hence
the integral above is concentrated in ${\mathrm{L}}_{+}$, and for
$(t,t)\in{\mathrm{L}}_{+}$ it holds that ${\hbox{\rm Ad\,}}U(\kappa t,\kappa
t)(x\otimes{\mathbbm{1}})={\hbox{\rm Ad\,}}{\mathbbm{1}}\otimes
U_{0}(\sqrt{2}\kappa t)(x\otimes{\mathbbm{1}})=x\otimes{\mathbbm{1}}$. Then
the integral simplifies as follows:
$\displaystyle(x\otimes{\mathbbm{1}})_{\Theta_{\kappa}}(\xi\otimes\Omega_{0})$
$\displaystyle=$
$\displaystyle\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B\cap{\mathrm{L}}_{+}}{\hbox{\rm
Ad\,}}U(\kappa t,\kappa t)(x\otimes{\mathbbm{1}})\cdot F\cdot
dE(t,t)(\xi\otimes\Omega_{0})$ $\displaystyle=$
$\displaystyle\underset{F\nearrow{\mathbbm{1}}}{\lim_{B\nearrow{\mathbb{R}}^{2}}}\int_{B\cap{\mathrm{L}}_{+}}x\otimes{\mathbbm{1}}\cdot
dE(t,t)(\xi\otimes\Omega_{0})$ $\displaystyle=$ $\displaystyle
x\xi\otimes\Omega_{0}.$
This is what we had to prove.
## References
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|
arxiv-papers
| 2011-07-13T18:54:49 |
2024-09-04T02:49:20.507511
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Yoh Tanimoto",
"submitter": "Yoh Tanimoto",
"url": "https://arxiv.org/abs/1107.2629"
}
|
1107.2673
|
# On the prior dependence of constraints on the tensor-to-scalar ratio
Marina Cortês Andrew R. Liddle and David Parkinson
###### Abstract
We investigate the prior dependence of constraints on cosmic tensor
perturbations. Commonly imposed is the strong prior of the single-field
inflationary consistency equation, relating the tensor spectral index $n_{\rm
T}$ to the tensor-to-scalar ratio $r$. Dropping it leads to significantly
different constraints on $n_{\rm T}$, with both positive and negative values
allowed with comparable likelihood, and substantially increases the upper
limit on $r$ on scales $k=0.01{\rm\,Mpc}^{-1}$ to $0.05{\rm\,Mpc}^{-1}$, by a
factor of ten or more. Even if the consistency equation is adopted, a uniform
prior on $r$ on one scale does not correspond to a uniform one on another;
constraints therefore depend on the pivot scale chosen. We assess the size of
this effect and determine the optimal scale for constraining the tensor
amplitude, both with and without the consistency relation.
## 1 Introduction
The inflationary proposal [1] is a compelling scenario for the origin of
primordial density fluctuations. Its prescription for the spectrum of density
perturbations has passed the scrutiny of high precision measurements of the
Cosmic Microwave Background (CMB), being in agreement with small deviations
from a purely scale-invariant spectrum [2]. In all models of inflation, a
small but potentially non-negligible stochastic background of primordial
gravitational waves is also generated alongside the density fluctuations. This
prediction remains to be confirmed, with only an upper bound on the density of
gravitational waves so far.
Although there are currently no useful theoretical lower bounds on the tensor
amplitude from inflation models (since we have classes of models that predict
undetectable levels of tensors [3]), determining the amplitude of
gravitational waves or, failing that, imposing a stringent upper bound is
still formally a requirement for completion of the underlying physical picture
behind the inflation mechanism. The amplitude of gravitational waves
determines the energy scale at which inflation took place, shedding light on
the particle physics sector in those regimes, and reveals the total excursion
of the scalar field during inflation. This provides information on symmetries
that may have been broken at the time, thereby probing energy scales many
orders of magnitude larger than those at the Large Hadron Collider.
One of the predictions of the simplest models of inflation is that the
spectrum of tensor perturbations depends on that of scalars. The spectra have
a common origin in the same function, the potential of the single scalar
field, and are therefore related. This prediction characterizes the simplest
models of inflation (single field and slowly rolling), and yields a set of
consistency relations all valid to a given order in the slow-roll regime [4].
These models give a good description of current data [2].
Nevertheless, other assumptions on the nature of tensor perturbations are
possible. More general classes of inflation models include those with multiple
scalar fields [5], where the consistency equation becomes an inequality [6],
and those with deviations from the slow-roll mechanism. In addition there are
models in which density fluctuations are seeded through a physical mechanism
making use of the known duality between expanding and contracting cosmologies.
In collapse-type models the rapidly expanding horizon in an inflationary era
is associated to contraction in a matter-dominated era [7]. In ekpyrotic
models the slow contraction takes place during domination by a stiff fluid
with equation of state $w>1$ [8]. Alternatively, tensor perturbations might be
seeded by cosmic defects [9], or one might simply ask what can be learnt from
observations without imposing specific theoretical preconceptions. There are
therefore a variety of possible assumptions as to how the tensor perturbations
might behave, and some examples have been investigated in the literature.
Finelli, Rianna, and Mandolesi [10] tested the validity of the consistency
equation by sampling freely from the plane $16\,\epsilon-r$. They probed
tensors at a scale close to the scalar pivot, $k=0.01{\rm\,Mpc}^{-1}$, and not
at $k=0.002{\rm\,Mpc}^{-1}$. Camerini et al [11] dropped the assumption of
single field inflation and searched at $k=0.002{\rm\,Mpc}^{-1}$ for
compatibility with blue tilted tensors in current data. The applied prior on
$n_{\rm T}$ was $-1<n_{\rm T}<20$. This apparently strongly conservative prior
still misses a large fraction of allowed red tilted models at $n_{\rm T}<-1$.
Also they probed solely at $k=0.002{\rm\,Mpc}^{-1}$ and thus didn’t detect the
weakening of constraints as we move towards smaller scales. Valkenburg,
Krauss, and Hamann [12] considered a prior density on $r$ which differs from
the common choice of a flat prior. They select for a theoretically motivated
uniform prior on the energy scale of inflation. They also probe at a single
scale, $k=0.002{\rm\,Mpc}^{-1}$. Even assuming the consistency equation they
detect that constraints on $r$ reflect this change on the tensor prior. Zhao,
Baskaran and Zang, [13] searched for the best multipole to probe tensors at.
Here they dropped the consistency equation assumption and probed tensors at
more than one scale. They then draw forecasts under this setup, for a variety
of ground- and space-based upcoming B-mode experiments. The chosen fiducial
model had small tensor amplitude and as result they don’t detect the large
variation allowed at the short wavelengths. In Ref. [14] Gjerløw and Elgarøy
relax the consistency equation relation but retain the assumption of a nearly
scale invariant tensor spectrum $n_{\rm T}\sim n_{\rm S}-1$. Finally Powell
[15] investigates the sensitivity of upcoming B-mode experiments for detecting
the tensor tilt, both for blue and red $n_{\rm T}$.
In our work we allow for full variability in these sets of assumptions, and
make a systematic exploration of how constraints on tensor fluctuations
respond to changes of prior. We relax the imposition of the simplest inflation
models, vary the cosmological scale probed, and study different prior
densities on both $r$ and $n_{\rm T}$. This permits us to unveil the
underlying variation of constraints on tensor parameters, that result from
shifts in one’s set of priors, and we observe a larger range of behaviour of
the tensor spectra than any of the studies in the literature so far.
## 2 Methodology
We assume throughout a flat $\Lambda$CDM model and parameterize our set of
primordial spectra as
$\displaystyle A_{\rm S}^{2}(k)$ $\displaystyle=$ $\displaystyle A_{\rm
S}^{2}(k_{0})k^{n_{\rm S}(k_{0})-1+\alpha(k_{0})\ln k/k_{0}}\,,$ (2.1)
$\displaystyle A_{\rm T}^{2}(k)$ $\displaystyle=$ $\displaystyle A_{\rm
T}^{2}(k_{0})k^{n_{\rm T}}\,,$ (2.2)
where $\alpha\equiv dn_{\rm S}/d\ln k$, and $k_{0}$ is the pivot scale where
observables are specified. We define the ratio of tensor to scalar amplitude
of perturbations as
$r(k)\equiv 16\frac{A_{\rm T}^{2}(k)}{A_{\rm S}^{2}(k)}\,.$ (2.3)
The usual assumption when fitting primordial fluctuations for parameter
estimation is the imposition of the first consistency relation between the
scalar and tensor power spectra,
$r=-8n_{\rm T}\,.$ (2.4)
For inflationary models the first and second derivatives of the scalar field
potential are usually expressed in terms of the slow-roll parameters [16],
$\epsilon=\frac{m_{{\rm
Pl}}^{2}}{16\pi}\left(\frac{V^{\prime}}{V}\right)^{2}\quad;\quad\eta=\frac{m_{{\rm
Pl}}^{2}}{8\pi}\,\frac{V^{\prime\prime}}{V}\,,$ (2.5)
where prime indicates $d/d\phi$. In the context of the slow-roll expansion and
a single field, we can write the observables in terms of the derivatives of
the potential [16]
$\displaystyle n_{\rm S}-1$ $\displaystyle=$
$\displaystyle-6\epsilon+2\eta\,;$ (2.6) $\displaystyle n_{\rm T}$
$\displaystyle=$ $\displaystyle-2\epsilon\,;$ (2.7) $\displaystyle r$
$\displaystyle=$ $\displaystyle 16\epsilon\,,$ (2.8)
from which the first consistency relation immediately follows. A consequence
of the consistency equation Eq. (2.4) is to enforce the amplitude of tensors
to be a decreasing function of wavenumber, since $r$ is by construction always
positive.
In multi-field models, tensor perturbations are still given by the usual
formula but extra scalar perturbations can be generated by conversion of
isocurvature perturbations in the additional degrees of freedom. The
consistency equation then weakens to an inequality, $r<-8n_{\rm T}$ with
$n_{\rm T}$ still constrained to be negative [6]. Other models may give yet
further variation, e.g. collapsing Universe inflation models such as the pre-
big-bang models [7] give a positive tensor spectral index.
In this paper we carry out an extensive exploration of the prior dependence of
constraints on the tensor-to-scalar ratio. The prior dependence takes two
forms: (a) restrictions on the parts of the $r$–$n_{\rm T}$ plane that are
considered, e.g. by enforcing the above consistency equation or inequality,
and (b) the prior probability distribution adopted on the permitted parameter
region. For the latter, commonly a uniform prior is chosen at whichever
‘pivot’ scale has been selected to specify the parameters, but this picks out
a special scale for which this is being assumed true. A related question is to
ask on what scale the tensor-to-scalar ratio is actually best constrained by a
given dataset, given prior assumptions.
In order to explore these dependencies, we will consider three different
cases:
1. 1.
No consistency relation, i.e. $r$ and $n_{\rm T}$ able to vary independently
including positive $n_{\rm T}$.
2. 2.
Single-field consistency equation imposed, i.e $r=-8n_{\rm T}$.
3. 3.
Multi-field consistency inequality imposed, i.e. $r\leq-8n_{\rm T}$.
In each case we need to consider the effect of imposing priors at different
scales; we choose $0.002{\rm\,Mpc}^{-1}$ as used in WMAP papers [17, 2] and
$k=0.02{\rm\,Mpc}^{-1}$ which is about the scale at which scalar perturbations
are optimally constrained [18].
## 3 Constraints on the tensor amplitude
[ caption = Uniform priors assumed in this article., label = prior, ]llcc
[a]at scale $k=0.002{\rm\,Mpc}^{-1}$ [b]at scale $k=0.02{\rm\,Mpc}^{-1}$ &
lower upper
Physical baryon density $\Omega_{\rm b}h^{2}$ 0.005 0.1
Physical dark matter density $\Omega_{\rm cdm}h^{2}$ 0.01 0.99
Sound horizon $\theta$ 0.5 10
Optical depth $\tau$ 0.01 0.8
Scalar amplitude $\log(10^{10}A_{\rm S}^{2})$ 2.7 4
Scalar index $n_{\rm S}$ 0.5 1.5
Scalar running $dn_{\rm S}/d\ln k$ $-0.2$ 0.2
SZ-amplitude $A_{\rm SZ}$ 0 2
Tensor-to-scalar ratio $r$ 0 $1\tmark[a]$
0 $16\tmark[b]$
Tensor spectral index $n_{\rm T}$ $-4$ $4\tmark[a]$
$-4$ $8\tmark[b]$
To probe parameter space we use the Markov Chain Monte Carlo method as
implemented in the CosmoMC package [19], and assume our cosmology can be
described by the set of parameters in Table LABEL:prior where we specify the
uniform priors imposed. All our MCMC runs use the combination of datasets from
WMAP-7 year [2], matter power spectrum measurements from SDSS-DR7 [20], and
$H_{0}$ from the Hubble Science Telescope (HST) [21]. Though the non-CMB
experiments (SDSS and HST) cannot detect the tensor perturbations directly,
these extra data are necessary to break some of the parameter degeneracies
between $r$ and other cosmological parameters.
$\begin{array}[]{cc}\includegraphics[width=220.0pt]{noce_lpnt44_r01_mar4_0002_infl_line_2D.ps}&\includegraphics[width=220.0pt]{noce_lpnt48_r16_mar4_002_2D.ps}\\\
\end{array}$
Figure 1: The $n_{\rm T}$–$r$ two-dimensional distributions obtained at
different scales, without imposition of the consistency equation, and uniform
priors over the regions plotted. Left panel: constraints obtained at
$k=0.002{\rm\,Mpc}^{-1}$; right panel: constraints at scale
$k=0.02{\rm\,Mpc}^{-1}$. The colour scale represents the mean likelihood of
samples, going to higher likelihood towards redder regions, and the contours
denote the marginalized distributions at 68% and 95% confidence limits. The
straight line shows the consistency equation.
### 3.1 No consistency relation
We begin by discussing the most general case where $r$ and $n_{\rm T}$ vary
freely, limited only by priors that we have chosen to be so wide that the
posterior distributions are limited by the data, weak though they are in
constraining the tensors, particularly their scale dependence. As well as
displaying these ‘model independent’ constraints, we use this case to explore
the effect of pivot scale transformations.
Figure 1 (left panel) shows combined constraints on $n_{\rm T}$ and $r$, with
uniform priors imposed at $k=0.002{\rm\,Mpc}^{-1}$; similar results were
obtained by Camerini et al [11]. The consistency equation would have
restricted models to the line, and obviously dropping this assumption greatly
increases the parameter space that is available. As a byproduct, this figure
shows how far we are from a meaningful test of the consistency relation, which
would require observational constraints tightly associated to a point on the
consistency equation line or excluding it entirely.
We find that the data do constrain $n_{\rm T}$, albeit very weakly, both from
above and from below. The constraints weaken as $r$ becomes small, because
$A_{\rm T}^{2}$ is very small there and $n_{\rm T}$ is irrelevant if the
amplitude is too small to detect on all scales probed. Large very positive
$n_{\rm T}$ would eventually contribute to the temperature modes at large
$\ell$ and conversely large negative $n_{\rm T}$ would disrupt the Sachs–Wolfe
effect beyond cosmic variance.
Note that the data allow the spectral tilt of tensor perturbations to be
positive, contrary to standard inflationary predictions. Indeed the one-
dimensional marginalized confidence limits on $n_{\rm T}$ are approximately
symmetric about zero:
$n_{\rm T}(k=0.002{\rm\,Mpc}^{-1})=0.0^{+1.1\,+2.0}_{-0.9\,-2.0}\,,$ (3.1)
where uncertainties quoted are 68% and 95%.
We now wish to investigate the constraints if the tensors are specified at a
shorter scale, $k=0.02{\rm\,Mpc}^{-1}$. The right panel of Figure 1 shows the
$n_{\rm T}$–$r$ plane when we change the pivot scale to
$k=0.02{\rm\,Mpc}^{-1}$ and otherwise maintain the same setup, i.e. the same
datasets and priors on remaining cosmology. The principal difference is that
the priors on parameters are now assumed uniform at this scale; we also
widened the ranges of $r$ and $n_{\rm T}$ in order to be able to reach regions
constrained by the data. Motivated by the fact that $\epsilon=1$ signals the
end of slow-roll inflation, and that $r=16\epsilon$ from the consistency
equation, we allow for $0<r<16$ and choose the prior on $n_{\rm T}$ such that
95% limits are well within the prior range $-4<n_{\rm T}<8$.
One might have expected that at least the constraints on $n_{\rm T}$ would
have been unchanged, since it is unaffected by a transformation of pivot
scale. However we see that this is not the case; the distribution is shifted
significantly to the right. The reason is that a uniform prior on one scale
does not transform to a uniform prior on another, and because the tensors are
so weakly constrained (at least in absence of consistency relations) the
effect of this is quite dramatic. Figure 2 (left panel) shows the prior
induced on parameters at $0.02{\rm\,Mpc}^{-1}$ by a uniform prior at
$k=0.002{\rm\,Mpc}^{-1}$. We stress there is no data at all in this figure; we
just draw uniform points on one scale and analytically transform $r$ to the
second scale ($n_{\rm T}$ does not change). As we see, the resulting prior is
extremely non-uniform. Note that it is the region where the prior is
compressed that leads to a high prior (since the fraction of the prior in each
interval of $n_{\rm T}$ is constant).
$\begin{array}[]{cc}\includegraphics[width=220.0pt]{nt_r_prior_chain_2D.ps}&\includegraphics[width=220.0pt]{ntr_ovrl.ps}\\\
\end{array}$
Figure 2: Left panel: The induced prior at $k=0.02{\rm\,Mpc}^{-1}$ from a
uniform prior at $k=0.002{\rm\,Mpc}^{-1}$. Right Panel: superposition of two-
dimensional distributions of $n_{\rm T}$–$r$ at $k=0.02{\rm\,Mpc}^{-1}$. The
dashed lines represent the marginalized distributions of $n_{\rm T}$–$r$
obtained by running a chain with uniform prior $k=0.02{\rm\,Mpc}^{-1}$. The
solid lines and colour code represent the distributions of $n_{\rm T}$–$r$
obtained in a chain run with uniform prior at $k=0.002{\rm\,Mpc}^{-1}$ and
shifted to $k=0.02{\rm\,Mpc}^{-1}$.
The consequence is shown in the right panel of Figure 2, which overlays the
chain run at $k=0.02{\rm\,Mpc}^{-1}$ with the chain run at the pivot scale
$k=0.002{\rm\,Mpc}^{-1}$ and transposed to the smaller scale
$k=0.02{\rm\,Mpc}^{-1}$. The dashed lines are the posterior assuming a uniform
prior, matching Fig. 1 (right panel), and hence also indicate the likelihood.
The posterior of the transformed chain is obtained by multiplying this
likelihood with the transformed prior of Fig. 2 (left panel), which shifts the
preferred region downwards and leftwards yielding the solid lines. The choice
of scale to impose the uniform prior clearly substantially modifies the
constraints on each parameter.
Note that on this shorter scale the constraint on $r$ is considerably weakened
with the $2\sigma$ constraints increased by a factor of more than ten (see
below in Eq. (3.1)). The primordial amplitude of tensor fluctuations is
allowed to be as large as six times that of scalar fluctuations, and steeply-
rising tensor spectra of tilt $n_{\rm T}=5$ are allowed at the $95\%$ level.
We show the TT and BB CMB spectra for such a model in Fig. 3. Even at $n_{{\rm
T}}=0$, corresponding to scale-invariant tensors, there is a significant
change in the upper limit on $r$ due to this change in prior, investigated in
more detail later in this section.
Figure 3: Angular power spectra for a model selected from the 2-dimensional
$n_{\rm T}-r$ distribution of Fig. 1 (right panel), allowed at 95% confidence
level, and comparison with the single-field inflationary prediction. Parameter
values are $n_{\rm T}=5$, $n_{\rm S}=0.956$, $r=0.05$ and $dn_{\rm S}/d\ln
k=-0.009$, all specified at scale $k=0.02{\rm\,Mpc}^{-1}$. Solid blue and pink
lines show TT and BB total power modes. Yellow long-dashed shows the scalar
contribution to TT power, and green short-dashed the tensor contribution to
TT. Blue dotted line shows the corresponding BB total power for the
corresponding single-field model that would have been obtained by imposition
of the consistency equation.
The 1D (95%) marginalized limits on $n_{\rm T}$ and $r$ at the two scales are:
$\displaystyle-2.0<$ $\displaystyle n_{\rm T}$ $\displaystyle<2.0\,,\quad
r<0.35\quad{\rm at}\quad k=0.002{\rm\,Mpc}^{-1}\,,$ $\displaystyle 0.76<$
$\displaystyle n_{\rm T}$ $\displaystyle<3.6\,,\quad r<6.6\quad{\rm at}\quad
k=0.02{\rm\,Mpc}^{-1}$ (3.2)
which reveals a significant shift towards more positive values in the
preferred $n_{\rm T}$ at smaller scales, with the best fit changing by about
$1\sigma$. At this scale the interesting region for inflation, $-1\lesssim
n_{\rm T}<0$, lies around the $1\sigma$ lower limit. Naive examination of
these contours would lead to the conclusion that data are favouring a larger
value of $n_{\rm T}$ at the smaller scale, and give evidence of non-zero
running of the tensor spectral index, given the stark difference in confidence
regions. However, none of the models fitting the data in the left panel of
Fig. 1 fail the prior imposed in the right panel, i.e. no model with $r$,
$n_{\rm T}$ at $k=0.002{\rm\,Mpc}^{-1}$ (left panel) is outside the confidence
limits at $k=0.02{\rm\,Mpc}^{-1}$ (right panel in the same figure). This means
that the same models are represented in the two distributions. It is rather
the effect of change in prior, i.e. the change in sampling at the new scale,
which is causing the change in the posterior. This is true for any parameter
that transforms under cosmological scale, but is more relevant if in addition
the parameter is poorly constrained as in the case of tensor spectra.
### 3.2 With consistency relations
We now turn to examining constraints on $r$ under different assumptions for
the inflation model. We assess the effect of imposing the single-field
equality, $r=-8n_{\rm T}$, and that of restricting to the multi-field region,
$r\lesssim-8n_{\rm T}$, obtained by clipping out from the full case the models
that don’t obey the inequality. We then obtain constraints for these models at
different scales, and their response to variations in the prior imposed at
each scale. Figure 4 shows the one-dimensional marginalized constraints on $r$
grouped in two different ways.
$\begin{array}[]{ccc}\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_0002.ps}&\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_002.ps}&\includegraphics[width=132.00134pt]{r_1D_ce_multi_noce_transported_002.ps}\\\
\end{array}$
$\begin{array}[]{ccc}\includegraphics[width=132.00134pt]{r_1D_ce_0002_002_transported.ps}&\includegraphics[width=132.00134pt]{r_1D_multi_0002_002_transported.ps}&\includegraphics[width=132.00134pt]{r_1D_noce_0002_002_transported.ps}\\\
\end{array}$
Figure 4: One-dimensional constraints on $r$. In the upper panels they are
grouped by scale and in the lower by class of models.
In the upper row of Fig. 4 the grouping is by scale: the first two panels show
constraints obtained by sampling from the likelihood under a uniform prior at
$k=0.002{\rm\,Mpc}^{-1}$ and $k=0.02{\rm\,Mpc}^{-1}$, and the last column
shows the chain transported from $k=0.002{\rm\,Mpc}^{-1}$ to
$k=0.02{\rm\,Mpc}^{-1}$, which results in a non-uniform prior at the new
scale. We see that at the scale $k=0.002{\rm\,Mpc}^{-1}$ used by WMAP, the
constraint is only modestly dependent on the prior. By constrast, at
$k=0.02{\rm\,Mpc}^{-1}$ the constraint on $r$ changes greatly under the
different prior assumptions, the centre panel showing the effect of the choice
of model type, and the right panel then showing further modification when the
alternative prior, induced from a uniform prior at $k=0.002{\rm\,Mpc}^{-1}$,
is used instead.
The lower row shows the same results grouped by model type. In the single-
field case (left panel) the constraint on $r$ does not much change when we
alter the pivot scale or vary our prior. This is a consequence of $n_{\rm T}$
being less than zero by assumption, preventing the tensors from growing
towards shorter scales, in combination with the observed near scale-invariance
of the scalar spectrum across the scales we are considering. As we have
already seen, the data alone do not significantly constrain the tensor
amplitude at $0.02{\rm\,Mpc}^{-1}$, without this additional model assumption.
Interestingly, sampling under the consistency equation assumption at
$k=0.002{\rm\,Mpc}^{-1}$ actually gives weaker bounds on $r$ than the case of
freely-varying $r$ and $n_{\rm T}$, as it happens that the models not
satisfying the consistency equation typically cannot fit the data with $r$
values as high as in the single-field case.
The multi-field constraints on $r$ are given by the middle panel in the bottom
row and are tighter than in single-field. That is because here we are sampling
from the region below the consistency equation line, which corresponds to
smaller $r$, as in Figs. 1. Also, we see that the region passing the multi-
field inequality at the tensor pivot, Fig. 1 (left panel), is much larger than
the region at the smaller scale $k=0.02{\rm\,Mpc}^{-1}$ (right panel). At
$k=0.002{\rm\,Mpc}^{-1}$ about 50% of the models are multi-field, against a
scarce 2% at $k=0.02{\rm\,Mpc}^{-1}$. This also results in lower statistics at
$k=0.02{\rm\,Mpc}^{-1}$ in our procedure for selecting multi-field models from
the full case, and accounts for the fluctuations seen in the middle bottom
panel of Fig. 4 (red line). The same panel shows that constraints for multi-
field models are, as in single-field, rather robust to changes in our prior.
The variation here is more evident however: $n_{\rm T}$ is more negative than
in the single-field case, so going to smaller scales leads to smaller $r$ and
tightens the contours as compared to the case at the tensor pivot (green, red
and blue lines in bottom middle panel).
Overall, Fig. 4 illustrates how constraints on tensor quantities with present-
day data are rather sensitive to one’s choice of prior and cosmological scale
probed. This means that at current sensitivities, in addition to uncertainty
in the data, one must admit an uncertainty arising from our prior assumptions,
for which the variation we see in Fig. 4 is an indication. This is not the end
of story, as priors leading to more extreme variations could be envisaged too,
and we have investigated only the transformation of a uniform prior.
## 4 Transformation of observables with cosmological scale
We now return to the issue of choice of scale to impose constraints on the
tensors. This issue in fact splits into two separate ones:
1. 1.
On what scale is the prior probability distribution imposed (for example the
choice of a uniform prior on some specific scale)?
2. 2.
For a given choice of prior, on what scale are the tensors optimally
constrained?
### 4.1 Scale transformations and the choice of priors
Our discussion above focussed on the first of these. Only when parameters
transform linearly with scale will the prior density be preserved when
transforming between cosmological scales $k$, because linear transformations
then amount to a rotation in the 2D plane of the parameters considered. This
is the case for $\ln A_{\rm T}^{2}$ and $n_{\rm S}$, for example, which
transform linearly with scale,
$\displaystyle\ln A_{\rm T}^{2}(k)$ $\displaystyle=$ $\displaystyle\ln A_{\rm
T}^{2}(k_{0})+n_{\rm T}(k_{0})\ln\frac{k}{k_{0}}\,,$ (4.1) $\displaystyle
n_{\rm S}(k)$ $\displaystyle=$ $\displaystyle n_{\rm S}(k_{0})+\frac{dn_{\rm
S}}{d\ln k}(k_{0})\ln\frac{k}{k_{0}}\,.$ (4.2)
For parameters that do not transform linearly this is not usually true. In
this case the density of models in parameter space is modified as parameters
do not keep their relative proportions when we shift with $k$. This is what
happens with $r$, where the transformation is exponential, depending on
$n_{\rm S}$, $n_{\rm T}$ and scalar running according to
$r(k)=r(k_{0})\left(\frac{k}{k_{0}}\right)^{n_{\rm T}(k_{0})-\left[n_{\rm
S}(k_{0})-1\right]-\alpha(k_{0})\ln{k/k_{0}}}.$ (4.3)
As we saw in Fig. 2 (left panel), this can substantially modify the prior
relative to assuming a uniform prior at the new scale. The net effect on the
prior in $r$, when going towards smaller scales, is a compression in $r$
regions with negative $n_{\rm T}$, raising its prior density, and expansion of
regions that correspond to positive $n_{\rm T}$ lowering their prior density.
If the parameters were well constrained data could overcome this change in
prior, but unfortunately they are not.
For our choice of smaller scale, $k=0.02{\rm\,Mpc}^{-1}$, this differentiated
sampling amounts to the posterior preferring a best-fit $n_{\rm T}$ different
by one-sigma from the one at $k=0.002{\rm\,Mpc}^{-1}$, as shown in Fig. 1. We
would expect an even larger variation in the posterior if we were to sample
$r$ at smaller scales. This change is a consequence of the variation in prior
alone, since we are not including any extra information when we sample at
smaller scales.
From a purely theoretical point of view a prior imposed at one scale is as
plausible as that imposed at the next. Unless the theoretical framework
selects for a particular scale as preferred to specify parameters at, our
results for the significantly different constraints on $r$ and $n_{\rm T}$ at
each scale are equally reasonable: a large positive tilt of the tensor power
spectrum is just as plausible as a nearly scale invariant spectrum.
One might wonder whether sampling from a log prior on $r$, or $A_{\rm T}^{2}$,
would help.111One might even suspect that $A_{\rm T}^{2}$ is already
effectively being sampled from a log prior by inheritting the prior on
$\log(A_{\rm S}^{2})$ in combination with the uniform prior on $r$. But since
$A_{\rm S}^{2}$ is tightly constrained by data its prior is fairly flat in the
region where the likelihood peaks, so the fact that $r$ has a uniform prior
means that $A_{\rm T}^{2}$ is effectively being sampled from a uniform prior
as well. However with a log prior any well-motivated lower cut-off is likely
to be at an extremely low value of $r$, and already puts most of the prior
parameter range well below future observational sensitivity. While this
doesn’t prevent a detection, in absence of a detection any observational
limits are going to be dominated by the prior rather than the data.
### 4.2 Variation of constraints with pivot — implications for inflationary
models
Figure 5: Variation of the 95% confidence limits on $r$ with scale. We compare
the variation of chains which have the consistency equation imposed (blue),
with those that have not (red). The prior on $r$ is uniform at
$0.002{\rm\,Mpc}^{-1}$ in both cases.
We now turn to the question of the scale on which constraints should be
imposed once the prior is fixed. Figure 5 shows the change in 95% confidence
limits on the tensor-to-scalar ratio when we probe at different $k$ scales. We
compare this effect on chains which have the consistency equation imposed with
those that haven’t.
The non-inflationary runs have a quite well-defined pivot point, outside of
which constraints rapidly deteriorate. The strongest constraints are obtained
at $k=0.002{\rm\,Mpc}^{-1}$, with the WMAP constraints on tensors coming
mainly from the amplitude of TT modes at low multipoles [17, 2]. Due to cosmic
variance the constraints are dominated by statistical uncertainty up to
multipoles $\ell\sim 10$ corresponding to $k\sim 0.002{\rm\,Mpc}^{-1}$. The
upper limits quickly go up as we move away from this pivot, towards smaller
scales, and allow for $r$ to become as large as 6 at $k\simeq
0.02{\rm\,Mpc}^{-1}$.
However, for the consistency equation runs, the situation is very different.
Constraints on $r$ appear robust to variation with scale in this case. This is
because enforcing the consistency relation imposes a strong correlation on
$n_{\rm S}$ and $r$, since it makes the assumption these can be described
jointly in terms of the slow-roll parameters, given in Eq. (2.5) via Eq.
(2.6). This imposition mimics apparent robust constraints on tensor quantities
at all angular scales, when in fact it is the prior rather than the data that
is ruling out sizable tensor mode contributions at all but the very largest
scales.
Figure 6: An overlay of two $n_{\rm S}$–$r$ distributions shown at
$k=0.02{\rm\,Mpc}^{-1}$, the scalar pivot scale. Green contours show
constraints when the consistency equation is enforced and red when it is not.
Relaxing the consistency equation prior decorrelates $n_{\rm S}$ and $r$ and
as a result, in the absence of the inflationary prior, $n_{\rm S}$ actually
becomes better determined.
Enforcing the consistency equation on small scales, around
$k=0.02{\rm\,Mpc}^{-1}$, excludes significant parameter space that would
otherwise be available in the $n_{\rm S}$–$r$ plane, as shown in Fig. 6. It
may be that some of this space is available in models with non-inflationary
sources for primordial gravitational waves, or in the more general set of
inflationary proposals (multiple-field, higher-order corrections) that deviate
from single-field slow roll. These are expected to have their own internal
consistency relations that do not necessarily comply with the single-field
one.
## 5 Conclusions
The simplest models of single-field slow-roll inflation predict a simple
connection between the scalar and tensor contributions to the spectra of
perturbations, given by the consistency equation. Aside from those there is a
wide variety of models that predict deviations from the simplest models, as
well as non-inflationary proposals for the origin of perturbations.
The tensor-to-scalar ratio, and therefore connections between the scalar and
tensor perturbations, are a powerful discriminator between models. We have
shown that, with the current state of knowledge, extraction of constraints on
$r$ depends significantly on our prior assumptions for the form of scalar and
tensor perturbations. We show that enforcing the consistency relation leads to
a reduction of available parameter space by a factor of 10 or larger when
quoting constraints at the usual scales around $k=0.01{\rm\,Mpc}^{-1}$ to
$0.05{\rm\,Mpc}^{-1}$ for the $n_{\rm S}$–$r$ plane. As a result, constraints
obtained under the assumption of this relation should be used for studying
models of single-field slow roll alone, for which it is valid. For other
inflation proposals an analysis based on the imposition of the consistency
equation can lead to artificial inferences.
In particular, the allowed values for the tensor-to-scalar ratio can be
significantly different from those one would expect from traditional fits.
Even when combining multiple datasets, data still allows for tensor mode
amplitudes which are several times larger than the amplitude of scalar modes
at scales around $k=0.02{\rm\,Mpc}^{-1}$. Furthermore, the ekyprotic- and
collapse-type models prediction for positive spectral index of tensor
perturbations, $n_{\rm T}>0$, is as valid as the inflationary equivalent which
predicts $n_{\rm T}<0$.
We conclude that constraints on tensors presently have significant prior
dependence, and must be interpreted with care in light of the particular
models to be studied. Even the apparently innocuous assumption of placing
uniform priors at one scale rather than another can significantly modify the
constraints obtained, whether or not the consistency equation or inequality is
imposed.
## Acknowledgments
We thank Eiichiro Komatsu and Antony Lewis for helpful discussions. M.C.
thanks the Astronomy Centre at the University of Sussex for hospitality during
this work. M.C. was supported by the Director, Office of Science, Office of
High Energy Physics, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. A.R.L. was supported by the Science and Technology Facilities
Council [grant numbers ST/F002858/1 and ST/I000976/1]. D.P. was supported by
the Australian Research Council through a Discovery Project grant.
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|
arxiv-papers
| 2011-07-13T20:44:37 |
2024-09-04T02:49:20.525356
|
{
"license": "Public Domain",
"authors": "Marina Cort\\^es, Andrew R. Liddle, David Parkinson",
"submitter": "Marina Cort\\^es",
"url": "https://arxiv.org/abs/1107.2673"
}
|
1107.2675
|
# Ignition from a Fire Perimeter in a WRF Wildland Fire Model111WRF Summer
Workshop 2011. This research was supported by the National Science Foundation
under grant AGS-0835579, and by U.S. National Institute of Standards and
Technology Fire Research Grants Program grant 60NANB7D6144.
Volodymyr Y. Kondratenko222Department of Mathematical and Statistical
Sciences, University of Colorado Denver, Denver, CO, Jonathan D.
Beezley222Department of Mathematical and Statistical Sciences, University of
Colorado Denver, Denver, CO, Adam K. Kochanski333Department of Meteorology,
University of Utah, Salt Lake City, UT, and Jan Mandel222Department of
Mathematical and Statistical Sciences, University of Colorado Denver, Denver,
CO
###### Abstract
The current WRF-Fire model starts the fire from a given ignition point at a
given time. We want to start the model from a given fire perimeter at a given
time instead. However, the fuel balance and the state of the atmosphere depend
on the history of the fire. The purpose of this work is to create an
approximate artificial history of the fire based on the given fire perimeter
and time and an approximate ignition point and time. Replaying the fire
history then establishes a reasonable fuel balance and outputs heat fluxes
into the atmospheric model, which allow the atmospheric circulation to
develop. Then the coupled atmosphere-fire model takes over. In this
preliminary investigation, the ignition times in the fire area are calculated
based on the distance from the ignition point to the perimeter, assuming that
the perimeter is convex or star-shaped. Simulation results for an ideal
example show that the fire can continue in a natural way from the perimeter.
Possible extensions include algorithms for more general perimeters and running
the fire model backwards in time from the perimeter to create a more realistic
history. The model used extends WRF-Fire and it is available from openwfm.org.
## 1 Introduction
Fire models generally start the fire from a given ignition point at a given
time, and sophisticated ignition parameterizations exist, including line
ignition and submesh ignition procedures Mandel et al. (2011). However, in
practice one is often faced with the need to start a fire model from a fire
already in progress. Often only perimeter data pertaining to some time are
available, such as from the US Forest Service at activefiremaps.fs.fed.us.
This need arizes in analyses of existing fires currently, and it will become
even more important for forecasting of the behavior of fires in progress in
future.
With models that do not include two-way interaction with the atmosphere,
continuing from an existing fire state is essentially straightforward. While
the fuel can be partially burned in some areas, fuel in locations untouched by
the fire is unchanged, and the model can simply progress to the new areas
regardless of the fire history. (A model that may include long-range effects
such as preheating in front of the fire, would be an exception.) In a coupled
atmosphere-fire model, however, the situation is very different. First, simply
igniting the whole area inside the given perimeter is not an option, because
the large instantaneous heat release will cause the model to break down. More
importantly, the state of the atmosphere evolves in interaction with the fire,
and the buyoancy caused by the heat flux causes significant changes to the
wind field, which in turn influences the future progress of the fire.
Starting the fire from a state already developed is essentially a data
assimilation problem, and it could be treated as such by a shooting method:
the fire starts from a ignition point at a time in the past, then at the given
simulation time, the state of the fire is compared with the given perimeter,
and adjustments can be made to the ignition time and location, much as in
variational data assimilation methods such as 4DVAR. We plan to study such
approaches in future as a part of our effort in the area of data assimilation
for wildland fires Beezley (2009); Beezley and Mandel (2008); Mandel et al.
(2008, 2009, 2010).
The approach adopted here is different. Given a fire perimeter, we create an
artificial history. Then the fire history is replayed, which produces heat
output into the atmosphere, and the atmospheric model spins up to a state that
is plausible for the fire at the stage given by the perimeter. The artificial
history is essentially a parameterization of the process that leads to the
development of the fire perimeter. Parameterizations of various levels of
sophistication can be considered, up to and including running a fire model
backwards in time to find an ignition point and iterating to find a matching
atmospheric state. In this initial study, we consider a very simple artificial
history model, and show that it results in acceptable fire and atmosphere
states for the perimeter. Approximate perimeter states obtained by such method
could provide also a good starting point for data assimilation in future.
## 2 The model
Fire models range from tools based on Rothermel (1972) fire spread rate
formulas, such as BehavePlus (Andrews 2007) and FARSITE (Finney 1998),
suitable for operational forecasting, to sophisticated 3-D computational fluid
dynamics and combustion simulations suitable for research and reanalysis, such
as FIRETEC (Linn et al. 2002) and WFDS (Mell et al. 2007). BehavePlus, the PC-
based successor of the calculator-based BEHAVE, determines the fire spread
rate at a single point from fuel and environmental data; FARSITE uses the fire
spread rate to provide a 2-D simulation on a PC; while FIRETEC and WFDS model
combustion in 3D, which is much more expensive. See the survey by Sullivan
(2009) for a number of other models.
The model considered here couples the mesoscale atmospheric code WRF-ARW
Skamarock et al. (2008) with a fire spread module, based on the Rothermel
model Rothermel (1972) and implemented by the level set method. In each time
step, the fire model inputs the atmospheric winds and outputs surface sensible
and latent heat fluxes into the atmosphere. Only the finest domain in WRF is
coupled with the fire model. The fire model works in conjunction with WRF land
use models, and it interpolates horizontal winds from the ideal logarighmic
wind profile to appropriate heights above the surface, for each fuel.
The model has grown out of NCAR’s CAWFE code Clark et al. (1996b, a, 2004);
Coen (2005), which couples the Clark-Hall atmospheric model with fire spread
implemented by tracers, and it got its start from a prototype code coupling
the fire model in CAWFE with WRF in LES mode Patton and Coen (2004). The
tracers, however, were replaced by a level set method, which we considered
more flexible and more suitable for data assimilation and WRF parallel
infrastructure. The coupled model is capable of running faster than real time
in LES mode, with resolution of tens of meters for the atmosphere, and meters
for the fire, with the matching time step of a fraction of a second, on the
innermost modeling domain of many kilometers in size Jordanov et al. (2011).
Fuel data and topography can be obtained from government databases in the
United States and from satellite images and GIS elsewhere Jordanov et al.
(2011). See Mandel et al. (2009, 2011) for futher details and references. The
model is currently available from the Open Wildland Fire Modeling environment
at openwfm.org, along with utilities for data preparation, visualization, and
diagnostics and a wiki with many user guides for the specific features and
utilities, discussion, and support. A code containing a subset of the features
is distributed with WRF as WRF-Fire.
## 3 Encoding and replaying the fire history
The state of the fire model consists of a level set function, $\Phi$, given by
its values on the nodes of the fire model mesh, and time of ignition
$T_{\mathrm{i}}$. The level set function is interpolated linearly. At a given
simulation time $t$, the fire area is the set of all points $\left(x,y\right)$
where $\Phi\left(t,x,y\right)\leq 0$. The level set function and the ignition
time satisfy the consistency condition
$\Phi\left(t,x,y\right)\leq 0\Longleftrightarrow
T_{\mathrm{i}}\left(x,y\right)\leq t,$ (1)
as both of these inequalities express the condition that the location
$\left(x,y\right)$ is burning at the time $t$. In every time step of the
simulation, the level set function is advanced by one step of a Runge-Kutta
scheme for the level set equation
$\frac{d\Phi}{dt}=-R\left\|\nabla\Phi\right\|,$
where $R=R\left(t,x,y\right)$ is the fire rate of spread, which depends on the
fuel, wind speed, and slope. The ignition time at nodes is then computed for
all newly ignited nodes, and it satisfies the consistency condition (1).
The fire history is encoded as an array of ignition times
$T_{\mathrm{i}}\left(x,y\right)$, prescribed at all fire mesh nodes. To replay
the fire in the period $0\leq t\leq T_{\mathrm{per}}$, the numerical scheme
for advancing $\Phi$ and $T_{\mathrm{i}}$ is suspended, and instead the level
set function is set to
$\Phi\left(t,x,y\right)=T_{\mathrm{i}}\left(x,y\right)-t.$
After the end of the replay is reached, the numerical scheme of the level set
method is started from the level set function $\Phi$ at $t=T_{\mathrm{per}}$.
For reasons of numerical accuracy and stability, the level set function needs
to have approximately uniform slope. For example, a very good level set
function, which has slope equal to one, is the signed distance from a given
closed curve $\Gamma$,
$\Phi\left(x,y\right)=\pm\operatorname*{dist}\left(\left(x,y\right),\Gamma\right),$
where the sign is taken to be negative inside the region limited by $\Gamma$,
and positive outside Osher and Fedkiw (2003). Thus, the ignition times
$T_{\mathrm{i}}$ need to be given on the whole domain and they need to be such
that $T_{\mathrm{i}}$ decreases with the distance from the given perimeter
inside the fire region, and increases outside. The ignition times
$T_{\mathrm{i}}$ outside of the given fire perimeter are perhaps best thought
of as what the ignition times might be in future as the fire keeps burning.
## 4 Creating an artificial fire history
The purpose of this algorithm is to create the artificial values of the time
of ignition on the fire model mesh, given ignition point ($x_{\mathrm{ign}}$,
$y_{\mathrm{ign}}$), ignition time $T_{\mathrm{ign}}$, fire perimeter
$\Gamma$, and the time when the fire reached the perimeter $T_{\mathrm{per}}$,
assuming that the fire perimeter is convex, or at least star-shaped with
respect to the ignition point. The fire perimeter is given as a set of points
$(x_{k},y_{k})$ in the fire model domain, $k=1,\ldots,n$ which form a closed
curve consisting of line segments
$\left[(x_{k},y_{k}),(x_{k+1},y_{k+1})\right]$ between each two successive
points. We take $(x_{1},y_{1})=(x_{n+1},y_{n+1})$ so that the starting and the
ending point are identical. The coordinates of the point of ignition and of
the points defining of the fire perimeter do not need to coincide with mesh
points of the grid.
The method consists of linear interpolation of the ignition time between
$T_{\mathrm{ign}}$ at the ignition point and $T_{\mathrm{per}}$ on the
perimeter, along straight lines connecting the ignition point with points on
the perimeter. The ignition time is also extrapolated beyond the perimeter in
the same manner to provide a suitable level set function, as discussed in the
previous section. Given a mesh point with coordinates $\left(x,y\right)$, the
algorithm to determine the ignition time $T_{\mathrm{i}}\left(x,y\right)$
consists of the following steps.
1. 1.
Find the intersection $\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)$ of the fire
perimeter and the half-line starting at the ignition point and passing through
the point $\left(x,y\right)$ (Fig. 1). For this purpose, we use the function
$F(x,y,x_{b},y_{b})=(y_{b}-y_{\mathrm{ign}})(x-x_{\mathrm{ign}})-(x_{b}-x_{\mathrm{ign}})(y-y_{\mathrm{ign}}),$
which is zero if point $\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)$ lies on
the line defined by $(x,y)$ and $(x_{\mathrm{ign}},y_{\mathrm{ign}})$, and it
is positive in one half-plane and negative in the other. We then find segment
$\left[(x_{k},y_{k}),(x_{k+1},y_{k+1})\right]$ such that
$F(x,y,x_{k},y_{k})F(x,y,x_{k+1},y_{k+1})<0$, that is, the points
$(x_{k},y_{k})$ and $(x_{k+1},y_{k+1})$ lie on opposite sides of the line
passing through $(x,y)$ and $(x_{\mathrm{ign}},y_{\mathrm{ign}})$. Since the
line intersects the fire perimeter at two points, one on each side of the
ignition point, we choose correct segment as follows:
* •
If $\left(x,y\right)$ is inside $\Gamma$, that is, closer to the ignition
point than to the intersection, then the desired segment is the one that lies
on the same side from the ignition point as the point $\left(x,y\right)$;
* •
If $\left(x,y\right)$ is outside of $\Gamma$, then the needed segment lies on
the same side from the ignition point as $\left(x,y\right)$.
2. 2.
Calculate the time of ignition of the mesh point, based on the ratio of the
distances of the mesh point and the perimeter point to the ignition point,
$T_{\mathrm{i}}(x,y)=T_{\mathrm{ign}}+\frac{\left\|\left(x,y\right)-\left(x_{\mathrm{ign}},y_{\mathrm{ign}}\right)\right\|}{\left\|\left(x_{\mathrm{b}},y_{\mathrm{b}}\right)-\left(x_{\mathrm{ign}},y_{\mathrm{ign}}\right)\right\|}\left(T_{\mathrm{per}}-T_{\mathrm{ign}}\right).$
Figure 1: Construction of intersection of the fire perimeter and the half line
originating from the Ignition point and passing through a given mesh point.
## 5 Computational results
We have tested this algorithm on an ideal example to measure the difference in
the atmospheric winds between a simulation propagated naturally from a point
and another one advanced artificially. In this example, the topography was
flat except for a small hill roughly $500$ m in diameter and $100$ m high in
the center of a domain of size $2.4$ km $\times$ $2.4$ km. The atmospheric and
fire grid resolutions used were $60$ m and $6$ m respectively, with a $0.25$ s
time step. The background winds were approximately $9.5$ m/s traveling
southwest at the lowest atmospheric layer $30$ m above the surface. The first
simulation was ignited from a point in the northeast corner of the domain $2$
seconds from the start, and the fire perimeter was recorded after $40$
minutes. This perimeter and ignition location were used to generate an
artificial history for the first $40$ minutes, which was replayed in the
second simulation. Therefore, the fire perimeters in both simulations are
identical at $40$ minutes. Both simulations were then allowed to advance
another $28$ minutes, using the standard coupled model. The outputs were then
collected for analysis.
Any differences in the simulations after this time are a result of the error
of the artificial fire propagation. In Fig. 3, we show 3D renderings of the
simulation. The streamlines near the surface show the updraft created as a
result of the heat output from the fire. In Fig. 3a, the fire is affecting the
atmosphere despite being propagated artificially. A semi-transparent volume
rendering of QVAPOR was added to simulate the smoke release. In Fig. 2, the
differences in the wind between the two simulations at $68$ minutes and the
fire perimeter are shown. Fig. 2a, shows the difference of the wind from the
direct fire propagation minus the wind from the artificial propagation. Fig.
2b shows the relative error in the wind speed defined as the norm of the
difference from Fig. 2a, divided by the wind speed from the direct simulation.
This shows that the maximum error at the end of this $68$ minute simulation is
less than $2.5\%$. In this case, the Froude number is about $F_{c}=0.79$,
showing that the heating from the fire may significantly affect ambient wind,
therefore small differences caused by using the artificial history have an
effect. The effect is concentrated downwind from the fire, as it could be
expected.
(a) The difference in the winds of the direct simulation minus the artificial
propagation at $68$ minutes.
(b) The relative error in the speed of the wind at $68$ minutes.
Figure 2:
(a) The artificial fire simulation at $40$ minutes.
(b) The direct fire simulation at $68$ minutes.
Figure 3:
## 6 Conclusion
We have presented a parameterization of the fire up to a point in time when a
fire perimeter is given. The parameterization allows for the changes in the
atmospheric circulation to develop, caused by the heat flux from the fire.
This provides appropriate starting conditions for the computation to continue
with the full coupled fire-atmosphere simulation. We have shown on an ideal
example that the differences in the state of the atmosphere between a complete
fire simulation and when the parameterization is used are not significant. In
the studied case, the coupling between the fire and the atmosphere was strong.
It would be interesting to observe how the the differences change if the
problem moves from the wind-driven ($F_{c}>1$) to the plume driven regime
($F_{c}<1$) regime. This will be studied elsewhere. We plan to study also
algorithms for more general domains, not just star-shaped, and to take into
account different rates of fire propagation due to fuel nonhomogeneity.
## References
* Andrews (2007) Andrews, P. L., 2007: BehavePlus fire modeling system: past, present, and future. Paper J2.1, 7th Symposium on Fire and Forest Meteorology, http://ams.confex.com/ams/pdfpapers/126669.pdf.
* Beezley (2009) Beezley, J. D., 2009: High-Dimensional Data Assimilation and Morphing Ensemble Kalman Filters with Applications in Wildfire Modeling. Ph.D. thesis, University of Colorado Denver.
* Beezley and Mandel (2008) Beezley, J. D. and J. Mandel, 2008: Morphing ensemble Kalman filters. Tellus, 60A, 131–140, doi:10.1111/j.1600-0870.2007.00275.x.
* Clark et al. (2004) Clark, T. L., J. Coen, and D. Latham, 2004: Description of a coupled atmosphere-fire model. International Journal of Wildland Fire, 13, 49–64, doi:10.1071/WF03043.
* Clark et al. (1996a) Clark, T. L., M. A. Jenkins, J. Coen, and D. Packham, 1996a: A coupled atmospheric-fire model: Convective feedback on fire line dynamics. J. Appl. Meteor, 35, 875–901, doi:10.1175/1520-0450(1996)035$<$0875:ACAMCF$>$2.0.CO;2.
* Clark et al. (1996b) Clark, T. L., M. A. Jenkins, J. L. Coen, and D. R. Packham, 1996b: A coupled atmosphere-fire model: Role of the convective Froude number and dynamic fingering at the fireline. International Journal of Wildland Fire, 6, 177–190, doi:10.1071/WF9960177.
* Coen (2005) Coen, J. L., 2005: Simulation of the Big Elk Fire using coupled atmosphere-fire modeling. International Journal of Wildland Fire, 14, 49–59, doi:10.1071/WF04047.
* Finney (1998) Finney, M. A., 1998: FARSITE: Fire area simulator-model development and evaluation. Res. Pap. RMRS-RP-4, Ogden, UT: U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, http://www.fs.fed.us/rm/pubs/rmrs_rp004.html.
* Jordanov et al. (2011) Jordanov, G., J. D. Beezley, N. Dobrinkova, A. K. Kochanski, and J. Mandel, 2011: Simulation of the 2009 Harmanli fire (Bulgaria). 8th International Conference on Large-Scale Scientific Computations, Sozopol, Bulgaria, June 6-10, 2011, lecture notes in Computer Science, Springer, to appear.
* Linn et al. (2002) Linn, R., J. Reisner, J. J. Colman, and J. Winterkamp, 2002: Studying wildfire behavior using FIRETEC. Int. J. of Wildland Fire, 11, 233–246, doi:10.1071/WF02007.
* Mandel et al. (2009) Mandel, J., J. D. Beezley, J. L. Coen, and M. Kim, 2009: Data assimilation for wildland fires: Ensemble Kalman filters in coupled atmosphere-surface models. IEEE Control Systems Magazine, 29, 47–65, doi:10.1109/MCS.2009.932224.
* Mandel et al. (2011) Mandel, J., J. D. Beezley, and A. K. Kochanski, 2011: Coupled atmosphere-wildland fire modeling with WRF-Fire version 3.3. Geoscientific Model Development Discussions, 4, 497–545, doi:10.5194/gmdd-4-497-2011.
* Mandel et al. (2010) Mandel, J., J. D. Beezley, and V. Y. Kondratenko, 2010: Fast Fourier transform ensemble Kalman filter with application to a coupled atmosphere-wildland fire model. Computational Intelligence in Business and Economics, Proceedings of MS’10, A. M. Gil-Lafuente and J. M. Merigo, eds., World Scientific, 777–784.
* Mandel et al. (2008) Mandel, J., L. S. Bennethum, J. D. Beezley, J. L. Coen, C. C. Douglas, M. Kim, and A. Vodacek, 2008: A wildland fire model with data assimilation. Mathematics and Computers in Simulation, 79, 584–606, doi:10.1016/j.matcom.2008.03.015.
* Mell et al. (2007) Mell, W., M. A. Jenkins, J. Gould, and P. Cheney, 2007: A physics-based approach to modelling grassland fires. Intl. J. Wildland Fire, 16, 1–22, doi:10.1071/WF06002.
* Osher and Fedkiw (2003) Osher, S. and R. Fedkiw, 2003: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York.
* Patton and Coen (2004) Patton, E. G. and J. L. Coen, 2004: WRF-Fire: A coupled atmosphere-fire module for WRF. Preprints of Joint MM5/Weather Research and Forecasting Model Users’ Workshop, Boulder, CO, June 22–25, NCAR, 221–223, http://www.mmm.ucar.edu/mm5/workshop/ws04/Session9/Patton_Edward.pdf.
* Rothermel (1972) Rothermel, R. C., 1972: A mathematical model for predicting fire spread in wildland fires. USDA Forest Service Research Paper INT-115, http://www.treesearch.fs.fed.us/pubs/32533.
* Skamarock et al. (2008) Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, M. G. Duda, X.-Y. Huang, W. Wang, and J. G. Powers, 2008: A description of the Advanced Research WRF version 3. NCAR Technical Note 475, http://www.mmm.ucar.edu/wrf/users/docs/arw_v3.pdf.
* Sullivan (2009) Sullivan, A. L., 2009: A review of wildland fire spread modelling, 1990-present, 1: Physical and quasi-physical models, 2: Empirical and quasi-empirical models, 3: Mathematical analogues and simulation models. International Journal of WildLand Fire, 18, 1: 347–368, 2: 369–386, 3: 387–403, doi:10.1071/WF06143, 10.1071/WF06142, 10.1071/WF06144.
|
arxiv-papers
| 2011-07-13T20:49:47 |
2024-09-04T02:49:20.533565
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Volodymyr Y. Kondratenko, Jonathan D. Beezley, Adam K. Kochanski, and\n Jan Mandel",
"submitter": "Volodymyr Kondratenko Y",
"url": "https://arxiv.org/abs/1107.2675"
}
|
1107.2708
|
# Generation-recombination processes via acoustic phonons in a disorded
graphene
F. T. Vasko ftvasko@yahoo.com V. V. Mitin Department of Electrical
Engineering, University at Buffalo, Buffalo, NY 1460-1920, USA
###### Abstract
Generation-recombination interband transitions via acoustic phonons are
allowed in a disordered graphene because of violation of the energy-momentum
conservation requirements. The generation-recombination processes are analyzed
for the case of scattering by a short-range disorder and the deformation
interaction of carriers with in-plane acoustic modes. The generation-
recombination rates were calculated for the cases of intrinsic and heavily-
doped graphene at room temperature. The transient evolution of nonequilibrium
carriers is described by the exponential fit dependent on doping conditions
and disorder level. The characteristic relaxation times are estimated to be
about 150 - 400 ns for sample with the maximal sheet resistance $\sim$5
k$\Omega$. This rate is comparable with the generation-recombination processes
induced by the thermal radiation.
###### pacs:
72.80.Vp, 73.61.-b, 78.60.-b
## I Introduction
Transport 1 and optical 2 properties of graphene as well as noise phenomena
in this material 2a are not completely understood for the regime of nonlinear
response. The treatment of nonequilibrium carriers requires not only
verification the momentum and energy relaxation processes but also
understanding of the interband generation-recombination processes which
determine electron and hole concentrations far from equilibrium (similar
transport conditions take place for the bulk gapless materials, see review 3
and references therein). Effective interband transitions via optical phonons
of energy $\hbar\omega_{0}$ take place for the energy of carriers greater than
$\hbar\omega_{0}/2$, see Refs. 5 and 6 where the cases of optical excitation
and heating by dc current were analyzed. At lower energies, the generation-
recombination processes become ineffective because the Auger transitions are
forbidden due to the symmetry of electron-hole states 6 (c.f. with 7 ). Since
the carrier’s velocity $\upsilon\simeq 10^{8}$ cm/s exceeds significantly the
sound velocity $s$, the interband transitions via acoustic phonons are also
forbidden due to the momentum-energy conservation laws. Only slow generation-
recombination processes induced by the thermal radiation are allowed in a
perfect graphene. 8 To the best of our knowledge, consideration of a disorder
effect on the interband transitions via acoustic phonons in the low-energy
region, $\varepsilon<\hbar\omega_{0}/2$, is not performed yet. Thus, the
evaluation of the generation-recombinaton rate caused by the interaction of
carriers with the acoustic phonon thermostat under violation of the momentum-
energy laws in a disordered graphene (allowed electron-hole transitions are
depicted in Fig. 1) is timely now.
In this paper, the calculations are performed for the model of short-range
disorder whose parameters are taken from the mobility data, 1 ; 9 for samples
with the maximal resistance of 2 - 6 k$\Omega$ per square. The probability of
electron-hole transitions is expressed through the averaged spectral density
functions and is calculated taking into account the contribution of interband
interference. Due to slowness of the interband transitions, the
quasiequilibrium distributions of electrons and holes with the same
temperature are used for the description of a temporal evolution of
nonequilibrium concentrations of carriers. The electron and hole
concentrations are also connected through the electroneutrality condition with
the surface charge controlled by a gate voltage.
Figure 1: Interband generation-recombination transitions via acoustic phonons
with energies $\sim\hbar\omega_{ac}$ (thick arrows) between broadened
electron-hole (e - h, shown by grey) states in the low-energy region,
$\varepsilon<\hbar\omega_{0}/2$. Thin lines show the ideal dispersion law.
The results were obtained for the cases of intrinsic and heavily-doped
graphene at temperature $T$ and can be briefly summarized as follows. The
concentration balance equation is written through the chemical potential
normalized to $T$ and the characteristic rate, which is proportional to a
carrier-phonon coupling and increases with temperature as $T^{2}$. The
transient evolution of nonequilibruim population can be fitted by an
exponential decay with the relaxation time 150 - 400 ns at room temperature
and a typical disorder level corresponding to the maximal sheet resistance
$\sim$5 k$\Omega$. This time scale appears to be comparable to the
recombination rate via thermal radiation and the mechanism under consideration
can be verified by temperature and temporal measurements.
The paper is organized as follows. In the next section we present the basic
equations which describe the generation-recombination processes under
consideration. In Sec. III we evaluate the generation-recombination rates and
analyze their dependencies on temperature, disorder level, and doping
conditions. The last section includes the discussion of the approximations
used and conclusions. In Appendix we consider the generation-recombination
mechanism caused by the interaction with the thermal radiation.
## II Basic Equations
Temporal evolution of carriers in a random potential, which are weakly
interacting with the acoustic phonon modes, is described by the distribution
$f_{\alpha t}$ over the states $|\alpha)$ with energies
$\varepsilon_{\alpha}$. An exact with respect to a disordere effect kinetic
equation takes the form 10
$\displaystyle\frac{\partial f_{\alpha t}}{\partial
t}=\sum\limits_{\alpha^{\prime}}\left[W_{\alpha^{\prime}\alpha}f_{\alpha^{\prime}t}\left(1-f_{\alpha
t}\right)\right.$ $\displaystyle\left.-W_{\alpha\alpha^{\prime}}f_{\alpha
t}\left(1-f_{\alpha^{\prime}t}\right)\right].$ (1)
The transition probability $W_{\alpha\alpha^{\prime}}$ is written within the
Born approximation with respect to the carrier-phonon interaction with $q$th
phonon mode of frequency $\omega_{q}$:
$\displaystyle
W_{\alpha\alpha^{\prime}}=\frac{2\pi}{\hbar}\sum\limits_{q}\left|(\alpha|\hat{\chi}_{q}|\alpha^{\prime})\right|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2)
$\displaystyle\times\left[(N_{q}+1)\delta\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}-\hbar\omega_{q}\right)+N_{q}\delta\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}+\hbar\omega_{q}\right)\right].$
Here the operator $\hat{\chi}_{q}$ determines the carrier-phonon interaction
$\sum_{q}\left(\hat{\chi}_{q}\hat{b}_{q}+H.c.\right)$ where $\hat{b}_{q}$ is
the annihilation operator of $q$th mode and $N_{q}$ is the Planck distribution
of phonons at the equilibrium temperature $T$. Note, that the transition
probabilities $W_{\alpha^{\prime}\alpha}$ and $W_{\alpha\alpha^{\prime}}$ are
connected by
$W_{\alpha^{\prime}\alpha}=\exp\left[-\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}\right)/T\right]W_{\alpha\alpha^{\prime}}$
and
$\sum\limits_{\alpha\alpha^{\prime}}\left[W_{\alpha^{\prime}\alpha}f_{\alpha^{\prime}t}\left(1-f_{\alpha
t}\right)-W_{\alpha\alpha^{\prime}}f_{\alpha
t}\left(1-f_{\alpha^{\prime}t}\right)\right]=0$ (3)
due to the particle conservation law.
The concentrations of electrons and holes, $n_{t}$ and $\overline{n}_{t}$,
which are averaged over random disorder (such averaging is denoted as
$\langle\ldots\rangle$), are given by
$\left|\begin{array}[]{*{20}c}{n_{t}}\\\
\overline{n}_{t}\end{array}\right|=\frac{4}{L^{2}}\left\langle\sum\limits_{\alpha}\left|{\begin{array}[]{*{20}c}{\theta\left(\varepsilon_{\alpha}\right)f_{\alpha
t}}\\\ {\theta\left(-\varepsilon_{\alpha}\right)\left(1-f_{\alpha
t}\right)}\end{array}}\right|\right\rangle,$ (4)
where $L^{2}$ is the normalization area and the step function
$\theta(\pm\varepsilon)$ appears due to the symmetry of electron-hole spectrum
[see Eq. (10) below]. Since effective intraband scattering is caused by the
phonon thermostat and carrier-carrier interaction, the quasiequilibrium
distributions over the conduction and valence bands are imposed during a
short-time scales and below we use
$\tilde{f}_{\varepsilon
t}=\left\\{{\begin{array}[]{*{20}c}{\left[{\exp\left({\frac{{\varepsilon-\mu_{t}^{>}}}{T}}\right)+1}\right]^{-1},}&{\varepsilon>0}\\\
{\left[{\exp\left({\frac{{\varepsilon-\mu_{t}^{<}}}{T}}\right)+1}\right]^{-1},}&{\varepsilon<0}\\\
\end{array}}\right..$ (5)
Due to effective energy relaxation the same temperatures are established in
both bands. At the same time the electon and hole concentrations are
determined through the different chemical potentials, $\mu_{t}^{>}$ and
$\mu_{t}^{<}$, respectively. The chemical potentials are connected by the
electroneutrality condition $n_{t}-\overline{n}_{t}=n_{s}$, where the surface
charge $en_{s}$ is controlled by the gate voltage, $V_{g}$, according to
$n_{s}=aV_{g}$ with $a\simeq 7.2\times 10^{10}$ cm-2/V written for the SiO2
substrate of thickness 0.3 $\mu$m.
The concentration of electrons is governed by the balance equation
$dn_{t}/dt=(dn/dt)_{ac}$ with the generation-recombination rate
$\displaystyle\left(\frac{dn}{dt}\right)_{ac}=\int\limits_{0}^{\infty}d\varepsilon\int\limits_{-\infty}^{0}d\varepsilon^{\prime}W(\varepsilon,\varepsilon^{\prime})~{}~{}~{}~{}~{}$
(6)
$\displaystyle\times\left[{\exp\left({\frac{\varepsilon^{\prime}-\varepsilon}{T}}\right)\left(1-\tilde{f}_{\varepsilon
t}\right)\tilde{f}_{\varepsilon^{\prime}t}-\left({1-\tilde{f}_{\varepsilon^{\prime}t}}\right)\tilde{f}_{\varepsilon
t}}\right].$
We take into account that the intraband transitions (when
$\varepsilon,\varepsilon^{\prime}>0$) vanish in (6) and transform the
transition probability as follows
$W(\varepsilon,\varepsilon^{\prime})=\frac{4}{{L^{2}}}\left\langle{\sum\limits_{\alpha\alpha^{\prime}}{\delta\left({\varepsilon-\varepsilon_{\alpha}}\right)\delta\left({\varepsilon^{\prime}-\varepsilon_{\alpha^{\prime}}}\right)W_{\alpha\alpha^{\prime}}}}\right\rangle.$
(7)
Further, we introduce the exact spectral density function
$A_{\varepsilon}\left({l{\bf x},l^{\prime}{\bf
x}^{\prime}}\right)=\sum\limits_{\alpha}{\delta\left(\varepsilon-\varepsilon_{\alpha}\right)\Psi_{l{\bf
x}}^{(\alpha)}\Psi_{l^{\prime}{\bf x}^{\prime}}^{(\alpha)*}},$ (8)
which is determined through the double-row wave function $\Psi_{l{\bf
x}}^{(\alpha)}$ with $l=$1,2. The column $\Psi_{\bf x}^{(\alpha)}$ is a
solution of the eigenvalue problem $(\hat{h}+V_{\bf x})\Psi_{\bf
x}^{(\alpha)}=\varepsilon_{\alpha}\Psi_{\bf x}^{(\alpha)}$ written through the
single-particle Hamiltonian $\hat{h}$ and a random potential $V_{\bf x}$.
Using the definition (2) one obtains the probability
$W(\varepsilon,\varepsilon^{\prime})$ as follows
$\displaystyle W(\varepsilon,\varepsilon^{\prime})=\frac{8\pi}{\hbar
L^{2}}\sum\limits_{\bf
q}|C_{q}|^{2}\left(N_{q}+1\right)\delta\left(\varepsilon-\varepsilon^{\prime}-\hbar\omega_{q}\right)$
(9) $\displaystyle\times\int d{\bf x}\int d{\bf x}^{\prime}e^{i{\bf
q}\cdot({\bf x}-{\bf x}^{\prime})}{\rm
tr}\left\langle\hat{A}_{\varepsilon^{\prime}}\left({\bf x},{\bf
x}^{\prime}\right)\hat{A}_{\varepsilon}\left({\bf x}^{\prime},{\bf
x}\right)\right\rangle,$
where $\bf q$ is the in-plane wave vector and $|C_{q}|^{2}$ is the matrix
element of deformation interaction. 9 As a result, $(dn/dt)_{ac}$ is
expressed through the two-particle correlation function. Since the main
contributions to (6) appears from $\varepsilon\neq\varepsilon^{\prime}$, this
correlation function can be decoupled according to
$\left\langle{\hat{A}_{\varepsilon^{\prime}}\left({{\bf x},{\bf
x}^{\prime}}\right)\hat{A}_{\varepsilon}\left({{\bf x}^{\prime},{\bf
x}}\right)}\right\rangle\approx\hat{A}_{\varepsilon^{\prime},\Delta{\bf
x}}\hat{A}_{\varepsilon,-\Delta{\bf x}}$, where
$\hat{A}_{\varepsilon,\Delta{\bf
x}}=\left\langle{\hat{A}_{\varepsilon}\left({{\bf x},{\bf
x}^{\prime}}\right)}\right\rangle$ is the averaged spectral function given by
2$\times$2 matrix.
Below, we calculate the probability (9) using the model of the short-range
disorder described by the Gaussian correlator $\langle V_{\bf x}V_{\bf
x^{\prime}}\rangle=\bar{V}^{2}\exp[-({\bf x}-{\bf
x^{\prime}})^{2}/2l_{c}^{2}]$, where $\bar{V}$ is the averaged amplitude,
$l_{c}$ is the correlation length, and the cut-off energy
$E_{c}=\upsilon\hbar/l_{c}$ exceeds the energy scale under consideration.
According to Refs. 12 and 13 the retarded Green’s function in the momentum
representation takes form:
$\displaystyle\hat{G}_{\varepsilon,{\bf p}}^{R}=\hat{P}_{\bf
p}^{(+)}G_{\varepsilon,p}+\hat{P}_{\bf
p}^{(-)}G_{\varepsilon,-p},~{}~{}~{}~{}~{}~{}$ (10) $\displaystyle
G_{\varepsilon,p}\approx\left[\varepsilon(1+\Lambda_{\varepsilon}+ig)-\upsilon
p\right]^{-1},~{}~{}~{}\Lambda_{\varepsilon}=\frac{g}{\pi}\ln\left(\frac{E_{c}}{|\varepsilon|}\right)$
where $\hat{P}_{\bf
p}^{(\pm)}=\left[1\pm(\hat{\mbox{\boldmath$\sigma$}}\cdot{\bf p})/p\right]/2$
are the projection operators on the conduction ($+$) and valence ($-$) bands,
$\hat{\mbox{\boldmath$\sigma$}}$ is the isospin Pauli matrix, and
$g=(\bar{V}^{2}l_{c}/\hbar\upsilon)^{2}\pi/2$ is the coupling constant. Here
we restrict ourselves by the Born approximation when the self-energy
contribution $\varepsilon(\Lambda_{\varepsilon}+ig)$ is written through the
logarithmically-divergent real correction and the damping factor. Note, that
these corrections vanish at $\varepsilon\to 0$. From a comparision with the
mobility data 1 ; 9 one obtains that the parameters $g\simeq$0.45, 0.3, and
0.15 correspond to the sheet resistances $\sim$6, $\sim$4, and $\sim$2
k$\Omega$ per square, respectively. The density of states,
$\rho_{\varepsilon}=-4{\rm Im}\sum_{\bf p}{\rm tr}\hat{G}_{\varepsilon,{\bf
p}}^{R}$ is shown in Fig. 2a and $\rho_{\varepsilon}=\rho_{-\varepsilon}$, i.
e. the electron-hole symmetry is not violate due to disorder. Since
$\rho_{\varepsilon}$ increases in comparision to the ideal case,
$\overline{\rho}_{\varepsilon}=2|\varepsilon|/[(\hbar\upsilon)^{2}\pi]$, the
energy-dependent renormalized velocity, $\upsilon V_{\varepsilon}$ decreases
up to 10% if $g\leq$0.5 and the concentration of carriers in an intrinsic
graphene increases up to 2 times, see Figs. 2b and 2c, respectively (here
$n_{T}\simeq 8.1\times 10^{10}$ cm-2 is the equilibrium concentration at room
temperature and at $g\to$0).
Figure 2: (Color online) (a) Density of states $\rho_{\varepsilon}$ versus
energy at $g=$0 (1), 0.15 (2), 0.3 (3) and 0.45 (4). (b) Ratio
$V_{\varepsilon}=\sqrt{\overline{\rho}_{\varepsilon}/\rho_{\varepsilon}}$
versus $g$ for $\varepsilon=$20 meV (1), 30 meV (2), 40 meV (3), and 50 meV
(4). (c) Equilibrium concentration of non-doped graphene $n_{eq}$ versus $g$
normalized to $n_{T}\simeq 0.52(T/\hbar\upsilon)^{2}$.
Figure 3: Contour plots of dimensionless kernels
$w(\varepsilon/T,\varepsilon^{\prime}/T)$ for $g=$0.4 (a) and $g=$0.2 (b).
Further, we use the standard relation $\hat{A}_{\varepsilon,{\bf
p}}=i\left(\hat{G}_{\varepsilon,{\bf p}}^{R}-\hat{G}_{\varepsilon,{\bf
p}}^{R~{}+}\right)/2\pi$ and transform the probability (9) taking into account
the energy conservation law:
$\displaystyle
W(\varepsilon,\varepsilon^{\prime})\approx\left\\{|C_{q}|^{2}(N_{q}+1)\right\\}_{\hbar\omega_{q}=\varepsilon-\varepsilon^{\prime}}$
(11) $\displaystyle\times\frac{8\pi}{\hbar L^{2}}\sum\limits_{\bf
pp^{\prime}}\delta\left(\varepsilon-\varepsilon^{\prime}-s|{\bf p}-{\bf
p^{\prime}}|\right){\rm tr}\left(\hat{A}_{\varepsilon^{\prime},{\bf
p}^{\prime}}\hat{A}_{\varepsilon,{\bf p}}\right).$
The trace here should be taken using ${\rm tr}\left(\hat{P}_{\bf
p^{\prime}}^{(\pm)}\hat{P}_{\bf p}^{(\pm)}\right)=[1+({\bf p}\cdot{\bf
p^{\prime}})/pp^{\prime}]/2$ and ${\rm tr}\left(\hat{P}_{\bf
p^{\prime}}^{(\pm)}\hat{P}_{\bf p}^{(\mp)}\right)=[1-({\bf p}\cdot{\bf
p^{\prime}})/pp^{\prime}]/2$. This result differs from the standard
consideration, 13 because interference of electron and hole states gives an
essential contribution to $W(\varepsilon,\varepsilon^{\prime})$ due to the
matrix structure of the spectral density functions. After the integrations
over $\bf p$-plane, one transforms (11) into
$\displaystyle
W(\varepsilon,\varepsilon^{\prime})\equiv\Theta_{GR}w(\varepsilon/T,-\varepsilon^{\prime}/T)/T^{2},$
(12)
$\displaystyle\Theta_{GR}=\frac{\upsilon_{ac}s}{\upsilon^{2}}\frac{T}{\hbar}\left(\frac{T}{\pi\hbar\upsilon}\right)^{2},~{}~{}~{}~{}~{}~{}\upsilon_{ac}=\frac{D^{2}T}{4\hbar^{2}\rho_{s}\upsilon
s^{2}},$
where we separated the dimensionless kernel, $w(\xi,\xi^{\prime})$, and the
factor, $\Theta_{GR}$, which is written for the case of the deformation
interaction of carriers with the in-plane acoustic modes, see Refs. 10 and 15.
Here $D$ is the deformation potential, $s$ is the sound velocity, and
$\rho_{s}$ is the sheet density of graphene. At room temperature and typical
other parameters 9 we obtain $\upsilon_{ac}\simeq 0.96\times 10^{6}$ cm/s and
$\Theta_{GR}\simeq 5.06\times 10^{19}$ cm-2s-1 (notice, that
$\upsilon_{ac}\propto D^{2}$ and we used $D\simeq$ 12 eV). The dimensionless
kernel is plotted in Fig. 3 and the probability
$W(\varepsilon,\varepsilon^{\prime})$ is suppressed fast if
$(\varepsilon-\varepsilon^{\prime})/T\geq$0.15. This cut-off factor is
determined by the weak ratio $s/\upsilon\simeq$1/137 mainly while parameter
$g$ determines a peak value of $W(\varepsilon,\varepsilon^{\prime})$, c.f.
Figs. 3a and 3b.
The generation-recombination rate (6) is written through (11) and (12) with
the use of the dimensionless variables $\xi=\varepsilon/T$ and
$\xi^{\prime}=\varepsilon^{\prime}/T$:
$\displaystyle\left(\frac{dn}{dt}\right)_{ac}=\Theta_{GR}\int\limits_{0}^{\infty}d\xi\int\limits_{0}^{\infty}d\xi^{\prime}w(\xi,\xi^{\prime})$
(13)
$\displaystyle\times\frac{e^{-\xi^{\prime}-\mu_{t}^{>}/T}-e^{-\xi^{\prime}-\mu_{t}^{<}/T}}{\left[\exp\left(\xi-\frac{\mu_{t}^{>}}{T}\right)+1\right]\left[\exp\left(\xi^{\prime}-\frac{\mu_{t}^{<}}{T}\right)+1\right]}$
For typical concentrations of carriers, $\mu_{t}^{<}$ and $\mu_{t}^{>}$ exceed
0.15$T$ and one can simplify the rate as follows:
$\displaystyle\left(\frac{dn}{dt}\right)_{ac}\approx\Theta_{GR}\frac{w_{g}\left[e^{-\mu_{t}^{>}/T}-e^{-\mu_{t}^{<}/T}\right]}{\left[e^{-\mu_{t}^{>}/T}+1\right]\left[e^{-\mu_{t}^{<}/T}+1\right]},$
$\displaystyle
w_{g}=\int\limits_{0}^{\infty}d\xi\int\limits_{0}^{\infty}d\xi^{\prime}w(\xi,\xi^{\prime})~{}~{}~{}~{}~{}~{}~{}$
(14)
where the averaged over energies kernel $w_{g}$ is plotted versus $g$ in Fig.
4, together with a simple parabolic fit. For the disorder level corresponding
to the resistance $\sim$5 k$\Omega$ per square, one obtains $w_{g}\simeq$0.02.
Figure 4: (Color online) Averaged kernel $w_{g}$ versus coupling constant $g$.
## III Results
In this section we analyze the concentration balance equation
$dn_{t}/dt=(dn/dt)_{ac}$, where the right-hand side of Eq. (14) is written
through $\psi_{t}^{>}=\mu_{t}^{>}/T$ and $\psi_{t}^{<}=\mu_{t}^{>}/T$,
together with the initial conditions $\psi_{t=0}^{>}=\psi_{0}^{>}$ and
$\psi_{t=0}^{<}=\psi_{0}^{<}$ determined through the initial concentrations
$n_{t=0}$ and $\overline{n}_{t=0}$ according to Eq. (4). Variables
$\psi_{t}^{>}$ and $\psi_{t}^{<}$ are connected through the electroneutrality
condition
$\int\limits_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left(\frac{1}{e^{\varepsilon/T-\psi_{t}^{>}}+1}-\frac{1}{e^{\varepsilon/T+\psi_{t}^{<}}+1}\right)=n_{s}$
(15)
and below we consider the cases of an intrinsic graphene ($n_{s}=0$) and a
$n$-type heavily-doped graphene ($n_{s}>0$).
### III.1 Intrinsic graphene
For the case under consideration, $\psi_{t}^{>}=-\psi_{t}^{<}\equiv\psi_{t}$
and the concentration of electrons (or holes, because now
$n_{t}=\overline{n}_{t}$) is given by
$n_{t}=\int_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left[\exp(\varepsilon/T-\psi_{t})+1\right]^{-1}$,
so that $n_{t}$ and $\psi_{t}$ are connected through
$\frac{dn_{t}}{dt}=\frac{d\psi_{t}}{dt}\int\limits_{0}^{\infty}\frac{d\varepsilon\rho_{\varepsilon}}{1+\cosh\left(\varepsilon/T-\psi_{t}\right)}.$
(16)
As a result, the concentration balance equation is derived from Eq. (13) as
$\frac{dn_{t}}{dt}=-\Theta_{GR}w_{g}\tanh\left(\frac{\psi_{t}}{2}\right)$ (17)
and Eqs. (16) and (17) are transformed into the first-order differential
equation for $\psi_{t}$ with the initial condition $\psi_{t=0}=\psi_{0}$ where
$\psi_{0}$ is determined through the $n_{t=0}$. The implicit solution of this
equation takes form:
$\displaystyle\nu_{GR}t=\int\limits_{\psi_{t}}^{\psi_{0}}d\psi
F(\psi),~{}~{}~{}~{}\nu_{GR}=w_{g}\frac{\upsilon_{ac}s}{\pi\upsilon^{2}}\frac{T}{\hbar},$
(18) $\displaystyle
F(\psi)=\tanh\left(\frac{\psi}{2}\right)\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}}{1+\cosh(\xi-\psi)}.$
Here $r_{\xi}=\rho_{\xi T}/\overline{\rho}_{T}$ is the dimensionless density
of states and the temporal evolution of $n_{t}$ is described through the
characteristic rate $\nu_{GR}$ and the dimensionless function $F(\psi)$. At
room temperature and at $w_{g}\simeq$0.02, one obtains $\nu_{GR}\simeq
1.85\times 10^{7}$ s-1 for the parameters used.
Figure 5 shows the transient evolution of $n_{t}$ normalized to the
equilibrium concentration $n_{eq}$, for the cases of recombination or
generation of carriers, if $n_{t=0}>n_{eq}$ or $n_{t=0}<n_{eq}$, respectively.
The relaxation becomes suppressed if the disorder level decreases both due to
a slowness of dependency on $\nu_{GR}t$, c.f. curves for $g=$0.5 and 0.25 in
Fig. 5, and, mainly, due to the relation $\nu_{GR}\propto w_{g}$, see Fig. 4.
Within a 5% accuracy, the evolution of $n_{t}$ can be fitted by the
exponential dependencies
$n_{t}\approx n_{0}+n_{T}\left[1-\exp(-\alpha\nu_{GR}t)\right]$ (19)
with the parameter $\alpha$ varying between 0.16 and 0.28 depending on the
initial conditions. The corresponding times, $(\alpha\nu_{GR})^{-1}$, vary
between 310 and 180 ns for $g\simeq$0.5. This time scale is comparable to the
radiative recombination times, see Appendix.
Figure 5: (Color online) Transient evolution of concentration $n_{t}$ at
different initial conditions: $n_{t=0}=3n_{eq}$ (1,2), $n_{t=0}=2n_{eq}$,
(3,4) and $n_{t=0}=0.5n_{eq}$ (5,6) for coupling parameters $g=$0.25 (1, 3, 5)
and $g=$0.5 (2, 4, 6). Dotted curves correspond to exponential fits (19), with
$\alpha=$0.16 (1), 0.125 (2), 0.19 (3), 0.155 (4), 0.22 (5), and 0.275 (6).
### III.2 Heavily-doped graphene
We turn now to the case of a heavily doped graphene, when $\psi_{t}^{>}\gg 1$
and it is convenient to introduce a weak variation
$\delta\psi_{t}=\psi_{t}^{>}-\psi_{s}$ where $\psi_{s}$ corresponds to the
equilibrium case. Neglecting a hole concentration and using the step function
in $c$-band, one obtains $n_{s}$ from Eq. (15):
$n_{s}\approx\int\limits_{0}^{\psi_{s}T}d\varepsilon\rho_{\varepsilon}.$ (20)
As a result, $\delta\psi_{t}$ and $\psi_{t}^{<}$ are connected by the
electroneutrality condition (15) as follows
$\delta\psi_{t}\approx\frac{\overline{\rho}_{T}}{\rho_{\varepsilon=\psi_{s}T}}\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}}{1+\exp\left(\xi+\psi_{t}^{<}\right)},$ (21)
where the ratio $\overline{\rho}_{T}/\rho_{\varepsilon=\psi_{s}T}$ can be
found from Fig. 2a. Using Eq. (14) we transform the concentration balance
equation into the form:
$\frac{d\delta\psi_{t}}{dt}=-\frac{\nu_{GR}\overline{\rho}_{T}}{2\rho_{\varepsilon=\psi_{s}T}[1+\exp\left(\psi_{t}^{<}\right)]}.$
(22)
Substituting the relation (21) into Eq. (22) one obtains the first-order
differential equation for $\psi_{t}^{<}$, with the implicit solution
$\nu_{GR}t=\int\limits_{\overline{\psi}_{0}}^{\psi_{t}^{<}}d\psi\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}\left(1+e^{\psi}\right)}{1+\cosh\left(\xi+\psi\right)},$ (23)
where $\overline{\psi}_{0}=\psi_{t=0}^{<}$ appears from the initial condition.
Notice, that the factor $\overline{\rho}_{T}/\rho_{\varepsilon=\psi_{s}T}$
drops out from the solution (23), i. e. the transient process under
consideration does not depend on the doping level because the only low-energy
states are involved in the interband transitions.
Figure 6: (Color online) Transient evolution of concentration $n_{t}-n_{s}$ at
different initial conditions: $n_{t=0}-n_{s}=3n_{T}$ (1),
$n_{t=0}-n_{s}=2n_{T}$, (2) and $n_{t=0}-n_{s}=0.5n_{T}$ (3). Solid and dashed
curves correspond to coupling parameters $g=$0.5 and $g=$0.25, respectively.
Dotted curves correspond to the exponential fits.
Further, we plot the transient evolution of the hole concentration
$\overline{n}_{t}=n_{t}-n_{s}$ determined through $\psi_{t}^{<}$ according to
$\overline{n}_{t}=\int_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left[\exp(\varepsilon/T+\psi_{t}^{<})+1\right]^{-1}$.
Figure 6 shows the concentration $\overline{n}_{t}$ versus dimensionless time,
$\nu_{GR}t$, for the initial conditions written through $n_{T}$. Similarly to
the undoped case, the exponential fits
$(n_{t}-n_{s})/(n_{t=0}-n_{s})\approx\exp(-\beta\nu_{GR}t)$ with
$\beta\simeq$0.12 (1), 0.15 (2), and 0.2 (3) describe the transient evolution
with an accuracy $\sim$10% if $\nu_{GR}t<$10\. An enhancement of recombination
takes place at tails of transient evolution, if $\nu_{GR}t>$10\. Since the
relaxation rate increases with the disorder level, $\nu_{GR}\propto w_{g}$,
the recombination process becomes faster in spite of an opposite dependency on
$\nu_{GR}t$ in Fig. 6. The relaxation times, $\sim(\beta\nu_{GR})^{-1}$, vary
between 410 and 240 ns for $g=$0.5 and different initial conditions. Once
again, the recombination scale is comparable to the radiative recombination
process shown in Fig. 7b, Appendix.
## IV SUMMARY AND CONCLUSIONS
We have examined the new channel for interband generation-recombination
process of carriers in a disordered graphene via acoustic phonons. The
efficiency of transitions increases with the disorder level and concentration
of nonequilibrium carriers as well as with temperature. We have found that the
relaxation rate belongs to submicrosecond range for the samples with typical
disorder level at room temperature.
Let us discuss the assumptions used in the presented calculations. The main
restriction of the results is the description of the response in the framework
of the quasiequilibrium approach, with different chemical potentials in $c$\-
and $v$-bands but the same temperature due to the fast energy relaxation
caused by phonon and carrier-carrier scattering processes. We also restrict
ourselves by the simplest model of the short-range disorder. By analogy with
the description of transport phenomena, 1 ; 9 ; 11 more complicated
calculations for finite-range disorder should give similar results. But the
case of impurities with a low-energy resonant level, which was discussed
recently in Refs. 16, requires a special consideration. We considered the
deformation interaction of carriers with longitudinal acoustic modes 9 ; 14
neglecting scattering by surface phonons of the substrate in agreement with
the experimental data. 16 Such a contribution can only restrict the energies
under consideration because of the lower surface phonon energy ($\sim$55 meV
for the SiO2 substrate). Since the non-diagonal components give a weak
contribution to the concentration balance equation under consideration, 10 we
take into account only diagonal components of the density matrix $f_{\alpha
t}$ while evaluating of the generation-recombination rate. The simplifications
mentioned above do not change either the peculiarities of the generation-
recombination processes or the numerical estimates of relaxation times given
in Sec. III.
Next, we briefly consider some possibilities for experimental verification of
the mechanism of interband transitions suggested. It is clear from a
comparison of the results in Sec. III and in Appendix that interband
transitions via acoustic phonons and via thermal radiation can be separated
due to different temperature and concentration dependencies of damping. A
possible contribution of the disorder-induced Auger process is beyond of our
consideration and requires a special study. In contrast to the ultrafast
optical measurements applied for the study of the relaxation and recombination
of high-energy carriers, 2 a transient evolution of concentration over time
scales $\sim$100 ns can be measured directly (e.g. in Ref. 5 the transient
response under abrupt switching on of a dc field lasts up to hundreds of
nanoseconds). But under a verification of the slow process examined, a
possible contact injection or a trapping into substrate states should be
analyzed.
To conclude, we believe that the generation-recombination via acoustic phonons
can be verified experimentally and more detailed numerical calculations are
necessary in order to separate this mechanism from other contributions. The
results obtained will stimulate a further study of the generation-
recombination processes which are essential in many transport and optical
phenomena far from equilibrium.
## ACKNOWLEDGMENT
This paper is based upon work supported by the National Science Foundation
under Grant No DMR 0907126.
*
## Appendix A Radiative transitions
Below we describe the generation-recombination processes which are associated
with the interband transitions induced by the thermal radiation and evaluate
the radiative relaxation rate for the weak disorder case, $g\ll 1$. The
corresponding collision integral was evaluated in Ref. 9 and the kinetic
equation for the electron distribution $f_{ept}$ takes the form
$\frac{\partial f_{ept}}{\partial t}=\nu_{p}^{(R)}\left[N_{2\upsilon
p/T}\left(1-f_{ept}-f_{hpt}\right)-f_{ept}f_{hpt}\right],$ (24)
where $N_{2\upsilon p/T}$ describes the Planck distribution of the thermal
photons at temperature $T$. The hole distribution can be obtained from the
condition $\partial(f_{ept}+f_{hpt})/\partial t=0$. The interband absorption
or emission of photons are described by the first or second terms in the
right-hand side of Eq. (A.1) and are responsible for the generation or
recombination processes. The rate of spontaneous radiative transitions is
given by $\nu_{p}^{(R)}=\upsilon_{R}p/\hbar$ where we have introduced the
characteristic velocity $\upsilon_{R}\simeq$41.6 cm/s for graphene surrounded
by SiO2 layers. Similar to Eq. (6) contribution of the radiative collision
integral from (A.1) into the concentration balance equation takes the form
$(dn/dt)_{ac}=(4/L^{2})\sum\nolimits_{\bf p}\nu_{p}^{(R)}\left[N_{2\upsilon
p/T}\left(1-f_{ept}-f_{hpt}\right)-f_{ept}f_{hpt}\right]$.
For the case of an intrinsic graphene, the balance equation is written by
analogy with Sect. III A through $\psi_{t}=\mu_{t}/T$:
$\frac{d\psi_{t}}{dt}=-\nu_{R}\frac{F_{i}(\psi_{t})}{N(\psi_{t})},~{}~{}~{}~{}\nu_{R}=\frac{2\upsilon_{R}}{\upsilon}\frac{T}{\hbar}$
(25)
where $\nu_{R}^{-1}\approx$30 ns is the radiative recombination time at room
temperature. The functions $F_{i}(\psi)$ and $N(\psi)$ are given by
$\displaystyle
F_{i}(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi\left(1-e^{-2\psi}\right)}{\left(1-e^{-2\xi}\right)\left(e^{\xi-\psi}+1\right)^{2}}$
$\displaystyle
N(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi}{1+\cosh(\xi-\psi)}$ (26)
and the implicit solution of Eqs. (A.2), (A.3) is given by the similar to Eq.
(18) formula:
$\nu_{R}t=\int_{\psi_{0}}^{\psi_{t}}d\psi\frac{N(\psi)}{F_{i}(\psi)}.$ (27)
In Fig. 7a we plot the transient evolution of concentration versus the
dimensionless time, $\nu_{R}t$ for the same initial conditions as in Fig. 5.
These transient dependencies are described by the exponential decay given by
Eq. (19) with $\alpha\approx$0.25 for all cases. Thus, one obtains the
radiative recombination time $\sim 4/\nu_{R}\approx$120 ns which does not
depend on an initial concentration.
Figure 7: (Color online) Transient evolution of concentration due to interband
radiative transitions for intrinsic (a) and heavily-doped (b) graphene at
different initial conditions: $n_{t=0}=3n_{T}$ (1), $n_{t=0}=2n_{T}$ (2) and
$n_{t=0}=0.5n_{T}$ (3). Dotted curves correspond to exponential fits.
For the case of doped graphene, the concentration balance equation (22) should
be replaced by
$\displaystyle\frac{d\delta\psi_{t}}{dt}=-\frac{\nu_{R}}{2}F_{d}(\psi_{t}^{<}),$
(28) $\displaystyle
F_{d}(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi^{2}}{\left(1-e^{-2\xi}\right)\left(e^{\xi+\psi}+1\right)}$
while the relation between $\delta\psi_{t}$ and $\psi_{t}^{<}$ takes form
[c.f. Eq. (21)]
$\delta\psi_{t}\approx\frac{1}{\psi_{s}}\int\limits_{0}^{\infty}\frac{d\xi\xi}{1+\exp\left(\xi+\psi_{t}^{<}\right)}.$
(29)
As a result, the equation for $\psi_{t}^{<}$ has the only difference from Eq.
(2) due to the replacement $F_{i}(\psi)$ by $F_{d}(\psi)$. The implicit
solution of Eqs. (A.5) and (A.6) is given by (A.4) with the same replacement.
In Fig. 7b we plot the transient evolution of hole concentration,
$\overline{n}_{t}/n_{T}$, for the same initial conditions as in Fig. 6. The
corresponding exponential fits are determined by the coefficients
$\beta\simeq$0.15 (1), 0.125 (2), and 0.06 (3), i.e. the relaxation rate
depends on hole concentration. At room temperature the radiative recombination
time $(\beta\nu_{R})^{-1}$ corresponds to the time interval between 190 and
480 ns.
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* (17) A. Barreiro, M. Lazzeri, J. Moser, F. Mauri, and A. Bachtold, Phys. Rev. Lett. 103, 076601 (2009); S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008).
|
arxiv-papers
| 2011-07-14T01:47:45 |
2024-09-04T02:49:20.540372
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. T. Vasko and V. V. Mitin",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/1107.2708"
}
|
1107.2868
|
# The effect of electrostatic shielding using invisibility cloak
Ruo-Yang Zhang zhangruoyang@gmail.com Theoretical Physics Division, Chern
Institute of Mathematics, Nankai University, Tianjin, 300071, P.R.China Qing
Zhao qzhaoyuping@bit.edu.cn Department of Physics, College of Science,
Beijing Institute of Technology, Beijing, 100081, P.R. China Mo-Lin Ge
geml@nankai.edu.cn Theoretical Physics Division, Chern Institute of
Mathematics, Nankai University, Tianjin, 300071, P.R.China Department of
Physics, College of Science, Beijing Institute of Technology, Beijing, 100081,
P.R. China
###### Abstract
The effect of electrostatic shielding for a spherical invisibility cloak with
arbitrary charges inside is investigated. Our result reveals that the charge
inside the cloak is a crucial factor to determine the detection. When charged
bodies are placed inside the cloak with an arbitrary distribution, the
electric fields outside are purely determined by the total charges just as the
fields of a point charge at the center of the cloak. As the total charges
reduce to zero, the bodies can not be detected. On the other hand, if the
total charges are nonzero, the electrostatic potential inside an ideal cloak
tends to infinity. For unideal cloaks, this embarrassment is overcome, while
they still have good behaviors of shielding. In addition, the potential across
the inner surface of an ideal cloak is discontinuous due to the infinite
polarization of the dielectric, however it can be alternatively interpreted as
the dual Meissner effect of a dual superconductive layer with a surface
magnetic current.
###### pacs:
41.20.Cv, 42.79.-e
## I Introduction
Since the pioneering works of Pendry et al. Pendry _et al._ (2006) based on
coordinate transformation and Leonhardt Leonhardt (2006) based on conformal
mapping method, invisibility cloak has attracted much attention and widely
research. Noteworthy, in 2003, Greenleaf et al. have suggested to design
anisotropic conductivities that cannot be detected by electrical impedance
tomography through coordinate transformation of Poisson’s equation which is
similar to Pendry’s method Greenleaf _et al._ (2003a, b). The effectiveness
of transformation based cloak with passive objects inside has been verified
through ray tracing approach Schurig _et al._ (2006); Niu _et al._ (2009),
full-wave simulations Cummer _et al._ (2006), and analysis based on
scattering models Chen _et al._ (2007); Ruan _et al._ (2007); Zhang _et
al._ (2007); Luo _et al._ (2008). On the other hand, the case that active
sources are located inside the hidden area has also been investigated both in
physics Zhang _et al._ (2008) and mathematical senses Greenleaf _et al._
(2007). These investigations revealed the ideal invisibility cloak at
particular frequency can prevents the electromagnetic waves generated by the
sources inside the cloak from propagating out, meanwhile an extra surface
electric and magnetic voltage are induced by the infinite polarization on the
inner surface. As a result, it seems that the invisibility cloak with passive
or active objects inside can not be detected by means of electromagnetic
waves, while another method was suggested to detect the cloak by shooting a
fast-moving charged particle through it Zhang and Wu (2009).
In previous researches, the hidden passive objects and active sources are all
electric neutral. As far as we know, the situation that the hidden objects are
charged has never been discussed. However, the charges inside the cloak would
be an important factor influencing the detection of the hidden objects. In
this paper, we will only discuss the situation that the distribution of
charges does not change with time, so the shielding effect of the invisibility
cloak reduces to an electrostatic problem. The electrostatic problem could be
regarded as the limit of harmonically varying fields as $\omega\rightarrow 0$.
However, some special property of harmonic fields could remain and the
electric and magnetic fields would not decouple completely even at the limit
$\omega\rightarrow 0$, such as $|E/B|\equiv c$ does not varying with $\omega$
for a plane wave propagating in vacuum. So we treat the problem through
solving the Poisson’s equation strictly rather than taking the limit of zero
frequency.
In this paper, we fist construct the electrostatic cloak according to the
invariance of Poisson’s equation under coordinate transformation. Actually,
the method to design electrostatic cloak has been proposed by Greenleaf et al.
Greenleaf _et al._ (2003a, b). However, Greenleaf’s device is made of
conductivity , while the cloak suggested here is composed of dielectric. Then,
we calculate the fields in the whole space where the charges with arbitrary
distribution are located inside a spherical cloak which are constructed by
arbitrary radial transformation $f(r)$. The result reveals the fields out of
the cloak are only determined by the total charges inside the cloak and the
shielding behavior of the cloak are very like the effect of charge confinement
by a dual superconductor under the monopole-existing supposition, which model
is often used in color confinement Ripka (2005). Moreover, for ideal cloaks,
the result suggests an infinite electric energy when the total charges are
nonzero. To overcome the embarrassment, we investigate the unideal cloaks and
find that they can also realize a good effect of shielding.
## II Electrostatic cloak
In vacuum, the Electrostatic fields obey Poisson’s equation Greenleaf _et
al._ (2003b, a):
$\nabla^{2}\psi\ =\
\frac{1}{\sqrt{\gamma}}\partial_{i}(\sqrt{\gamma}\gamma^{ij}\partial_{j}\psi)\
=\ -\frac{\rho}{\varepsilon_{0}},$ (1)
where $\psi$ is the electrostatic potential, $\gamma^{ij}$ is the
contravariant component of the spatial metric under an arbitrary curvilinear
coordinate system (marked as S system) of the 3-D vacuous virtual space (VS),
and $\gamma$ is the determinant of $\gamma_{ij}$. In Eq. (1),
$\sqrt{\gamma}\gamma^{ij}$ can be alternatively interpreted as the
permittivity $\varepsilon^{ij}$ of a real material expressed in another
coordinate system of a real physical space (PS). Meanwhile, the physical
quantities in VS also can be expressed in the same system as in PS, which is
marked as S’ system. Concerning an arbitrary radial transformation between S’
and S system: $r^{\prime}=f(r)$, $\theta^{\prime}=\theta$,
$\phi^{\prime}=\phi$, where S’ is a spherical coordinate system, we obtain the
relative permittivity of the spherical electrostatic cloak through the same
process as in optical cloaking:
$\varepsilon_{\langle
ij\rangle}=\mathrm{diag}\left(\frac{f^{2}(r)}{r^{2}f^{\prime}(r)},\ \
f^{\prime}(r),\ \ f^{\prime}(r)\right).$ (2)
where we use $\varepsilon_{\langle ij\rangle}$ to denote the uncoordinate
components of the permittivity in orthonormal bases.
We can see that the expression of permittivity of an electrostatic cloak is
identical with it of an optical cloak Luo _et al._ (2008). So the
electrostatic cloak can be regarded as an optical cloak at zero frequency. We
will verify that the spherical electrostatic cloak expressed in Eq. (2) with
“invisibility conditions” $f(a)=0$, $f(b)=b$ can not be detected in
electrostatic sense, where $a,\ b$ are the inner and outer radiuses of the
cloak respectively.
Consider an electrostatic cloak wrapping a homogenous medium $\varepsilon_{1}$
is located in vacuum endowed with an uniform electric field. A spherical
coordinate system is constructed whose origin is at the center of the cloak
and $z$ axis is along the direction of the field. So the original field is
$\psi_{0}=-E_{0}r\cos\theta+C$, where $E_{0}$ is the intensity of the uniform
field. For charge-free anisotropic media, the potential satisfies
$\nabla\cdot(\tensor{\varepsilon}\cdot\nabla\psi)=0$. Substituting Eq. (2)
into the formula, we have
$\frac{\partial}{\partial f}\left(f^{2}\frac{\partial\psi}{\partial
f}\right)+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\phi^{2}}=0,$
(3)
which is just the Poisson’s equation in S’ system of VS. Through separating
variables, $\psi$ can be represented by series
$\displaystyle\psi^{\mathrm{out}}\ =\ $
$\displaystyle\sum_{n=0}^{\infty}B^{\mathrm{out}}_{n}\frac{1}{r^{n+1}}P_{n}(\cos\theta),$
(4a) $\displaystyle\psi^{\mathrm{cl}}\ =\ $
$\displaystyle\sum_{n=0}^{\infty}\left(A^{\mathrm{cl}}_{n}f(r)^{n}+B^{\mathrm{cl}}_{n}\frac{1}{f(r)^{n+1}}\right)P_{n}(\cos\theta),$
(4b) $\displaystyle\psi^{\mathrm{int}}\ =\ $
$\displaystyle\sum_{n=0}^{\infty}A^{\mathrm{int}}_{n}r^{n}P_{n}(\cos\theta),$
(4c)
where $\psi^{\mathrm{out}}$, $\psi^{\mathrm{cl}}$, $\psi^{\mathrm{int}}$
represent the reflective field outside the cloak, the field in the cloak shell
and the field inside the internal hidden area respectively, and
$P_{n}(\cos\theta)$ denotes the $n$-order Legendre function.
According to the boundary conditions, which is the continuity of $\psi$ and
the normal component of $\vec{D}$ across the inner and outer surface of the
cloak, we obtain
$\displaystyle A^{\mathrm{cl}}_{0}=A^{\mathrm{int}}_{0}=c,$ (5a)
$\displaystyle A^{\mathrm{cl}}_{1}=\alpha E_{0},\quad
B^{\mathrm{cl}}_{1}=\beta E_{0},$ (5b) $\displaystyle
A^{\mathrm{int}}_{1}=\frac{1}{a}\big{[}\alpha
f(a)+\frac{\beta}{f(a)^{2}}\big{]}E_{0},$ (5c) $\displaystyle
B^{\mathrm{out}}_{1}=b^{2}\big{[}\alpha
f(b)+\frac{\beta}{f(b)^{2}}+b\big{]}E_{0},$ (5d)
and other coefficients are all zero, where
$\displaystyle\alpha=\frac{3f(b)^{2}b^{2}(\frac{\varepsilon_{1}}{\varepsilon_{0}}a+2f(a))}{2f(a)^{3}(f(a)-\frac{\varepsilon_{1}}{\varepsilon_{0}}a)(f(b)-b)-f(b)^{3}(\frac{\varepsilon_{1}}{\varepsilon_{0}}a+2f(a))(2b+f(b))},$
(6a)
$\displaystyle\beta=\frac{3f(a)^{3}f(b)^{2}b^{2}(f(a)-\frac{\varepsilon_{1}}{\varepsilon_{0}}a)}{2f(a)^{3}(f(a)-\frac{\varepsilon_{1}}{\varepsilon_{0}}a)(f(b)-b)-f(b)^{3}(\frac{\varepsilon_{1}}{\varepsilon_{0}}a+2f(a))(2b+f(b))}.$
(6b)
When $f(a)=0$, $f(b)=b$, they reduce to $\alpha=-1$, $\beta=0$. Therefore,
$\psi^{\mathrm{out}}=0$, $\psi^{\mathrm{int}}=C$, and the field in the cloak
layer is
$\psi^{\mathrm{cl}}\ =\ -E_{0}f(r)\cos\theta+C.$ (7)
The result indicates the perfect cloak and its hidden medium can not be
detected in that there is no reflective field. Furthermore, the hidden area
can not perceive the fields outside the cloak either.
In terms of Eq. (7), we get the electric intensity and the electric
displacement vector in the cloak layer:
$\displaystyle\vec{E}^{\mathrm{cl}}\ =\ $ $\displaystyle
E_{0}\left[f^{\prime}(r)\cos\theta\hat{e}_{r}-\frac{f(r)}{r}\sin\theta\hat{e}_{\theta}\right],$
(8a) $\displaystyle\vec{D}^{\mathrm{cl}}\ =\ $
$\displaystyle\frac{\varepsilon_{0}E_{0}f(r)}{r}\left[\frac{f(r)}{r}\cos\theta\hat{e}_{r}-f^{\prime}(r)\sin\theta\hat{e}_{\theta}\right].$
(8b)
The electric field intensity can be regarded as the tangential vector of the
electric lines of force, that is $d\vec{r}/d\lambda=\vec{E}^{\mathrm{cl}}$
where $\lambda$ is the parameter. Thus, we can derive the analytical
expression of the electric lines of force
$\exp\left[\int\frac{f(r)}{r^{2}f^{\prime}(r)}dr\right]\sin\theta\ =\
\mbox{const},\quad\phi\ =\ \phi_{0},$ (9)
which employ the same expression of wave-normal rays in optical cloak exposed
in a plane wave. Similarly, the lines of electric displacement are expressed
by
$f(r)\sin\theta\ =\ \text{const},\quad\phi\ =\ \phi_{0}.$ (10)
It is just the expression of straight lines in S’ system of VS, and it
corresponds to the light-ray in optical cloak. The distribution of the fields
are shown in Fig. 1, where the constant of the potential are selected to be
$C=0$.
Figure 1: Electrostatic cloak in uniform electric field. The distribution of
potential are illustrated in $y=0$ plane. Red lines denote equipotential
surfaces, (a) green lines denote electric lines of force, and (b) blue lines
denote lines of electric displacement. The inner and outer radiuses of cloak
are $a=1\mathrm{m}$ and $b=2\mathrm{m}$ respectively.
## III electrostatic shielding
Now we discuss the problem of electrostatic shielding of the cloak with an
arbitrary distribution of charges inside. The simplest situation is that a
point charge lies on the $z$ axis with a displacement $a_{0}(<a)$ from the
origin. The potential in each region holds the same form as in Eqs. (4) except
that the total potential inside the hidden area should add the potential
produced by the point charge
$\psi^{\mathrm{int}}=\frac{\tilde{q}}{\sqrt{r^{2}-2a_{0}r\cos\theta+a_{0}^{2}}}+\sum_{n=0}^{\infty}A^{\mathrm{int}}_{n}r^{n}P_{n}(\cos\theta),$
(11)
where $\tilde{q}=q/(4\pi\varepsilon_{1})$ and $q$ is the quantity of the point
charge. The zero-point of potential is selected at infinity. When $r>a_{0}$,
the potential of point charge can be expanded as
$\frac{\tilde{q}}{\sqrt{r^{2}-2a_{0}r\cos\theta+a_{0}^{2}}}=\tilde{q}\sum_{n=0}^{\infty}\frac{a_{0}^{n}}{r^{n+1}}P_{n}(\cos\theta).$
(12)
Substitution the potentials into the boundary conditions at $r=a$ and $b$
yields
$\displaystyle\frac{B^{\mathrm{out}}_{n}}{b^{n+1}}-f(b)^{n}A^{\mathrm{cl}}_{n}-\frac{B^{\mathrm{cl}}_{n}}{f(b)^{n+1}}=0,$
(13a)
$\displaystyle\frac{n+1}{b^{n+2}}B^{\mathrm{out}}_{n}+\frac{nf(b)^{n+1}}{b^{2}}A^{\mathrm{cl}}_{n}-\frac{n+1}{b^{2}f(b)^{n}}B^{\mathrm{cl}}_{n}=0,$
(13b) $\displaystyle
f(a)^{n}A^{\mathrm{cl}}_{n}+\frac{B^{\mathrm{cl}}_{n}}{f(a)^{n+1}}-\frac{\tilde{q}a_{0}^{n}}{a^{n+1}}-a^{n}A^{\mathrm{int}}_{n}=0,$
(13c)
$\displaystyle\begin{split}&\frac{nf(a)^{n+1}}{a^{2}}A^{\mathrm{cl}}_{n}-\frac{n+1}{a^{2}f(a)^{n}}B^{\mathrm{cl}}_{n}+\frac{\varepsilon_{1}}{\varepsilon_{0}}\frac{(n+1)\tilde{q}a_{0}^{n}}{a^{n+2}}\\\
&-\frac{\varepsilon_{1}}{\varepsilon_{0}}na^{n-1}A^{\mathrm{int}}_{n}=0,\end{split}$
(13d)
By solving the equations, all coefficients are determined
$\displaystyle A^{\mathrm{cl}}_{n}=\alpha_{n}\tilde{q}a_{0}^{n},\quad
B^{\mathrm{cl}}_{n}=\beta_{n}\tilde{q}a_{0}^{n},$ (14a) $\displaystyle
A^{\mathrm{int}}_{n}=\frac{1}{a^{n}}\big{[}\alpha_{n}f(a)^{n}+\frac{\beta_{n}}{f(a)^{n+1}}-\frac{1}{a^{n+1}}\big{]}\tilde{q}a_{0}^{n},$
(14b) $\displaystyle
B^{\mathrm{out}}_{n}=b^{n+1}\big{[}\alpha_{n}f(b)^{n}+\frac{\beta_{n}}{f(b)^{n+1}}\big{]}\tilde{q}a_{0}^{n},$
(14c)
where
$\displaystyle\alpha_{n}=\frac{\frac{\varepsilon_{1}}{\varepsilon_{0}}(n+1)(2n+1)(f(b)-b)f(a)^{n+1}}{a^{n}\big{\\{}n(n+1)f(a)^{2n+1}(\frac{\varepsilon_{1}}{\varepsilon_{0}}a-f(a))(f(b)-b)+f(b)^{2n+1}[(n+1)b+nf(b)][\frac{\varepsilon_{1}}{\varepsilon_{0}}na+(n+1)f(a)]\big{\\}}},$
(15a)
$\displaystyle\beta_{n}=\frac{\frac{\varepsilon_{1}}{\varepsilon_{0}}(2n+1)f(b)^{2n+1}[(n+1)b+nf(b)]f(a)^{n+1}}{a^{n}\big{\\{}n(n+1)f(a)^{2n+1}(\frac{\varepsilon_{1}}{\varepsilon_{0}}a-f(a))(f(b)-b)+f(b)^{2n+1}[(n+1)b+nf(b)][\frac{\varepsilon_{1}}{\varepsilon_{0}}na+(n+1)f(a)]\big{\\}}}.$
(15b)
For the ideal situation $f(a)=0$, $f(b)=b$, the results are simplified to
$\alpha_{n}=0$ and $\beta_{0}=\varepsilon_{1}/\varepsilon_{0}$, $\beta_{n}=0$
($n\neq 0$). Thus the potentials reduce to
$\displaystyle\psi^{\mathrm{out}}\ =\
\frac{q}{4\pi\varepsilon_{0}r},\quad\psi^{\mathrm{cl}}\ =\
\frac{q}{4\pi\varepsilon_{0}f(r)},$ (16a) $\displaystyle\begin{split}\\\
\psi^{\mathrm{int}}=&\frac{\tilde{q}}{\sqrt{r^{2}-2a_{0}r\cos\theta+a_{0}^{2}}}+\tilde{q}\left(\frac{\varepsilon_{1}}{\varepsilon_{0}}\frac{1}{f(a)}-\frac{1}{a}\right)\\\
&+\tilde{q}\sum_{n=1}^{\infty}\frac{n+1}{n}\frac{a_{0}^{n}}{a^{2n+1}}r^{n}P_{n}(\cos\theta),\end{split}$
(16b)
The field outside the cloak is precisely equal to the field produced by a
point charge $q$ at the origin in vacuum. While the potential in the whole of
the hidden area tends to infinite since the term
$\varepsilon_{1}\tilde{q}/(\varepsilon_{0}f(a))$ exists as $f(a)\rightarrow
0$.
We can further solve the Green’s function to this set of boundary conditions
in light of the above solutions. The Green’s function
$G(\vec{r},\vec{r}\,^{\prime})$ is regarded as the potential produced by an
unit point charge located at $(r^{\prime},\,\theta^{\prime},\,\phi^{\prime})$.
Thus $G(\vec{r},\vec{r}\,^{\prime})$ can be directly transformed from Eqs.
(16) through the substitution
$\displaystyle q\ \rightarrow\ 1,\quad a_{0}\ \rightarrow\ r^{\prime},$
$\displaystyle\cos\theta\ \rightarrow\
\cos\vartheta=\sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime})+\cos\theta\cos\theta^{\prime}.$
For an arbitrary distribution $\rho(\vec{r})$ of charges inside the cloak, the
potential in the whole space is written as
$\psi(\vec{r})=\int G(\vec{r},\vec{r}\,^{\prime})\rho(\vec{r}\,^{\prime})\
dV^{\prime},$ (18)
where the domain of integration is the whole hidden area. The potential
outside the hidden area still takes the expression shown in Eq. (16a), which
means the fields outside is not affected by the distribution of charges inside
the cloak but is only determined by the total charges $q$. Taking account of
Gauss’s theorem, it is actually not marvelous that the electrostatic cloak can
not screen the outside space from the electric field of charged bodies inside,
as a result the charged bodies can be detected outside the cloak.
Nevertheless, apart from the total charges, no more information about the
charge distribution can be detected, which is the least information allowed to
gain under the restriction of Gauss’s theorem. When the total charges tend to
zero, no field can go out of the cloak which is in complete agreement with the
result for an active source inside the cloak as its radiation frequency goes
to zero Zhang _et al._ (2008).
The behavior of electrostatic shielding of the cloak is identical with a
spherical conducting shell in the screened area no matter where the sources
are located, inside or outside the cloak. However, the electric field in the
source-existing area is different between the two systems. It is interesting
to generalize the two systems as two different limitation of an unified
system, a simple system with a dielectric sphere with permittivity
$\varepsilon_{1}$ placed in another medium $\varepsilon_{2}$. For simplicity,
we also consider that a point charge $q$ is located in the sphere with a
distance $a_{0}$ from the center. The electric potential inside the sphere can
be written as Batygin and Toptygin (1978)
$\begin{split}\psi^{\mathrm{int}}=\
&\frac{\tilde{q}}{\sqrt{r^{2}-2a_{0}r\cos\theta+a_{0}^{2}}}+\tilde{q}\frac{\varepsilon_{1}-\varepsilon_{2}}{\varepsilon_{1}}\\\
&\cdot\sum_{n=0}^{\infty}\frac{n+1}{n+\frac{\varepsilon_{2}}{\varepsilon_{0}}(n+1)}\frac{a_{0}^{n}}{a^{2n+1}}r^{n}P_{n}(\cos\theta).\end{split}$
(19)
When $\varepsilon_{2}\rightarrow\infty$, the result is the same as the case of
conducting shell. On the other hand, in the limit $\varepsilon_{2}\rightarrow
0$, the result is identical with the solution for the cloak shown in Eq.
(LABEL:potential2_ideal) except the different constant term which in Eq.
(LABEL:potential2_ideal) is
$\tilde{q}[\varepsilon_{1}/(\varepsilon_{0}f(a))-1/a]$ and which in the limit
of Eq. (19) is the $n=0$ term $\tilde{q}\varepsilon_{0}/(\varepsilon_{2}a)$,
while they both tend to infinity indeed. Actually, for spherical interface,
only the radial component $\varepsilon_{\langle rr\rangle}$ has been used in
the continuity of the normal component of $\vec{D}$. Thus Eq. (19) with
$\varepsilon_{2}=0$ and the ideal cloak, whose radial component
$\varepsilon_{\langle rr\rangle}$ tends to zero as $r\rightarrow a$, give the
same result of field inside the hidden area.
For the system of ideal cloak, the electric field in the hidden area is
finite, despite the potential in the whole hidden area is infinite. While the
field in the cloak shell tends to infinity as $r\rightarrow a$. In addition,
because the material of cloak is linear, the electrostatic energy of the
system can be calculated by Jackson (1999)
$W\ =\
\frac{1}{2}\int\psi^{\mathrm{int}}(\vec{r}\,^{\prime})\rho(\vec{r}\,^{\prime})\
dV^{\prime}.$ (20)
According to the infinite constant inside the cloak, the electrostatic energy
of the system tends to infinity for a nonzero total charges even if the term
of self-energy is not included. The result reveals it would cost an infinite
work to put charged bodies into the cloak, which is impossible in practice.
However, if the total charges are zero, the infinite constant of potential
tends to zero. The trouble of infinite work would disappear either.
Figure 2: Electric field of two point charges with opposite quantity located
at a distance $a/2$ from origin on positive and negative semiaxis of $z$ (a)
in free space, (b) inside an ideal cloak, and (c) inside a spherical
conducting shell.
Figure 2 shows an example of two point charges with opposite quantity located
on positive and negative semiaxis of $z$ with a same distance $a/2$ from the
origin respectively. When the two point charges are inside the ideal cloak,
all electric lines of force do not intersect with the inner surface of the
cloak, while the equipotential surface are orthogonal to the inner surface
(Fig. 2b). In contrast, as in the spherical conducting shell, all electric
lines of force are orthogonal to the inner surface which itself becomes an
equipotential surface (Fig. 2c).
## IV surface voltage and equivalent surface magnetic current
When checking the solution for an ideal cloak in Eqs. (16), one can find that
the potential is discontinuous across the inner interface of the cloak. For an
arbitrary distribution of charges, the potential difference across the
interface $r=a$ can be calculated by $\Delta\psi=\int\Delta
G(a\hat{r},\vec{r}\,^{\prime})\rho(r)\,^{\prime}\,dV^{\prime}$, where $\Delta
G(a\hat{r},\vec{r}\,^{\prime})$ is the difference of Green’s function across
the interface:
$\begin{split}\Delta
G(a\hat{r},\vec{r}\,^{\prime})=&\frac{-1}{4\pi\varepsilon_{1}}\left\\{\frac{1}{\sqrt{a^{2}-2r^{\prime}a\cos\vartheta+r^{\prime
2}}}-\frac{1}{a}\right.\\\
&\left.+\sum_{n=1}^{\infty}\frac{n+1}{n}\frac{r^{\prime
n}}{a^{n+1}}P_{n}(\cos\vartheta)\right\\}.\end{split}$ (21)
This discontinuity of potential comes from the terms
$B^{\mathrm{cl}}_{n}f(r)^{-(n+1)}$ in $\psi^{\mathrm{cl}}$. For the ideal case
$f(a)\rightarrow 0$, the coefficients $B^{\mathrm{cl}}_{n}\rightarrow 0$, so
the terms of $f(r)^{-(n+1)}$ should be vanish in the cloak layer. While a
meticulous calculation reveals that the limit
$B^{\mathrm{cl}}_{n}f(a)^{-(n+1)}$ is towards to a finite quantity which acts
as a surface voltage to balance the potential difference $\Delta\psi$ across
the interface. The discontinuity also appears on the tangent component of
electric fields. As we have pointed out in Fig. 2, the electric fields are
tangent to the interface $r=a$ on the inner side, yet are zero on the cloak
side. Similarly, in the case of a cylindrical cloak with a transverse-electric
(TE) incident wave, the tangent component of electric field is also
discontinuous at the inner interface of the cloak Zhang _et al._ (2007).
However, the discontinuities are caused by different reasons in the two cases.
In cylindrical cloak, the tangent component of magnetic field
$B_{\langle\phi\rangle}\rightarrow\infty$ at the inner surface $r=a$ of cloak,
and the integral $\frac{d}{dt}\int_{a-0}^{a+0}B_{\langle\phi\rangle}dr$ is
equal to a finite value which acts as the surface magnetic displacement
current to balance the difference of $E_{\langle z\rangle}$ in the Maxwell’s
Eq. $\oint\vec{E}\cdot d\vec{l}=-\frac{d}{dt}\int\vec{B}\cdot d\vec{s}$ for an
infinitesimal contour across the interface Zhang _et al._ (2007). While, for
the ideal case of the shielding phenomenon, the normal component of electric
field $E_{\langle r\rangle}$ becomes a delta function compressed on the
interface and contributes to the surface voltage
$\Delta\psi=-\int_{a-0}^{a+0}E_{\langle r\rangle}dr$ Zhang _et al._ (2008).
In the sense of dielectric materials, the surface voltage is caused by the
infinite electric polarization of the material with $\varepsilon_{\langle
rr\rangle}=0$ on the interface Zhang _et al._ (2008). However the surface
voltage and the shielding effect of charges can also be caused by the the
surface magnetic current in the dual superconductor. As we all known, the
Meissner effect says that a superconductor expels magnetic fields from its
interior and therefore the equivalent permeability of superconductor is equal
to zero. Similarly, in dual superconductor model, magnetic monopoles are
supposed to exist and the roles of electric and magnetic fields are
interchanged. Moreover, the dual Meissner effect tries to expel electric
fields out of the dual superconductors and gives a zero permittivity
equivalently Ripka (2005). In this case, the discontinuity of the tangent
electric fields across the surface of the dual superconductor comes from the
surface magnetic current $\vec{\alpha}$, which satisfies the boundary
condition
$\vec{\alpha}\ =\ \left.\vec{E}^{\mathrm{int}}\times\hat{e}_{r}\right|_{r=a}.$
(22)
Figure 3: Distributions of (a) surface voltage $\Delta\psi$ and (b) surface
magnetic current $\vec{\alpha}$ (only $\phi$ component exists) on the inner
surface of the cloak for the charge distribution shown in Fig. 2 with
normalized $\tilde{q}$.
Figure 4: Electric field of two point charges with opposite quantity located
at a distance $a/2$ from origin on $z$ axis and $x$ axis respectively (a) in
free space, (b) inside an ideal cloak; and the distributions of (c) surface
voltage $\Delta\psi$ and (d) surface magnetic current $\vec{\alpha}$ on the
inner surface for the charge distribution with normalized $\tilde{q}$
As a result, the effect of electrostatic shielding of the ideal cloak is very
like the charge confinement by the dual superconductors, so the inner surface
of the cloak can be also interpreted as a dual superconductor layer. Although
the medium of cloak is anisotropic, which is different from the simple dual
superconductor, we still can construct the ideal electrostatic cloak with dual
superconductive components as same as the way to construct magnetostatic cloak
using superconductors suggested by B. Wood and J. Pendry Wood and Pendry
(2007). Figure 3 shows the curves of the surface voltage and the equivalent
surface magnetic current varying with $\theta$ for the charge distribution
shown in Fig. 2. Since the distribution is symmetric about $z$ axis, the
voltage is also symmetric, and the surface current is along $\phi$ direction.
Figure 4 presents another example of charge distribution (Fig. 4a,b), Fig. 4c
shows the voltage distributing on the inner surface, and Fig. 4d shows the
surface magnetic current on the inner surface.
## V unideal cloak
In previous section, we have mentioned that the electric energy tends to
infinite for the case of nonzero charges inside an ideal cloak, and it is
impossible in practice. To ease this embarrassment, we would take into account
the case of unideal cloak which is also obtained from the radial
transformation $f(r)$. We still let $f(r)$ satisfy $f(b)=b$, however its zero
point is not at $r=a$ but at $a^{\prime}=(1-\delta)a$, when $\delta\rightarrow
0$ the cloak approaches to an ideal one. In this case, the potential inside
the cloak no longer has the infinite constant term, and is continuous across
the inner interface. A further calculation reveals that the unideal cloaks
still have good property of electrostatic shielding both when the total
charges is zero and nonzero. Figure 5 shows the case of unideal cloak designed
by the linear transformation $f(r)=b(r-a^{\prime})/(b-a^{\prime})$ with
$\delta=0.2$. For the case of a point charge located inside the cloak but not
at the center, the fields outside the cloak still retain a highly spherical
symmetry just as the fields generated by the point charge located at the
center of the cloak, as shown in Fig. 5a. For the case of two equal and
opposite charges located inside the cloak, the fields are mainly concentrated
inside the cloak, while the fields leaking out is extremely few, as shown in
Fig. 5b.
Figure 5: Electric field of (a) a point charge located on positive semiaxis of
$z$ with a distance $a/2$ from origin, (b) two equal and opposite point
charges located as same as in Fig. 2 for an unideal cloak constructed by the
linear transformation function with $\delta=0.2$. Figure 6: (a) Linear
transformation $f(r)$ and (b) the corresponding radial component of
permittivity $\varepsilon_{\langle rr\rangle}$ varying with $r$ in the cloak
region under different value of $\delta$ from 0 to 1. Figure 7: The absolute
value of electric flux $|\Psi|$ divided by $\Psi_{0}$ flowing out of the
hemispherical surface split by $z=0$ plane varying with $\delta$.
To measure the effect of shielding accurately, we consider the electric flux
propagating out of the cloak. Concerning the system of two equal and opposite
charges inside the cloak as shown in Fig. 2b, Gauss’ theorem tells the total
flux out of the cloak is zero. However, if we calculate the flux out of two
hemispherical surfaces of a sphere $r=r_{0}>b$ split by $z=0$ plane
respectively, the pair of flux must have an equal magnitude $|\Psi|$ but
opposite sign because of the symmetry of charge distribution. Thus the
absolute value of electric flux out of each hemisphere would be a suitable
quantity to measure the effect of shielding for this system. We still consider
the linear radial transformation $f(r)$ with different $\delta$ (Fig. 6a) and
corresponding permittivities (Fig. 6b). In this situation, $|\Psi|/\Psi_{0}$
varying with $\delta$ is shown in Fig. 7, where $|\Psi|$ is the absolute value
of flux flowing out of the hemisphere and $\Psi_{0}$ is the absolute value of
flux out of the hemisphere with no cloak existing. The slope of
$|\Psi|/\Psi_{0}$ varying with $\delta$ is equal to zero as $\delta\rightarrow
0$. The result manifests that a nearly ideal cloak would still realize a good
effect of shielding. In fact, the derivative of $\varepsilon_{\langle
rr\rangle}$ of ideal cloak with respect to $r$ is equal to zero for arbitrary
$f(r)$, therefore a nearly ideal cloak with small value of $\delta$ would
always have a good behavior of shielding. The effect of shielding can be
measured from another aspect of energy density
$\frac{1}{2}\vec{E}\cdot\vec{D}$, as shown in Fig. 8. We can see that the
energy density outside a nearly ideal cloak tends to zero when $\delta$ is
very small and is much smaller than the case of no cloak existing. Along with
the increase of $\delta$, the energy density increases nearly stable and goes
to the limit of no cloak existing when $\delta\rightarrow 1$. To sum up, the
slight breaking of the perfection would not change the property of
electrostatic shielding for the cloak significantly.
Figure 8: Electric field energy density varying on $z$ axis under different
value of $\delta$, where the red line denotes the case of no cloak existing
($\delta=1$).
## VI conclusion
To summarize, we have verified that electrostatic spherical cloak takes the
same form of permittivity as the invisibility cloak under nonzero frequency,
and demonstrated its behavior of electrostatic shielding is identical with a
spherical conducting shell in the screened region. If the electrostatic
sources are outside the cloak, the field can not propagate into the screened
region inside the cloak, on the other hand, if the charges with arbitrary
distribution are inside the cloak, the field outside the cloak is just as the
field generated by a point charge located at the center of the cloak, so the
only information which can be detected is the total charges inside the cloak.
For ideal case, the potential across the inner interface of the cloak is not
continuous in that $\varepsilon_{\langle rr\rangle}=0$ causes the infinite
polarization on the inner interface. However, $\varepsilon=0$ can be also
interpreted as the property of a dual superconductor, and the behavior of
shielding are also very like the property of charge confinement caused by dual
Meissner effect. Another problem existing in ideal case is the infinite field
energy when the total charges inside the cloak is not zero. Nevertheless, the
problem no longer exists for unideal case, in addition the nearly ideal cloak
also have a good effect of electrostatic shielding.
###### Acknowledgements.
This work was supported in part by NSF of China (Grants No. 11075077).
## References
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* Cummer _et al._ (2006) S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith, and J. Pendry, Phys. Rev. E 74, 036621 (2006).
* Chen _et al._ (2007) H. Chen, B.-I. Wu, B. Zhang, and J. A. Kong, Phys. Rev. Lett. 99, 063903 (2007).
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* Ripka (2005) G. Ripka, _Dual Superconductor Models of Color Confinement_ (Springer-Verlag, 2005).
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|
arxiv-papers
| 2011-07-14T16:58:18 |
2024-09-04T02:49:20.550379
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruo-Yang Zhang, Qing Zhao, and Mo-Lin Ge",
"submitter": "Ruo-Yang Zhang",
"url": "https://arxiv.org/abs/1107.2868"
}
|
1107.2896
|
# Destruction of valence-bond order in a $S=1/2$ sawtooth chain with a
Dzyaloshinskii-Moriya term
Zhihao Hao Yuan Wan Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, Maryland 21218, USA Ioannis Rousochatzakis Max-
Planck-Institut für Physik komplexer Systeme, Nöhtnitzer Straße 38, D-01187
Dresden, Germany Julia Wildeboer A. Seidel Department of Physics and Center
for Materials Innovation, Washington University, St. Louis, Missouri 63136,
USA F. Mila Institute of Theoretical Physics, École Polytechnique Fédérale
de Lausanne, CH-1015 Lausanne, Switzerland O. Tchernyshyov Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218,
USA
###### Abstract
A small value of the spin gap in quantum antiferromagnets with strong
frustration makes them susceptible to nominally small deviations from the
ideal Heisenberg model. One of such perturbations, the anisotropic
Dzyaloshinskii-Moriya interaction, is an important perturbation for the
$S=1/2$ kagome antiferromagnet, one of the current candidates for a quantum-
disordered ground state. We study the influence of the DM term in a related
one-dimensional system, the sawtooth chain that has valence-bond order in its
ground state. Through a combination of analytical and numerical methods, we
show that a relatively weak DM coupling, $0.115J$, is sufficient to destroy
the valence-bond order, close the spin gap, and turn the system into a
Luttinger liquid with algebraic spin correlations. A similar mechanism may be
at work in the kagome antiferromagnet.
## I Introduction
Antiferromagnets with $S=1/2$ and on non-bipartite lattices are considered
viable candidates for exotic ground states and excitations. Geometrical
frustration and strong quantum fluctuations tend to suppress long-range
magnetic order. The resulting ground state does not break the symmetry of
global spin rotations, but its exact properties remain subject of vigorous
debate, with proposals ranging from valence-bond crystals that break some
lattice symmetriesMarston and Zeng (1991); Nikolić and Senthil (2005);
Hastings (2000) to valence-bond liquids that fully preserve the symmetry of
the Hamiltonian.Ryu et al. (2007); Ran et al. (2007); Hermele et al. (2008);
Seidel (2009) A spin-liquid state with an energy gap to all excitations may
further possess a hidden topological order. Several antiferromagnetic
materials without long-range magnetic order well below the characteristic
Curie-Weiss temperature scale have been discovered recently, most notably
herbertsmithite Cu3Zn(OH)6Cl2,Braithwaite et al. (2004) where no magnetic
order has been detected down to 50 mK,Helton et al. (2007); de Vries et al.
(2008); Imai et al. (2008); Ofer et al. (unpublished); Olariu et al. (2008)
even though the exchange interaction is estimated to be $J=180$ K. The
material is a “structurally perfect”Shores et al. (2005); Helton et al. (2007)
realization of the $S=1/2$ Heisenberg antiferromagnet on kagome, a network of
corner-sharing triangles, Fig. 1(a).
Figure 1: (a) Kagome lattice. (b) and (c) In-plane and out-of-plane components
of the DM vector $\mathbf{D}_{ij}$ shown for directed links $(i\to j)$ on
kagome.
While most of the theoretical studies of quantum antiferromagnets deal with
the pure Heisenberg model with nearest-neighbor exchange, real systems
inevitably deviate from this idealization. Frustrated magnets in particular
are sensitive to various nominally weak perturbations. In this paper, we deal
with the Dzyaloshinskii-Moryia (DM) interaction,Dzyaloshinsky (1958); Moriya
(1960) the antisymmetric version of the Heisenberg exchange induced by the
spin-orbit coupling. The Hamiltonian of such a system is
$H=\sum_{\langle
ij\rangle}[J\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})].$
(1)
In herbertsmithite, the DM term is allowed by the crystal symmetry. The in-
plane and out-of-plane components of the DM vector $\mathbf{D}_{ij}$ on kagome
are shown in Fig. 1(b) and (c). From eSR measurements,Zorko et al. (2008) the
DM vector has the magnitude $D=0.08J$ and is dominated by the out-of-plane
component, whereas the in-plane component is small, $D_{\mathrm{in}}=0.01J\pm
0.02J$. The DM term can be gauged away by an appropriate rotation of the local
spin axes,Perk and Capel (1976); Shekhtman et al. (1992) provided that its
“line integral” vanishes for any closed loop $abc\ldots yza$:
$\mathbf{D}_{ab}+\mathbf{D}_{bc}+\ldots+\mathbf{D}_{yz}+\mathbf{D}_{za}=0.$
(2)
It can be seen from Fig. 1(b) that the in-plane component satisfies Eq. (2)
and thus can be gauged away. The out-of-plane component cannot be removed in
this way and thus represents a physical perturbation. In this work, we
concentrate on the out-of-plane component of $\mathbf{D}$.
A growing evidence from numerical studiesCh. Waldtmann et al. (1998); Singh
and Huse (2007); Evenbly and Vidal (2010); Jiang et al. (2008); Yan et al.
(2011) indicates that the pure Heisenberg model, $D=0$, has a $S=0$ ground
state with a small but finite energy gap for $S=1$ excitations, with estimates
ranging from $\Delta=0.05J$ to $0.15J$. These values are comparable to the
strength of the DM term, so it is plausible that the low-energy properties of
herbertsmithite are influenced by the DM interaction.
The effects of the DM interaction on the kagome antiferromagnet were first
studied by Rigol and SinghRigol and Singh (2007a, b) in order to explain low-
temperature paramagnetism in herbertsmithite: an upturn in magnetic
susceptibility at low temperaturesHelton et al. (2010) seems to indicate the
absence of a spin gap. Tovar et al.Tovar et al. (2009) concluded that a finite
DM term could be responsible for the non-zero susceptibility observed in
experiment even if the spin gap remains finite. A study employing exact
diagonalization Cépas et al. (2008) showed that a sufficiently strong DM term,
$D>D_{c}\approx 0.10J$, induces long-range magnetic order in the ground state,
with magnetic moments lying in the plane. This was later confirmed by
employing the Schwinger-boson approach.Messio et al. (2010); Huh et al. (2010)
The ordering tendency is easy to understand by turning to the classical
variant of the Heisenberg model. There, the out-of-plane $\mathrm{D}$ vectors
shown in Fig. 1(c) lift the extensive degeneracy of the classical ground
states leaving a $\mathbf{q}=0$ ground state that spontaneously breaks the
remaining O(2) symmetry of the DM Hamiltonian (1). Later numerical
workRousochatzakis et al. (2009) turned up some evidence that the system may
have an intermediate phase between $D_{c1}\approx 0.05J$ and $D_{c2}\approx
0.10J$, where $S_{z}=1$ excitations become gapless but the spin O(2) symmetry
remains intact. In the absence of an obvious order parameter that would
uniquely identify the intermediate phase, the authors of Ref. Rousochatzakis
et al., 2009 concluded that the appearance of an intermediate phase might be a
finite-size effect. Further work in this direction is required to elucidate
the nature—and even the existence—of the intermediate phase and its possible
relevance to herbertsmithite.
In our previous work,Hao and Tchernyshyov (2009) we have shown that the
$S=1/2$ Heisenberg antiferromagnet on kagome can be viewed as a collection of
fermionic spinons—topological defects with $S=1/2$—moving in an otherwise
inert vacuum of valence bonds. The spinons interact with an emerging compact
U(1) gauge field whose quantized electric flux is related to the valence-bond
configuration through Elser’s arrow representation.Elser and Zeng (1993)
Spinons carry one unit of the U(1) charge against a negatively charged
background. These features are reminiscent of the picture of fermionic spinons
proposed earlier by Marston et al.Marston and Zeng (1991); Ma and Marston
(2008) and HastingsHastings (2000), who used the Abrikosov-fermion
representation for spin operators. It is worth pointing out that the Fermi
statistics of spinons is not postulated ad hoc in our approach but rather
arises naturally as the Berry phase of valence bonds that are moved in th e
process of spinon exchange. We have further shown that strong, exchange-
mediated attraction binds spinons into small and heavy $S=0$ pairs and that
low-energy $S=1$ excitations result from breaking up a pair into “free”
spinons. Thus the spin gap is determined mostly by the binding energy of a
pair, which we estimated to be $0.06J$.
From this perspective, one potential route to the closing of the spin gap
could be via the destruction of the two-spinon bound state in the presence of
a sufficiently strong DM term. That, however, appears unlikely for two
reasons. First, the factors setting the pair binding energy—the spinon hopping
amplitude and the strength of exchange-mediated attraction–are both of order
$J$, so it is hard to see how a fairly weak coupling $D=0.05J$ to $0.10J$ can
disrupt the pairing. Second, a quantum phase transition to a state with long-
range magnetic order can be viewed as Bose condensation of magnons,Giamarchi
et al. (2008) quasiparticles with $S_{z}=1$ and there are no low-energy
excitations of this kind in the pure Heisenberg model. Although one could
think of condensing pairs of spinons with $S_{z}=1$, this route runs into
another difficulty: such an object would carry a double U(1) charge, whereas a
magnon is expected to be neutral. Put simply, a pair of spinons is a
topological defect whose motion affects the valence-bond background, which is
uncharacteristic of magnon motion.
A possible way out is to postulate that the condensing objects are pairs
consisting of a spinon and its antiparticle. Such a composite object would
have zero U(1) charge and be topologically trivial, like a magnon. In the pure
Heisenberg model, the energy cost of creating a spinon and its antiparticle is
approximately $0.25J$.Singh (2010) As we will see, the DM term lowers the
kinetic energy of both spinons and their antiparticles. It is thus reasonable
to expect that, at some critical coupling strength $D_{c}$, the energy cost of
adding a pair vanishes.
To test this scenario, we have studied a toy version of the kagome
antiferromagnet known as the sawtooth spin chain,Nakamura and Kubo (1996); Sen
et al. (1996) a one-dimensional lattice of corner-sharing triangles, Fig.
2(a). To make a connection with kagome, exchange couplings are set equal for
all bonds. At $D=0$, the chain has two valence-bond ground states, Fig. 2(b)
and (c), that spontaneously break the mirror reflection symmetry. Spin
excitations are topological defects: domain walls with spin $S=1/2$, Fig.
2(d). The domain walls come in two flavors: kinks have zero energy and are
localized, whereas antikinks are mobile and have a minimum energy of
$0.215J$.Nakamura and Kubo (1996) These excitations can only be created in
pairs by a local perturbation acting in the bulk. As we discussed
elsewhere,Hao and Tchernyshyov (2010) spinons of the kagome antiferromagnet
have similar properties, with one notable exception: the ground state of the
sawtooth chain is free from the defects, whereas kagome has a finit e
concentration of antikinks (1/3 per site) bound into $S=0$ pairs.
Figure 2: (a) The sawtooth chain. (b) and (c) Its valence-bond ground states.
(d) Spin-1/2 excitations: kink (left) and antikink (right). (e) Orientation of
the DM vectors $\mathbf{D}_{ij}$. (f) The ground state of the classical model
has a commensurate magnetic order with the wavenumber $q/2\pi=-1/3$.
We have studied the sawtooth spin chain with exchange and Dzyaloshinskii-
Moriya interactions, Eq. (1). The $\mathbf{D}_{ij}$ vectors had the same
length and a uniform out-of-plane orientation preserving the translational
symmetry of the chain as shown in Fig. 2(e). Qualitatively similar results
were obtained for the staggered choice of $\mathbf{D}_{ij}$, but we will not
provide the details here. The introduction of the DM term preserves the mirror
symmetry of the Hamiltonian (it inverts the $x$ coordinate of the lattice and
the $S_{y}$ and $S_{z}$ components of the spins), so that the notion of a
valence-bond order that spontaneously breaks this symmetry is still valid. The
valence-bond order survives to a finite value of the DM coupling.
As described below, kinks become mobile in the presence of a DM term. Their
minimal energy becomes negative, growing linearly with $D$. The minimal energy
of an antikink remains unchanged to the first order in $D$, so one can expect
that the minimum energy of a kink-antikink pair will vanish when $D$ reaches a
critical value $D_{c}$ of the order of the initial spin gap, $0.215J$. In Sec.
II, we describe a calculation of the spinon spectrum in the presence of a
nonzero $D$, from which we obtained an estimate of the critical DM strength,
$D_{c}=0.087J$. For $D>D_{c}$, spontaneous creation of kink-antikink pairs
leads to a finite concentration of topological defects, which obliterates the
valence-bond order and restores the reflection symmetry of the lattice. This
scenario is reminiscent of quantum phase transition at the end of
magnetization plateaus in the $S=1/2$ Ising-Heisenberg chainFowler and Puga
(1978) and in a frustrated two-leg ladder.Fouet et al. (2006). In both of
those models, the condensation of domain walls turns a state with a broken
translational symmetry and gapped excitations into a gapless phase with
incommensurate spin correlations decaying as a power of the distance. Exact
diagonalization calculation for the sawtooth chain with DM interactions,
described in Sec. III, are consistent with this scenario.
## II Spinon dispersions
### II.1 $D=0$
We briefly review the physics of the sawtooth chain in the pure Heisenberg
model without the DM term.Nakamura and Kubo (1996); Sen et al. (1996); Hao and
Tchernyshyov (2010) The Hamiltonian of the system is
$H=J\sum_{\langle
ij\rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j}=\frac{J}{2}\sum_{\Delta}\left(\mathbf{S}_{\Delta}^{2}-9/4\right),$
(3)
where the $\mathbf{S}_{\Delta}$ is the total spin of triangle $\Delta$. The
energy is minimized when $S_{\Delta}=1/2$ for every triangle, which can be
achieved by putting a singlet bond on every triangle. The ground state is
doubly degenerate. The two ground states shown in Fig. 2(b) and (c) violate
the symmetry of reflection.
Two types of domain walls interpolate between the ground states: the kink and
the antikink, Fig.2(d). A kink is an excitation with zero energy that happens
to be an exact eigenstate of the Hamiltonian (3). Thus kinks are localized in
the exchange-only model. The localized nature of kinks can be traced to an
accidental degeneracy of the ground state of the exchange Hamiltonian on a
triangle with half-integer spins in addition to the two-fold Kramers
degeneracy. The two degenerate states with Sz=1/2 have spin current going
clockwise or counter clockwise around the triangle. The states also carry
electric currents of opposite directions.Bulaevskii et al. (2008) An
alternative set of basis states would have distinct valence-bond averages
$\langle\mathbf{S}_{i}\cdot\mathbf{S}_{j}\rangle$ on the three bonds, which
translates to nonzero electric charge on the three sites.Bulaevskii et al.
(2008)
In contrast, an antikink is mobile. The motion of an antikink is accompanied
by the emission and absorption of kink-antikink pairs. The existence of a
finite spin gap guarantees that these excitations are virtual. Polarization
effects can be taken into account by using a variational approach. At the
crudest level, the Hamiltonian (3) is projected onto the Hilbert space with a
single antikink to obtain an effective hopping Hamiltonian for an antikink:
$H^{(1)}|x\rangle=\frac{5J}{4}|x\rangle-\frac{J}{2}|x+1\rangle-\frac{J}{2}|x-1\rangle.$
(4)
where $|x\rangle$ is a state with an antikink on triangle $x$. The energy
dispersion of the antikink is
$E_{a}(k)=5J/4-J\cos{k},$ (5)
with the minimum energy $\Delta=0.25J$. In view of the zero energy of a kink,
this value is the spin gap.
This estimate can be further improved by enlarging the Hilbert space to
include virtual excitations in the immediate neighborhood of an antikink. This
yields an improved estimate of the spin gap, $\Delta=0.219J$,Hao and
Tchernyshyov (2010) which is quite close to the result obtained by exact
diagonalization, $\Delta=0.215J$.Nakamura and Kubo (1996)
It seems clear from the above that the variational approach provides a
reliable description of the low-energy spin excitations in the pure Heisenberg
model. We will use the lowest-order approximation for $D\neq 0$, without
correcting for the vacuum polarization, to obtain a rough estimate for the
critical coupling $D_{c}$.
### II.2 $D\neq 0$
In the presence of a nonzero DM term, kinks become mobile. For a single
triangle, this means the splitting of the accidental degeneracy mentioned
previously: the energy of a state with $S_{z}=+1/2$ now depends on the orbital
momentum, reflecting the spin-orbit origin of the DM term.
For an infinite chain, we follow the variational method described above and
work in the Hilbert space spanned by states $|x\rangle$ with a single kink
located between triangles $x$ and $x+1$. These states are not orthogonal to
each other because they are not eigenstates of the same Hermitian operator.
The overlap is
$\langle x_{1}|x_{2}\rangle=2^{-|x_{1}-x_{2}|}.$ (6)
As with antikinks,Hao and Tchernyshyov (2010) a simple rotation can be made to
obtain an orthonormal basis $\\{|\tilde{x}\rangle\\}$:
$|\tilde{x}\rangle=\frac{2}{\sqrt{3}}|x\rangle-\frac{2}{\sqrt{3}}|x-1\rangle.$
(7)
The matrix elements of the effective Hamiltonian in this subspace are
$\langle\tilde{x}_{1}|H|\tilde{x}_{2}\rangle=-\frac{3iD}{2}\,2^{-|x_{1}-x_{2}|}\,\mathrm{sgn}(x_{1}-x_{2}),$
(8)
where the sign function is defined in such a way that $\mathrm{sgn}(0)=0$. A
Fourier transform of the matrix element yields the energy dispersion of the
kink:
$E_{\mathrm{k}}(k)=\frac{6D\sin{k}}{5-4\cos{k}}.$ (9)
The bottom of the band is at $E_{\mathrm{k}}^{\mathrm{min}}=-2|D|$. For $D>0$,
it is reached for an incommensurate wavenumber
$k/2\pi=-\mathrm{acos}{(4/5)}/2\pi\approx-0.10$.
The calculation of the antikink case proceeds in a similar way. The basis
states $\\{|x\rangle\\}$, with an antikink located at triangle $x$, can be
orthogonalized in the same way to yield an orthonormal basis
$\\{|\tilde{x}\rangle\\}$. The matrix element of the DM term is
$\langle\tilde{x}_{1}|H_{DM}|\tilde{x}_{2}\rangle=-iD\,2^{-|x_{1}-x_{2}|}\,\mathrm{sgn}(x_{1}-x_{2})\left[\frac{3}{2}-\frac{2}{3}(\delta_{x_{1},x_{2}+1}+\delta_{x_{1},x_{2}-1})\right].$
(10)
The resulting antikink dispersion is
$E_{\mathrm{a}}(k)=5J/4-J\cos{k}+\frac{5D}{6}\sin{k}+\frac{3D(4\cos{k}-1)\sin{k}}{10-8\cos{k}}.$
(11)
For $D\ll J$, the lowest energy of an antkink
$E_{\mathrm{a}}^{\mathrm{min}}=J/4-14D^{2}/J+\mathcal{O}(D^{4}/J^{3})$. The
bottom of the band is located at $k/2\pi=-8D/3\pi J+\mathcal{O}(D^{3}/J^{2})$.
The above energy dispersions were computed for spinons with $S_{z}=+1/2$. The
dispersions for $S_{z}=-1/2$ can be obtained by changing $k\mapsto-k$.
The bottom edge of the two-particle continuum as a function of total momentum
is shown as solid lines in Fig. 3 for $S_{z}=0$ and in Fig. 4 for $S_{z}=+1$.
(The former is a combination of two continua, one for a kink with $S_{z}=+1/2$
and an antikink with $S_{z}=-1/2$, the other for a kink with $S_{z}=-1/2$ and
an antikink with $S_{z}=+1/2$.) The edge dispersion mostly tracks the
dispersion of the heavier particle, in this case the kink (9). The minimum
energy of a kink-antikink pair
$E^{\mathrm{min}}=J/4-2|D|-14D^{2}/J+\mathcal{O}(D^{4}/J^{3})$ (12)
vanishes when the DM coupling reaches the critical strength $D_{c}=0.09J$. The
total momentum of a $S_{z}=+1$ spinon pair with the lowest energy is
$k/2\pi\approx-0.15$. The gapless state arising at this critical point is
expected to have transverse spin fluctuations with this wavenumber. The
wavenumber of longitudinal spin fluctuations is determined by the bottom of
the two-spinon continuum with $S_{z}=0$, which occurs at $k/2\pi\approx\pm
0.06$.
## III Exact diagonalization
To test the theory, we have performed an exact diagonalization study of the
sawtooth chain with exchange and DM interactions. We worked with finite chains
containing $2L$ sites in a system with $L$ triangles with periodic boundary
conditions. The length varied from $L=5$ to 15. Both uniform and staggered DM
interactions were investigated, with qualitatively similar results. Here we
report on the uniform case only. For the largest system sizes, we employed the
Lanczos algorithm, which provides convergent results for the ground state
energy and a limited number of low-lying excitations. To reduce the size of
the Hilbert space, we used the symmetry of translations along the chain and
the O(2) symmetry of spin rotations around the z-axis.
Figure 3: Low-energy spectra of the sawtooth chain with a uniform DM term in
the $S_{z}=0$ sector. Energy levels, measured relative to the ground state,
are shown as a function of total momentum. Circles are the results of exact
diagonalization for a periodic chain of length $L=15$. Solid curves show the
bottoms of the two-spinon continua computed analytically. Dashed straight
lines show a linear dispersion with the speed $v=0.36J$.
Figure 3 shows the low-energy portions of the spectra in the $S_{z}=0$ sector
for a chain with length $L=15$ (30 sites), for several values of the DM
coupling $D$. The invariance of the Hamiltonian (1) under time reversal
symmetry ($S_{z}\mapsto-S_{z}$, $k\to-k$) guarantees that the $S_{z}=0$
spectra are symmetric under mirror reflection ($k\to-k$). The lowest-energy
excitations in the $S_{z}=0$ sector are expected to be spinon pairs in two
channels: a kink with $S_{z}=-1/2$ and an antikink with $S_{z}=+1/2$ or vice
versa. The calculated edges of the two-particle continua reproduce the shape
of the dispersing bottom reasonably well. However, the calculated edge shifts
downward with $D$ faster than the numerical data do.
Figure 4: Low-energy spectra in the $S_{z}=+1$ sector. Notations are the same
as in Fig. 3.
In the $S_{z}=+1$ sector, the spectra are not symmetric under the mirror
symmetry (the $S_{z}=1$ spectrum maps onto that of the $S_{z}=-1$ sector),
Fig. 4. The lowest-energy excitations are expected to be spinon pairs
consisting of a kink and an antikink, both with $S_{z}=+1/2$. Again, the
calculated bottom edge of the excitation continuum has the right shape but
advances downward with $D$ somewhat too fast. In the two-spinon approximation,
both the $S_{z}=0$ and $S_{z}=1$ continua touch zero energy at $D_{c}=0.09J$.
However, the numerical energy spectra appear to still have a gap at that
point, see Fig. 3.
Figure 5: The splitting of the ground-state doublet as a function of the
system length $L$ for (a) $D<D_{c}=0.115J$ and (b) for $D>D_{c}$. (c) The
dependence of the inverse tunneling length $1/\xi$ and the wavenumber $k$ in
the scaling form (13) on the DM coupling strength $D$.
To locate the critical point, we turned to a scaling analysis of the ground-
state splitting. In the phase with valence-bond order, the ground state is
doubly degenerate in the limit $L\to\infty$. In finite systems, the ground-
state doublet is split thanks to quantum tunneling. Both members of the
doublet have momentum $k=0$ because the valence-bond order preserves
translational symmetry. The tunneling amplitude decays exponentially with the
system length $L$ and so does the splitting.
Fig. 5(a) shows the splitting of the ground state for $D\leq 0.11J$. All of
the data sets, with the exception of the largest coupling, are well fit by the
scaling expression
$\Delta E=AL^{-5/4}e^{-L/\xi}\cos{(kL)}$ (13)
with the same prefactor $A$. The dependence of the tunneling length $\xi$ and
the wavenumber $k$ is shown in Fig. 5(c). The tunneling length diverges, or at
least greatly exceeds the maximum attainable system length $L=15$, for
$D>D_{c}=0.115J$. For $0.11J\leq D\leq 0.15J$, the finite-size dependence of
the splitting was best fit by Eq. (13) with $\xi=\infty$ and a $D$-dependent
amplitude $A$, Fig. 5(b). Apart from the oscillating factor, Eq. (13) suggests
a scale-invariant ground state for $D\geq D_{c}$. The oscillations presumably
come from the interference of instantons as discussed in the Appendix.
For $D>D_{c}$, we expect a gapless phase with quasi-long-range incommensurate
spin correlations decaying as a power of the distance. For a sufficiently
large $D$, the classical model should become a good starting point. In the
classical limit, the sawtooth chain has a spiral order for any nonzero value
of $D$, Fig. 2(f). Low-energy excitations are spin waves with a speed
$v\approx 2.7S\sqrt{JD}.$ (14)
Quantum fluctuations disrupt the long-range spin order, restoring
translational invariance and the O(2) symmetry. Such a phase would be a
Luttinger liquid, whose lowest-energy $S_{z}=+1$ excitations are spin waves
with a sound-like spectrum at $k_{0}/2\pi=-1/3$. The numerically determined
$S_{z}=+1$ spectra for $D\geq 0.15J$ are consistent with spin waves. At
$D=0.19J$, the soft spot is located at $k_{0}/2\pi\approx-0.25$, not far from
the classical value. The speed of sound (estimated from the slope of the
dashed lines in Fig. 3 and 4) is $v=0.36J$, is not far from the classical
estimate (14) obtained below.
## IV Spin correlations in the ground state
To verify the location of the quantum critical point $D_{c}$ and to confirm
the critical nature of the ground state for $D>D_{c}$, we examined the long-
distance behavior of spin correlations, $G^{\alpha\beta}(r)=\langle
S^{\alpha}(0)S^{\beta}(r)\rangle$, in the ground state. In the Luttinger-
liquid regime, transverse spin correlations are expected to decay as a power
of the distance,Giamarchi (2004)
$|G^{+-}(r)|\sim\frac{C}{r^{1/2K}}.$ (15)
The stiffness constant $K$ varies between 1 (gas of dilute magnons) and 1/4
(gas of dilute spinons).Haldane (1980); Fouet et al. (2006)
In a finite system of length $L$ with periodic boundary conditions, the
Green’s function depends in the same way on the chord distance Haldane (1981)
$d(r)=(L/\pi)\sin{(\pi r/L)}.$ (16)
In a system with $2L$ spins, this distance varies from $d\approx 1$ to
$L/\pi$. In view of that, the range of distances in a system with $2L=30$
spins is not sufficient to reliably observe the critical behavior of the spin
correlation function.
Figure 6: The amplitude of transverse spin correlations (15) as a function of
the chord distance (16) on a log-log plot (left) and a simple log plot
(right).
To observe the critical behavior, we used the density-matrix renormalization
group (DMRG) method implemented through the Matrix Product ToolkitMcCulloch
(2007) to obtain the ground-state wavefunction in a periodic chain with up to
$2L=100$ spins. The system has a U(1) symmetry which we took into account to
reduce CPU time. The number $m$ of states kept varied from 800 to 1200 states.
Our results for the ground state energy per site for all values of DM coupling
$D$ investigated are consistent with the energy per site obtained from the ED
calculations.
The resulting transverse spin correlations $|G^{+-}(r)|$ in a system of length
$L=50$ are shown in Fig. 6 as a function of the chord distance (16). At
largest distances $d$, the data for $D=0.12J$ follow a power law $C/d^{2}$,
which is consistent with the value $K=1/4$ at the spinon condensation point.
For $D>0.12J$, spin correlations follow power laws with smaller slopes,
indicating $K>1/4$. For $D<0.12J$, the power-law scaling breaks down at large
$d$ changing to an exponential dependence. The estimated critical point,
$D_{c}=0.12J$, is in reasonable agreement with the value $D_{c}=0.115J$
obtained from the splitting of the ground-state doublet.
## V Discussion
Analytical arguments and numerical evidence presented above supports the
following scenario. In the absence of the Dzyaloshinskii-Moriya term, the
sawtooth chain has a doubly degenerate ground state with valence-bond order
spontaneously breaking the reflection symmetry of the lattice. Elementary
excitations are spinons of two flavors, localized kinks and mobile antikinks.
The gap to spin-1 excitations, $\Delta=0.215J$ is determined by the edge of
the two-spinon continuum. The introduction of a DM term with the $\mathbf{D}$
vector pointing along the same axis for all bonds, Fig. 2, lowers the spin-
rotation symmetry down to an O(2). At weak coupling $D$, the lattice
reflection symmetry remains spontaneously broken. At the same time, a finite
$D$ lowers the excitation energies of both kinks and antikinks and the spin
gap (understood as the lowest energy of $S_{z}=1$ excitations) begins to
close. A fairly crude analytical calculation indicates that the main factor
affecting the spin gap is the minimum energy of the kink, $-2|D|$. The gap
closes roughly when that energy equals the initial gap in absolute terms,
$|D|=D_{c}\approx\Delta/2\approx 0.1J$. This is confirmed by numerical work
involving exact diagonalization of finite chains, with the result
$D_{c}=0.115J$. Beyond the critical coupling, the spinons proliferate. Since
they act as domain walls in the valence-bond order parameter, the valence-bond
order is lost and the lattice symmetry is fully restored. The resulting state
is likely a Luttinger liquid with incommensurate spin correlations and spin-
wave excitations. Similar transitions between Ising-ordered phases and
Luttinger liquids have been found in other one-dimensional systems.Fowler and
Puga (1978); Fouet et al. (2006) The strength of the DM coupling
$D\approx(\delta g/g)J$ where $\delta g$ is the deviation of gyromagnetic
ratio from its free-electron value $g$.Moriya (1960) In kagome
antiferromagnets herbertsmithite and volborthite, $\delta g/g\approx
0.1$.Hiroi et al. (2009)
It is tempting to speculate that a somewhat similar transition may occur in
the $S=1/2$ Heisenberg model on kagome with a DM coupling. While the existence
of the transition is not in doubt—at a large enough $D$ the system should
develop magnetic orderCépas et al. (2008); Rousochatzakis et al. (2009);
Messio et al. (2010); Huh et al. (2010)—the nature of the transition remains
to be determined.
In the kagome antiferromagnet, spinon excitations are very similar to those of
the sawtooth chain.Hao and Tchernyshyov (2010) In the absence of the DM term,
kinks are localized and have zero energy, whereas antikinks follow one-
dimensional trajectories with the same energetics as on the sawtooth chain.
Adding the DM term thus has similar consequences, namely delocalization of
kinks is the main factor lowering the edge of the kink-antikink continuum. If
anything, the gap may close even faster than on the sawtooth chain because on
kagome kinks move in two dimensions and thus can lower their energy through
delocalization more effectively than on a one-dimensional chain. For this
reason, the critical DM coupling for kagome may be even lower than for the
sawtooth chain.
The kagome antiferromagnet differs from the sawtooth chain in one important
respect: it has a finite concentration of antikinks in the ground state. The
antikinks form tightly bound $S=0$ pairs, whose binding energy
$\Delta_{\mathrm{aa}}\approx 0.06J$ is lower than the threshold energy of
kink-antikink creation $\Delta_{\mathrm{ka}}\approx 0.25J$. Therefore the spin
gap in the Heisenberg antiferromagnet on kagome is determined by binding
energy of an antikink pair. Although the binding energy $\Delta_{\mathrm{aa}}$
is no doubt influenced by the introduction of the DM term, it is unlikely that
this energy is very sensitive to the presence of a small perturbtion like $D$
as $\Delta_{\mathrm{aa}}$ is determined by a competition of two high-energy
processes: the antikink hopping amplitude and the antikink attraction in the
singlet channel, both with a strength of order $J$. It seems more likely that
the larger gap $\Delta_{ka}$ will be quickly driven to zero as it is on the
sawtooth c hain.
The nature of the phase transition at the conjectured condensation of kinks
and antikinks is an open question. It is not even known whether the $D=0$
ground state is a valence-bond liquid or solid, with contradictory indications
from different numerical techniques.Singh and Huse (2007); Evenbly and Vidal
(2010); Jiang et al. (2008); Yan et al. (2011) (In our view, even a small
amount of bond disorder will turn the system into a disordered valence-bond
solid.) Adding the DM term will tend to melt the delicate valence-bond order
turning the valence-bond crystal into a liquid before the magnetic
condensation and thus inducing another phase transition along the way. The
nature of the condensed phase is not clear, either. Usually, ordering of the
transverse components of magnetization is associated with a proliferation of
$S_{z}=1$ objects, as is the case in magnon condensation,Giamarchi et al.
(2008) whereas here the condensing particles are spinons with half-integer
spin. This obsrvation lends support to the scenario with an intermediate
gapless phase lacking long-range spin order,Rousochatzakis et al. (2009) which
is some sort of an algebraic spin liquid.Ryu et al. (2007)
## Acknowledgments
Work at JHU was supported in part by the U.S. Department of Energy, Office of
Basic Energy Sciences, Division of Materials Sciences and Engineering under
Award No. DE-FG02-08ER46544. OT and ZH acknowledge hospitality of the Max-
Planck-Institute for Physics of Complex Systems, where they were part of the
Advanced Study Group on Unconventional Magnetism in High Fields. YW was
supported by William Gardner fellowship. AS and JW were supported by the
National Science Foundation under NSF Grant No. DMR-0907793, and would like to
thank I. McCulloch for patiently explaining the Matrix Product Toolkit.
## Appendix A Oscillations in the ground-state splitting
To understand the oscillatory behavior of the ground-state splitting, Eq.
(13), we turn to a much simpler model: the antiferromagnetic XXZ chain with DM
interaction described by the Hamiltonian $H=H_{\mathrm{XXZ}}+H_{\mathrm{DM}}$,
where
$H_{\mathrm{XXZ}}=\sum_{n}\left[J\cos{\alpha}(S^{x}_{n}S^{x}_{n+1}+S^{y}_{n}S^{y}_{n+1})+J_{z}S^{z}_{n}S^{z}_{n+1}\right]$
(17)
and
$H_{\mathrm{DM}}=J\sin{\alpha}\sum_{n}(S^{x}_{n}S^{y}_{n+1}-S^{y}_{n}S^{x}_{n+1}).$
(18)
In the easy-axis limit, $J_{z}\gg J$, the ground state is doubly degenerate
and exhibits Néel order. In a finite chain with periodic boundary condition,
quantum tunneling splits the doublet into eigenstates with momenta 0 and
$\pi$. Below we discuss the effect of the DM term, $\alpha\neq 0$, on the
splitting.
By rotating local axes at site $n$ through angle $n\alpha$ in the $xy$ plane,
the DM term in the Hamiltonian can be removed, producing the standard XXZ
model:
$H^{\prime}=\sum_{n}\left[J(S^{x}_{n}S^{x}_{n+1}+S^{y}_{n}S^{y}_{n+1})+J_{z}S^{z}_{n}S^{z}_{n+1}\right].$
(19)
For a closed chain of length $L$, the transformation yields twisted periodic
boundary conditions:
$\displaystyle S^{x}_{N}$ $\displaystyle=$ $\displaystyle
S^{x}_{0}\cos{(L\alpha)}+S^{y}_{0}\sin{(L\alpha)},$ $\displaystyle S^{y}_{N}$
$\displaystyle=$
$\displaystyle-S^{x}_{0}\sin{(L\alpha)}+S^{y}_{0}\cos{(L\alpha)}.$ (20)
The twist is absent if $L\alpha=2\pi m$, where $m$ is an integer. Then the
system has the same spectrum as in the absence of the DM term, $\alpha=0$. At
a fixed chain length $L$, the splitting is a periodic function of $\alpha$
with a period of $2\pi/L$.
To see that the splitting should have an oscillatory character, consider the
special case of a $\pi$ twist, $L\alpha=(2m+1)\pi$. As Haldane argued,Haldane
(1983) the tunneling between the two Néel states is mediated by instantons
with quantized winding numbers $n$, classical action $S_{n}$, and a Berry
phase $\exp{(2\pi inS)}$. For boundary conditions with a $\pi$ twist, the
winding numbers are half-integer, $n=\pm 1/2,\pm 3/2,\ldots$ Instantons with
opposite winding numbers have the same classical action, $S_{n}=S_{-n}$.
However, their Berry phases are exactly opposite, $\exp{(2\pi
inS)}=-\exp{(-2\pi inS)}$, when both the winding numbers $n$ and spin $S$ are
half-integer. As a result of destructive interference of instantons with
opposite winding numbers, the tunneling amplitude vanishes when
$L\alpha=(2m+1)\pi$. We thus expect an oscillatory dependence of the splitting
on $\alpha$ at a constant $L$ in the XXZ chain with half-integer spins and
periodic b oundary conditions. The exponential dependence of the splitting on
the length will acquire an oscillating prefactor $\cos{(\alpha L)}$. This
inspired Eq. (13).
## Appendix B Spin wave in sawtooth chain
We compute the spin-wave spectrum on the sawtooth chain in the classical
limit, $S\to\infty$. The Hamiltonian is
$H=\sum_{\langle
ij\rangle}[\mathbf{S}_{i}\cdot\mathbf{S}_{j}+\mathbf{D}_{ij}\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})].$
(21)
For brevity, we set $J=1$.
In equilibrium, spins lie in the plane normal to the DM vectors
$\mathbf{D}_{ij}$, with the angle of $120^{\circ}$ between nearest neighbors,
Fig. 2(f). It is convenient to choose reference frames in such a way that
spins point along the local $z$ axes, the $x$ axes are in the plane of the
spins, and the $y$ axes are parallel to $\mathbf{D}_{ij}$. For small
deviations from equilibrium,
$\mathbf{S}_{i}\approx
S(\alpha_{i},\,\beta_{i},\,1-\alpha_{i}^{2}/2-\beta_{i}^{2}/2)$ (22)
where $\alpha_{i}$ and $\beta_{i}$ are small deviations from the $120^{\circ}$
pattern.
In the harmonic approximation, the energy (21) reads
$H=S^{2}\sum_{\langle
ij\rangle}(\chi\beta_{i}\beta_{j}+\alpha_{i}\alpha_{j})-S^{2}\sum_{i}K_{i}\chi(\alpha_{i}^{2}+\beta_{i}^{2})$
(23)
where $\chi=-1/2-\sqrt{3}D/2$. $K_{i}=1$ if $i$ is an apex (A) site and
$K_{i}=2$ if it is a base (B) site.
The dynamics can be obtained from the Lagrangian, which includes a Berry phase
term in addition to the potential energy:
$L=S\sum_{i}(\cos\theta_{i}-1)\dot{\phi}_{i}-H.$ (24)
After expressing the angles $\theta$ and $\phi$ in terms of $\alpha$ and
$\beta$,
$\tan\phi=\beta/\alpha,\qquad\cos\theta\approx 1-(\alpha^{2}+\beta^{2})/2,$
(25)
we obtain the following Lagrangian:
$L=S\sum_{i}(\dot{\alpha}_{i}\beta_{i}-\alpha_{i}\dot{\beta}_{i})/2-H.$ (26)
It yields the equations of motion for spins on sublattices $A$ and $B$:
$\displaystyle\dot{\alpha}^{A}_{i}$ $\displaystyle=$ $\displaystyle
S\chi(\beta^{B}_{i+1/2}+\beta^{B}_{i-1/2})-2S\chi\beta^{A}_{i},$ (27a)
$\displaystyle\dot{\alpha}^{B}_{i}$ $\displaystyle=$ $\displaystyle
S\chi(\beta^{A}_{i+1/2}+\beta^{A}_{i-1/2}+\beta^{B}_{i+1}+\beta^{B}_{i-1})-4S\chi\beta^{B}_{i},$
(27b) $\displaystyle\dot{\beta}^{A}_{i}$ $\displaystyle=$
$\displaystyle-S(\alpha^{B}_{i+1/2}+\alpha^{B}_{i-1/2})+2S\chi\alpha^{A}_{i},$
(27c) $\displaystyle\dot{\beta}^{B}_{i}$ $\displaystyle=$
$\displaystyle-S(\alpha^{A}_{i+1/2}+\alpha^{A}_{i-1/2}+\alpha^{B}_{i+1}+\alpha^{B}_{i-1})+4S\chi\alpha^{B}_{i}.$
(27d)
Note that $i$ is half-integer on sublattice A and integer on sublattice B.
Plane waves with frequency $\omega$ and wavevector $k$ satisfy the equation
$\left(\begin{array}[]{cccc}-i\omega&0&2S\chi&-2S\chi\cos(k/2)\\\
0&-i\omega&-2S\chi\cos(k/2)&4S\chi-2S\chi\cos{k}\\\
-2S\chi&2S\cos(k/2)&-i\omega&0\\\
2S\cos(k/2)&-4S\chi+2S\cos{k}&0&-i\omega\end{array}\right)\left(\begin{array}[]{c}\alpha^{A}\\\
\alpha^{B}\\\ \beta^{A}\\\ \beta^{B}\end{array}\right)=0.$ (28)
At $D=0$, we have one zero mode and one mode with a finite frequency
$\omega=S\sqrt{2-\cos(2k)}$. For a finite $D$, the zero mode acquires a
dispersion linear in $k$ in the limit $k\to 0$. The wave velocity is
$v=3S\sqrt{\frac{\sqrt{3}D+7D^{2}+5\sqrt{3}D^{3}+3D^{4}}{2+8\sqrt{3}D+18D^{2}}}.$
(29)
Restoring $J$ as a coupling constant, we find the following behavior for the
velocity. As $D\to 0$, $v\sim 2.79S\sqrt{DJ}$. For $D=0.19J$, $v=1.05SJ$.
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|
arxiv-papers
| 2011-07-14T18:47:50 |
2024-09-04T02:49:20.559358
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhihao Hao, Yuan Wan, Ioannis Rousochatzakis, Julia Wildeboer, A.\n Seidel, F. Mila, O. Tchernyshyov",
"submitter": "Zhihao Hao",
"url": "https://arxiv.org/abs/1107.2896"
}
|
1107.2914
|
# A 95 GHz Class I Methanol Maser Survey Toward GLIMPSE Extended Green Objects
(EGOs)
Xi Chen11affiliation: Key Laboratory for Research in Galaxies and Cosmology,
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
200030, China; chenxi@shao.ac.cn 2 2affiliationmark: , Simon P.
Ellingsen33affiliation: School of Mathematics and Physics, University of
Tasmania, Hobart, Tasmania, Australia , Zhi-Qiang Shen11affiliation: Key
Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China;
chenxi@shao.ac.cn 2 2affiliationmark: , Anita Titmarsh 33affiliation: School
of Mathematics and Physics, University of Tasmania, Hobart, Tasmania,
Australia , and Cong-Gui Gan11affiliation: Key Laboratory for Research in
Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of
Sciences, Shanghai 200030, China; chenxi@shao.ac.cn 2 2affiliationmark:
###### Abstract
We report the results of a systematic survey for 95 GHz class I methanol
masers towards a new sample of 192 massive young stellar object (MYSO)
candidates associated with ongoing outflows (known as extended green objects
or EGOs) identified from the _Spitzer_ GLIMPSE survey. The observations were
made with the Australia Telescope National Facility (ATNF) Mopra 22-m radio
telescope and resulted in the detection of 105 new 95 GHz class I methanol
masers. For 92 of the sources our observations provide the first
identification of a class I maser transition associated with these objects
(i.e. they are new class I methanol maser sources). Our survey proves that
there is indeed a high detection rate (55%) of class I methanol masers towards
EGOs. Comparison of the GLIMPSE point sources associated with EGOs with and
without class I methanol maser detections shows they have similar mid-IR
colors, with the majority meeting the color selection criteria
-0.6$<$[5.8]-[8.0]$<$1.4 and 0.5$<$[3.6]-[4.5]$<$4.0. Investigations of the
IRAC and MIPS 24 $\mu$m colors and the associated millimeter dust clump
properties (mass and density) of the EGOs for the sub-samples based on which
class of methanol masers they are associated with suggests that the stellar
mass range associated with class I methanol masers extends to lower masses
than for class II methanol masers, or alternatively class I methanol masers
may be associated with more than one evolutionary phase during the formation
of a high-mass star.
masers – stars:formation – ISM: molecules – radio lines: ISM – infrared: ISM
22affiliationtext: Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, China
## 1 Introduction
Methanol masers are quite common in massive star forming regions (MSFRs).
Historically they have been empirically classified into two categories (class
I and class II). The initial classification was based on the sources towards
which the different transitions were detected (Batrla et al. 1988; Menten
1991a). Class II methanol masers are often found to be associated with strong
centimeter continuum emission within 1′′ (e.g. ultracompact (UC) regions;
Walsh et al. 1998), infrared sources and OH masers. The strongest and best
known class II transition is the 5${}_{1}-6_{0}$ A+ line at 6.7 GHz. Since its
discovery (Menten 1991b), this class bright maser has become a reliable tool
for detecting and studying regions where massive stars form and are in their
very early stages of evolution (e.g. Minier et al 2003; Bourke et al 2005;
Ellingsen 2006; Xu et al 2008; Pandian et al 2008). A number of surveys have
been conducted for the 6.7 GHz class II methanol maser, resulting in the
detection of $\sim$ 900 sources in the Galaxy to date. The surveys include
those that are summarized in the compilation of Pestalozzi et al. (2005) and
the recent searches of Pandian et al. (2007), Ellingsen (2007), Xu et al.
(2008; 2009) and the Parkes Methanol Multibeam (MMB) blind survey ($\sim$ 300
sources (Green et al. 2009), but so far only part published by Caswell et al.
2010 and Green et al. 2010). In contrast class I methanol masers are usually
found offset by 0.1 - 1.0 pc from UC Hii regions, infrared sources and OH
masers (e.g. Plambeck & Menten 1990; Kurtz et al. 2004). A number of
observations of class I methanol masers showed that they were located at the
interface between molecular outflows and the parent cloud (Plambeck & Menten
1990; Johnston et al. 1997; Kurtz et al. 2004; Voronkov et al. 2006). These
early results suggested that class I and class II methanol masers favored very
different environments. These observational findings were supported by early
theoretical models of methanol masers which found that class I masers are
collisionally excited, in contrast to class II masers which have a radiative
pumping mechanism (Cragg et al. 1992). Observations made since the mid-1990s
(e.g. Slysh et al. 1994) have shown that at single dish resolutions class I
and class II masers are often associated. Observations at high spatial
resolution (e.g. Cyganowski et al. 2009) show that while the two types of
masers are typically not co-spatial on arcsecond scales, they are usually
driven by the same young stellar object.
Compared to class II masers, class I methanol masers are relatively poorly
studied and understood. There have only been a few large surveys for class I
masers (mainly at 44 and 95 GHz), primarily undertaken with single-dish
telescopes (e.g. Haschick et al. 1990; Slysh et al. 1994; Val’tts et al. 2000;
Ellingsen 2005) along with a few smaller scale interferometric searches (e.g.
Kurtz et al 2004; Cyganowski et al. 2009). These have resulted in the
detection of about 200 class I maser sources in the Galaxy to date (Val’tts et
al. 2010).
Observations and theoretical calculations suggest that class I methanol masers
can form in the interface region between outflows and the ambient molecular
gas. The methanol abundance is significantly increased in the shocked
interface regions (e.g. Gibb & Davis 1998) and the gas is heated and
compressed providing more frequent collisions, which results in more efficient
pumping (Voronkov et al. 2006). Interferometric observations have shown the
location of some class I methanol masers correlates closely with the shocked
gas in outflows as traced by 2.12 $\mu$m H2 and SiO (e.g. Plambeck & Menten
1990 ; Voronkov et al. 2006). However, such a close physical association
between the shocks driven by outflows and the class I masers has only been
firmly established in a small number of sources. One of the problems
encountered in studying the relationship between outflows (or shocks) and
class I methanol masers has been in finding an appropriate outflow tracer (one
which is frequently associated with class I maser emission). Recently
Cyganowski et al. (2008) have suggested that the 4.5 $\mu$m band of the
_Spitzer Space Telescope’s_ Infrared Array Camera (IRAC) offers a promising
new approach for identifying massive young stellar objects (MYSOs) with
outflows. The strong, extended emission in this band is usually thought to be
produced by shock-excited molecular H2 and CO in protostellar outflows (e.g.
Noriega-Crespo et al. 2004; Reach et al. 2006; Smith et al. 2006; Davis et al.
2007; Ybarra & Lada 2009, 2010). These extended 4.5 $\mu$m emission features
are commonly known as “extended green objects” (EGOs; Cyganowski et al. 2008)
or “green fuzzies” (Chambers et al. 2009) due to the common color-coding of
the 4.5 $\mu$m band as green in three-color images. These objects are
providing a new and powerful outflow tracer for MSFRs. Cyganowski et al.
(2008) have catalogued over 300 EGOs from the Galactic Legacy Infrared Mid-
Plane Survey Extraordinaire (GLIMPSE) I survey (Churchwell et al. 2009), and
they divided cataloged EGOs into “likely” and “possible” MYSO outflow
candidates based primarily on the angular extent and morphology of the 4.5
$\mu$m. Recent mid-infrared spectroscopic observations of two EGOs identified
by Cyganowski et al. showed the presence of strong shocked H2 in the 4-5
$\mu$m wavelength range from one of the sources, but no evidence for shocked
gas in the second (De Buizer & Vacca 2010). They suggest that some EGOs may be
due to spatial variations in the mid-infrared extinction and exaggerated color
stretches rather than shocked gas, however, the rate of mis-classification of
EGOs is at present very poorly determined. Based on the mid-IR colors of EGOs
and their correlation with infrared dark clouds (IRDC) and known 6.7 GHz
methanol masers, Cyganowski et al. (2008) further suggests that most EGOs
trace a population of actively accreting MYSO outflow candidates. Utilizing
the published high resolution masers known at the time, Cyganowski et al.
found that 6.7 GHz class II methanol masers are associated with 73% (35
detections from 48 EGOs) of “likely” and 27% (11 detections from 67 EGOs) of
“possible” MYSO outflow candidate EGOs.
Chen et al. (2009) have analyzed 61 EGOs from the Cyganowski et al. catalog
that have been included in class I methanol maser surveys. Four previous class
I methanol maser surveys (for the 44 GHz $7_{0}$–$6_{1}$ A+ transition by
Slysh et al. (1994) and Kurtz et al. (2004) and for the 95 GHz $8_{0}$–$7_{1}$
A+ transition by Val’tts et al. (2000) and Ellingsen (2005)) were included in
their statistical analysis. They found that 41 EGOs are associated with one or
both of the 95 and 44 GHz class I methanol masers within 1 arcmin, thus the
expected detection rate of class I masers in EGOs is 67% at this resolution.
Based on this high predicted detection rate, they suggested that GLIMPSE-
identified EGOs might provide one of the best targeted samples for searching
for class I methanol masers. However their statistical study is also subject
to unknown influences produced by the target selection effects in the four
class I maser surveys they have utilised in the study. For example, most of
the currently known class I masers are associated with class II masers (72% as
reported by Valt́ts & Larinov 2007), for which a good association with EGOs
has already been demonstrated by Cyganowski et al. (2008). Table 3 of Chen et
al. (2009) shows that most EGOs (57/61) used in their statistical analysis are
also associated with 6.7 GHz class II masers. According to Cyganowski et al.
(2008), the majority of EGOs (35/48=73% in the likely sample) are associated
with 6.7 GHz class II masers, and the sample of known class I masers is
largely a subset of the sample of known class II masers, so a high detection
rate for class I masers towards these EGOs is not unexpected. Recent 6.7 GHz
class II and 44 GHz class I methanol maser surveys toward $\sim$20 EGOs in the
northern hemisphere with the VLA reported by Cyganowski et al. (2009) also
show that $>$64% and 89% of EGOs are associated with class II and class I
methanol masers, respectively. However, their observations only targeted a
small number ($\sim$ 20) of “likely” outflow candidate EGOs. And their 44 GHz
survey sample is essentially a 6.7 GHz methanol maser selected sample (19
sources observed at 44 GHz (17/19 with class I masers), of which 18 had
associated 6.7 GHz methanol masers (16/18 with class I masers)).
In order to determine whether there is indeed a high association rate of class
I methanol masers in EGOs and to investigate the relationship between them, it
is necessary to perform a class I maser search towards a full EGO-based target
sample. In this paper, we report a 95 GHz class I methanol maser survey
towards an EGO-selected sample which has been undertaken with the Australia
Telescope National Facility (ATNF) 22-m millimetre antenna at Mopra. In
Section 2 we describe the sample and observations, in Section 3 we present the
results of the survey, the discussion is given in Section 4, followed by our
conclusions in Section 5.
## 2 Source selection and Observations
### 2.1 Source selection
The EGO catalog compiled by Cyganowski et al. (2008) lists a total of 302
sources, of which 137 are classified as “likely” and 165 as “possible” outflow
sources. All EGOs with declinations in the range from -62∘ to +25∘ are
accessible with Mopra. However, there are some sources which have an angular
separation of less than $\sim$20′′ (corresponding the half beam size of the
Mopra antenna; see below), from their nearest EGO. Where this occurred, we
observed at the position of those classified as “likely” outflow sources, or
when neither fell into this category, at the location with the larger 4.5
$\mu$m flux density. Chen et al. (2009) reported that 61 EGOs have previously
been observed in 44 and 95 GHz class I methanol maser surveys. For these
sources we re-observed 10 (of 16) EGOs which had not been observed at 95 GHz
in previous searches (see Table 1). Note that we did not make observations of
6 of the sources without 95 GHz information in Chen et al. sample in present
survey (see Table 6 for this). After applying these selection criteria we were
left with a total of 191 EGOs to be observed. Among these sources, 65, 17 and
3 sources were selected from Tables 1, 2 and 5 of Cyganowski et al. (2008),
respectively – thus 85 “likely” outflow candidates; 55 and 51 sources were
selected from Tables 3 and 4, respectively – thus 106 “possible” outflow
candidates. In addition, a well studied MYSO IRDC 18223-3 which shows
distinctly extended 4.5 $\mu$m emission (Beuther & Steinacker 2007) was also
included as a “likely” outflow candidate. In this paper we report the results
of our 95 GHz class I methanol maser survey toward these 192 sources.
Table 1 lists the target sample source parameters including the source name
(derived from the Galactic coordinates), the equatorial coordinates and
whether the EGO is associated with an IRDC, 6.7 GHz class II methanol maser,
1665 or 1667 MHz OH maser, UC Hii region or 1.1 millimeter (mm) continuum
source. The positional accuracies of the 6.7 GHz methanol maser catalogs
(Cyganowski et al. 2009; Caswell 2009; Xu et al. 2009; Caswell et al. 2010;
Green et al. 2010), OH maser catalog (Caswell 1998) and UC Hii region catalogs
(Wood & Churchwell 1989; Becker et al. 1994; Kurtz et al. 1994; Walsh et al.
1998; Cyganowski et al. 2009) used in our analysis, are usually better than
1′′. The positional uncertainty of the 1.1 mm continuum sources in the BOLOCAM
Galactic Plane Survey (BGPS) is also of the order of several arcseconds
(Rosolowsky et al. 2010). The Spitzer GLIMPSE point source catalog has also a
high positional accuracy (better than 1″), however, EGOs are extended objects
with angular extents between a few and $>$30′′ (Cyganowski et al. 2008). Thus
we have assumed an association between EGOs and other tracers for separations
of less than 30′′. For BGPS sources, we used their peak positions (rather than
centroid positions) available in the BGPS catalog (Rosolowsky et al. 2010) and
did not account for their sizes in the cross-matching. We list the
characteristics (including targeted/untargeted, area covered, angular
resolution, sensitivity and extent of overlap with the GLIMPSE I survey area
from which the EGO targets are drawn) for the maser, UC Hii region and BGPS
1.1 mm datasets used in our analysis in Table 2. To more clearly show the
associations among these tracers and EGOs, we have also ploted the positions
of the masers, UC Hii regions, and mm sources on the _Spitzer_ 3-color IRAC
images for all 192 targeted sources in our Mopra survey in Figure 1. From this
figure, it can be seen that most of class II methanol (marked by black
crosses) and OH masers (marked by red small circles) are close to the EGO
positions (marked by blue pluses) with typical separations between them of
less than 5$\arcsec$. But for UC Hii regions (marked by blue squares) and mm
sources (marked by yellow diamonds), they usually have a slightly larger
separation from the targeted EGO position. One possible reason is that the UC
Hii regions and mm sources have larger angular sizes, of order a few to a few
tens of arcseconds. Even though there are larger separations between EGOs and
UC Hii regions and mm sources, we still consider them to be associated in our
analysis. However we recognize that in some cases there may be no true
physical association between them.
### 2.2 Observations
The observations were made using the Mopra 22m telescope during 2009 August
9-20. Mopra is located near Coonabarabran, New South Wales, Australia. The
telescope is at latitude 31 degrees south and has an elevation of 850 metres
above the sea level. A 3 mm Monolithic Microwave Integrated Circuits (MMICs)
receiver with a frequency range from 76 to 117 GHz is installed. The UNSW
Mopra spectrometer (MOPS) provides four 2.2 GHz bands which overlap slightly
to give a total instantaneous bandwidth of 8 GHz. MOPS was used in zoom mode
for the observations reported here. In this mode, up to 4 zoom windows can be
selected within each 2.2 GHz band thus allowing up to 16 spectral lines to be
observed simultaneously at high spectral resolution. Each zoom window provides
a bandwidth of 137 MHz with 4096 channels for each polarisation, which leads
to a total velocity range of $\sim$ 430 km s-1 and a velocity resolution of
0.11 km s-1 per channel in the 3 mm band. The 8 GHz bandpass was centred at
94.3 GHz to provide a complete coverage over the 90.3–98.3 GHz range. Within
this range we configured the individual zoom windows to simultaneously cover
the strong lines in the 3 mm band, e.g. $1-0$ HNC (90.6635680 GHz),
$5_{k}-4_{k}$ CH3CN ($\sim$ 92 GHz), $1-0$ N2H+ ($\sim$ 93.17 GHz), $2-1$ CS
(97.9809533 GHz), $2-1$ C34S (96.4129495 GHz), $8_{0}-7_{1}$ A+ CH3OH maser
(95.1694630 GHz) and a series of lines of $2-1$ CH3OH ($\sim$ 96.7 GHz). In
this paper we will focus on the results of the 95 GHz class I methanol masers
band. The results from the other lines will be reported in subsequent
publications.
Each source was observed in a position-switching mode with 1 minute spent at
the on-source position and 1 minute at a reference position. This procedure
was repeated a number of times to yield a total of 11–13 minutes of on-source
integration for most sources. Two different reference positions offset by +8
and -8 arcminutes in declination from the targeted EGO sites were used. The
antenna pointing was checked at hourly intervals by observing nearby bright 86
GHz SiO masers with known positions. The nominal pointing accuracy is
estimated to be better than 5″. The system temperature was typically between
180-250 K depending on weather conditions and telescope elevation, resulting
in an rms noise level of $\sim$30 mK per channel after averaging the two
polarizations and Hanning smoothing. The half-power main-beam size and the
main-beam efficiency were 36′′ and 0.5, respectively at 3 mm band (Ladd et al.
2005). The measured antenna temperatures (T${}_{A}^{*}$) were further
calibrated on to a main beam temperature scale (TMB) by dividing by the main
beam efficiency. Then a conversion factor of 9.3 Jy K-1 can be estimated from
Equ. 8.19 of Rohlfs & Wilson (2004) to convert the main beam temperature to a
flux density. Thus a direct conversion factor from T${}_{A}^{*}$ to Jy would
be 18.6 Jy K-1. The resulting flux density detection limit was $\sim 1.6$ Jy
(3 $\sigma_{rms}$; the typical rms noise is $\sim$0.55 Jy) per channel. The
sensitivity of 18.6 Jy K-1 for our 95 GHz class I methanol maser survey is
significantly better than that of 40 Jy K-1 in previous similar surveys with
Mopra (e.g. Val’tts et al. 2000; Ellingsen 2005), largely because the
effective antenna diameter has been increased from 15 to 22 meters in the 3 mm
band since 1999.
The spectral line data were reduced and analyzed with the ATNF spectral line
reducing software (ASAP) and the GILDAS/CLASS package. During the reduction,
quotient spectra were formed for each on/off pair of observations which were
then averaged together. A low-order polynomial baseline fitting and
subtraction, and Hanning smooth were performed for the averaged spectra. For
some sources the resulting spectra still contained baseline ripples due to bad
weather and for these cases, we Fourier transformed the spectrum and flagged
the data in frequency space to reduce the influence of the baseline ripple.
This procedure was also undertaken with ASAP (for more details see the Mopra
Website111http://www.narrabri.atnf.csiro.au/mopra/obsinfo.html). Usually the
95 GHz methanol spectra do not have a particularly Gaussian profile, possibly
because the spectra are blended with multiple maser features within a similar
velocity range. We converted all the data into CLASS format and performed
Gaussian fitting of each possible maser feature for each detected source with
the CLASS package.
## 3 Results
### 3.1 Nature of Detected Emission
95 GHz class I methanol emission was detected towards 105 sites from the 192
searched positions, yielding a detection rate of 55% for this survey. The
detected sources are listed, along with the parameters of Gaussian fits to
their 95 GHz spectral features in Table 3. The detected emissions range in
strength from $\sim$ 0.2 to 108 Jy (corresponding to main beam temperatures
TBM $\sim$ 0.02 to 11.6 K), derived from the Gaussian fits. Spectra of each of
the detected 95 GHz emission after Hanning smooth are shown in Figure 2. The
sources for which no 95 GHz emission was detected are listed in Table 4, along
with the 1 $\sigma$ noise of the observation (typically less than 0.7 Jy). We
consider a source to have been detected if it exhibits emission stronger than
3 $\sigma$. All of the 105 detected 95 GHz methanol sources are new detections
in this transition. Among them, only 13 sources have previously been detected
in the 44 GHz class I methanol transition in recent VLA surveys (G34.26+0.15
detected by Kurtz et al. 2004; G10.29-0.13, G10.34-0.14, G11.92-0.61,
G18.67+0.03, G18.89-0.47, G19.01-0.03, G19.36-0.03, G22.04+0.22, G23.96-0.11,
G24.94+0.07, G25.27-0.43 and G39.10+0.49 detected by Cyganowski et al. 2009).
Therefore 92 new class I methanol emission sources have been found in this
survey. From the spectra of the class I methanol emission sources shown in
Figure 2 it can be seen that the class I emission often consists of one or
more narrow spectral features which can be treated as maser emission,
apparently superimposed on broader emission features. Examination of Table 3
shows that spectral features with widths $>$ 1 km s-1 are common in the
detected sources. The characteristic of broader emission features in class I
transition is very similar to that reported by Ellingsen (2005). From single-
dish observations we cannot determine whether these broad features are quasi-
maser/quasi-thermal emission, or due to blending of a number of weaker narrow
maser features (since the Mopra spectra almost certainly include multiple
class I maser spots blended within the beam, multiple weak masers at different
velocities could in aggregate produce some of the broad emission features
seen). However broad, quasi-thermal like spectral profiles are commonly
observed in class I methanol masers (see for example Kurtz et al. 2004, or
Voronkov et al. 2006). The majority of the broad emission in these sources
when observed interferometrically is observed to be maser emission. Thus,
although we cannot conclusively rule out the possibility that some of the
sources may have thermal emission contributions, comparison with previous
observations makes us very confident that the majority of the broad emission
is maser emission. Moreover some of the EGOs with previous 44 GHz maser
detections (Kurtz et al. 2004; Cyganowski et al. 2009) also show similar broad
emission profiles at 44 GHz (for Cyganowski et al. 2009, the 44 GHz maser
spectra were not directly presented, but they can be determined from the
fitted intensities of the 44 GHz emission listed in Table 8 of their work).
The similar broad emission features from class I methanol masers associated
with EGOs are also detected and spatially unresolved in 44 GHz VLA
observations (Kurtz et al. 2004; Cyganowski et al. 2009), which suggests that
they are generally masers. It has yet to be confirmed through interferometric
observations that the broad 95 GHz spectral features are also maser emission,
although given the results of numerous past interferometric observations of
class I masers this appears likely for the majority.
We also checked the flux densities of 44 GHz maser emission for the 13 EGOs
which were observed as 95 GHz masers in our Mopra survey, and found that the
peak flux densities of the 44 GHz masers are typically 2-3 times stronger than
that of 95 GHz masers, consistent with the statistical result from Val’tts et
al. (2000). However the primary beam of the VLA (1$\arcmin$ at 44 GHz) is
larger than that of Mopra (36$\arcsec$ at 95 GHz). Thus the detection of
stronger 44 GHz emission may be because for some sources with extended maser
emission the VLA can detect additional maser emission which is within the VLA
primary beam, but outside the Mopra beam (see Cyganowski et al. 2009). To
account for this potential affect, we compared only the 8 sources with all 44
GHz maser emission detected by the VLA locating in a compact region and within
Mopra beam. We found that the peak flux densities of 44 GHz emission are still
2-3 times stronger than that of 95 GHz emission for these sources.
In Table 3 we also list the distance and the integrated maser luminosity of
each of the 105 detected methanol maser sources. The distance was estimated
from the Galactic rotation curve of Reid et al. (2009), with the Galactic
constants, R⊙= 8.4 kpc and $\Theta_{\odot}$= 254 km s-1. Class I masers are
generally found near the VLSR as measured from the thermal gas (e.g.
Cyganowski et al. 2009). We adopted the velocity of the brightest feature in
the 95 GHz maser spectrum (listed in column 3 of Table 2) in the distance
calculation. The near kinematic distance was adopted for sources with a
near/far distance ambiguity. Given that EGOs are by definition extended
sources, there will be bias towards nearby sources in their identification and
also the nearby sources will be brighter and easier to detect. Another
argument for EGOs being more likely at the near kinematic distance is their
associations with IRDCs, because the identification of IRDCs is greatly biased
toward the near kinematic distance (see Jackson et al. 2008). Thus the near
distances adopted for these sources are likely to be more reasonable for most
sources. The distances to G49.07-0.33, G309.91+0.32, G310.15+0.76 which could
not be derived from the Galactic rotation curve were assumed to be 5 kpc.
### 3.2 Detection Rates
The results of our single-dish survey prove that there is indeed a high
detection rate ($\sim$55%) of class I methanol masers towards EGOs. It expands
the number of published class I methanol masers by about 50% (an additional 92
on top of the 198 listed by Val’tts et al. 2010). However given the large beam
size of Mopra and clustered massive star formation, a detection in a pointing
towards an EGO does not necessarily mean that the detected class I maser is
truly associated with the EGO, though recent VLA observations towards $\sim$20
EGOs reported by Cyganowski et al. (2009) revealed that nearly all 44 GHz
class I masers trace the diffuse green 4.5 $\mu$m features of EGOs except for
one case (G28.28-0.36). If this is true of the sample as a whole then it
suggests that the false association rate between EGOs and detected class I
masers is likely to be very low with a range of a few to (at most) 10%. Where
there were multiple EGOs included in the large beam of Mopra, we pointed at
the position of the EGO classified as a “likely” outflow source, or where
there were two such sources, at the position with the largest 4.5 $\mu$m flux
density (see Section 2.1). It is not clear which of the EGOs (all, or only
one) included in such observations are associated with the detected maser
emission. Thus it requires further high resolution observations toward the
detected class I masers to determine whether they are truly physical
associations with EGOs or which EGOs are associated with them. In subsequent
analysis, we have assumed that the detected maser emission is associated with
the EGOs used as the targeted point.
The detection rate of 55% achieved in this survey is slightly lower than the
predicted value of 67% from our previous statistical analysis (Chen et al.
2009). One possible reason for this is that our previous analysis combined 44
and 95 GHz class I maser searches, and emission from the 44 GHz transition is
generally 3 times stronger than that at 95 GHz (Val’tts et al. 2000). Thus it
may be possible to detect a significant number of additional sources if we
were to make a search with similar or better sensitivity at 44 GHz. To test
this conjecture, we checked all 20 EGOs with detected 44 GHz masers in
previous surveys (Kurtz 2004 and Cyganowski et al. 2009), and found that only
13 EGOs have detected 95 GHz emission at our Mopra survey sensitivity of
$\sim$2 Jy (i.e. the detection rate of 95 GHz methanol masers in these EGOs is
13/20$\sim$65%). All the remaining 7 EGOs detected at 44 GHz for which we did
not detect 95 GHz emission have peak flux densities of less than 6 Jy for the
44 GHz masers. This is consistent with our expectation that these 7 EGOs might
not be detected class I masers with the current detection limit (corresponding
to $\sim$ 6 Jy at 44 GHz when considering the 44/95 GHz correlation from
Val’tts et al. 2000). If we extrapolate this result to our survey it suggests
that we have detected around 2/3 of the true number of class I maser sources
in our search. If this were correct then our detection rate would be well in
excess of the prediction of Chen et al. (2009). This extrapolation naively
ignores the likely presence of selection biases in the VLA comparison sample
and the effect of the different primary beam of the VLA and Mopra as stated
above, but these are not readily assessed. The true rate of association of
class I methanol masers with EGOs can perhaps only be accurately determined
through targeted sensitive 44 GHz observations towards the non-detected
sources from this survey.
### 3.3 Associations with Class II Methanol Masers
Table 1 gives information on the association between EGOs and class II
methanol masers. For those sources in the Galactic longitude range covered by
sections of the MMB survey published to date (Caswell et al. 2010; Green et
al. 2010), or other similarly sensitive observations (Caswell 2009, Cyganowski
et al. 2009, Xu et al. 2009) we have used that data. For the remaining EGOs we
have made additional 6.7 GHz observations with the Mt Pleasant 26m telescope.
The Mt Pleasant observations will be described in detail in a separate
publication, but the basic characteristics are summarised here. At a frequency
of 6.7 GHz the Mt Pleasant 26m telescope has a 7$\arcmin$ beam FHWM and a
system equivalent flux density (SEFD) of around 800 Jy in each of two
orthogonal circular polarizations. The correlator was configured with 4 MHz
bandwidth and 4096 spectral channels for each circular polarization yielding a
velocity range for the observations of approximately 180 km s-1 and a velocity
resolution (after Hanning smoothing) of 0.09 km s-1. Observations were made
using 10 minute integrations towards each EGO target position observed at 95
GHz and the 3 $\sigma$ detection limit of the observations is around 1.5 Jy,
comparable to that achieved for the 95 GHz Mopra search. Although it is
possible that some of the EGOs not detected in the Mt Pleasant observations
may have a weak associated 6.7 GHz maser, this is likely to effect only a
small number of sources. From comparison with the MMB observations we estimate
this to be less than 10% of non-detections. While the class II methanol maser
data used in our comparison with the class I masers are not derived from a
homogeneous set of observations, their relative sensitivities differ only by
approximately a factor of 2 and are unlikely to introduce significant
additional uncertainty into our statistical analysis.
We have compared the cataloged EGOs with the subset of class II methanol
masers for which accurate positions have been published (see Section 2.1). Of
the 192 cataloged EGOs, 49 are found to be associated with class II masers and
81 were not. The results for the EGOs which are not associated with class II
masers were primarily determined from the surveys for 6.7 GHz methanol masers
with the Mt Pleasant 26m (see above). The remaining 62 EGOs were detected 6.7
GHz methanol masers by the Mt Pleasant 26m telescope, but lack a high-
precision position, thus we consider them as sources for which we have “no”
information with respect to the class II masers in our work. Among the 105
EGOs with detected class I methanol masers, we found that 39 EGOs are, and 31
EGOs are not associated with class II methanol masers, respectively. For the
other 35 EGOs we have insufficient information on the class II masers due to
the lack accurate position observations of the class II masers for these
sources. The sources associated with only class I methanol masers, but without
class II methanol masers are especially important for our understanding of the
properties of class I methanol masers, because most previous class I methanol
maser surveys were targeted towards known class II methanol masers. We have
undertaken further analysis and discussion of this issue in Section 4.2.
On the basis of one of the earliest class I methanol maser searches to detect
a large number of sources, Slysh et al. (1994) suggested that there exists an
anti-correlation between the flux densities of class I and class II methanol
masers towards the same sources. Ellingsen (2005) further investigated this
finding for a sample of class I methanol masers detected towards a
statistically complete sample of class II masers and in that case found no
evidence for an anticorrelation. We have also compared the peak flux densities
of class I and class II methanol masers in 39 sources associated with both
class masers. The peak flux densities of the class II masers are from
interferometric studies (Cyganowski et al. 2009; Caswell 2009; Xu et al. 2009;
Caswell et al. 2010; Green et al. 2010). The logarithms of the peak flux
densities of class I masers vs. that of class II masers are shown in Figure 3.
From this figure it can be seen that there is no statistically significant
correlation (or anticorrelation) between the flux densities of class I and
class II methanol masers in our observed EGO sample, consistent with the
result of Ellingsen (2005). The flux densities of the class II methanol masers
used in our comparison were all obtained from interferometric studies, whereas
the previous comparisons of Slysh et al. (1994) and Ellingsen (2005) used
single-dish data. However, the class II methanol maser emission is usually
distributed over a compact region typical $<$0.2′′ and are not resolved by
connected element interferometry (e.g. the VLA). This means that their flux
density is the same in interferometric and single dish studies and hence this
difference produces no new or additional biases to our statistical comparison.
### 3.4 Associations of Class I methanol masers with other star formation
tracers
In this section, we compare the detection rates of class I methanol masers in
different subsamples, including “likely” and “possible” outflow candidates,
IRDC and non-IRDC, those associated with class II methanol masers, OH masers,
and UC Hii region subsamples, etc. In order to more clearly compare the
various samples we have listed the information relating to the class I
methanol maser detections for all 192 sample sources in the last column of
Table 1. The detection rates for each of the different categories are
summarized in Table 5. It should be remembered that the detection rate in some
categories will be affected by the limitations of single dish surveys with a
large beam size and the clustering which occurs in high-mass star formation
regions (as described in Section 3.1).
A total of fifty three 95 GHz methanol masers were detected towards the 86
likely outflow sources targeted by our observations. The 53 detections include
40, 10 and 2 sources from Tables 1, 2 and 5 of Cyganowski et al. (2008),
respectively, and also IRDC 18223-3. This corresponds to a detection rate of
62% for the “likely” outflow sources. The remaining 52 class I masers
detections were made towards the 106 “possible” outflow candidate EGOs
observed. For the possible outflow sources, 27 and 25 were from Tables 3 and 4
of Cyganowski et al. (2008), respectively. This corresponds to a detection
rate of 49%. A z-test finds that the difference in these detection rates is
significant at the 90% level (i.e. it is marginally significant). Regardless
of this, it is apparent that the detection rate of class I methanol masers in
the “likely” outflow subsample is only slightly higher than that in the
“possible” outflow subsample. This suggests that the class I methanol maser
emission may be not very sensitive to the outflow classifications (i.e.
“likely” and “possible”) seen from the IRAC images. Alternatively, if we
assume that our finding for “likely” outflow sources is the true rate of
association between EGOs and class I methanol masers then this suggests that
approximately 49/62=79% of the “possible” outflow candidates are indeed
outflow sources.
Dividing our sample of EGOs searched for class I methanol masers on the basis
of their association (or otherwise) with IRDCs, we found that 71 of the 128
EGOs associated with an IRDC exhibit class I methanol maser emission (a
detection rate of 55%). Whereas, 34 of the 64 EGOs without an IRDC were
detected as class I methanol maser sources (a detection rate of 53%). IRDCs
are generally thought to host an early stage of the high-mass star formation
process, so it is somewhat surprising that our results show no difference
between the detection rate of class I methanol maser in those EGOs which are
and are not associated with an IRDC. The visibility of an IRDC is dependent on
both the strength of the mid-infrared background emission and the amount of
foreground emission, particularly at 8 $\mu$m (Cyganowski et al. 2008; Peretto
& Fuller 2009). If there is no, or weak 8 $\mu$m background emission in a
particular region, an IRDC may not be apparent even where dense molecular gas
and very young MYSOs are present. Moreover, MYSOs and YSOs of a range of
masses and evolutionary states are also found in IRDCs (e.g. Pillai et al.
2006; Ragan et al. 2009; Rathborne et al. 2010). So sources not associated
with IRDCs do not necessarily host a later evolutionary stage than those which
are.
Our target EGO sample included 49 sources associated with known class II
methanol masers for which the position is accurately known and 81 sources
which have been searched for class II methanol maser emission with no
detection by the Mt Pleasant 26 m telescope (see Table 1 and Section 3.2). We
found that 39 of the 49 sources associated with class II methanol masers were
detected in the 95 GHz class I methanol transition. Thus there is a very high
detection rate ($\sim$ 80%) of class I methanol masers towards EGOs which are
also associated with class II methanol masers, which is somewhat lower than
the 89% reported by Cyganowski et al 2009, but likely not significant given
the sensitivity differences between the observations. In comparison only 31 of
the 81 sources not associated with a class II methanol maser were found to
have an associated 95 GHz class I methanol maser (a detection rate of 38%). A
more in-depth discussion about the lower detection rate of class I methanol
masers towards those EGOs without an associated class II methanol maser and
the evolutionary relationship between class I and class II methanol masers is
given in Section 4.2.
Comparing the positions of the EGOs we observed with the spatial region of the
Parkes/ATCA 1665/1667 MHz OH maser survey (see Table 2) (Caswell 1998), we
found that there were 104 targeted EGO sources within the survey area of the
Parkes OH maser survey. Amongst these, 14 EGOs were, and 90 EGOs were not
associated an OH masers. Nearly all (13 of 14) of the EGOs associated with an
OH maser, while approximately one-half (43 of 90) of the sources not
associated with an OH maser were found to have a class I methanol maser.
We also note that there is a very high detection rate (11/13=85%) of class I
methanol masers towards those EGOs which are associated with an UC Hii region.
For most EGOs no deep centimeter continuum data is available. There are two
Hii region survey datasets that we have used to compile a category of “without
associated UC Hii region” for comparison. The survey of Becker et al. (1994)
covered the region of $|b|<0.4^{\circ}$ and $l=350^{\circ}-40^{\circ}$ (see
Table 2). The other observations are those by Cyganowski et al. (2009) towards
$\sim$20 EGOs in the northern hemisphere, who found that no 44 GHz continuum
emission was detected toward 95% of their surveyed EGO sample. Using the
published data from these two sets of observations, we compiled a sample of 34
EGOs that are not associated with an UC Hii region. From this sample, 21
sources were found to have an associated class I maser. Thus the detection
rate of class I masers of 21/34=62% towards EGOs without associated UC Hii
regions is lower than that in sources which are associated with UC Hii
regions. The size of the subsample associated with UC Hii regions used in our
statistical analysis is small, and they may be biased since many of the large-
scale UC Hii region surveys cited were targeted based on IRAS colors. In
addition, high-mass star formation usually occurs in a cluster environment, so
for an EGO associated with an UC HII region, it is not clear a priori whether
the 4.5 $\mu$m outflow is driven by the UC Hii region or by another
(potentially lower-mass or less evolved) source in a (proto)cluster. However,
given that UC ii regions are relatively rare towards EGOs, the high detection
rate of class I methanol masers in those few sources where they have been
observed to date is suggestive.
There are 63 EGOs which fall within the 1.1 mm BGPS survey area (Rosolowsky et
al. 2010). Fifty four of these are associated with BGPS sources within 30′′,
and 9 are not (see Table 2). We find that the detection rate of class I maser
in the EGOs with an associated BGPS source (35/54=65%) is higher than that in
those without an associated BGPS source (1/9=11%; only one EGO G34.39+0.22 in
this category was detected to have class I maser emission). Conversely, all
class I maser sources except G34.39+0.22 which fall within the BGPS survey
region have an associated 1.1 mm BGPS source.
## 4 Discussion
### 4.1 The mid-IR colors of the class I methanol masers
Based on their mid-IR colors ([3.6]-[5.8] versus [8.0]-[24]), Cyganowski et
al. (2008) have suggested that most EGOs fall in the region of color-color
space occupied by the youngest MYSOs and are surrounded by substantial
accreting envelopes (see Figure 13 in their work). We have performed
additional color-color analysis for EGOs to further investigate the
distinguishing mid-IR properties of EGOs with and without an associated class
I methanol masers. Our mid-IR color analysis includes the 192 EGOs targeted
for our class I maser observations and an additional 51 EGOs that are listed
in our previous work (Chen et al. 2009), but were not observed in our Mopra
survey. We list the information (including their associations with class I
masers and high-precision position class II masers) of these 51 EGOs in Table
6. There are 6 EGOs without 95 GHz class I maser information listed in this
table since they were omitted in our Mopra survey (see also Section 2.1).
Adopting the integrated mid-IR flux densities in the four IRAC bands presented
in Tables 1 and 3 of Cyganowski et al. (2008), we have plotted a diagram of
the [3.6]-[4.5] versus [5.8]-[8.0] colors of these selected EGOs in Figure 4
(note that we do not consider the flux density limitation on each IRAC band
denoted by column 12 of Tables 1 and 3 of Cyganowski et al. in our analysis).
In total, 81 and 58 EGOs with and without an associated class I methanol maser
(see Tables 1 and 6) are shown in this figure represented by red and blue
triangles, respectively. The regions of color-color space for sources at
different evolutionary Stages I, II and III, derived from the 2D radiative
transfer model by Robitaille et al. (2006) are also marked in Figure 4. We
found that most EGOs fall in the region occupied by the youngest protostar
models (Stage I), consistent with the conclusions from Cyganowski et al.
(2008). There is significant overlap in colors between sources with and
without class I methanol masers. Figure 4 also shows that many EGOs lie in the
upper-left of the color-color diagram, and outside the Stage I evolutionary
zone. One possible reason for this is that the colors of these sources are
effected by reddening. The reddening vector for Av$=$20 derived from the
Indebetouw et al. (2005) extinction law is shown in Figure 4\. An
A${}_{v}\sim$80 can produce reddening of approximately 1.4 mag in the
[3.6]-[4.5] color, which would be sufficient to return most of these source to
the Stage I region. However, such a large extinction value Av$\sim$80 is of
dubious plausibility, since the path to the Galactic center has a total
Av$\sim$25 (Indebetouw et al. 2005). Another possibility is that because the
mid-IR flux density measurements were determined from an extended region (i.e.
extended green region), with a typical scale of a few to 30′′, they may
include emission from many GLIMPSE point sources which are not physically
associated with the EGO, shifting the colors for some sources outside the
Stage I region. Moreover the integrated fluxes likely include emission
mechanisms (H2 and PAH line emission in particular) which were not included in
the Robitaille et al. models. This also may result in some sources with
extended 4.5 $\mu$m and PAH emission lying outside the Stage I region.
We have also undertaken additional similar color-color analysis using flux
measurements for all EGOs extract from the highly reliable GLIMPSE point
source catalog (rather than the less reliable GLIMPSE point source archive).
This analysis may allow us to determine whether it is possible to refine the
criteria for targeting class I methanol masers using mid-IR colors of GLIMPSE
point sources, similar to the analysis undertaken by Ellingsen (2006) for
class II methanol masers. The sample includes the 192 EGOs in the current
Mopra observations and an additional 51 EGOs listed in our previous work (Chen
et al. 2009). Although EGOs are by definition extended objects, the GLIMPSE
point catalog allows us to study the characteristics of the possible driving
source of the EGOs and class I methanol masers. To decrease the contamination
of our investigation of the EGO driving sources from chance associations, we
assumed that the driving source is the closest point source to the cataloged
EGO position (within 5$\arcsec$), with flux measurements in all four IRAC
bands. Using these criteria we identified the assumed driving GLIMPSE point
sources for 126 EGOs (including 74 associated with detected class I methanol
masers and 52 without class I methanol masers; see Tables 1 and 6). In Figure
5 we have marked these closest associated GLIMPSE point sources using red and
blue triangles for EGOs which are and are not with associated class I methanol
masers respectively. Examination of Figure 5 clearly shows that the GLIMPSE
point sources which lie closest to the EGOs are predominantly inside the
color-color region representative of Robitaille et al. evolutionary Stage I.
Qualitatively Figures 4 and 5 show similar color-color distributions, except
perhaps for a greater spread in the [3.6]-[4.5] color for the point source
data compared to the integrated fluxes. We have used the sample of 126 GLIMPSE
point sources identified in the manner outlined above to perform our
subsequent color-color analysis for EGOs with/without class I methanol masers.
It is almost certain that there are some false associations between the
GLIMPSE point sources assumed to be the driving sources and the true driving
sources of the EGOs. For example, in some cases the true driving sources for
EGOs may not appear in the GLIMPSE point source catalog, either due to
saturation, the presence of bright diffuse emission (which limits point source
extraction), or inherently extended morphology in the IRAC bands with the IRAC
resolution, e.g. due to extended PAH emission or extended H2 emission from
outflows (see Robitaille et al. 2008, Povich et al. 2009, and Povich & Whitney
2010). Moreover the EGO position cataloged by Cyganowski et al. (2008) adopted
the position of the brightest 4.5 $\mu$m emission within the extended region
of the EGO, but it is not clear that the brightest 4.5 $\mu$m emission
associated with an EGO must necessarily lie close to the driving source.
However, given the extended nature of EGOs such identifications will always be
somewhat problematic. The presence of false associations will add confusion to
attempts to identify any color-color differences between EGOs with and without
class I methanol masers; however, provided the mis-identification rate is not
too large (it is not possible to make a quantitative estimate for this at
present) it is unlikely to mask the difference completely, if it is present.
We find that the color-color regions occupied by GLIMPSE point sources which
are and are not associated with class I methanol masers are not significantly
different. For both groups they predominantly lie in a box region
-0.6$<$[5.8]-[8.0]$<$1.4 and 0.5$<$[3.6]-[4.5]$<$4.0. This color-color region
is very similar to that occupied by 6.7 GHz class II methanol masers
identified by Ellingsen (2006). But interestingly, we note that the class I
methanol masers extend to smaller [3.6]-[4.5] colors than do the class II
methanol masers for which the [3.6]-[4.5] color is usually greater than 1.3.
Similar to what is seen in Figure 4, Figure 5 shows that many GLIMPSE point
sources which are closest to an EGO lie in the upper-left of the color-color
diagram, and outside the Stage I evolutionary zone derived from 2D radiative
transfer models (Robitaille et al. 2006). Some of the GLIMPSE point sources
have redder [3.6]-[4.5] colors ($>$3), compared to those seen in Figure 4
which typically have [3.6]-[4.5] $<$ 3\. As discussed above, these sources may
be those which suffer larger reddening, however it requires extreme extinction
(A${}_{v}>100$) to produce such large color shifts ($>$2) in [3.6]-[4.5]. The
typical Av of an IRDC simply estimated from the parameters provide by Peretto
& Fuller (2009) is 25 (here we adopted an average optical depth at 8 $\mu$m of
1.15 for IRDCs, and A8μm/Av=0.045 from that work). Since most EGOs are
associated with IRDCs (Cyganowski et al. 2008), this shows that typical Av for
IRDCs can not account for the entirety of redder colors of these sources. From
our calculations of Av for the 1.1 mm BGPS sources associated with EGOs which
have class I maser detections (see Table 7, with more details described in
Section 4.3), the estimated Av of the 1.1 mm clumps associated with EGOs
outside stage I region ranges from 10 to 40 (with an average of 20), similar
to that for sources inside the Stage I region which range from 5 to 50 (with
an average of 23). This also suggests that the reddening does not play an
important role in explaining the redder colors of these sources. Another
possible reason for this is that the GLIMPSE point source photometry (similar
to the integrated flux measurements discussed above) may be affected by
contributions from extended H2 emission, while the classifications of Stage
I-III by Robitaille et al. (2006) did not consider these emission mechanisms.
The EGOs associated with GLIMPSE point sources outside Stage I region have a
significantly higher detection rate for class I methanol masers (21/28=75%;
see Tables 1 and 6) than that observed in the full sample (55%). We have also
checked the masses of 1.1 mm BGPS sources associated with EGOs which have
class I methanol maser detections (see Table 7, and the additional details
described in Section 4.3), and found that the mass range of the BGPS sources
associated EGOs outside stage I region (covering the range 1000-6000 M⊙ with
an average of 2500 M⊙) is significant higher than those associated with EGOs
in Stage I region (73-2000 M⊙, with an average of 1200 M⊙). Thus they may
correspond to MYSOs with an extremely high mass envelope which is more deeply
embedded causing redder colors. Moreover, the detection rate of class I
methanol maser in EGOs tends to be higher (30/40=75%) in the color-color
region with [3.6]-[4.5] $>$ 2.4. Given that EGOs are identified and defined by
their excess 4.5 $\mu$m emission, the high detection rate of class I masers
towards EGOs that fall in the left-upper regions make sense if the excess 4.5
$\mu$m emission is due to shocked H2 in outflows, and so sources in the left-
upper region may have particularly strong/active outflows which can readily
produce maser emission. This suggests that GLIMPSE point sources with redder
[3.6]-[4.5] color are the best target population for class I methanol maser
searches. However, the small number ($\sim$30) of class I methanol maser and
small number ($\sim$40) of GLIMPSE point sources in this region of color space
should be taken into account when drawing any conclusions.
### 4.2 An evolutionary sequence for class I and II masers
Ellingsen et al. (2007) suggested that the common maser species (class I and
II methanol, water, and OH masers) may help identify the evolutionary phase of
a high-mass star, and proposed a possible evolutionary sequence for these
common maser species. This proposed sequence has recently been refined and
quantified by Breen et al. (2010a) in their Figure 6. However, there remains
significant uncertainty about where within star formation regions the
different maser species arise and the evolutionary phase they are associated
with. In this work, we focus on the evolutionary sequence for class I and II
methanol masers. In previous work, one of the main difficulties in determining
the relative evolutionary sequence for class I and II masers has been the lack
of a large sample of sources associated with class I masers but not class II
masers. Ellingsen (2006) investigated the mid-IR colors of the associated
GLIMPSE point sources for a relatively small sample of class II methanol
masers associated with and without class I methanol masers numbering $\sim$10
for each group, and found there is a tendency for the sources with an
associated class I methanol maser to have redder GLIMPSE colors than those
without class I methanol masers. Based on the assumption that the redder
colors are associated with more deeply embedded and hence youngest stellar
objects, Ellingsen (2006) suggested that some class I methanol masers may
precede the earliest class II methanol maser evolutionary stage. However, the
absence of a comparison sample of class I methanol masers with no associated
class II methanol masers presents a significant limitation to the Ellingsen
(2006) work. Our class I methanol maser survey towards EGOs has identified 31
sources which are associated with class I, but not class II, methanol masers.
To test the proposed evolutionary scenario for class I and II methanol masers,
the EGOs were split into three subsamples on the basis of which class methanol
masers they were associated with (see Tables 1 and 6): 1) associated only with
class I methanol masers (32 members in total; 31 from our surveyed sample and
1 from Chen et al. 2009 sample); 2) associated only with class II methanol
masers (20 members in total; 10 from our surveyed sample and 10 from Chen et
al. 2009); 3) associated with both class I and class II methanol masers (72
members in total; 39 from our surveyed sample and 32 from Chen et al. 2009).
In compiling the second and third subsamples we only considered sources for
which the position of the class II maser emission is known to high accuracy
(i.e. the sources with class II methanol maser information marked by “Y” in
Tables 1 and 6). IRAC and Multiband Imaging Photometer for Spitzer (MIPS) 24
$\mu$m colors provide a diagnosis for YSO evolutionary state (Robitaille et
al. 2006). We plot [3.6]-[5.8] versus [8.0]-[24] color diagram using the flux
measurements from Tables 1 and 3 of Cyganowski et al. (2008) for the above
three subsamples (note that we do not consider the flux density limitations on
the IRAC and MIPS bands in this plot) in Figure 6 with different symbols. The
regions of color-color space for sources at different evolutionary stages I,
II and III derived from Robitaille et al. (2006) are also marked in Figure 6.
In total, we have 26 EGOs containing both class I and II methanol masers, 7
EGOs associated with only class II methanol masers and 25 EGOs associated with
only class I methanol masers in this figure. Comparing the color distributions
of these three subsamples with the color-color space occupied by the
evolutionary stages derived from Robitaille et al. (2006), we find that all
class II maser EGOs (including both class I and II maser subsample and the
only class II maser subsample) are located in the region of Stage I, i.e. the
easiest evolutionary stage, while all but one class I maser only EGOs are also
located in the Stage I region (the exception is G317.88-0.25 which lies in
Stage II). However, as seen in Figure 6, despite the significant overlap of
the various subsamples, EGOs which are associated with only class I methanol
masers extend to less red colors than those associated with only class II
methanol masers and both class I and II methanol masers.
Here we propose a number of possible explanations as to why the mid-IR sources
associated with only class I methanol masers have less red colors than those
associated with class II methanol masers:
1\. The EGOs associated with only class I maser are less heavily extincted
than those associated with class II masers.
2\. The stellar mass range of objects with associated class I methanol masers
extends to lower masses than that of objects with class II methanol masers.
The lower mass sources may be generally less deeply embedded and hence have a
less red colors, than the higher mass objects.
3\. There may be two epochs of class I methanol maser emission associated with
high-mass star formation. An early epoch which overlaps significantly with the
class II methanol maser phase and a second phase which occurs after class II
methanol maser emission has ceased.
The first possibility is supported by the evidence that the mid-IR color
differences among the three subsamples shown in Figure 6 are mostly along the
direction of the reddening vector. But it needs a very large Av to produce the
color shifts observed between the EGOs associated with only class I masers and
those associated with class II masers (e.g. A${}_{v}\sim$80 corresponds to a
color shift of 3 in [8.0]-[24]). Even though at present we can not accurately
determine Av for our full sample sources, as discussed in Secion 4.1 from
estimations of Av for a few sources associated with 1.1 mm BGPS listed in
Table 7 and the typical Av for IRDCs which are often associated with EGOs, it
appears that reddening alone is unlikely to be responsible for the observed
color differences among the three subsamples.
The second possibility is also consistent with other recent observations of
class I methanol masers. Class I methanol masers are known to be associated
with some regions that are forming only low-mass stars (Kalenskii et al. 2006,
2010). This suggests that class I methanol masers can be associated with lower
stellar mass sources than class II methanol masers and hence supports this
possibility. To further test this we have compared the detected 95 GHz class I
maser luminosity distributions of four subsamples. The subsamples of class I
masers are (a) those not associated with class II masers; (b) those associated
with class II masers for which an accurate position has been measured; (c)
those associated with an UC Hii and (d) those associated with an OH maser. The
distribution of the 95 GHz class I maser luminosity for each of these
subsamples is shown in Figure 7. This shows that most (25/31$\approx$80%) of
the sources which are not associated with class II methanol masers are located
in the lowest luminosity bin (less than 5$\times$10-6 L⊙); whereas those
associated with class II masers, UC Hii regions and OH masers have a
relatively small fraction (typically 40%) in this lowest luminosity bin. One
explanation for the observed distributions is that class I masers can be
associated with lower stellar mass sources than class II masers or the other
two tracers (OH and UC Hii), since class I maser excited by outflows from low
stars are expected to be less luminous. Recalling Section 3.3, the detection
rate of class I methanol masers in the sources without an associated class II
maser (37%) is lower than that in the sources with an associated class II
maser (80%). This statistical result also seems to support the hypothesis that
class I methanol masers can extend lower stellar mass sources since less
luminous class I masers excited from lower mass stars are harder to detect
with the same sensitivity. This hypothesis is also supported by our further
analysis of the relationship between class I methanol masers and 1.1 mm BGPS
sources (discussed in Section 4.3).
The last hypothesis is more speculative, as it requires the mechanism through
which class I methanol masers are produced to be switched off and then at a
later time on again. It is generally considered that sources with associated
OH masers and UCHii regions lie towards the later evolutionary phases (Breen
et al. 2010a). The very high rate of association we have found for class I
methanol masers towards OH masers (93%; see Table 5) demonstrates that some
class I methanol masers may be present at these later stages. However the
current single dish survey with large beam size is not sufficient to argue
that the driving source of OH maser is also responsible for exciting the class
I maser emission. Further high-resolution observations are needed to
definitively establish whether the class I masers are truly associated with
the same MYSO as the OH masers, although the results of Cyganowski et al.
(2009) suggests that most of the detected class I masers will be associated
with the targeted EGOs. Recently Voronkov et al. (2010) presented new high
resolution observations which strengthen the case that some class I methanol
masers are produced in shocks driven into molecular clouds from expanding Hii
regions. The 9.9 GHz class I methanol masers (detected towards 4 of 48 class I
maser sources observed by Voronkov et al.), are all associated with relatively
old sources, e.g. Hii regions and OH masers. They also tentatively report a
detection rate of greater than 50% for 44 GHz class I methanol masers towards
OH masers which are not associated with class II methanol masers. This
indicates that the class I masers can extend beyond the time when class II
masers are destroyed, and overlap well into the time when OH masers are
active. Voronkov et al. suggest that these findings are consistent with the
cloud-cloud collision hypothesis for class I methanol masers which has been
realized in some sources (Sobolev 1992; Mehringer & Menten 1996; Salii,
Sobolev & Kalinina 2002), but are in contrast with the generally held view of
class I methanol masers derived from sources such as DR 21(OH) (Plambeck &
Menten 1990) and G343.12-0.06 (Voronkov et al. 2006) where they are clearly
associated with outflow-molecular cloud interaction regions.
Our findings and those reported by Voronkov et al. (2010) are inconsistent
with some aspects of the evolutionary sequence presented by Breen et al.
(2010a) which has both the appearance and disappearance of class I methanol
masers preceding that of the class II methanol masers. Breen et al. also have
no overlap between the class I methanol masers and OH maser stages. In our
survey the EGOs which are associated with class I methanol masers, but not
class II methanol masers are not found to be associated with any known
published Hii regions or OH masers (actually there is absence of any systemic
surveys of Hii regions or OH masers towards EGOs at present). This suggests
that these class I methanol masers are likely to be excited by shocks driven
from outflows, rather than in the shocks driven by expanding Hii regions at
later evolutionary stage. However, the true nature of these class I only EGOs
can only definitively be resolved by high resolution observations which can
determine the location of the maser emission with respect to the EGOs. Because
high-mass star formation regions are crowded and frequently contain objects at
a range of evolutionary phases, chance associations are possible at low
angular resolutions. Examination of the results of Cyganowski et al. (2009)
for the source G28.28-0.36 illustrates how this can occur. For the majority of
the EGOs imaged by Cyganowski et al. (2009) in the 6.7 GHz class II and 44 GHz
class I methanol maser transitions, both types of masers are clearly
associated with the targeted EGOs. However, in the case of G28.28-0.36, while
the 6.7 GHz class II methanol masers are associated with the EGO, the class I
masers are offset and clearly trace the interface between an Hii region and
the surrounding molecular gas. With single dish spatial resolution both maser
transitions and the EGO would be considered likely to be coincident, although
with the benefit of high resolution data this is clearly not the case. This is
only one object from $\sim$20 observed by Cyganowski et al., so it is not
likely that all of the class I only EGO sources we have identified are chance
detections unrelated to the EGO, however, it is possible that some may be.
The evolutionary scheme outlined by Ellingsen et al. (2007) and Breen et al.
(2010a) assume that each maser species arises only once during the evolution
of an individual massive star. However, it appears that this assumption may
require revision for class I methanol masers. One possible manifestation of a
two evolutionary phase scenario for class I methanol masers (as discussed
above) would be that they initially arise at a relatively early phase of the
star formation process when powerful outflows interact with surrounding
molecular gas and that they are typically accompanied by class II maser
emission during this phase (the birthplace (disk or outflow) of class II
methanol masers remains uncertain, but at least in some sources 6.7 GHz class
II methanol masers can be excited in the inner regions of outflows e.g. De
Buizer 2003). As the source evolves and the outflows diminish the class I
maser emission fades and ceases, but the class II maser emission continues,
before it too fades rapidly soon after the creation of the UC Hii region. As
the ionized bubble rapidly expands it creates a second phase of class I maser
emission at the interface with the ambient molecular gas. Of course, there is
the possibility that the lifetime of class I masers associated with outflows
may also continue as far as the stage when Hii regions are detectable.
Molecular line observations of several massive star-forming regions show
evidence that outflows (and infall) can continue once ionization turns on, and
an UC Hii region is formed (e.g. Keto & Klaassen 2008, Chen et al. 2010).
However, high-mass star formation occurs in a cluster environment which may
include sequential or triggered star formation, allowing a YSO associated with
a well developed Hii region and a YSO associated with young outflow may
coexist as near neighbours. The extended spatial distribution of class I
masers compared to the other common maser species makes it more difficult to
determine which object the emission is associated with, outflow or expanding
Hii region or both astrophysical phenomena (particularly without high-
resolution data). At present we cannot determine whether the class I methanol
masers associated with outflows survive through to the stage when Hii region
appear, although it appears possible from our results that this is the case.
At present we cannot confidently determine why the MIR colors of class I only
sources extends to less red colors than those associated with class II masers.
It seems unlikely to be purely the result of less reddening in these sources,
but both association with lower mass stars and their being more than one epoch
of class I maser emission remain plausible hypotheses. Further observations an
millimetre and submillimetre wavelengths, combined with high resolution
observations of the class I masers will be required to answer this question.
### 4.3 The properties of mm dust clumps associated with methanol masers
Table 1 shows that there are 63 EGOs in our observations which are within the
1.1 mm continuum BGPS surveyed area (Rosolowsky et al. 2010). Among them, 54
are associated with a 1.1 mm BGPS sources, while 9 are not (see also Table 2).
A 1.1 mm BGPS source was considered to be associated with an EGO if the
separation between the peak position of the BGPS source and the EGO position
is less than 30′′. We did not take the size of the mm continuum source into
account when cross-matching (see Section 2.1).
In the two-evolutionary phase hypothesis for class I methanol masers discussed
in Section 4.2, we might expect similar trends to that seen in the Mid-IR
colors (e.g. Figure 6) to be present in other physical tracers of the source
evolution, such as the density of the associated gas and dust. For example,
Breen et al. (2010a; 2010b) suggest that the density of the associated dust
and gas decreases as the sources evolve for class II methanol masers and water
masers. So in the two-evolutionary phase hypothesis we would predict class I
methanol only sources should have a lower gas density than class II methanol
maser only sources, which in turn would have a lower density than those
sources with both class I and II methanol masers. To test this, we perform an
investigation of the properties of 1.1 mm BGPS dust clumps associated with
class I methanol masers in our surveyed sample and Chen et al. (2009) sample
(37 sources in total). For each of the associated 1.1 mm BGPS source, we have
assumed that the 1.1 mm emission detected toward EGOs is from optically thin
dust. We can then calculate the gas mass using the equation:
$M_{gas}=\frac{S_{\nu}(int)D^{2}}{\kappa_{d}B_{\nu}(T_{dust})R_{d}},$ (1)
where $S_{\nu}$(int) is the 1.1 mm continuum integrated flux density, $D$ is
the distance to the source, $\kappa_{d}$ is the mass absorption coefficient
per unit mass of dust, $B_{\nu}(T_{dust})$ is the Planck function for a
blackbody at temperature $T_{dust}$, and $R_{d}$ is the dust-to-gas mass
ratio. Here we adopt $\kappa_{d}$$=$1 cm2 g-1 (Ossenkopf & Henning 1994) for
1.1 mm emission, and assume a dust-to-gas ratio ($R_{d}$) of 1:100.
$B_{\nu}(T_{dust})$ was derived for an assumed dust temperature of 20 K. The
H2 column and volume densities of each dust clump were then derived from its
mass and radius (Robj), assuming a spherical geometry and a mean mass per
particle of $\mu=2.29$ mH, which takes into account a 10% contribution from
helium (Faúndez et al. 2004). We list the parameters of the 1.1 mm continuum
integrated flux density, $S_{\nu}$(int) and 1.1 mm source radius, Robj
obtained from the BGPS catalog (Rosolowsky et al. 2010) for all the 37 class I
maser sources with an associated 1.1 mm BGPS source in Table 7. Dunham et al.
(2010) suggested that a correction factor of 1.5 must be applied to the
Rosolowsky et al. BGPS catalog flux densities. In this paper, we also apply
this flux calibration correction factor to the integrated flux density
$S_{\nu}$(int) listed in Table 7. All the associated 1.1 mm BGPS sources are
resolved with the BGPS beam, with the exception of G34.28+0.18. For this
source we assumed half the beam size (17′′) as an upper limit for the object
radius. Thus the derived gas density of this source should be seen as a lower
limit. The masses and gas densities for all 1.1 mm dust clumps associated with
class I methanol masers determined using the methods outlined above are listed
in Table 7.
Figure 8 presents a log-log plot of the luminosity of the class I methanol
maser emission versus of the gas mass (left panel) and H2 density (right
panel) of the associated 1.1 mm dust clump. We have used different symbols in
the plot to show whether the class I masers are associated with class II
methanol masers or not (21 with associated high-precision position class II
masers; 9 without high-precision position class II maser data; 7 without
associated class II masers; see Table 7). Figure 8 shows that a weak, but
statistically significant positive correlation exists in both cases. We have
performed a linear regression analysis for each distribution, and plotted the
relevant line of best fit in each panel in Figure 8. The best fit linear
equation for each distribution is as follows:
log(Lm/Lsun)$=$0.50[0.14]log(M/Msun)$-$6.34[0.435]
(correlation coefficient of 0.60 and p-value of 8.75e-04) and,
log(Lm/Lsun)$=$0.57[0.21]log(n(H2))$-$7.25[0.75]
(correlation coefficient of 0.44 and p-value of 9e-03). These fits demonstrate
that the luminosity of the class I methanol masers depends on the physical
properties of the associated clump: the more massive and denser the clump, the
stronger the class I methanol emission. We checked for correlations between
the mass and density of clump with the distance, but found no significant
correlation in either source property (mass or density) with distance. This
suggests that the observed dependence between the 1.1 mm source properties and
class I maser luminosity is intrinsic and not an observational artefact.
Interestingly, the dependence between the luminosity of class I methanol
masers and the gas density of the associated mm dust clump is opposite to the
relationship observed between the luminosity of class II methanol masers and
the gas density of mm dust clump reported by Breen et al. (2010a). Breen et
al. found the more luminous 6.7 GHz class II methanol masers to be associated
with mm dust clumps with lower H2 density, i.e. there is a negative
correlation between them. Even though their results were derived with the peak
luminosity rather than integrated luminosity of class II methanol masers, the
peak luminosity and integrated luminosity are positively correlated. The
simplest picture which fits the different dependence between the luminosity of
class I and class II methanol masers with the gas density of the clump, is if
the intensity of class II methanol masers increases as the source
evolves/warms, while the class I maser intensity decreases as the outflow
broadens. If this is the case then we would expect to see an anti-correlation
in the class I/II flux densities. However, comparing the peak flux density of
class I and class II methanol masers for our observed EGO sample, we have
already shown that there is no statistically significant correlation between
them (see Figure 3).
We performed a similar analysis to that undertaken for the class I masers
associated with EGOs to check the relationship between the peak luminosity of
6.7 GHz class II methanol maser and the gas mass/density of the associated
dust clump for the 21 EGOs associated with both class I and II methanol masers
(listed in Table 7). We also found that there is a significant positive
correlation between the peak luminosity of the 6.7 GHz class II methanol maser
and mass of the associated dust clump (a slope of 1.21 with a standard error
of 0.32 and a p-value of 0.001 which allows us to reject the null hypothesis
of zero slope). The correlation coefficient between the points was measured to
be 0.66 for this linear regression analysis. But there is no statistical
correlation between the peak luminosity of 6.7 GHz maser and the gas density
(the linear regression analysis shows a slope of -0.38 with a large p-value of
0.57 and a small correlation coefficient of 0.15). The absence of correlation
between the peak luminosity of 6.7 GHz masers and gas density measured in our
analysis is not consistent with that of the anti-correlation between these
quantities found by Breen et al. (2010a). However the sample size for this
analysis is small and the class II masers are clustered in a small range of
parameter space with lower gas density (log(n(H2))$<$4), whereas the sample
shown in Figure 2 of Breen et al. covers a much wider range of gas densities.
Thus our class II maser sample is likely not representative of the larger
population. The correlation between the class I maser luminosity and gas
density is tighter than that measured for the 6.7 class II masers in our
analysis. The most likely explanation for this is that the gas density of the
clump is measured over a large spatial scale (the angular resolution of the
observations was typically 30′′), which can not accurately reflect the
properties of the smaller compact regions (of the order of 1′′) associated
with class II methanol masers. In contrast the class I methanol maser spots
usually have much larger angular and spatial distributions (usually at the
order of 10′′) and are associated with shocked regions (e.g. Cyganowski et al.
2009). Thus we might expect that the gas densities derived from dust clumps
should correlate more closely with class I methanol maser properties. The
dependence of class I maser luminosity on clump properties makes sense if the
class I masers are excited at the interface between outflows and surrounding
material. Future arcsecond resolution mm continuum imaging of these sources
will be necessary to determine if the relationship between the class II maser
properties and the associated gas and dust is tighter when the spatial scales
are comparable.
It is interesting to note that all of the sources associated with only class I
methanol masers are located in the bottom-left corner of the right panel in
Figure 8. This location corresponds to sources with a lower maser luminosity
and lower density of the dust clump. The left panel of Figure 8 shows that
median mass properties are also different between the population associated
with only class I masers and that associated with both class I and II masers:
the population associated with both class I and II masers can extend to higher
clump masses than that associated with only class I masers. The most widely
accepted mechanism for massive star formation suggests that high-mass sources
are believed to originate from massive clumps in the fragmentation of the
giant molecular cloud. The stellar mass of the sources is set by the
fragmentation process and the reservoir of material available to accrete is
determined by that as well (e.g. Hennebelle & Chabrier 2008). Therefore the
measured distribution of 1.1 mm clump mass among different populations
indicates that these class II methanol maser EGO sources may be associated
with a higher stellar mass range than those where only class I methanol masers
are also observed. However, it is also consistent with the predictions of the
two-evolutionary phase hypothesis for the class I only sources as the density
of the associated gas and dust decreases as the sources evolve (Breen et al.
2010a; 2010b). So in the two-evolutionary phase hypothesis the class I only
sources (i.e. the later evolutionary sources) should have a lower density than
sources with only class II masers or with both class I and II methanol masers
(see the more detailed discussions of this in Section 4.2). The assumed
constant dust temperature (Tdust) of 20 K in our calculations will affect the
results. However if the sources which are thought to be more evolved (i.e.
only class I maser sources) have a higher dust temperature, it would result in
a larger $B_{\nu}(T_{dust})$, thus a smaller mass and lower density of dust
clump associated with only class I masers. So this is also credible in the
two-evolutionary phase hypothesis.
However it must be noted that only 7 of the 31 class I maser-only sources have
estimates for the mass and density parameters from mm dust continuum
observations. The small size of this sample means that it may not be
representative of the entire population of class I maser-only EGOs. Moreover
some sources associated with both class I and class II methanol masers also
extend to the left-bottom corner of each panel in Figure 8. This suggests that
a small fractional of the class II methanol maser population can also appear
in an environment of comparable mass and dust clump density to that seen in
the class I only associated EGOs. However, Figure 8 clearly shows that the
class II methanol masers are usually associated with more massive and dense
dust clumps than those associated with only class I methanol masers.
## 5 Conclusions
Using the Mopra telescope, we have performed a systematic search for 95 GHz
class I methanol masers toward EGOs. EGOs are new MYSO candidates with ongoing
outflows identified from the _Spitzer_ GLIMPSE I survey. We detected 105 new
95 GHz masers from a sample of 192 targets. Of these, 92 have no previously
observed class I methanol maser activity, while the remaining 13 sources have
been detected in the 44 GHz transition. Thus our single-dish survey proves
that there is indeed a high detection rate ($\sim$55%) of class I methanol
masers in EGOs. Our findings increase the number of published class I methanol
masers to 290 (an additional 92 on top of the $\sim$198 from Val’tts & Larinov
2010). Mid-IR color analysis shows that the color-color region occupied by the
GLIMPSE point sources for EGOs which are and are not associated with class I
methanol masers are very similar, and mostly located in ranges
-0.6$<$[5.8]-[8.0]$<$1.4 and 0.5$<$[3.6]-[4.5]$<$4.0 (see section 4.1 for
detailed discussion of the uncertainties involved in this analysis). We find
that the detection rate of class I methanol maser is likely to be higher in
those sources with redder GLIMPSE point source colors.
Comparison of the [3.6]-[5.8] vs. [8.0]-[24] colors determined with integrated
fluxes from Cyganowski et al. (2008) for the subsamples of the EGOs based on
which class of methanol masers they are associated with, shows that those
which are only associated class I methanol masers extend to less red colors
than those associated with both classes of methanol maser. We suggest that the
less red colors of class I methanol maser only EGOs is either because the
class I only EGOs are associated with lower stellar mass objects, or because
class I maser emission arises at more than one evolutionary phase of the high-
mass star formation process. On the basis of current observations both
scenarios can be plausibly argued and further observations will be required to
determine which, if either of these hypotheses is correct. The thermal
molecular line observations taken in conjunction with our maser search will be
useful for trying to determine which of these scenarios is more likely. It
will also be important to undertake high resolution observations of the class
I maser emission in EGOs which are only associated with class I methanol
masers to determine where the maser emission arises relative to the EGO. These
observations are required to rule out the possibility that these sources
represent a sample of chance associations between the EGOs and class I masers.
Analysis of the properties of mm dust clumps associated with class I methanol
masers (for a subset of the EGOs in the class I maser survey sample which have
available millimeter continuum data) shows that the luminosity of the class I
methanol masers is correlated with the both the mass and density of the
associated dust clump. The more massive and denser the clump, the stronger the
class I methanol emission will be. We also find that the EGOs which are only
associated with class I methanol masers have a lower maser luminosity and
mass/density of dust clump. This finding supports either the hypothesis that
the class I maser can trace a population with lower stellar masses, or that
class I methanol masers may be associated with more than one evolutionary
phase during the formation of a high-mass star.
We thank Dr. Karl Menten and an anonymous referee for their helpful comments
on this paper. We are grateful to the staff of the ATNF for their assistance
in the observation. The Mopra telescope is operated through a collaborative
arrangement between the University of New South Wales and the CSIRO. This
research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France, and the data products from the GLIMPSE survey, which is a legacy
science program of the Spitzer Space Telescope, funded by the National
Aeronautics and Space Administration. This work was supported in part by the
National Natural Science Foundation of China (grants 11073041, 10803017,
10625314, 10633010 and 10821302), the Knowledge Innovation Program of the
Chinese Academy of Sciences (Grant No. KJCX2-YW-T03), the National Key Basic
Research Development Program of China (No. 2007CB815405), and the CAS/SAFEA
International Partnership Program for Creative Research Teams.
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Table 1: Sample parameters
| Positiona | | | | | | |
---|---|---|---|---|---|---|---|---
Source | R.A. (2000) | Dec. (2000) | IRDCb | Class IIc | OHd | UC Hiid | mm d | Remarke | class If
| (h m s) | (° ′ ″) | | | | | | |
G10.29$-$0.13l | 18 08 49.3 | $-$20 05 57 | Y | Y | – | N | Y | 2 | Y
G10.34$-$0.14l | 18 09 00.0 | $-$20 03 35 | Y | Y | – | N | Y | 2 | Y
G11.11$-$0.11${}^{i^{\ast}}$${}^{m^{\dagger}}$ | 18 10 28.3 | $-$19 22 31 | Y | Y | – | N | Y | 3 | N
G11.92$-$0.61g${}^{h^{\ast}}$${}^{l^{\dagger}}$ | 18 13 58.1 | $-$18 54 17 | Y | Y | – | – | – | 1 | Y
G12.02$-$0.21ik | 18 12 40.4 | $-$18 37 11 | Y | N | – | N | Y | 1 | N
G12.20$-$0.03jl | 18 12 23.6 | $-$18 22 54 | N | Y | – | Y | Y | 4 | Y
G12.42$+$0.50n | 18 10 51.1 | $-$17 55 50 | N | N | – | – | Y | 4 | Y
G12.68$-$0.18l | 18 13 54.7 | $-$18 01 47 | N | Y | Y | N | Y | 4 | Y
G12.91$-$0.03${}^{h^{\ast}}$${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 13 48.2 | $-$17 45 39 | Y | Y | – | N | Y | 1 | Y
G12.91$-$0.26l | 18 14 39.5 | $-$17 52 00 | N | Y | Y | N | Y | 5 | Y
G14.63$-$0.58hjl | 18 19 15.4 | $-$16 30 07 | Y | Y | – | – | Y | 1 | Y
G16.58$-$0.08k | 18 21 15.0 | $-$14 33 02 | Y | N | – | N | Y | 4 | N
G17.96$+$0.08k | 18 23 21.0 | $-$13 15 11 | N | N | – | N | Y | 4 | N
IRDC18223$-$3n | 18 25 08.5 | $-$12 45 23 | Y | N | – | N | Y | – | Y
G18.67$+$0.03h${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 24 53.7 | $-$12 39 20 | N | Y | – | N | Y | 1 | Y
G18.89$-$0.47h${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 27 07.9 | $-$12 41 36 | Y | Y | – | – | Y | 1 | Y
G19.01$-$0.03h${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 25 44.8 | $-$12 22 46 | Y | Y | – | N | Y | 1 | Y
G19.36$-$0.03l | 18 26 25.8 | $-$12 03 57 | Y | Y | – | Y | Y | 2 | Y
G19.61$-$0.12m | 18 27 13.6 | $-$11 53 20 | N | Y | – | N | N | 2 | N
G19.88$-$0.53hj${}^{l^{\dagger}}$ | 18 29 14.7 | $-$11 50 23 | Y | Y | – | – | Y | 1 | Y
G20.24$+$0.07g${}^{k^{\ast}}$m | 18 27 44.6 | $-$11 14 54 | N | Y | – | N | Y | 4 | N
G21.24$+$0.19k | 18 29 10.2 | $-$10 18 11 | Y | N | – | N | Y | 4 | N
G22.04$+$0.22${}^{h^{\ast}}$${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 30 34.7 | $-$09 34 47 | Y | Y | – | N | Y | 1 | Y
G23.82$+$0.38 | 18 33 19.5 | $-$07 55 37 | N | – | – | N | Y | 4 | Y
G23.96$-$0.11${}^{h^{\ast}}$${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 35 22.3 | $-$08 01 28 | N | Y | – | N | Y | 1 | Y
G24.00$-$0.10${}^{h^{\ast}}$${}^{j^{\ast}}$${}^{l^{\dagger}}$ | 18 35 23.5 | $-$07 59 32 | Y | Y | – | N | Y | 1 | Y
G24.11$-$0.17n | 18 35 52.6 | $-$07 55 17 | Y | N | – | N | Y | 4 | Y
G24.17$-$0.02i | 18 35 25.0 | $-$07 48 15 | Y | – | – | N | N | 1 | N
G24.33$+$0.14g${}^{j^{\ast}}$l | 18 35 08.1 | $-$07 35 04 | Y | Y | – | N | Y | 4 | Y
G24.63$+$0.15${}^{h^{\ast}}$j${}^{n^{\dagger}}$ | 18 35 40.1 | $-$07 18 35 | Y | N | – | N | Y | 3 | Y
G24.94$+$0.07h${}^{l^{\dagger}}$ | 18 36 31.5 | $-$07 04 16 | N | Y | – | N | Y | 1 | Y
G25.27$-$0.43${}^{h^{\ast}}$j${}^{l^{\dagger}}$ | 18 38 57.0 | $-$07 00 48 | Y | Y | – | – | Y | 1 | Y
G25.38$-$0.15j | 18 38 08.1 | $-$06 46 53 | Y | – | – | N | Y | 2 | Y
G27.97$-$0.47h${}^{n^{\dagger}}$ | 18 44 03.6 | $-$04 38 02 | Y | N | – | – | Y | 1 | Y
G28.28$-$0.36m | 18 44 13.2 | $-$04 18 04 | N | Y | – | Y | Y | 2 | N
G28.85$-$0.23m | 18 44 47.5 | $-$03 44 15 | N | Y | – | N | N | 4 | N
G29.89$-$0.77 | 18 48 37.7 | $-$03 03 44 | Y | N | – | – | N | 4 | N
G29.91$-$0.81 | 18 48 47.6 | $-$03 03 31 | N | N | – | – | Y | 4 | N
G29.96$-$0.79ik | 18 48 50.0 | $-$03 00 21 | Y | N | – | – | Y | 3 | N
G34.26$+$0.15g | 18 53 16.4 | $+$01 15 07 | N | – | – | Y | Y | 5 | Y
G34.28$+$0.18h | 18 53 15.0 | $+$01 17 11 | Y | – | – | N | Y | 3 | Y
G34.39$+$0.22${}^{j^{\ast}}$ | 18 53 19.0 | $+$01 24 08 | Y | – | – | N | N | 2 | Y
G34.41$+$0.24h${}^{j^{\ast}}$ | 18 53 17.9 | $+$01 25 25 | Y | – | – | N | Y | 1 | Y
G35.04$-$0.47hj${}^{n^{\dagger}}$ | 18 56 58.1 | $+$01 39 37 | Y | N | – | – | Y | 1 | Y
G35.13$-$0.74h | 18 58 06.4 | $+$01 37 01 | N | – | – | – | – | 1 | Y
G35.15$+$0.80i | 18 52 36.6 | $+$02 20 26 | N | N | – | – | – | 1 | N
G35.20$-$0.74h${}^{l^{\dagger}}$ | 18 58 12.9 | $+$01 40 33 | N | Y | – | – | – | 1 | Y
G35.68$-$0.18ik | 18 57 05.0 | $+$02 22 00 | Y | N | – | N | Y | 1 | N
G35.79$-$0.17hj | 18 57 16.7 | $+$02 27 56 | Y | – | – | N | Y | 1 | Y
G36.01$-$0.20${}^{i^{\ast}}$${}^{k^{\ast}}$ | 18 57 45.9 | $+$02 39 05 | Y | – | – | N | Y | 1 | N
G37.48$-$0.10ik${}^{m^{\dagger}}$ | 19 00 07.0 | $+$03 59 53 | N | Y | – | N | Y | 1 | N
G37.55$+$0.20 | 18 59 07.5 | $+$04 12 31 | N | – | – | N | N | 5 | N
G39.10$+$0.49${}^{h^{\ast}}$j${}^{l^{\dagger}}$ | 19 00 58.1 | $+$05 42 44 | N | Y | – | – | Y | 1 | Y
G39.39$-$0.14 | 19 03 45.3 | $+$05 40 43 | N | – | – | Y | Y | 4 | Y
G40.28$-$0.22hj | 19 05 41.3 | $+$06 26 13 | Y | – | – | – | Y | 3 | Y
G40.28$-$0.27ik | 19 05 51.5 | $+$06 24 39 | Y | N | – | – | Y | 1 | N
G40.60$-$0.72jn | 19 08 03.3 | $+$06 29 15 | N | N | – | – | – | 4 | Y
G44.01$-$0.03ik | 19 11 57.2 | $+$09 50 05 | N | N | – | – | N | 1 | N
G45.47$+$0.05gi | 19 14 25.6 | $+$11 09 28 | Y | – | – | Y | Y | 1 | N
G45.50$+$0.12k | 19 14 13.0 | $+$11 13 30 | N | – | – | – | N | 4 | N
G45.80$-$0.36h | 19 16 31.1 | $+$11 16 11 | N | – | – | – | Y | 3 | Y
G48.66$-$0.30 | 19 21 48.0 | $+$13 49 21 | Y | N | – | – | Y | 2 | N
G49.07$-$0.33${}^{h^{\ast}}$j${}^{n^{\dagger}}$ | 19 22 41.9 | $+$14 10 12 | Y | N | – | – | Y | 3 | Y
G49.42$+$0.33m | 19 20 59.1 | $+$14 46 53 | N | Y | – | – | Y | 2 | N
G53.92$-$0.07i | 19 31 23.0 | $+$18 33 00 | N | N | – | – | Y | 3 | N
G54.11$-$0.04k | 19 31 40.0 | $+$18 43 53 | N | N | – | – | Y | 4 | N
G54.11$-$0.08ik | 19 31 48.8 | $+$18 42 57 | N | N | – | – | Y | 3 | N
G54.45$+$1.01hj${}^{n^{\dagger}}$ | 19 28 26.4 | $+$19 32 15 | N | N | – | – | – | 3 | Y
G57.61$+$0.02 | 19 38 40.8 | $+$21 49 35 | N | – | – | – | N | 4 | N
G59.79$+$0.63hj | 19 41 03.1 | $+$24 01 15 | Y | – | – | – | – | 1 | Y
G298.90$+$0.36ik | 12 16 43.2 | $-$62 14 25 | N | N | – | – | – | 1 | N
G304.89$+$0.64ik | 13 08 12.1 | $-$62 10 22 | Y | N | – | – | – | 3 | N
G305.48$-$0.10h | 13 13 45.8 | $-$62 51 28 | N | – | – | – | – | 1 | Y
G305.52$+$0.76${}^{h^{\ast}}$j${}^{n^{\dagger}}$ | 13 13 29.3 | $-$61 59 53 | Y | N | – | – | – | 1 | Y
G305.57$-$0.34i${}^{k^{\ast}}$ | 13 14 49.1 | $-$63 05 38 | Y | – | – | – | – | 1 | N
G305.62$-$0.34i | 13 15 11.5 | $-$63 05 30 | N | – | – | – | – | 1 | N
G305.77$-$0.25 | 13 16 30.0 | $-$62 59 09 | N | N | – | – | – | 4 | N
G305.80$-$0.24gjl | 13 16 43.4 | $-$62 58 29 | N | Y | Y | Y | – | 4 | Y
G305.82$-$0.11hj | 13 16 48.6 | $-$62 50 35 | Y | – | – | – | – | 1 | Y
G305.89$+$0.02h${}^{l^{\dagger}}$ | 13 17 15.5 | $-$62 42 24 | Y | Y | – | – | – | 1 | Y
G309.15$-$0.35ik | 13 45 51.3 | $-$62 33 46 | Y | N | – | – | – | 1 | N
G309.90$+$0.23k | 13 51 00.4 | $-$61 49 53 | Y | – | – | – | – | 2 | N
G309.91$+$0.32${}^{h^{\ast}}$j | 13 50 53.9 | $-$61 44 22 | Y | – | – | – | – | 3 | Y
G309.97$+$0.50k | 13 51 05.2 | $-$61 33 20 | Y | – | – | – | – | 4 | N
G309.97$+$0.59ik | 13 50 52.6 | $-$61 27 46 | Y | – | – | – | – | 3 | N
G309.99$+$0.51i${}^{k^{\ast}}$ | 13 51 12.2 | $-$61 32 09 | Y | – | – | – | – | 1 | N
G310.08$-$0.23i | 13 53 23.0 | $-$62 14 13 | Y | N | – | – | – | 1 | N
G310.15$+$0.76l | 13 51 59.2 | $-$61 15 37 | N | Y | – | – | – | 4 | Y
G310.38$-$0.30n | 13 56 01.0 | $-$62 14 19 | Y | N | – | – | – | 4 | Y
G311.04$+$0.69n | 13 59 18.1 | $-$61 06 33 | Y | N | – | – | – | 4 | Y
G311.51$-$0.45hj${}^{n^{\dagger}}$ | 14 05 46.1 | $-$62 04 49 | Y | N | – | – | – | 3 | Y
G312.11$+$0.26ik${}^{m^{\dagger}}$ | 14 08 49.3 | $-$61 13 25 | Y | Y | N | – | – | 1 | N
G313.71$-$0.19m | 14 22 37.4 | $-$61 08 17 | Y | Y | Y | – | – | 4 | N
G313.76$-$0.86l | 14 25 01.3 | $-$61 44 57 | Y | Y | Y | Y | – | 4 | Y
G317.44$-$0.37 | 14 51 03.0 | $-$59 49 58 | Y | – | N | – | – | 4 | N
G317.46$-$0.40ik | 14 51 19.6 | $-$59 50 51 | Y | – | N | – | – | 1 | N
G317.87$-$0.15hj${}^{n^{\dagger}}$ | 14 53 16.3 | $-$59 26 36 | Y | N | N | – | – | 1 | Y
G317.88$-$0.25hj${}^{n^{\dagger}}$ | 14 53 43.5 | $-$59 31 35 | Y | N | N | – | – | 1 | Y
G320.23$-$0.28gl | 15 09 52.6 | $-$58 25 36 | Y | Y | Y | – | – | 2 | Y
G321.94$-$0.01hj${}^{n^{\dagger}}$ | 15 19 43.3 | $-$57 18 06 | Y | N | N | – | – | 1 | Y
G324.11$+$0.44ik | 15 31 05.0 | $-$55 43 39 | Y | N | N | – | – | 3 | N
G324.17$+$0.44 | 15 31 24.6 | $-$55 41 30 | N | N | N | – | – | 2 | N
G324.19$+$0.41hj${}^{n^{\dagger}}$ | 15 31 38.0 | $-$55 42 36 | N | N | N | – | – | 1 | Y
G325.52$+$0.42 | 15 39 10.6 | $-$54 55 40 | N | N | N | – | – | 4 | N
G326.27$-$0.49i | 15 47 10.8 | $-$55 11 12 | Y | N | N | – | – | 1 | N
G326.31$+$0.90ik | 15 41 35.9 | $-$54 03 42 | Y | N | N | – | – | 1 | N
G326.32$-$0.39${}^{j^{\ast}}$ | 15 47 04.8 | $-$55 04 51 | N | – | N | – | – | 2 | Y
G326.36$+$0.88ik | 15 41 55.4 | $-$54 02 55 | Y | – | N | – | – | 3 | N
G326.37$+$0.94 | 15 41 44.1 | $-$54 00 00 | Y | N | N | – | – | 4 | N
G326.41$+$0.93hj | 15 41 59.4 | $-$53 59 03 | Y | – | N | – | – | 3 | Y
G326.57$+$0.20ik | 15 45 53.4 | $-$54 27 50 | N | N | N | – | – | 3 | N
G326.61$+$0.80hj${}^{n^{\dagger}}$ | 15 43 36.2 | $-$53 57 51 | Y | N | N | – | – | 3 | Y
G326.78$-$0.24gi | 15 48 55.2 | $-$54 40 37 | N | N | N | – | – | 1 | N
G326.79$+$0.38ik | 15 46 20.9 | $-$54 10 45 | Y | N | N | – | – | 1 | N
G326.80$+$0.51ik | 15 45 48.6 | $-$54 04 30 | Y | N | N | – | – | 3 | N
G326.92$-$0.31hj${}^{n^{\dagger}}$ | 15 49 56.2 | $-$54 38 29 | Y | N | N | – | – | 3 | Y
G326.97$-$0.03hj${}^{n^{\dagger}}$ | 15 49 03.2 | $-$54 23 37 | Y | N | N | – | – | 1 | Y
G327.57$-$0.85 | 15 55 47.3 | $-$54 39 09 | Y | – | N | – | – | 4 | Y
G327.65$+$0.13 | 15 52 00.5 | $-$53 50 41 | N | N | N | – | – | 4 | N
G327.72$-$0.38h | 15 54 32.3 | $-$54 11 55 | N | – | N | – | – | 3 | Y
G327.86$+$0.19ik | 15 52 49.2 | $-$53 40 07 | N | N | N | – | – | 3 | N
G327.89$+$0.15hj${}^{n^{\dagger}}$ | 15 53 10.3 | $-$53 40 28 | Y | N | N | – | – | 3 | Y
G328.16$+$0.59jn | 15 52 42.5 | $-$53 09 51 | N | N | N | – | – | 4 | Y
G328.55$+$0.27hj${}^{n^{\dagger}}$ | 15 56 01.5 | $-$53 09 44 | Y | N | N | – | – | 3 | Y
G328.60$+$0.27k | 15 56 15.8 | $-$53 07 50 | Y | N | N | – | – | 4 | N
G329.16$-$0.29hj | 16 01 33.6 | $-$53 11 15 | Y | – | N | – | – | 3 | Y
G329.47$+$0.52${}^{h^{\ast}}$${}^{j^{\ast}}$ | 15 59 36.6 | $-$52 22 55 | Y | – | N | – | – | 1 | Y
G330.88$-$0.37gl | 16 10 19.9 | $-$52 06 13 | Y | Y | Y | Y | – | 2 | Y
G331.08$-$0.47ik | 16 11 46.9 | $-$52 02 31 | Y | N | N | – | – | 3 | N
G331.12$-$0.46k | 16 11 55.3 | $-$52 00 10 | Y | N | N | – | – | 4 | N
G331.37$-$0.40hj | 16 12 48.1 | $-$51 47 30 | Y | – | N | – | – | 3 | Y
G331.51$-$0.34 | 16 13 11.7 | $-$51 39 12 | Y | – | N | – | – | 4 | N
G331.62$+$0.53h${}^{n^{\dagger}}$ | 16 09 56.8 | $-$50 56 25 | N | N | N | – | – | 3 | Y
G331.71$+$0.58hj | 16 10 06.3 | $-$50 50 29 | Y | – | N | – | – | 3 | Y
G331.71$+$0.60hj | 16 10 01.9 | $-$50 49 33 | Y | – | N | – | – | 3 | Y
G332.12$+$0.94${}^{k^{\ast}}$ | 16 10 30.4 | $-$50 18 05 | Y | N | N | – | – | 2 | N
G332.28$-$0.07ik | 16 15 35.1 | $-$50 55 36 | Y | – | N | – | – | 3 | N
G332.28$-$0.55${}^{j^{\ast}}$n | 16 17 41.8 | $-$51 16 04 | Y | N | N | – | – | 4 | Y
G332.33$-$0.12ik | 16 16 03.3 | $-$50 55 34 | Y | N | N | – | – | 3 | N
G332.35$-$0.12h${}^{l^{\dagger}}$ | 16 16 07.0 | $-$50 54 30 | Y | Y | Y | – | – | 1 | Y
G332.36$+$0.60hj${}^{n^{\dagger}}$ | 16 13 02.4 | $-$50 22 39 | Y | N | N | – | – | 3 | Y
G332.47$-$0.52i | 16 18 26.5 | $-$51 07 12 | Y | N | N | – | – | 3 | N
G332.58$+$0.15hj | 16 16 00.6 | $-$50 33 30 | Y | – | N | – | – | 3 | Y
G332.59$+$0.04ik | 16 16 30.1 | $-$50 37 50 | Y | N | N | – | – | 3 | N
G332.81$-$0.70hj | 16 20 48.1 | $-$51 00 15 | N | – | N | – | – | 1 | Y
G332.91$-$0.55${}^{h^{\ast}}$j${}^{n^{\dagger}}$ | 16 20 32.6 | $-$50 49 46 | Y | N | N | – | – | 3 | Y
G333.08$-$0.56i${}^{k^{\ast}}$ | 16 21 20.9 | $-$50 43 05 | Y | – | N | – | – | 3 | N
G333.32$+$0.10${}^{j^{\ast}}$l | 16 19 28.9 | $-$50 04 40 | Y | Y | Y | – | – | 4 | Y
G334.04$+$0.35ik | 16 21 36.9 | $-$49 23 28 | N | N | N | – | – | 3 | N
G335.43$-$0.24${}^{i^{\ast}}$k | 16 30 05.8 | $-$48 48 44 | Y | – | N | – | – | 3 | N
G335.59$-$0.30j | 16 31 02.5 | $-$48 44 07 | Y | – | N | – | – | 4 | Y
G336.02$-$0.83l | 16 35 09.7 | $-$48 46 44 | N | Y | Y | Y | – | 4 | Y
G336.03$-$0.82 | 16 35 09.6 | $-$48 45 55 | N | – | N | Y | – | 4 | Y
G336.87$+$0.29ik | 16 33 40.3 | $-$47 23 32 | Y | N | N | – | – | 3 | N
G336.96$-$0.98hj${}^{n^{\dagger}}$ | 16 39 37.5 | $-$48 10 58 | Y | N | N | – | – | 3 | Y
G337.16$-$0.39i | 16 37 49.6 | $-$47 38 50 | Y | – | N | – | – | 3 | N
G337.30$-$0.87${}^{h^{\ast}}$ | 16 40 31.3 | $-$47 51 31 | Y | – | N | – | – | 1 | Y
G338.32$-$0.41${}^{h^{\ast}}$j | 16 42 27.5 | $-$46 46 57 | Y | – | N | – | – | 3 | Y
G338.39$-$0.40${}^{i^{\ast}}$ | 16 42 41.2 | $-$46 43 40 | Y | – | N | – | – | 1 | N
G338.42$-$0.41${}^{k^{\ast}}$ | 16 42 50.5 | $-$46 42 29 | Y | – | N | – | – | 4 | N
G339.58$-$0.13l | 16 45 59.5 | $-$45 38 44 | Y | Y | N | – | – | 4 | Y
G339.95$-$0.54h${}^{l^{\dagger}}$ | 16 49 07.9 | $-$45 37 59 | N | Y | N | – | – | 1 | Y
G340.05$-$0.25l | 16 48 14.7 | $-$45 21 52 | Y | Y | Y | Y | – | 4 | Y
G340.07$-$0.24 | 16 48 15.1 | $-$45 20 57 | Y | – | N | – | – | 4 | N
G340.10$-$0.18 | 16 48 07.0 | $-$45 17 06 | N | – | N | – | – | 4 | N
G340.75$-$1.00ik | 16 54 04.0 | $-$45 18 50 | Y | N | N | – | – | 3 | N
G340.77$-$0.12ik | 16 50 17.5 | $-$44 43 54 | Y | – | N | – | – | 3 | N
G340.97$-$1.02${}^{h^{\ast}}$${}^{j^{\ast}}$ | 16 54 57.3 | $-$45 09 04 | Y | – | N | – | – | 1 | Y
G341.20$-$0.26ik | 16 52 27.8 | $-$44 29 29 | Y | – | N | – | – | 3 | N
G341.22$-$0.26 | 16 52 32.2 | $-$44 28 38 | Y | – | N | – | – | 2 | Y
G341.24$-$0.27hj | 16 52 37.3 | $-$44 28 09 | Y | – | N | – | – | 1 | Y
G341.99$-$0.10${}^{i^{\ast}}$ | 16 54 32.8 | $-$43 46 45 | Y | N | N | – | – | 1 | N
G342.04$+$0.43 | 16 52 27.8 | $-$43 24 17 | N | N | N | – | – | 4 | N
G342.15$+$0.51hj${}^{n^{\dagger}}$ | 16 52 28.3 | $-$43 16 08 | N | N | N | – | – | 3 | Y
G342.48$+$0.18l | 16 55 02.6 | $-$43 13 01 | Y | Y | N | – | – | 2 | Y
G343.19$-$0.08ik | 16 58 34.9 | $-$42 49 46 | Y | N | N | – | – | 3 | N
G343.40$-$0.40ik | 17 00 40.4 | $-$42 51 33 | Y | N | N | – | – | 3 | N
G343.42$-$0.37k | 17 00 37.4 | $-$42 49 40 | Y | N | N | – | – | 4 | N
G343.50$+$0.03h${}^{j^{\ast}}$${}^{n^{\dagger}}$ | 16 59 10.7 | $-$42 31 07 | Y | N | N | – | – | 3 | Y
G343.50$-$0.47hj | 17 01 18.4 | $-$42 49 36 | N | – | N | – | – | 1 | Y
G343.53$-$0.51j | 17 01 33.5 | $-$42 49 50 | Y | – | N | – | – | 4 | Y
G343.72$-$0.18i | 17 00 48.3 | $-$42 28 25 | Y | – | N | – | – | 1 | N
G343.78$-$0.24${}^{i^{\ast}}$ | 17 01 13.1 | $-$42 27 48 | Y | N | N | – | – | 3 | N
G344.21$-$0.62ik | 17 04 17.8 | $-$42 21 09 | N | N | N | – | – | 3 | N
G345.13$-$0.17jl | 17 05 23.1 | $-$41 21 11 | N | Y | N | – | – | 4 | Y
G345.72$+$0.82ik | 17 03 06.4 | $-$40 17 09 | Y | N | N | – | – | 1 | N
G345.99$-$0.02${}^{h^{\ast}}$${}^{l^{\dagger}}$ | 17 07 27.6 | $-$40 34 45 | N | Y | N | – | – | 1 | Y
G346.04$+$0.05m | 17 07 19.9 | $-$40 29 49 | N | Y | N | – | – | 4 | N
G348.17$+$0.46hj${}^{n^{\dagger}}$ | 17 12 10.9 | $-$38 31 59 | N | N | N | – | – | 3 | Y
G348.55$-$0.98h${}^{l^{\dagger}}$ | 17 19 20.9 | $-$39 03 55 | N | Y | Y | – | – | 1 | Y
G348.58$-$0.92h${}^{l^{\dagger}}$ | 17 19 10.7 | $-$39 00 23 | Y | Y | Y | – | – | 1 | Y
G348.73$-$1.04gl | 17 20 06.5 | $-$38 57 08 | Y | Y | Y | Y | – | 4 | Y
aafootnotetext: The targeted positions for the observations are the EGO
positions given in the EGO catalog of Cyganowski et al. (2008), with the
exception of IRDC18223$-$3, the position for which is from Beuther &
Steinacker (2007).
bbfootnotetext: Association with IR dark clouds, : Y = Yes, N = No, given by
Cyganowski et al. (2008).
ccfootnotetext: Associations with 6.7 GHz class II methanol masers within
30″identified from the 6.7 GHz maser catalogs (Cyganowski et al. 2009; Caswell
2009; Xu et al. 2009; Caswell et al. 2010; Green et al. 2010) and our recent
class II methanol maser surveys with the University of Tasmania Mt. Pleasant
26 m (Titmarsh et al. in prep.): “–” are sources for which 6.7 GHz maser
emission is detected in the Mt Pleasant survey, but for which accurate
positional information is not available, “Y” are sources with 6.7 GHz masers
with accurate positions from high-resolution observations, “N” are sources
without 6.7 GHz maser detections in the Mt Pleasant survey.
ddfootnotetext: Associations with OH masers, UC Hii regions and 1.1 mm
continuum sources within 30″: Y = Yes, N = No,“–” = no information, identified
from 1665 and 1667 MHz OH maser catalog (Caswell 1998), UC Hii catalogs (Wood
& Churchwell 1989; Becker et al. 1994; Kurtz et al. 1994; Walsh et al. 1998;
Cyganowski et al. 2009), and 1.1 mm continuum BOLOCAM GPS (BGPS) archive
(Rosolowsky et al. 2010).
eefootnotetext: Remarks: 1 – 5 represent that the sources are selected from
Tables 1 – 5 of Cyganowski et al. (2008), respectively. The source
IRDC18223$-$3 is selected from Beuther & Steinacker (2007). Sources from
tables 1, 2, and 5 and IRDC18223$-$3 are classified as “likely” outflow
candidates, while those from tables 3 and 4 are classified as ”possible”
outflow candidates in our analysis.
fffootnotetext: Detections with 95 GHz class I methanol masers in our
observations: Y = Yes, N = No.
ggfootnotetext: The sources have been included in Chen et al. sample (10
members in total).
h, ih, ifootnotetext: Represent the sources with and without 95 GHz class I
methanol maser detections in our survey respectively which are shown in the
[3.6]-[4.5] vs. [5.8]-[8.0] color-color diagram (Figure 4) obtained with
integrated fluxes from Table 1 and 3 of Cyganowski et al. (2008). A “$\ast$”
in the superscript marks the source which locates in the left-upper of the
color region and outside Stage I evolutionary zone derived by Robitaille et
al. (2006) (see Section 4.1 and Figure 4).
j, kj, kfootnotetext: Represent the sources with and without 95 GHz class I
methanol maser detections in our survey respectively which are plotted in the
[3.6]-[4.5] vs. [5.8]-[8.0] color-color diagram obtained with GLIMPSE point
source fluxes in Figure 5. A “$\ast$” in the superscript denotes the source
which lies in the region outside Stage I (see Section 4.1 and Figure 5).
l, m, nl, m, nfootnotetext: Represent the three different subsamples for the
IRAC and MIPS 24 $\mu$m color analysis with integrated fluxes of EGOs: l is a
source associated with both a class I methanol maser and a class II methanol
maser with an accurate position, m is a source associated with only a class II
maser (with an accurate position), and n is a source associated with only
class I maser emission. A “$\dagger$” in the superscript denotes the source
plotted in [3.6]-[5.8] versus [8.0]-[24] color diagram (Figure 6; see Section
4.2).
Table 2: Summary of the other datasets utilised in this work
Category | Datasets | (Un)Targeted | Area covered | Angular resolution | Sensitivitya | Commentsb
---|---|---|---|---|---|---
OH maser | Caswell (1998) | Untargeted | $|b|<1^{\circ}$ & | $\sim$ 7′′ | $\sim$ 0.12 Jy | 101 EGOs fell within the area
| | | $l=312^{\circ}-356^{\circ}$ | | | 11 detected
| | Targeted | $l=230^{\circ}-13^{\circ}$ | $\sim$ 7′′ | $\sim$ 0.12 Jy | additional 3 EGOs were detected
Class II maser | Caswell (2009) | Targeted | $l=188^{\circ}-50^{\circ}$ | $\sim$ 2′′ | … | $\sim$ 60 EGOs were detected
| Cyganowski et al. (2009) | Targeted | … | $\sim$ 2′′ | $\sim$ 0.1 Jy | $\sim$ 20 EGOs were detected
| Xu et al. (2009) | Targeted | … | a few arcsecs | … | $\sim$ 30 EGOs were detected
| Caswell et al. (2010) | Untargeted | $|b|<2^{\circ}$ & | a few arcsecs | $\sim$ 0.2 Jy | 11 EGOs fell within the area,
| | | $l=345^{\circ}-6^{\circ}$ | | | 8 were detected
| Green et al. (2010) | Unargeted | $|b|<2^{\circ}$ & | a few arcsecs | $\sim$ 0.2 Jy | 22 EGOs fell within the area,
| | | $l=6^{\circ}-20^{\circ}$ | | | 16 were detected
| Titmarsh et al. (in prep.) | Targeted | … | $\sim$ 7′ | $\sim$ 1.5 Jy | searched $\sim$ 140 EGOs without
| | | | | | observed in previous survey,
| | | | | | $\sim$ 80 were not detected
UC Hii region | Wood & Churchwell (1989) | Targeted | … | 0.4′′ | $\sim$ 1 mJy/beam at 6 cm | 2 EGOs were detected
| Becker et al. (1994) | Untargeted | $|b|<0.4^{\circ}$ & | $\sim$ 4′′ | $\sim$ 7.5 mJy/beam at | $\sim$ 40 EGOs fell within the area,
| | | $l=350^{\circ}-40^{\circ}$ | | field center, 5 GHz | but only 4 were detected
| Kurtz et al. (1994) | Targeted | … | $<$ 1′′ | $\sim$ 1.2 mJy/beam at 2 cm | 1 EGO was detected
| | | | | $\sim$ 0.6 mJy/beam at 3.6 cm |
| Walsh et al. (1998) | Targeted | … | $\sim$ 1′′ | $\sim$ 3 mJy at 8.64 GHz | 9 EGOs were detected
| | | | | $\sim$ 30 mJy at 6.67 GHz |
| Cyganowski et al. (2009) | Targeted | … | $\sim$ 0.5′′ | $\sim$ 3 mJy/beam at 44 GHz | Among 19 surveyed EGOs,
| | | | | | only 1 was detected
1.1 mm source | Rosolowsky et al. (2010) | Untargeted | $|b|<0.4^{\circ}$ | 33′′ | $\sim$ 0.1-0.2 Jy | 63 EGOs fell within the area,
| | | $l=350^{\circ}-90^{\circ}$ | | | 54 were detected
aafootnotetext: The sensitivities quoted here are uniformly set to 3
$\sigma_{rms}$ for all surveys. For the Class II maser surveys of Caswell et
al. (2010) and Green et al. (2010), we quoted the sensitivity from the
subsequent ATCA observation for the Parkes MMB survey.
bbfootnotetext: Comments on the extent of overlap with our Morpa surveyed
EGOs, except for class II maser datasets that also include 51 EGOs listed in
our previous work (Chen et al. 2009) but which were not observed in our Mopra
survey (see Table 6).
Table 3: Observed properties of 95 GHz class I methanol maser sources
detected.
| Gaussian Fit | | | |
---|---|---|---|---|---
Source | $S$ | VLSR | $\Delta V$ | $P$ | $\sigma_{rms}$ | Sint | D | Lm
| (Jy km s-1) | (km s-1) | (km s-1) | (Jy) | (Jy) | (Jy km s-1) | (kpc) | (10-6 L⊙)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
G10.29-0.13 | 10.43(0.92) | 13.96(0.41) | 8.41(0.88) | 0.99 | 0.55 | 12.87 | 2.3 | 2.1
| 1.62(0.24) | 14.31(0.03) | 0.46(0.07) | 2.82 | | | |
| 0.82(0.18) | 15.13(0.03) | 0.28(0.07) | 2.30 | | | |
G10.34-0.14 | 39.05(0.81) | 12.24(0.05) | 5.30(0.12) | 6.92 | 0.62 | 69.21 | 2.3 | 11.5
| 30.16(0.26) | 14.74(0.00) | 0.45(0.00) | 63.05 | | | |
G11.92-0.61 | 22.55(0.33) | 33.94(0.01) | 0.82(0.01) | 25.70 | 0.62 | 69.51 | 3.8 | 31.5
| 28.63(0.45) | 35.10(0.00) | 0.52(0.01) | 52.18 | | | |
| 18.34(0.60) | 36.26(0.02) | 1.44(0.06) | 11.95 | | | |
G12.20-0.03 | 1.93(0.28) | 49.28(0.04) | 0.59(0.11) | 3.05 | 0.63 | 7.08 | 4.5 | 4.5
| 0.92(0.23) | 49.96(0.03) | 0.32(0.08) | 2.69 | | | |
| 1.26(0.43) | 51.11(0.04) | 0.47(0.12) | 2.52 | | | |
| 2.97(0.70) | 52.36(0.23) | 2.04(0.55) | 1.37 | | | |
G12.42+0.50 | 0.58(0.24) | 16.57(0.06) | 0.42(0.16) | 1.28 | 0.49 | 7.08 | 2.4 | 1.3
| 5.37(0.61) | 17.98(0.07) | 1.67(0.24) | 3.01 | | | |
| 1.14(0.45) | 19.67(0.15) | 0.99(0.33) | 1.08 | | | |
G12.68-0.18 | 7.97(0.37) | 55.25(0.06) | 2.64(0.14) | 2.84 | 0.45 | 7.97 | 4.7 | 5.5
G12.91-0.03 | 2.55(0.29) | 55.23(0.03) | 0.77(0.08) | 3.09 | 0.46 | 9.76 | 4.8 | 7.0
| 6.50(0.46) | 57.02(0.05) | 1.82(0.17) | 3.34 | | | |
| 0.71(0.20) | 58.47(0.04) | 0.39(0.10) | 1.73 | | | |
G12.91-0.26 | 15.63(0.76) | 36.99(0.09) | 3.57(0.23) | 4.12 | 0.53 | 19.55 | 3.8 | 8.8
| 1.52(0.22) | 36.58(0.02) | 0.35(0.05) | 4.01 | | | |
| 2.40(0.54) | 39.42(0.05) | 0.96(0.16) | 2.34 | | | |
G14.63-0.58 | 0.54(0.14) | 16.88(0.04) | 0.30(0.09) | 1.68 | 0.46 | 14.04 | 2.2 | 2.1
| 8.99(3.27) | 18.46(0.21) | 1.24(0.24) | 6.83 | | | |
| 4.50(0.83) | 18.72(0.01) | 0.37(0.03) | 11.40 | | | |
IRDC18223-3 | 3.32(0.82) | 45.55(0.70) | 6.25(1.87) | 0.50 | 0.60 | 6.32 | 3.6 | 2.6
| 1.36(0.41) | 44.83(0.09) | 0.73(0.23) | 1.75 | | | |
| 1.65(0.33) | 45.65(0.04) | 0.50(0.10) | 3.11 | | | |
G18.67+0.03 | 3.12(0.28) | 80.08(0.06) | 1.15(0.12) | 2.16 | 0.49 | 3.61 | 5.1 | 2.9
| 0.49(0.17) | 82.53(0.09) | 0.43(0.17) | 0.91 | | | |
G18.89-0.47 | 16.30(0.62) | 64.75(0.01) | 1.02(0.04) | 12.75 | 0.62 | 46.53 | 4.5 | 29.5
| 14.64(0.41) | 66.01(0.01) | 0.52(0.01) | 22.57 | | | |
| 2.78(0.34) | 66.85(0.02) | 0.44(0.05) | 5.04 | | | |
| 12.81(1.13) | 66.59(0.30) | 6.34(0.59) | 1.61 | | | |
G19.01-0.03 | 12.99(0.84) | 59.26(0.14) | 4.70(0.36) | 2.59 | 0.60 | 20.46 | 4.2 | 11.3
| 3.42(0.95) | 59.28(0.09) | 0.73(0.17) | 4.42 | | | |
| 3.10(0.86) | 59.88(0.04) | 0.50(0.07) | 5.78 | | | |
| 0.94(0.26) | 61.89(0.05) | 0.46(0.12) | 1.93 | | | |
G19.36-0.03 | 22.96(1.14) | 26.61(0.05) | 3.83(0.18) | 5.63 | 0.49 | 68.67 | 2.5 | 13.5
| 32.40(0.95) | 26.48(0.01) | 1.07(0.02) | 28.41 | | | |
| 13.32(0.45) | 27.59(0.01) | 0.66(0.02) | 18.95 | | | |
G19.88-0.53 | 8.11(0.80) | 41.49(0.11) | 2.31(0.26) | 3.30 | 0.62 | 70.77 | 3.4 | 25.6
| 30.06(3.20) | 43.34(0.03) | 0.91(0.04) | 31.03 | | | |
| 26.75(3.24) | 44.20(0.10) | 1.29(0.14) | 19.40 | | | |
| 1.37(0.47) | 45.45(0.03) | 0.37(0.09) | 3.49 | | | |
| 3.32(0.54) | 46.23(0.08) | 1.03(0.20) | 3.04 | | | |
| 1.16(0.26) | 47.77(0.07) | 0.64(0.17) | 1.71 | | | |
G22.04+0.22 | 20.41(1.87) | 50.61(0.04) | 1.04(0.06) | 15.63 | 0.75 | 57.90 | 3.7 | 24.8
| 13.29(1.60) | 51.57(0.01) | 0.51(0.04) | 20.96 | | | |
| 7.07(1.82) | 52.33(0.04) | 1.29(0.04) | 4.36 | | | |
| 4.10(1.14) | 54.03(0.10) | 0.85(0.17) | 3.86 | | | |
| 13.03(2.12) | 55.63(0.72) | 7.40(0.97) | 1.41 | | | |
G23.82+0.38 | 7.25(0.70) | 73.35(0.19) | 3.99(0.41) | 1.71 | 0.58 | 9.67 | 4.6 | 6.4
| 2.43(0.35) | 75.36(0.03) | 0.67(0.08) | 3.41 | | | |
G23.96-0.11 | 0.85(0.29) | 70.82(0.13) | 0.79(0.33) | 1.01 | 0.54 | 9.11 | 4.5 | 5.8
| 0.73(0.28) | 71.83(0.10) | 0.60(0.27) | 1.14 | | | |
| 5.91(0.47) | 73.05(0.02) | 0.81(0.06) | 6.85 | | | |
| 1.62(0.49) | 74.30(0.17) | 1.21(0.41) | 1.26 | | | |
G24.00-0.10 | 1.26(0.61) | 69.28(0.18) | 0.99(0.40) | 1.21 | 0.58 | 3.42 | 4.4 | 2.1
| 2.16(0.73) | 71.05(0.30) | 1.89(0.76) | 1.07 | | | |
G24.11-0.17 | 3.43(0.66) | 82.05(0.26) | 2.73(0.60) | 1.18 | 0.64 | 3.43 | 4.9 | 2.6
G24.33+0.14 | 24.13(0.71) | 113.32(0.00) | 0.67(0.01) | 33.67 | 0.66 | 95.70 | 6.2 | 115.3
| 63.13(0.89) | 114.04(0.02) | 2.23(0.03) | 26.63 | | | |
| 8.45(0.51) | 116.32(0.03) | 1.06(0.06) | 7.47 | | | |
G24.63+0.15 | 1.49(0.26) | 50.89(0.05) | 0.60(0.11) | 2.32 | 0.54 | 3.93 | 3.5 | 1.5
| 0.69(0.30) | 51.95(0.03) | 0.31(0.11) | 2.05 | | | |
| 1.76(0.58) | 52.75(0.27) | 1.64(0.57) | 1.01 | | | |
G24.94+0.07 | 1.50(0.64) | 40.55(0.12) | 0.71(0.20) | 2.00 | 0.46 | 7.66 | 3.0 | 2.2
| 3.73(0.94) | 41.50(0.06) | 0.96(0.25) | 3.66 | | | |
| 0.52(0.29) | 42.25(0.04) | 0.29(0.12) | 1.67 | | | |
| 1.90(0.52) | 43.18(0.22) | 1.66(0.53) | 1.08 | | | |
G25.27-0.43 | 0.98(0.32) | 58.85(0.06) | 0.54(0.14) | 1.71 | 0.47 | 6.73 | 3.9 | 3.2
| 5.40(0.45) | 59.87(0.03) | 1.09(0.12) | 4.67 | | | |
| 0.35(0.17) | 60.77(0.04) | 0.26(0.11) | 1.29 | | | |
G25.38-0.15 | 2.74(0.41) | 93.68(0.02) | 0.60(0.07) | 4.27 | 0.48 | 14.64 | 5.4 | 13.4
| 6.40(0.99) | 94.67(0.23) | 2.39(0.35) | 2.51 | | | |
| 0.22(0.10) | 95.06(0.06) | 0.24(0.18) | 0.88 | | | |
| 0.45(0.19) | 95.74(0.05) | 0.29(0.16) | 1.47 | | | |
| 0.83(0.45) | 96.22(0.06) | 0.35(0.15) | 2.21 | | | |
| 1.33(0.25) | 96.74(0.06) | 0.47(0.25) | 2.67 | | | |
| 0.80(0.33) | 97.29(0.12) | 0.46(0.23) | 1.65 | | | |
| 1.86(0.41) | 99.01(0.22) | 2.12(0.57) | 0.83 | | | |
G27.97-0.47 | 5.10(0.44) | 45.14(0.06) | 1.61(0.13) | 2.98 | 0.49 | 6.83 | 3.1 | 2.1
| 1.73(0.34) | 45.42(0.02) | 0.42(0.06) | 3.87 | | | |
G34.26+0.15 | 35.17(1.77) | 58.04(0.06) | 2.52(0.12) | 13.13 | 0.73 | 64.20 | 3.9 | 30.6
| 1.25(0.28) | 58.89(0.09) | 0.39(0.18) | 2.99 | | | |
| 4.25(1.90) | 59.40(0.06) | 0.62(0.24) | 6.46 | | | |
| 9.89(1.98) | 60.31(0.03) | 0.80(0.16) | 11.56 | | | |
| 8.27(1.20) | 60.96(0.01) | 0.41(0.03) | 18.84 | | | |
| 2.14(0.66) | 61.61(0.08) | 0.66(0.20) | 3.06 | | | |
| 3.23(0.68) | 63.50(0.23) | 2.29(0.59) | 1.32 | | | |
G34.28+0.18 | 3.40(0.54) | 54.84(0.09) | 1.21(0.20) | 2.64 | 0.48 | 24.72 | 3.6 | 10.0
| 11.43(0.50) | 55.75(0.01) | 0.55(0.02) | 19.50 | | | |
| 8.22(0.27) | 56.48(0.01) | 0.42(0.01) | 18.35 | | | |
| 1.68(0.25) | 57.16(0.04) | 0.60(0.11) | 2.63 | | | |
G34.39+0.22 | 6.54(2.51) | 57.15(0.33) | 2.26(0.81) | 2.72 | 0.48 | 11.95 | 3.7 | 5.1
| 1.87(0.95) | 58.28(0.06) | 0.81(0.28) | 2.15 | | | |
| 3.55(0.73) | 60.00(0.15) | 1.61(0.26) | 2.06 | | | |
G34.41+0.24 | 3.90(1.18) | 55.89(0.13) | 1.38(0.23) | 2.66 | 0.49 | 43.05 | 3.7 | 18.5
| 13.09(1.65) | 57.78(0.06) | 2.03(0.27) | 6.06 | | | |
| 1.85(0.69) | 59.49(0.07) | 0.87(0.20) | 1.99 | | | |
| 24.22(1.26) | 58.18(0.16) | 8.77(0.46) | 2.59 | | | |
G35.04-0.47 | 0.54(0.18) | 50.81(0.05) | 0.36(0.12) | 1.43 | 0.47 | 4.27 | 3.4 | 1.5
| 1.95(0.31) | 51.54(0.04) | 0.64(0.11) | 2.84 | | | |
| 1.13(0.54) | 52.77(0.17) | 1.04(0.60) | 1.02 | | | |
| 0.65(0.38) | 53.74(0.12) | 0.57(0.25) | 1.07 | | | |
G35.13-0.74 | 19.70(1.22) | 33.81(0.16) | 5.93(0.30) | 3.12 | 0.53 | 38.11 | 2.5 | 7.5
| 2.94(0.57) | 33.41(0.04) | 0.57(0.08) | 4.80 | | | |
| 4.25(1.53) | 34.39(0.10) | 0.94(0.35) | 4.25 | | | |
| 1.99(0.71) | 35.00(0.03) | 0.41(0.10) | 4.60 | | | |
| 5.40(0.84) | 35.64(0.03) | 0.77(0.13) | 6.58 | | | |
| 3.83(0.49) | 36.59(0.03) | 0.65(0.07) | 5.55 | | | |
G35.20-0.74 | 5.18(0.98) | 32.15(0.17) | 1.87(0.35) | 2.60 | 0.58 | 41.65 | 2.5 | 8.2
| 11.18(1.93) | 33.90(0.06) | 1.24(0.14) | 8.49 | | | |
| 1.96(0.38) | 34.87(0.01) | 0.30(0.04) | 6.17 | | | |
| 23.33(1.73) | 35.37(0.05) | 1.47(0.08) | 14.90 | | | |
G35.79-0.17 | 1.43(0.36) | 59.92(0.02) | 0.36(0.07) | 3.73 | 0.66 | 19.21 | 3.9 | 9.2
| 1.34(0.79) | 60.87(0.08) | 0.66(0.23) | 1.91 | | | |
| 3.17(1.55) | 62.16(0.09) | 1.18(0.30) | 2.53 | | | |
| 12.48(2.32) | 61.52(0.20) | 4.33(0.70) | 2.71 | | | |
| 0.79(0.41) | 64.65(0.11) | 0.66(0.30) | 1.12 | | | |
G39.10+0.49 | 1.10(0.24) | 21.59(0.04) | 0.47(0.10) | 2.21 | 0.58 | 17.26 | 2.0 | 2.2
| 5.47(0.82) | 23.12(0.37) | 4.67(0.82) | 1.10 | | | |
| 10.69(0.47) | 26.31(0.01) | 1.02(0.04) | 9.80 | | | |
G39.39-0.14 | 2.15(0.59) | 65.62(0.31) | 2.23(0.56) | 0.91 | 0.57 | 3.90 | 4.5 | 2.5
| 0.66(0.46) | 65.97(0.11) | 0.38(0.23) | 1.61 | | | |
| 1.09(0.45) | 66.42(0.06) | 0.38(0.13) | 2.71 | | | |
G40.28-0.22 | 30.10(0.54) | 72.77(0.00) | 0.53(0.01) | 53.21 | 0.48 | 124.20 | 5.3 | 109.4
| 79.45(0.66) | 73.31(0.01) | 1.57(0.01) | 47.61 | | | |
| 14.64(0.29) | 75.27(0.01) | 0.93(0.02) | 14.84 | | | |
G40.60-0.72 | 1.37(0.43) | 62.98(0.04) | 0.55(0.12) | 2.32 | 0.49 | 20.37 | 4.5 | 12.9
| 8.97(1.06) | 64.21(0.02) | 0.97(0.06) | 8.72 | | | |
| 10.03(1.41) | 64.52(0.15) | 2.91(0.23) | 3.24 | | | |
G45.80-0.36 | 8.22(0.41) | 58.06(0.06) | 2.45(0.15) | 3.15 | 0.48 | 9.89 | 4.8 | 7.1
| 0.47(0.17) | 59.96(0.05) | 0.33(0.12) | 1.34 | | | |
| 1.20(0.21) | 60.85(0.05) | 0.67(0.13) | 1.67 | | | |
G49.07-0.33 | 1.72(0.23) | 61.43(0.06) | 0.86(0.13) | 1.88 | 0.51 | 2.93 | 5.0 | 2.3
| 1.21(0.25) | 69.50(0.10) | 1.02(0.24) | 1.12 | | | |
G54.45+1.01 | 2.04(0.41) | 36.90(0.06) | 0.61(0.13) | 3.13 | 0.64 | 3.20 | 4.0 | 1.6
| 1.17(0.37) | 39.13(0.09) | 0.50(0.15) | 2.17 | | | |
G59.79+0.63 | 1.66(0.31) | 27.62(0.06) | 0.66(0.14) | 2.37 | 0.76 | 29.35 | 4.0 | 14.7
| 11.31(0.36) | 30.89(0.01) | 0.55(0.02) | 19.29 | | | |
| 16.38(1.01) | 33.53(0.16) | 5.33(0.37) | 2.89 | | | |
G305.48-0.10 | 6.28(0.57) | -38.23(0.11) | 2.53(0.27) | 2.33 | 0.72 | 7.05 | 4.6 | 4.7
| 0.77(0.20) | -35.86(0.03) | 0.28(0.08) | 2.58 | | | |
G305.52+0.76 | 1.50(0.46) | -29.91(0.13) | 1.01(0.29) | 1.40 | 0.61 | 6.04 | 2.8 | 1.5
| 4.54(0.55) | -28.09(0.10) | 1.79(0.26) | 2.38 | | | |
G305.80-0.24 | 4.65(1.02) | -34.21(0.21) | 2.00(0.46) | 2.18 | 0.69 | 13.56 | 3.2 | 4.4
| 4.40(1.48) | -32.17(0.12) | 1.37(0.44) | 3.02 | | | |
| 1.29(0.90) | -31.01(0.13) | 0.71(0.35) | 1.71 | | | |
| 0.97(0.28) | -30.24(0.04) | 0.34(0.10) | 2.68 | | | |
| 2.25(0.51) | -28.32(0.21) | 1.98(0.53) | 1.07 | | | |
G305.82-0.11 | 0.62(0.29) | -44.34(0.17) | 0.79(0.40) | 0.74 | 0.62 | 11.17 | 4.4 | 6.8
| 7.69(1.09) | -41.12(0.19) | 2.79(0.42) | 2.59 | | | |
| 2.48(0.77) | -39.51(0.05) | 0.84(0.17) | 2.77 | | | |
| 0.38(0.18) | -38.69(0.05) | 0.23(0.11) | 1.52 | | | |
G305.89+0.02 | 7.56(0.87) | -34.93(0.07) | 1.23(0.13) | 5.78 | 0.59 | 16.89 | 3.4 | 6.1
| 7.07(0.93) | -33.70(0.04) | 0.96(0.11) | 6.94 | | | |
| 1.11(0.30) | -32.63(0.06) | 0.52(0.14) | 2.00 | | | |
| 1.14(0.22) | -31.47(0.05) | 0.57(0.13) | 1.90 | | | |
G309.91+0.32 | 22.81(0.65) | -61.20(0.01) | 0.64(0.02) | 33.63 | 0.72 | 31.81 | 5.0 | 24.9
| 3.74(0.78) | -60.43(0.05) | 0.68(0.13) | 5.20 | | | |
| 5.26(0.49) | -59.14(0.05) | 1.17(0.13) | 4.22 | | | |
G310.15+0.76 | 1.04(0.23) | -58.42(0.04) | 0.40(0.10) | 2.42 | 0.62 | 19.45 | 5.0 | 15.2
| 3.44(0.29) | -57.61(0.03) | 0.67(0.07) | 4.82 | | | |
| 2.71(0.29) | -56.47(0.02) | 0.51(0.05) | 5.02 | | | |
| 8.93(0.50) | -55.33(0.02) | 1.08(0.07) | 7.74 | | | |
| 3.34(0.49) | -53.42(0.11) | 1.59(0.28) | 1.97 | | | |
G310.38-0.30 | 0.83(0.22) | -55.09(0.07) | 0.52(0.17) | 1.50 | 0.56 | 8.41 | 5.2 | 7.1
| 4.03(0.39) | -54.33(0.01) | 0.41(0.03) | 9.22 | | | |
| 3.55(0.45) | -53.55(0.06) | 0.96(0.14) | 3.46 | | | |
G311.04+0.69 | 0.90(0.15) | 24.66(0.03) | 0.38(0.07) | 2.21 | 0.50 | 0.90 | 12.7 | 4.5
G311.51-0.45 | 2.98(0.70) | -50.91(0.21) | 1.88(0.50) | 1.49 | 0.61 | 2.98 | 4.6 | 2.0
G313.76-0.86 | 7.49(0.85) | -50.55(0.01) | 0.54(0.04) | 12.94 | 0.70 | 36.28 | 4.0 | 18.2
| 8.81(2.01) | -49.78(0.11) | 1.67(0.24) | 4.95 | | | |
| 17.59(1.78) | -49.80(0.21) | 6.70(0.64) | 2.47 | | | |
| 0.51(0.22) | -47.45(0.04) | 0.25(0.11) | 1.91 | | | |
| 1.88(0.26) | -46.69(0.02) | 0.36(0.05) | 4.84 | | | |
G317.87-0.15 | 2.09(0.63) | -42.36(0.14) | 1.27(0.38) | 1.55 | 0.64 | 15.75 | 3.0 | 4.4
| 0.95(0.32) | -41.40(0.03) | 0.34(0.09) | 2.65 | | | |
| 2.60(0.72) | -40.19(0.13) | 1.50(0.38) | 1.63 | | | |
| 10.11(1.33) | -39.80(0.58) | 9.88(1.42) | 0.96 | | | |
G317.88-0.25 | 1.63(0.39) | -38.63(0.10) | 0.87(0.24) | 1.76 | 0.50 | 13.15 | 2.6 | 2.8
| 0.81(0.41) | -37.86(0.06) | 0.44(0.16) | 1.73 | | | |
| 5.29(0.49) | -36.88(0.03) | 1.00(0.11) | 4.96 | | | |
| 5.41(0.39) | -35.53(0.03) | 0.98(0.08) | 5.17 | | | |
G320.23-0.28 | 0.52(0.23) | -67.28(0.05) | 0.32(0.14) | 1.52 | 0.62 | 7.61 | 4.6 | 5.0
| 1.49(0.28) | -66.03(0.02) | 0.36(0.06) | 3.91 | | | |
| 5.15(0.65) | -65.60(0.14) | 2.40(0.36) | 2.02 | | | |
| 0.45(0.20) | -63.78(0.05) | 0.29(0.13) | 1.48 | | | |
G321.94-0.01 | 0.86(0.33) | -35.35(0.03) | 0.26(0.09) | 3.14 | 0.85 | 35.35 | 2.4 | 6.4
| 9.07(1.53) | -34.19(0.10) | 1.52(0.28) | 5.62 | | | |
| 0.94(0.34) | -33.37(0.05) | 0.35(0.16) | 2.52 | | | |
| 19.05(2.20) | -32.41(0.03) | 1.24(0.08) | 14.42 | | | |
| 5.43(2.50) | -30.50(0.77) | 3.40(1.44) | 1.50 | | | |
G324.19+0.41 | 2.96(0.34) | -52.48(0.06) | 1.18(0.17) | 2.35 | 0.56 | 3.65 | 3.5 | 1.4
| 0.70(0.21) | -51.41(0.04) | 0.34(0.10) | 1.93 | | | |
G326.32-0.39 | 24.40(0.79) | -72.48(0.04) | 3.31(0.12) | 6.92 | 0.59 | 35.14 | 4.5 | 22.3
| 2.04(0.31) | -73.08(0.02) | 0.39(0.05) | 4.86 | | | |
| 7.72(0.49) | -72.43(0.01) | 0.55(0.03) | 13.18 | | | |
| 0.72(0.20) | -71.33(0.03) | 0.26(0.07) | 2.61 | | | |
| 0.25(0.15) | -69.96(0.05) | 0.20(0.14) | 1.17 | | | |
G326.41+0.93 | 1.73(0.37) | -44.41(0.16) | 1.56(0.38) | 1.04 | 0.56 | 9.80 | 2.8 | 2.4
| 8.07(0.49) | -40.82(0.08) | 2.85(0.21) | 2.66 | | | |
G326.61+0.80 | 9.09(0.65) | -37.72(0.11) | 4.04(0.35) | 2.11 | 0.48 | 11.48 | 2.7 | 2.6
| 1.63(0.48) | -37.92(0.06) | 0.86(0.20) | 1.79 | | | |
| 0.76(0.22) | -37.03(0.04) | 0.39(0.10) | 1.85 | | | |
G326.92-0.31 | 1.61(1.12) | -46.76(0.58) | 1.68(1.06) | 0.90 | 0.53 | 4.19 | 3.1 | 1.3
| 1.45(0.91) | -45.81(0.05) | 0.64(0.21) | 2.12 | | | |
| 1.13(0.26) | -44.65(0.07) | 0.70(0.17) | 1.53 | | | |
G326.97-0.03 | 1.71(0.37) | -61.29(0.06) | 0.62(0.13) | 2.58 | 0.52 | 8.86 | 3.9 | 4.2
| 2.31(0.47) | -60.40(0.06) | 0.78(0.17) | 2.80 | | | |
| 3.11(0.40) | -58.78(0.08) | 1.43(0.23) | 2.04 | | | |
| 0.74(0.17) | -57.27(0.04) | 0.37(0.10) | 1.85 | | | |
| 0.99(0.25) | -55.16(0.12) | 0.96(0.28) | 0.97 | | | |
G327.57-0.85 | 13.76(0.35) | -37.04(0.00) | 0.52(0.01) | 24.96 | 0.67 | 24.28 | 2.6 | 5.1
| 10.51(0.71) | -35.95(0.11) | 3.14(0.21) | 3.14 | | | |
G327.72-0.38 | 1.63(0.27) | -73.31(0.06) | 0.73(0.14) | 2.10 | 0.64 | 1.63 | 4.5 | 1.0
G327.89+0.15 | 1.84(0.22) | -92.76(0.04) | 0.60(0.09) | 2.87 | 0.56 | 8.66 | 5.5 | 8.2
| 0.63(0.23) | -91.81(0.06) | 0.42(0.16) | 1.40 | | | |
| 6.19(0.38) | -90.49(0.04) | 1.40(0.11) | 4.14 | | | |
G328.16+0.59 | 1.85(0.19) | -93.03(0.02) | 0.47(0.05) | 3.72 | 0.57 | 1.85 | 5.7 | 1.9
G328.55+0.27 | 33.39(0.32) | -59.45(0.00) | 0.82(0.01) | 38.03 | 0.67 | 35.70 | 3.8 | 16.2
| 2.31(0.22) | -58.48(0.02) | 0.36(0.04) | 6.12 | | | |
G329.16-0.29 | 4.21(0.41) | -49.99(0.07) | 1.42(0.16) | 2.79 | 0.56 | 5.53 | 3.3 | 1.9
| 1.32(0.40) | -48.04(0.18) | 1.27(0.44) | 0.97 | | | |
G329.47+0.52 | 2.18(0.81) | -68.84(0.29) | 1.72(0.59) | 1.19 | 0.57 | 5.89 | 14.6 | 39.4
| 3.71(0.81) | -66.73(0.18) | 1.81(0.39) | 1.92 | | | |
G330.88-0.37 | 38.02(1.38) | -61.20(0.15) | 8.20(0.32) | 4.36 | 0.90 | 45.38 | 3.8 | 20.5
| 7.35(0.41) | -57.92(0.01) | 0.64(0.04) | 10.86 | | | |
G331.37-0.40 | 2.57(0.22) | -64.91(0.03) | 0.66(0.06) | 3.67 | 0.55 | 2.57 | 4.1 | 1.4
G331.62+0.53 | 1.97(0.65) | -53.10(0.32) | 2.04(0.79) | 0.90 | 0.59 | 3.68 | 3.5 | 1.4
| 0.81(0.32) | -51.91(0.04) | 0.38(0.12) | 2.03 | | | |
| 0.89(0.33) | -50.72(0.15) | 0.92(0.36) | 0.91 | | | |
G331.71+0.58 | 50.84(0.28) | -68.07(0.00) | 0.92(0.01) | 52.12 | 0.49 | 67.70 | 4.3 | 39.2
| 16.86(0.38) | -66.03(0.02) | 1.82(0.05) | 8.71 | | | |
G331.71+0.60 | 3.36(0.57) | -68.23(0.03) | 0.56(0.07) | 5.66 | 0.59 | 21.83 | 4.3 | 12.7
| 3.32(0.84) | -67.39(0.05) | 0.77(0.20) | 4.04 | | | |
| 1.72(0.59) | -66.48(0.06) | 0.59(0.16) | 2.75 | | | |
| 1.57(0.78) | -65.36(0.12) | 1.03(0.38) | 1.43 | | | |
| 11.86(1.70) | -66.89(0.22) | 5.74(0.73) | 1.94 | | | |
G332.28-0.55 | 12.77(0.71) | -53.11(0.11) | 4.74(0.32) | 2.53 | 0.54 | 88.89 | 3.6 | 36.1
| 5.25(0.31) | -52.37(0.01) | 0.57(0.03) | 8.59 | | | |
| 70.88(0.31) | -53.29(0.00) | 0.56(0.00) | 119.32 | | | |
G332.35-0.12 | 3.97(0.48) | -50.16(0.04) | 0.68(0.09) | 5.45 | 0.56 | 5.79 | 3.4 | 2.1
| 1.14(0.44) | -49.61(0.04) | 0.35(0.09) | 3.10 | | | |
| 0.68(0.17) | -49.08(0.04) | 0.30(0.09) | 2.11 | | | |
G332.36+0.60 | 3.56(0.35) | -43.10(0.02) | 0.50(0.05) | 6.64 | 0.75 | 12.97 | 3.1 | 3.9
| 9.41(0.81) | -42.11(0.17) | 3.89(0.35) | 2.27 | | | |
G332.58+0.15 | 1.04(0.32) | -42.29(0.07) | 0.44(0.29) | 2.23 | 0.54 | 1.04 | 3.0 | 0.3
G332.81-0.70 | 9.83(0.20) | -53.51(0.00) | 0.50(0.01) | 18.48 | 0.55 | 24.15 | 3.6 | 9.8
| 5.79(1.75) | -52.59(0.05) | 0.50(0.06) | 10.86 | | | |
| 5.25(2.17) | -52.07(0.08) | 0.63(0.19) | 7.83 | | | |
| 3.28(0.67) | -51.21(0.08) | 0.80(0.15) | 3.84 | | | |
G332.91-0.55 | 1.43(0.35) | -56.23(0.11) | 0.95(0.29) | 1.42 | 0.62 | 2.90 | 3.7 | 1.2
| 0.87(0.23) | -55.37(0.03) | 0.31(0.08) | 2.64 | | | |
| 0.60(0.20) | -54.69(0.07) | 0.42(0.16) | 1.35 | | | |
G333.32+0.10 | 1.68(0.40) | -49.35(0.12) | 1.12(0.29) | 1.41 | 0.62 | 13.63 | 3.3 | 4.7
| 5.91(0.53) | -47.19(0.07) | 1.82(0.21) | 3.06 | | | |
| 5.40(0.43) | -44.30(0.06) | 1.66(0.16) | 3.06 | | | |
| 0.63(0.18) | -42.25(0.05) | 0.37(0.12) | 1.60 | | | |
G335.59-0.30 | 2.07(0.40) | -47.34(0.10) | 1.14(0.24) | 1.71 | 0.52 | 6.04 | 3.3 | 2.1
| 1.11(0.62) | -46.06(0.08) | 0.65(0.23) | 1.61 | | | |
| 1.09(0.47) | -45.23(0.04) | 0.43(0.12) | 2.39 | | | |
| 1.76(1.29) | -44.36(0.89) | 2.45(1.66) | 0.68 | | | |
G336.02-0.83 | 62.43(0.39) | -48.43(0.00) | 0.87(0.01) | 67.77 | 0.53 | 97.07 | 3.5 | 37.3
| 34.64(0.90) | -47.17(0.09) | 6.96(0.20) | 4.68 | | | |
G336.03-0.82 | 3.71(0.79) | -46.24(0.08) | 0.75(0.15) | 4.66 | 0.64 | 6.17 | 3.3 | 2.1
| 1.29(0.76) | -45.61(0.09) | 0.50(0.17) | 2.44 | | | |
| 1.17(0.26) | -44.15(0.08) | 0.72(0.19) | 1.52 | | | |
G336.96-0.98 | 0.62(0.23) | -45.31(0.05) | 0.32(0.11) | 1.85 | 0.59 | 6.72 | 3.3 | 2.3
| 3.19(0.62) | -44.49(0.06) | 0.81(0.18) | 3.72 | | | |
| 2.16(0.59) | -43.64(0.05) | 0.59(0.14) | 3.45 | | | |
| 0.74(0.34) | -42.63(0.17) | 0.80(0.44) | 0.87 | | | |
G337.30-0.87 | 2.75(0.32) | -94.03(0.07) | 1.31(0.18) | 1.97 | 0.58 | 2.75 | 5.4 | 2.5
G338.32-0.41 | 0.40(0.16) | -39.11(0.06) | 0.32(0.14) | 1.18 | 0.44 | 5.36 | 3.1 | 1.6
| 2.52(0.22) | -38.56(0.01) | 0.43(0.04) | 5.56 | | | |
| 2.43(0.25) | -37.57(0.05) | 0.96(0.12) | 2.38 | | | |
G339.58-0.13 | 6.90(1.84) | -34.53(0.25) | 1.93(0.40) | 3.36 | 0.54 | 27.31 | 2.8 | 6.7
| 2.40(1.20) | -33.73(0.05) | 0.64(0.15) | 3.55 | | | |
| 8.48(1.15) | -32.76(0.04) | 1.08(0.12) | 7.34 | | | |
| 9.52(0.57) | -31.18(0.04) | 1.35(0.09) | 6.60 | | | |
G339.95-0.54 | 2.88(0.37) | -94.64(0.04) | 0.74(0.09) | 3.63 | 0.49 | 15.30 | 5.5 | 14.5
| 3.73(0.47) | -93.45(0.05) | 1.11(0.14) | 3.15 | | | |
| 8.69(0.78) | -94.40(0.32) | 7.97(0.87) | 1.02 | | | |
G340.05-0.25 | 0.55(0.16) | -57.01(0.06) | 0.44(0.15) | 1.17 | 0.50 | 8.38 | 3.9 | 4.0
| 2.43(0.42) | -55.02(0.11) | 1.29(0.27) | 1.77 | | | |
| 0.65(0.35) | -54.05(0.06) | 0.43(0.17) | 1.43 | | | |
| 2.69(0.60) | -52.99(0.08) | 1.18(0.24) | 2.15 | | | |
| 2.07(0.58) | -51.04(0.27) | 2.02(0.65) | 0.96 | | | |
G340.97-1.02 | 107.93(1.50) | -24.35(0.03) | 5.61(0.06) | 18.07 | 0.52 | 218.65 | 2.3 | 36.3
| 62.23(2.89) | -23.88(0.03) | 1.52(0.03) | 38.41 | | | |
| 42.05(2.24) | -23.29(0.00) | 0.71(0.01) | 55.52 | | | |
| 6.45(0.50) | -21.83(0.02) | 0.78(0.05) | 7.81 | | | |
G341.22-0.26 | 2.99(0.49) | -44.58(0.03) | 0.55(0.08) | 5.14 | 0.89 | 17.20 | 3.6 | 7.0
| 14.21(0.97) | -44.13(0.12) | 3.82(0.30) | 3.50 | | | |
G341.24-0.27 | 12.69(2.08) | -44.90(0.29) | 3.50(0.46) | 3.40 | 0.83 | 27.19 | 3.5 | 10.4
| 11.10(1.05) | -43.78(0.02) | 0.75(0.05) | 13.82 | | | |
| 1.86(0.54) | -43.02(0.04) | 0.41(0.10) | 4.23 | | | |
| 1.55(0.65) | -42.33(0.06) | 0.52(0.17) | 2.79 | | | |
G342.15+0.51 | 2.37(0.44) | -84.53(0.14) | 1.56(0.33) | 1.43 | 0.42 | 2.37 | 5.3 | 2.1
G342.48+0.18 | 7.01(0.80) | -42.39(0.09) | 1.56(0.20) | 4.21 | 0.54 | 13.98 | 3.6 | 5.7
| 0.84(0.25) | -42.43(0.02) | 0.27(0.06) | 2.93 | | | |
| 4.84(0.80) | -40.76(0.09) | 1.31(0.17) | 3.47 | | | |
| 1.29(0.27) | -37.34(0.11) | 1.08(0.26) | 1.12 | | | |
G343.50+0.03 | 1.94(0.23) | -31.78(0.04) | 0.73(0.10) | 2.48 | 0.53 | 4.89 | 3.1 | 1.5
| 0.75(0.28) | -30.63(0.05) | 0.42(0.13) | 1.68 | | | |
| 2.20(0.49) | -29.34(0.19) | 1.85(0.50) | 1.12 | | | |
G343.50-0.47 | 1.96(0.25) | -35.74(0.03) | 0.57(0.08) | 3.24 | 0.50 | 28.72 | 3.2 | 9.2
| 5.34(0.81) | -34.67(0.05) | 0.79(0.12) | 6.34 | | | |
| 11.59(0.87) | -33.95(0.01) | 0.54(0.03) | 19.99 | | | |
| 9.83(0.39) | -33.14(0.01) | 0.72(0.03) | 12.81 | | | |
G343.53-0.51 | 1.75(0.49) | -34.77(0.41) | 2.85(0.75) | 0.58 | 0.46 | 2.51 | 3.2 | 0.8
| 0.76(0.26) | -33.68(0.06) | 0.53(0.16) | 1.35 | | | |
G345.13-0.17 | 9.18(0.63) | -25.79(0.09) | 3.38(0.27) | 2.55 | 0.52 | 17.77 | 2.8 | 4.4
| 2.19(0.38) | -26.39(0.03) | 0.46(0.07) | 4.43 | | | |
| 6.41(0.49) | -25.74(0.02) | 0.62(0.04) | 9.74 | | | |
G345.99-0.02 | 1.34(0.22) | -82.83(0.04) | 0.52(0.10) | 2.42 | 0.65 | 1.91 | 5.6 | 1.9
| 0.57(0.15) | -81.43(0.03) | 0.24(0.07) | 2.27 | | | |
G348.17+0.46 | 8.90(2.30) | -8.35(0.25) | 1.93(0.39) | 4.33 | 0.72 | 54.21 | 1.3 | 2.9
| 26.04(1.94) | -7.43(0.01) | 0.70(0.03) | 35.13 | | | |
| 12.57(1.13) | -6.56(0.02) | 0.71(0.07) | 16.61 | | | |
| 1.70(0.79) | -5.83(0.06) | 0.48(0.15) | 3.31 | | | |
| 5.00(0.60) | -4.85(0.07) | 1.20(0.17) | 3.91 | | | |
G348.55-0.98 | 41.47(0.92) | -16.53(0.05) | 4.91(0.10) | 7.93 | 0.54 | 89.61 | 2.4 | 16.2
| 29.32(0.48) | -16.10(0.00) | 0.84(0.01) | 32.93 | | | |
| 18.82(0.45) | -14.86(0.01) | 0.85(0.02) | 20.87 | | | |
G348.58-0.92 | 3.92(1.40) | -15.30(0.41) | 2.32(0.72) | 1.59 | 0.73 | 15.23 | 2.2 | 2.3
| 6.57(0.97) | -14.22(0.02) | 0.68(0.06) | 9.12 | | | |
| 3.39(0.49) | -12.26(0.08) | 1.20(0.19) | 2.65 | | | |
| 1.35(0.29) | -9.63(0.07) | 0.69(0.17) | 1.84 | | | |
G348.73-1.04 | 18.17(1.59) | -12.13(0.31) | 7.95(0.66) | 2.15 | 0.73 | 42.84 | 1.6 | 3.4
| 9.79(1.85) | -11.64(0.10) | 1.28(0.19) | 7.16 | | | |
| 4.42(1.79) | -10.61(0.07) | 0.81(0.22) | 5.12 | | | |
| 2.52(1.06) | -9.83(0.04) | 0.44(0.11) | 5.33 | | | |
| 7.93(1.08) | -9.17(0.05) | 0.91(0.11) | 8.21 | | | |
Note. — Column (1): source name. Columns (2)-(5): the integrated flux density
S, the velocity at peak maser emission VLSR, the line width (FWHM) $\Delta$V,
and the peak flux density P, of each maser feature estimated from Gaussian
fits to 95 GHz class I methanol maser lines, the formal error from the
Gaussian fit is given in parenthesis. The corresponding values in main beam
temperature TMB (K) can be obtained by dividing the flux density by a factor
of 9.3 (see Section 2.2). Column (6): 1$\sigma$ noise in the observed maser
spectrum. Column (7): the total integrated flux density Sint of the maser
spectrum obtained from summing the integrated flux density of all maser
features in each source in column (2). Column (8): the kinematic distance to
source estimated from galactic rotation curve of Reid et al. (2009). The
distances to G49.07-0.33, G309.91+0.32, G310.15+0.76 which cannot be derived
from the galactic rotation curve are assumed to be 5 kpc. Column (9): the
integrated luminosity of 95 GHz methanol maser estimated with assuming maser
isotropic emission, i.e. Lm=4$\pi$$\cdot$D2$\cdot$Sint.
(This table is available in its entirety in a machine-readable form in the
online journal. A portion is shown here for guidance regarding its form and
content.)
Table 4: Sources undetected at 95 GHz. Source | $\sigma_{rms}$ (Jy) | Source | $\sigma_{rms}$ (Jy) | Source | $\sigma_{rms}$ (Jy) | Source | $\sigma_{rms}$ (Jy)
---|---|---|---|---|---|---|---
G11.11-0.11 | 0.71 | G12.02-0.21 | 0.61 | G16.58-0.08 | 0.56 | G17.96+0.08 | 0.60
G19.61-0.12 | 0.45 | G20.24+0.07 | 0.63 | G21.24+0.19 | 0.63 | G24.17-0.02 | 0.52
G28.28-0.36 | 0.48 | G28.85-0.23 | 0.45 | G29.89-0.77 | 0.45 | G29.91-0.81 | 0.52
G29.96-0.79 | 0.56 | G35.15+0.80 | 0.45 | G35.68-0.18 | 0.56 | G36.01-0.20 | 0.63
G37.48-0.10 | 0.60 | G37.55+0.20 | 0.56 | G40.28-0.27 | 0.48 | G44.01-0.03 | 0.47
G45.47+0.05 | 0.48 | G45.50+0.12 | 0.48 | G48.66-0.30 | 0.47 | G49.42+0.33 | 0.50
G53.92-0.07 | 0.58 | G54.11-0.04 | 0.74 | G54.11-0.08 | 0.60 | G57.61+0.02 | 0.74
G298.90+0.36 | 0.58 | G304.89+0.64 | 0.61 | G305.57-0.34 | 0.43 | G305.62-0.34 | 0.65
G305.77-0.25 | 0.71 | G309.15-0.35 | 0.39 | G309.90+0.23 | 0.69 | G309.97+0.50 | 0.56
G309.97+0.59 | 0.67 | G309.99+0.51 | 0.63 | G310.08-0.23 | 0.65 | G312.11+0.26 | 0.61
G313.71-0.19 | 0.41 | G317.44-0.37 | 0.63 | G317.46-0.40 | 0.65 | G324.11+0.44 | 0.71
G324.17+0.44 | 0.58 | G325.52+0.42 | 0.61 | G326.27-0.49 | 0.48 | G326.31+0.90 | 0.56
G326.36+0.88 | 0.67 | G326.37+0.94 | 0.54 | G326.57+0.20 | 0.71 | G326.78-0.24 | 0.86
G326.79+0.38 | 0.47 | G326.80+0.51 | 0.52 | G327.65+0.13 | 0.61 | G327.86+0.19 | 0.45
G328.60+0.27 | 0.60 | G331.08-0.47 | 0.43 | G331.12-0.46 | 0.50 | G331.51-0.34 | 0.50
G332.12+0.94 | 0.52 | G332.28-0.07 | 0.50 | G332.33-0.12 | 0.65 | G332.47-0.52 | 0.67
G332.59+0.04 | 0.45 | G333.08-0.56 | 0.61 | G334.04+0.35 | 0.63 | G335.43-0.24 | 0.50
G336.87+0.29 | 0.61 | G337.16-0.39 | 0.58 | G340.07-0.24 | 0.58 | G340.10-0.18 | 0.52
G340.75-1.00 | 0.45 | G340.77-0.12 | 0.48 | G338.39-0.40 | 0.50 | G338.42-0.41 | 0.48
G341.20-0.26 | 0.48 | G341.99-0.10 | 0.47 | G342.04+0.43 | 0.45 | G343.19-0.08 | 0.47
G343.40-0.40 | 0.48 | G343.42-0.37 | 0.54 | G343.72-0.18 | 0.56 | G343.78-0.24 | 0.56
G344.21-0.62 | 0.43 | G345.72+0.82 | 0.63 | G346.04+0.05 | 0.63 | |
Table 5: Detection rates of class I methanol maser in different subsamples
Source Properties | NDa | NTa | Detection rate
---|---|---|---
Likely outflow | 53 | 86 | 62%
Possible outflow | 52 | 106 | 49%
IRDC | 71 | 128 | 55%
Non-IRDC | 34 | 64 | 53%
With associated class II maser | 39 | 49 | 80%
Without associated class II maser | 31 | 81 | 38%
With associated OH maser | 13 | 14 | 93%
Without associated OH maser | 43 | 90 | 48%
With associated UC Hii region | 11 | 13 | 85%
Without associated UC Hii region | 21 | 34 | 62%
With associated 1.1 mm | 35 | 54 | 65%
Without associated 1.1 mm | 1 | 9 | 11%
aafootnotetext: ND and NT represent the numbers of the detected 95 GHz class I
maser sources and total sources, respectively.
Table 6: Sources used in Chen et al. (2009) analysis but not observed in the
present Mopra survey
Source | Class IIa | Class Ib | Remarkc | Source | Class IIa | Class Ib | Remarkc
---|---|---|---|---|---|---|---
G14.33-0.64d${}^{h^{\dagger}}$ | Y | Y | 1 | G331.34-0.35fh | Y | Y | 4
G16.59-0.05${}^{f^{\ast}}$h | Y | Y | 2 | G332.29-0.09h | Y | Y | 4
G23.01-0.41${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1 | G332.35-0.44i | Y | N | 4
G28.83-0.25e${}^{i^{\dagger}}$ | Y | N | 1 | G332.56-0.15e${}^{i^{\dagger}}$ | Y | N | 1
G43.04-0.45 | Y | – | 4 | G332.60-0.17h | Y | Y | 2
G298.26+0.74 | Y | – | 1 | G332.73-0.62 | Y | – | 2
G309.38-0.13${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1 | G332.94-0.69d${}^{h^{\dagger}}$ | Y | Y | 1
G318.05+0.09fi | Y | N | 2 | G332.96-0.68d${}^{f^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1
G323.74-0.26${}^{f^{\ast}}$h | Y | Y | 4 | G333.13-0.56h | Y | Y | 4
G324.72+0.34${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1 | G333.18-0.09${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1
G326.48+0.70h | Y | Y | 2 | G333.47-0.16h | Y | Y | 2
G326.86-0.67${}^{d^{\ast}}$${}^{f^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1 | G335.06-0.43h | Y | Y | 2
G327.12+0.51e${}^{i^{\dagger}}$ | Y | N | 1 | G335.59-0.29${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1
G327.30-0.58d${}^{f^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 3 | G335.79+0.18i | Y | N | 2
G327.39+0.20d${}^{h^{\dagger}}$ | Y | Y | 1 | G337.40-0.40h | Y | Y | 4
G327.40+0.44e${}^{i^{\dagger}}$ | Y | N | 1 | G337.91-0.48 | – | Y | 2
G328.25-0.53fh | Y | Y | 2 | G338.92+0.55h | Y | Y | 4
G328.81+0.63h | Y | Y | 4 | G340.06-0.23 | N | – | 4
G329.03-0.20h | Y | Y | 2 | G340.78-0.10 | Y | – | 3
G329.07-0.31df${}^{h^{\dagger}}$ | Y | Y | 3 | G343.12-0.06d${}^{j^{\dagger}}$ | N | Y | 1
G329.18-0.31${}^{d^{\ast}}$${}^{h^{\dagger}}$ | Y | Y | 1 | G344.23-0.57fh | Y | Y | 2
G329.41-0.46i | Y | N | 2 | G344.58-0.02 | Y | – | 1
G329.47+0.50h | Y | Y | 2 | G345.00-0.22h | Y | Y | 4
G329.61+0.11eg${}^{i^{\dagger}}$ | Y | N | 1 | G345.51+0.35h | Y | Y | 5
G330.95-0.18i | Y | N | 4 | G348.18+0.48f | N | N | 4
G331.13-0.24h | Y | Y | 2 | | | |
aafootnotetext: Associations with 6.7 GHz class II methanol masers within
30″identified from the 6.7 GHz maser catalogs (Cyganowski et al. 2009; Caswell
2009; Xu et al. 2009; Caswell et al. 2010; Green et al. 2010) and our recent
class II methanol maser surveys with the University of Tasmania Mt. Pleasant
26 m (Titmarsh et al. in prep.): “–” are sources for which 6.7 GHz maser
emission is detected in the Mt Pleasant survey, but for which accurate
positional information is not available, “Y” are sources with 6.7 GHz masers
with accurate positions from high-resolution observations, “N” are sources
without 6.7 GHz maser detections in the Mt Pleasant survey.
bbfootnotetext: Associations with class I methanol masers: Y = Yes, N = No,
“–” = no information (see also Table 3 of Chen et al. 2009). In our analysis,
we only focus on 95 GHz class I maser, thus for sources without 95 GHz class I
maser detections (even with 44 GHz detections), we marked them as no class I
masers.
ccfootnotetext: Remarks: 1 – 5 represent that the sources are selected from
Tables 1 – 5 of Cyganowski et al. (2008), respectively.
d, ed, efootnotetext: Represent the sources with and without 95 GHz class I
methanol maser detections respectively which are shown in the [3.6]-[4.5] vs.
[5.8]-[8.0] color-color diagram obtained with integrated fluxes from
Cyganowski et al. (2008) in Figure 4. A “$\ast$” in the superscript marks the
source which lies in the left-upper of the color region and outside zone
occupied by the Stage I model derived by Robitaille et al. (2006) (see Section
4.1 and Figure 4).
f, gf, gfootnotetext: Represent the sources with and without 95 GHz class I
methanol maser detections respectively which are plotted in the [3.6]-[4.5]
vs. [5.8]-[8.0] color-color diagram obtained with GLIMPSE point source fluxes
in Figure 5. A “$\ast$” in the superscript denotes the source which lies in
the region outside Stage I (see Section 4.1 and Figure 5).
h, i, jh, i, jfootnotetext: Represent the three different subsamples for the
IRAC and MIPS 24 $\mu$m color analysis with integrated fluxes of EGOs: h is a
source associated with both a class I methanol maser and a class II methanol
maser with an accurate position, i is a source associated with only a class II
maser (with an accurate position), and j is a source associated with only
class I maser emission. A “$\dagger$” in the superscript denotes the source
plotted in [3.6]-[5.8] versus [8.0]-[24] color diagram (Figure 6; see Section
4.2).
Table 7: Properties of mm dust clumps associated with methanol masers
EGO name | BGPS namea | Robj$b$ | Sv(int)b | Mc | n(H2)c | N(H2)d | Avd
---|---|---|---|---|---|---|---
| | (′′) | (Jy) | (M⊙) | (103 cm-3) | (1022 cm-2) |
G10.29$-$0.13h | G010.286$-$00.120 | 68 | 9.41 | 1100 | 10.5 | 3.3 | 33
G10.34$-$0.14h | G010.343$-$00.144 | 85 | 8.58 | 1000 | 5.0 | 1.9 | 19
G12.20$-$0.03fh | G012.201$-$00.034 | 53 | 2.06 | 940 | 3.2 | 1.2 | 12
G12.42$+$0.50j | G012.419+00.506 | 51 | 0.76 | 99 | 6.8 | 0.5 | 5
G12.68$-$0.18h | G012.681$-$00.182 | 86 | 11.82 | 5900 | 5.6 | 2.6 | 26
G12.91$-$0.03eh† | G012.905$-$00.030 | 65 | 3.56 | 1800 | 2.2 | 1.4 | 14
G12.91$-$0.26h | G012.909$-$00.260 | 79 | 15.85 | 5200 | 1.7 | 4.1 | 41
G14.63$-$0.58fh† | G014.633$-$00.574 | 72 | 10.20 | 1100 | 5.4 | 3.2 | 32
IRDC18223$-$3j | G018.608$-$00.074 | 71 | 2.18 | 640 | 2.5 | 0.7 | 7
G18.67$+$0.03eh† | G018.666$+$00.032 | 57 | 2.49 | 1500 | 1.5 | 1.3 | 13
G18.89$-$0.47eh† | G018.888$-$00.475 | 101 | 9.85 | 4500 | 3.2 | 1.6 | 16
G19.01$-$0.03eh† | G019.010$-$00.029 | 44 | 2.40 | 960 | 7.7 | 2.0 | 20
G19.36$-$0.03h | G019.364$-$00.031 | 96 | 6.82 | 960 | 7.6 | 1.2 | 12
G19.88$-$0.53fh† | G019.884$-$00.535 | 41 | 5.22 | 1400 | 3.2 | 5.1 | 51
G22.04$+$0.22eh† | G022.041$+$00.221 | 88 | 4.68 | 1400 | 1.7 | 1.0 | 10
G23.82$+$0.38i | G023.818$+$00.384 | 19 | 0.59 | 280 | 2.7 | 2.8 | 28
G23.96$-$0.11eh† | G023.968$-$00.110 | 57 | 3.35 | 1500 | 2.5 | 1.7 | 17
G24.00$-$0.10eh† | G023.996$-$00.100 | 36 | 1.87 | 820 | 2.0 | 2.4 | 24
G24.11$-$0.17j | G024.116$-$00.174 | 77 | 2.48 | 1300 | 2.2 | 0.7 | 7
G24.33$+$0.14eh | G024.329$+$00.142 | 42 | 4.38 | 3800 | 1.4 | 4.0 | 40
G24.63$+$0.15fj† | G024.632$+$00.155 | 50 | 1.98 | 550 | 18.5 | 1.3 | 13
G24.94$+$0.07h† | G024.943$+$00.075 | 49 | 1.40 | 280 | 16.1 | 0.9 | 9
G25.27$-$0.43fh† | G025.266$-$00.439 | 67 | 2.38 | 820 | 12.7 | 0.9 | 9
G25.38$-$0.15fi | G025.388$-$00.147 | 34 | 3.17 | 2100 | 14.8 | 4.6 | 46
G27.97$-$0.47j† | G027.969$-$00.474 | 60 | 1.32 | 290 | 5.1 | 0.6 | 6
G34.26$+$0.15i | G034.258+00.154 | 103 | 78.55 | 27000 | 3.4 | 11.9 | 119
G34.28$+$0.18i | G034.283$+$00.184 | $<$17 | 0.77 | 230 | $>$4.8 | 1.1 | 11
G34.41$+$0.24ei | G034.410$+$00.232 | 96 | 20.78 | 6400 | 5.6 | 3.7 | 37
G35.04$-$0.47fj† | G035.045$-$00.478 | 98 | 4.71 | 1200 | 12.7 | 0.8 | 8
G35.79$-$0.17fi | G035.794$-$00.176 | 54 | 2.60 | 890 | 9.7 | 1.4 | 14
G39.10$+$0.49fh† | G039.100$+$00.491 | 50 | 0.81 | 73 | 0.9 | 0.5 | 5
G39.39$-$0.14i | G039.389$-$00.143 | 35 | 1.37 | 630 | 3.7 | 1.8 | 18
G40.28$-$0.22fi | G040.283$-$00.221 | 35 | 3.50 | 2200 | 1.6 | 4.7 | 47
G45.80$-$0.36i | G045.805$-$00.355 | 26 | 0.97 | 500 | 1.2 | 2.4 | 24
G49.07$-$0.33fj† | G049.069$-$00.328 | 64 | 3.13 | 1800 | 2.0 | 1.2 | 12
G16.59$-$0.05gh | G016.586$-$00.051 | 32 | 3.42 | 1700 | 18.6 | 5.5 | 55
G23.01$-$0.41gh† | G023.012$-$00.410 | 96 | 12.6 | 9500 | 2.0 | 2.2 | 22
aafootnotetext: The associated 1.1 mm BGPS continuum source identified from
BGPS catalog (Rosolowsky et al. 2010).
bbfootnotetext: The associated 1.1 mm BGPS source radius and flux density used
in the calculation. Note that a flux calibration correction factor of 1.5 was
needed to apply to the flux density listed here when calculating the gas mass
(see Dunham et al. 2010). All sources are resolved with the BGPS beam expect
G34.28+0.18. We assumed the beam size as an upper limit on the object radius
of this source.
ccfootnotetext: The gas mass and beam-averaged volume density of clump derived
from the corresponding 1.1 mm continuum source. The beam size of BGPS is 33′′.
ddfootnotetext: The beam-averaged column density N(H2) and reddening vector Av
of clump. We adopt A${}_{v}=10^{-21}\times$ N(H2) from Bohlin et al. (1978).
e, fe, ffootnotetext: The sources associated with GLIMPSE point sources whose
colors locate outside and in the color region occupied by Stage I derived by
Robitaille et al. (2006) in Figure 5, respectively. See more details in
Section 4.1.
ggfootnotetext: The sources are selected from Chen et al. (2009) sample.
h, i, jh, i, jfootnotetext: Represent the three different subsamples: h is a
source associated with both a class I methanol maser and a high-precision
postion class II methanol maser (including 21 members), i is a source
associated with class I masers but without high-precision position class II
maser information (including 9 members), and h is a source associated with
only class I maser emission and without class II maser detection by Mt
Pleasant (including 7 members).
††footnotetext: Sources with integrated flux measurements from Cyganowski et
al. (2008) are also overlapped in the [3.6]-[5.8] vs. [8.0]-[24] color
analysis of Figure 6.
Figure 1: Overlays of the OH and 6.7 GHz class II masers, UC Hii regions, and
1.1 mm BGPS sources on the _Spitzer_ 3-color IRAC images with 8.0 $\mu$m
(red), 4.5 $\mu$m (green) and 3.6 $\mu$m (blue) for all 192 targeted EGOs. The
yellow contours are the 24 $\mu$m MIPSGAL data (Carey et al. 2009) (the
contour levels for each source are not presented). The positions of OH masers,
6.7 GHz class II methanol masers, UC Hii regions, and 1.1 mm BGPS sources are
denoted by small red circles, black crosses, blue squares and yellow diamonds,
respectively. The targeted point is marked by a blue plus. The large white
circle represents the region covered by the Mopra beam, with a solid circle
for detected and dashed circle for undetected 95 GHz class I methanol masers,
respectively. (A color and complete version of this figure is available in the
online journal.)
Figure 2: Spectra of the 95 GHz methanol maser sources detected in the EGO-
based searches. The left and right labels of Y-axis show the values in flux
density and main beam temperature, respectively for each panel. Note that the
Y-axis scale is not the same panel-to-panel. The velocity range covering 40 km
s-1 shown in X-axis is chosen to locate the emission approximately in the
middle for each panel.
Fig. 2.— Continued.
Fig. 2.— Continued.
Fig. 2.— Continued.
Fig. 2.— Continued.
Fig. 2.— Continued.
Fig. 2.— Continued.
Figure 3: Comparison of the distribution of the peak flux density of the 95
GHz class I methanol masers (represented by Flux 95) to that of the 6.7 GHz
class II methanol masers (represented by Flux 6.7) for the EGOs associated
with both classes of masers in our observing sample.
Figure 4: [3.6]-[4.5] vs. [5.8]-[8.0] color-color plot of EGOs. Only sources
listed in Table 1 and 3 of Cyganowski et al. (2008) for which there is flux
density measurements for all the four IRAC bands are plotted. The red and blue
triangles represent the EGOs which are and are not associated with class I
methanol masers, respectively. The solid lines mark the regions occupied by
various evolutionary-stage (Stages I, II and III) YSOs according to the models
of Robitaille et al. (2006). The hashed region in the color-color plot are
regions where models of all evolutionary stages can be present. The error bar
in the top left was derived from the average standard deviation of the
measurements of all data in the plot. The reddening vectors show an extinction
of Av=20, assuming the Indebetouw et al. (2005) extinction law.
Figure 5: Same as Figure 4, but for the nearest GLIMPSE point sources
associated with EGOs. Only sources for which there is flux density
measurements for all the four IRAC bands are plotted.
Figure 6: [3.6]-[5.8] vs. [8.0]-[24] color diagram of EGOs associated with
three subsamples based on which of class methanol masers they are associated
with (see Section 4.2): associated only with class I methanol masers (marked
by open circles), associated only with known 6.7 GHz class II methanol masers
with high accurate positions (marked by open triangles), and associated with
both class I and high accurate position 6.7 GHz class II methanol masers
(marked by filled squares). Only sources listed in Table 1 and 3 of Cyganowski
et al. (2008) for which there is flux density measurements for all the four
IRAC bands and MIPS 24 $\mu$m are plotted. The error bar in the top left of
each plot was derived from the average standard deviation of the measurements
of all data in the corresponding plot. The arrow in each plot represents a
reddening vector at an extinction of Av=20 derived from the Indebetouw et al.
(2005) extinction law.
Figure 7: Histogram of the luminosity of 95 GHz class I methanol maser
detected in our observations with various associations. The different color
bins represent the different associations marked in the right-top corner. The
class II methanol maser subsample includes the Mopra-surveyed EGOs (39 in
total) associated with high-precision position class II masers within 30′′
(see Table 1 and Section 4.2).
Figure 8: Logarithm of the 95 GHz class I methanol maser luminosity as a
function of the gas mass (left panel) and H2 density (right panel) of the
associated 1.1 mm dust clump. The filled squares, open squares and open
circles represent the class I maser sources which are with high-precision
position class II methanol maser associated (21 members), without high-
precision position class II maser information (9 members), and without an
class II maser detection by Mt Pleasant (7 members), respectively. The line in
each panel marks the best fit to the corresponding distribution. The upward
arrow in the right-hand panel indicates the lower limit for the gas density of
the source G34.28+0.18.
|
arxiv-papers
| 2011-07-14T01:41:42 |
2024-09-04T02:49:20.571550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xi Chen, Simon P. Ellingsen, Zhi-Qiang Shen, Anita Titmarsh and\n Cong-Gui Gan",
"submitter": "Xi Chen",
"url": "https://arxiv.org/abs/1107.2914"
}
|
1107.2931
|
# Oxford SWIFT IFS and multi-wavelength observations of the Eagle galaxy at
z=0.77
Susan A. Kassin,1,2 L. Fogarty,1 T. Goodsall,1,3 F. J. Clarke,1 R. W. C.
Houghton,1 G. Salter,1 N. Thatte,1 M. Tecza,1 Roger L. Davies,1 Benjamin J.
Weiner,4 C. N. A. Willmer,4 Samir Salim,5 Michael C. Cooper,6 Jeffrey A.
Newman,7 Kevin Bundy,8 C. J. Conselice,9 A. M. Koekemoer,10 Lihwai Lin,11
Leonidas A. Moustakas,3 Tao Wang12
1 Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK
2 currently: Astrophysics Science Division, Goddard Space Flight Center, Code
665, Greenbelt, MD 20771, USA
3 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, MS 169-327, Pasadena, CA 91109, USA
4 Steward Observatory, 933 N. Cherry St., University of Arizona, Tucson, AZ
85721, USA
5 Department of Astronomy, Indiana University, Bloomington, IN 47404, USA
6 Center for Galaxy Evolution, Department of Physics and Astronomy, University
of California, Irvine,
4129 Frederick Reines Hall, Irvine, CA 92697, USA; Hubble Fellow
7 Department of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara
Street, Pittsburgh, PA 1526, USA
8 Astronomy Department, University of California, Berkeley, CA 94705, USA;
Hubble Fellow
9 University of Nottingham, School of Physics and Astronomy, Nottingham NG7
2RD, UK
10 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218, USA
11 Institute of Astronomy & Astrophysics, Academia Sinica, Taipei 106, Taiwan
12 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge,
MA 02138, USA
E-mail: susan.kassin@nasa.gov
###### Abstract
The ‘Eagle’ galaxy at a redshift of 0.77 is studied with the Oxford Short
Wavelength Integral Field Spectrograph (SWIFT) and multi-wavelength data from
the All-wavelength Extended Groth strip International Survey (AEGIS). It was
chosen from AEGIS because of the bright and extended emission in its slit
spectrum. Three dimensional kinematic maps of the Eagle reveal a gradient in
velocity dispersion which spans $35-75\pm 10$ km s-1 and a rotation velocity
of $25\pm 5$ km s-1 uncorrected for inclination. Hubble Space Telescope images
suggest it is close to face-on. In comparison with galaxies from AEGIS at
similar redshifts, the Eagle is extremely bright and blue in the rest-frame
optical, highly star-forming, dominated by unobscured star-formation, and has
a low metallicity for its size. This is consistent with its selection. The
Eagle is likely undergoing a major merger and is caught in the early stage of
a star-burst when it has not yet experienced metal enrichment or formed the
mass of dust typically found in star-forming galaxies.
###### keywords:
galaxies: high-redshift, galaxies – galaxies: kinematics and dynamics –
galaxies: interactions – galaxies: irregular.
## 1 Introduction
Large redshift surveys at $0<z<1.2$ have revealed that the population of
galaxies is divided into red and blue rest-frame colours, and have measured
the evolution of these populations (e.g., Strateva et al., 2001; Bell et al.,
2004; Faber et al., 2007). In particular, blue galaxies in the past were
brighter by $B\simeq 1.3$ mag (e.g., Bell et al., 2004; Willmer et al., 2006),
more highly star-forming (e.g., Noeske et al., 2007), and more morphologically
irregular (e.g., Abraham et al., 1996; Oesch et al., 2010). In addition, blue
galaxies have an increasing contribution to their kinematics from disordered
motions the further one looks back in time (Weiner et al., 2006a; Kassin et
al., 2007).
These disordered motions, quantified by an integrated velocity dispersion,
appear to play an important role in galaxy kinematics (e.g., Weiner et al.,
2006b; Kassin et al., 2007; Cresci et al., 2009; Covington et al., 2010;
Förster Schreiber et al., 2009; Puech et al., 2010; Lemoine-Busserolle et al.,
2010; Lemoine-Busserolle & Lamareille, 2010). A study of $\sim 550$ galaxies
with slit spectroscopy over $0.1<z<1.2$ has shown that, for galaxies with
stellar masses greater than $10^{10}$ M⊙, there is increasing scatter in the
Tully-Fisher relation (which relates the rotation velocities of galaxies to
their stellar masses or magnitudes) to $z=1.2$ (Kassin et al., 2007). This
scatter is dominated by galaxies with disturbed morphologies (Kassin et al.,
2007). A large scatter in the Tully-Fisher relation is also found for galaxies
in integral field spectrograph (IFS) studies at $z\sim 0.6$ (Flores et al.,
2006; Puech et al., 2008, 2010) and $z\sim 2-3$ (e.g., Law et al., 2009;
Cresci et al., 2009; Lemoine-Busserolle & Lamareille, 2010).
When a kinematic estimator which incorporates both rotation velocity ($V$) and
integrated velocity dispersion ($\sigma$) is adopted,
$S_{K}\equiv\sqrt{KV^{2}+\sigma^{2}}$ with K=0.5 (Weiner et al., 2006a), the
resulting relation with stellar mass has remarkably small scatter at all
redshifts (Kassin et al., 2007; Cresci et al., 2009; Covington et al., 2010;
Puech et al., 2010; Lemoine-Busserolle et al., 2010; Lemoine-Busserolle &
Lamareille, 2010). This is such that disturbed galaxies have higher values of
$\sigma$ than normal disc-like galaxies. Undisturbed disc galaxies in the
local Universe have $\sigma$ values that range from 10–35 km s-1 and an
average value of $\sim$20–25 km s-1 (Epinat et al., 2010) which are due to the
relative motions of individual gas clouds in spiral arms or a thick disc. The
higher $\sigma$ values found for many high redshift galaxies likely represent
effective velocity dispersions caused by the blurring of velocity gradients on
scales at or below the seeing limit (Weiner et al., 2006a; Kassin et al.,
2007). These may not even have a preferred plane.
The nature of these kinematically peculiar objects cannot be fully determined
from slit spectroscopy alone since it is unable to probe the full 3D
kinematics of galaxies. 3D integral field observations can give further clues
as to whether the high $\sigma$ values are driven by phenomena such as
rotation, major or minor merger activity, and/or violent star-formation
processes.
The advent of IFSs on large telescopes has allowed for detailed studies of 3D
galaxy kinematics over $0.4\la z\la 3$ (e.g., Förster Schreiber et al., 2006,
2009; Puech et al., 2007; Yang et al., 2008; van Starkenburg et al., 2008;
Wright et al., 2009; Law et al., 2009; Epinat et al., 2009; Lemoine-Busserolle
et al., 2010; Lemoine-Busserolle & Lamareille, 2010). These studies have all
found galaxies with large $\sigma$ values and attribute it to star-formation
and major and minor merger activity, when active galactic nuclei are not
present. However, these samples are still small ($\la 100$) and are typically
too bright to be representative of typical galaxies, although there are
exceptions (e.g., Puech et al., 2008; Yang et al., 2008; Law et al., 2009).
A representative sample of 68 galaxies at redshifts $0.4<z<0.74$ has been
studied with the GIRAFFE IFS with the Intermediate MAss Galaxy Evolution
Sequence (IMAGES) Survey. Two main results of this survey are the discovery of
a low fraction of rotating disc galaxies (Flores et al., 2006; Neichel et al.,
2008; Yang et al., 2008) and the finding that more distant galaxies have
smaller ratios of $V$ to $\sigma$ than galaxies at lower redshifts (Puech et
al., 2007). These results are consistent with the findings of slit based
studies of larger samples of $\sim$500–1000 galaxies (Weiner et al., 2006a, b;
Kassin et al., 2007). Studies of IMAGES galaxies generally attribute disturbed
kinematics to major mergers.
It is highly desirable to study more galaxies with IFSs at $z\sim 1$, an epoch
probed by large redshift surveys but with few IFS observations. In this paper,
IFS observations of the 3D kinematics of a galaxy at $z=0.7686$ with a high
$\sigma$ value are presented along with a suite of multi-wavelength data from
the All-wavelength Extended Groth strip International Survey (AEGIS, Davis et
al., 2007). These data allow for the galaxy to be understood in terms of the
general population of galaxies at its epoch and offer insight into a likely
cause for its high $\sigma$ value. A $\Lambda$CDM cosmology is adopted
throughout: $H=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$. All
logarithms are base 10, and all magnitudes are on the AB system.
## 2 The Eagle galaxy
### 2.1 Selection
The Eagle galaxy, DEEP2 ID 13019195 ($\alpha$ 14:19:39.3, $\delta$
+52:55:48.8, J2000), was selected from the AEGIS Survey (for which the DEEP2
Survey provides redshifts and spectra; Davis et al., 2003) to be observed with
the Oxford Short Wavelength Integral Field Spectrograph (SWIFT; Thatte et al.,
2006). The Eagle galaxy is so named for its distinctive appearance in Hubble
Space Telescope images (Figures 1 and 2). It was primarily selected from the
AEGIS Survey to have very bright and extended line emission ($\ga 10^{-16}$
ergs s-1 cm-2 and $\ga 2$″ from a visual inspection of the spectrum,
respectively) which falls into the SWIFT bandpass. The aim of these
requirements was to provide a target which would result in observations with a
high signal-to-noise ratio and at least two independent spatial resolution
elements. The [OIII] $\lambda 5007$ emission line in the DEEP2 spectrum of the
Eagle has an integrated flux of $9.6(\pm 2.5)\times 10^{-16}$ ergs s-1 cm-2
and a spatial extent of $\ga 2$″from a visual inspection.
The target was further required to have at least one emission line not
contaminated by sky lines, a redshift greater than $\simeq 0.7$, an
inclination $\la 70$°, and a significant $\sigma$ ($>40$ km s-1). The limit on
inclination allows for kinematics to be studied across the face of the galaxy
without significant projection effects, and the requirement on $\sigma$ allows
for an investigation into the cause of the unusually high $\sigma$ values
found for galaxies at high redshift.
Figure 1: Broad-band and SWIFT IFS observations of the Eagle galaxy are shown
in the top and bottom panels, respectively. All panels are 3″ on a side and
are oriented such that north is up and east is to the left. Top row: Broad-
band images at $B$ (CFHT/CFH12K; 0.21″pixel-1), $V$ ($Hubble/ACS$;
0.1″pixel-1), $R$ (CFHT), and $I$ (CFHT) are shown from left to right. The
image FWHM for the CFHT images was $\sim 0.7$″. Bottom row: Maps of $V$,
$\sigma$, [OIII] line emission, and signal-to-noise ratio for the [OIII] line
are shown from left to right. These maps have been smoothed with a Gaussian
with a $\sigma$ of 1 spaxel (0.235″), which is much smaller than the $\sim 1$″
image FWHM during the observations. Due to similar image FWHMs, the most
direct comparison can be made between the SWIFT observations and the broad-
band images from CFHT.
### 2.2 Hubble morphology & size
#### 2.2.1 Morphology
$Hubble$/ACS $V$-band and $V+I$-band images are shown in Figures 1 and 2,
respectively. Quantitative morphological analyses were performed on the
$I$-band image with the CAS and Gini/M20 systems (Conselice et al. 2003 and
Lotz, Primack, & Madau 2004, respectively). Both classify the Eagle as a major
merger galaxy. Specifically, the Eagle has an asymmetry (A) value of 0.32 (Lin
et al., 2007), and although major mergers are defined in CAS as galaxies with
A$\geq$0.35, once the signal-to-noise in the ACS image is taken into account,
it is classified as such. In addition, Lin et al. (2007) classify galaxies
with A$>$0.25 as major mergers. The Eagle has Gini and M20 values of 0.46 and
-0.91, respectively (Lotz et al., 2008). These qualify it as a major merger
according to the definition in Lotz et al. (2008): Gini $>-0.14$M20 \+ 0.33.
Furthermore, according to the visual classification of $Hubble$ images in
Kassin et al. (2007) which categorised galaxies as ‘normal,’ ‘disturbed’, or
‘compact’, the Eagle is identified as ‘disturbed.’ This is mainly due to outer
ispohotes which are not elliptical and an asymmetric three-armed structure,
both of which are inconsistent with the morphology of an undisturbed disc
galaxy. In summary, quantitative and qualitative morphological classifications
of the Eagle are consistent with it being a major merger galaxy and
inconsistent with it being a disc galaxy.
#### 2.2.2 Size
It is difficult to measure the sizes of high redshift galaxies which are not
always smooth or elliptical. In order to avoid biases due to disturbed
morphologies, we look to measure a constant fraction of the total galaxy
light, independent of the morphology or orientation of the object. The
Petrosian radius comes closest to this ideal (e.g., Petrosian, 1976; Bershady
et al., 2000; Massey et al., 2004). Petrosian radii measured in elliptical
apertures are adopted from Lotz et al. (2008). The radii were measured from
$I$-band $Hubble$ images using the SExtractor software (Bertin & Arnouts,
1996) for $\eta=0.2$. The Petrosian radius of the Eagle is $1.2\arcsec\pm
0.2\arcsec$, which corresponds to 8.9 kpc at its redshift, and the position
angle of the major axis is 49°(east of north).
## 3 SWIFT observations and data reduction
The Eagle Galaxy was observed under clear conditions with SWIFT mounted on the
Hale Telescope at Palomar Observatory on May $8$th, $9$th, and $10$th of
$2009$. Observations were performed in natural seeing mode in the coarsest
spaxel scale of 0.235″, which results in a field of view of 10.3″$\times$
20.9″. The spectral resolution at the observed wavelength of 8855Å is $R\sim
4000$, which results in an instrumental broadening of $\simeq 1.2$ Å or 41 km
s-1 (Gaussian $\sigma=21$ km s-1), as measured from fits of line profiles to
isolated sky lines near the observed wavelength. To obtain an accurate
pointing, the telescope was offset blindly from a nearby bright star to the
galaxy. The large field of view allowed for dithering on source, and
integrations of 900s each were taken. During this commissioning run we were
unable to guide the telescope, and relied on tracking alone. A total of 6
exposures which had the best delivered image quality, or image FWHM, were
chosen to create the final cube, resulting in a total exposure time of 1.5
hours. The image FWHM was $\sim 1$″ for these observations; they were taken at
the beginnings of the nights of May $9$th and $10$th.
The SWIFT data were reduced with custom-purpose software written by R.
Houghton and T. Goodsall, based on the spred pipeline for the SINFONI IFS
(Schreiber et al., 2004; Abuter et al., 2006). The SWIFT reduction procedure
is described in detail in Fogarty et al. (2010), and is briefly reviewed here.
The galaxy was observed in one of the two CCDs, so only data from that CCD was
processed. All frames were reduced in the same manner. First the bias was
removed, the overscan region trimmed, and cosmic rays removed with the
L.A.Cosmic algorithm (van Dokkum, 2001). A wavelength solution was found from
Ar and Ne arc lamp exposures taken at the beginning of the observing run. Any
flexure in wavelength between observations was quantified by measuring the
shift of sky lines close to the [OIII] line in wavelength. For 2 of the 6
exposures used, there was flexure in the wavelength axis of 0.46 and 0.49 of a
pixel (1 pixel = 1Å). These cubes were shifted in wavelength to align them
with the other observations using a third order spline interpolation.
Since dithering was performed on source, sky subtraction could be accomplished
with the subtraction of AB pairs of observations. To obtain optimal sky
subtraction, “super skys” were created by taking the median of the 2 or 3 A or
B observations closest in time to each B and A observation, respectively. The
sky subtracted data cubes were aligned spatially on the peak flux of the
[OIII] line map of the galaxy and co-added.
### 3.1 Kinematic maps
To measure kinematics from the final co-added data cube, Gaussian functions
were fit to the [OIII] line in the spectrum of each spatial pixel in which the
galaxy was detected. The resulting rotation velocity ($V$), velocity
dispersion corrected for instrumental broadening ($\sigma$), [OIII] line, and
signal-to-noise maps are shown in Figure 1. The maps have been smoothed
spatially with a Gaussian $\sigma$ of 1 pixel (1 spaxel = 0.235″), which is
much less than the image FWHM of $\sim 1$″, to decrease pixel-to-pixel noise.
A signal-to-noise threshold of 2.5 was applied to these maps.
Also shown in Figure 1 are ground-based images from the Canada-France-Hawaii
Telescope (CFHT)/CFH12k camera at $B$, $R$, and $I$, and a $V$-band image from
$Hubble$/Advanced Camera for Surveys (ACS). The most direct comparison for the
SWIFT observations is to the CFHT images, rather than the $Hubble$ image,
since the image FWHM for the CFHT images is $\sim 0.7$″, similar to that
during the SWIFT observations ($\sim 1$″). Figure 1 shows that the morphology
of the Eagle galaxy does not vary significantly from observed $B$ to $I$ in
the CFHT images. The SWIFT O[III] linemap has a similar morphology to the CFHT
broad-band images, but is less irregular-shaped, consistent with a poorer
image FWHM. The number of spatial elements per resolution element (i.e., the
oversampling rate) for the SWIFT maps is $\sim 4$.
The $V$ map is consistent with a maximum value of $V\times$sin($i$) of $25\pm
5$ km s-1, where $i$ is the inclination of the galaxy. This is the maximum
rotation spread along a line with a position angle of 145°(measured east of
north). From a visual inspection of the $Hubble$ images, it is clear that the
Eagle has isophotes which are not ellipsoidal and an asymmetric three-armed
structure. In addition, the kinematic major axis is not aligned with the
morphological major axis. (The direction of the morphological major axis is
the same as the position angle of the slit for the DEEP2 observations; see
§4.5 and Figure 2.) Both of these findings are inconsistent with the Eagle
being an edge-on disc. It is more likely that this galaxy is viewed nearly
face-on, i.e. with an inclination $\la 30$°. We refrain from measuring the
inclination due to the large uncertainties inherent in measuring low
inclinations, especially for systems with isophotes which are not ellipsoidal.
For inclinations of 30°, 20°, 10°, and 5°, the inclination-corrected $V$
varies systematically from 50–287 km s-1, namely $V=50,73,144,$ and 287 km
s-1, respectively.
The $\sigma$ field shows a gradient such that the southern and northern
sections of the galaxy have typical $\sigma$’s of $35-50$ and $65-75$ km s-1,
respectively, with uncertainties of 10 km s-1. As mentioned in §1, undisturbed
disc galaxies in the local Universe have $\sigma$ values that range from 10–35
km s-1 and an average value of $\sim$20–25 km s-1 (Epinat et al., 2010) which
are due to the relative motions of individual gas clouds in spiral arms or a
thick disc. The much higher $\sigma$ values found for the Eagle likely
represent effective velocity dispersions caused by the blurring of velocity
gradients on scales at or below the seeing limit (Weiner et al., 2006a; Kassin
et al., 2007). These may not even have a preferred plane. In addition, the
northern edge of the Eagle, where the highest $\sigma$ values occur, is where
the three “arms” of the galaxy which are visible in the $Hubble$ images meet,
possibly indicative of tidal disturbances there.
For a rotating disc galaxy observed at high redshift, seeing (i.e., beam
smearing) smooths out the $V$ gradient and produces a strong peak in the
centre of the $\sigma$ map (e.g., Weiner et al., 2006a; Covington et al.,
2010). The $\sigma$ peak is produced in the central parts of the galaxy where
the velocity gradient is strongest and seeing smears the light from gas at
different velocities together. There is no seeing-induced peak observed in the
$\sigma$ field of the Eagle. This is likely due to a combination of its low
value of $V\times$sin($i$) and correspondingly shallow velocity gradient in
its inner parts, and low spatial resolution. For an image FWHM of $\sim 1$″,
the resulting decriment in $V$, which is typically $\sim 10-15$ km s-1 for a
galaxy the size of the Eagle (Weiner et al., 2006a; Kassin et al., 2007), is
significantly less than the range in $V$ allowed due to the uncertainty in the
inclination ($\pm\sim 95$ km s-1). Due to this, the low spatial resolution of
the 3D maps, and the misalignment of the kinematic and morphological major
axes, we refrain from creating a detailed kinematic model for this system.
### 3.2 Comparison with kinematics of galaxies at the same epoch
In Figure 3, the kinematics of the Eagle are compared with those of a sample
of galaxies at the same epoch in terms of the ratio of $V$ to $\sigma$, as
measured in Kassin et al. (2007). These galaxies are essentially selected on
emission line strength and are $\ga 80$% complete down to $\sim 10^{9.5}$ M⊙
(Kassin et al., 2007). For the Eagle, the range of allowed values of $V$ from
§3.1, and an average $\sigma$ over the entire galaxy of $55\pm 10$ km s-1, are
adopted. Galaxies in Figure 3 are coded according to the visual morphological
classification from Kassin et al. (2007). It is apparent that galaxies with
low $V/\sigma$ values are generally classified as disturbed, whereas those
with high values have normal disc-like morphologies. Galaxies in the IMAGES
Survey at $z\sim 0.6$ show a similar range in $V/\sigma$ (Puech et al., 2007).
Although the Eagle has a large allowable range in $V/\sigma$ due to the large
uncertainty in its inclination, it cannot be among the very $V$-dominated
systems because of its large $\sigma$ which, at its lowest possible value of
45 km s-1, is still greater than that found for local galaxies (§1). This is
consistent with the Eagle’s morphological classification as disturbed or major
merger (§2.2).
Table 1: Properties of the Eagle galaxy. $V$ sin($i$) | $\sigma$ | SFR (total) | $M_{*}$ | $M_{B}$ | $U-B$ | $12+log(O/H)$ | $R_{I,Petrosian}$
---|---|---|---|---|---|---|---
km s-1 | km s-1 | M⊙ yr-1 | log M⊙ | AB | AB | | $\arcsec$
$25\pm 5$ | $35-75\pm 10$ | $26.3\pm 0.4$ | $10.0\pm 0.2$ | $-21.51\pm 0.08$ | $0.46\pm 0.09$ | $8.66\pm 0.05$ | $1.2\pm 0.2$
Figure 2: $Hubble$ and DEIMOS observations of the Eagle galaxy (top) and fits
to the kinematics (bottom). In the upper left panel is a colour $Hubble$ image
created from $V$ and $I$-band exposures with the position of the 1″ wide
DEIMOS slit marked. It is oriented such that north is up and east is to the
left. The upper right panel shows the [OIII] line in the slit spectrum. Th
spectrum is shown as observed, so constant wavelength runs diagonally (black
line). The lower panels show the rotation and dispersion profiles created by
Gaussian fits to each row of pixels in the [OIII] line. The black points are
data used, and the green points are rejected. Note that not all points are
independent due to the $\sim 0.7$″seeing. The solid red lines are the best fit
kinematic models described in §4.1. Figure 3: The range of values allowed for
the ratio of rotation velocity to integrated velocity dispersion of the Eagle
(grey shading) is compared to values for emission line galaxies at
$0.65<z<0.85$ from Kassin et al. (2007) as a function of stellar mass. The
range in $V/\sigma$ for the Eagle is mainly due to the uncertainty in its
inclination. Other galaxies are coded according to their visual morphology as
deduced from $Hubble$ images by Kassin et al. (2007): normal (black points),
disturbed (magenta triangles), and compact (green squares). A typical error
bar for these galaxies is shown in black. Although the Eagle has a large
uncertainty in $V/\sigma$, it is not among the very $V$-dominated systems at
this epoch due to its large average $\sigma$ which, at its lowest possible
value of 45 km s-1, is still greater than that found for local galaxies.
## 4 Multi-wavelength data and spectral energy distribution from AEGIS
A few IFS studies have incorporated multi-wavelength data which span from the
UV or optical to the far-infrared (e.g., Fuentes-Carrera et al., 2010; Hammer
et al., 2009; Puech et al., 2009, 2010). Such data facilitate the formulation
of more complete pictures of the galaxies by allowing for more accurate
determinations of e.g., star-formation rates, dust content, and sometimes
metallicity.
Multi-wavelength photometry and a flux calibrated optical spectrum come from
the AEGIS Survey. The spectrum and redshift are from the DEEP2 Survey, a
portion of which is incorporated into AEGIS. A far-infrared flux (observed
$24\mu$m) is adopted from the FIDEL Survey (Dickinson et al. in preparation)
which overlaps AEGIS and has a greater depth than the AEGIS $24\mu$m imaging.
Data are taken from the following telescope/instrument combinations:
Keck-2/DEep Imaging Multi-Object Spectrograph (DEIMOS) for an optical
spectrum, Galaxy Evolution Explorer for ultraviolet photometry, Hubble Space
Telescope/Advanced Camera for Surveys (ACS) for optical imaging, Canada-
France-Hawaii Telescope (CFHT)/CFH12K for optical imaging and photometry,
Palomar/Wide Field Infrared Camera (WIRC) for $K_{s}$-band photometry,
$Spitzer$/Infrared Array Camera (IRAC) for mid-infrared photometry,
$Spitzer$/Multiband Imaging Photometry for Spitzer (MIPS) for 24µm photometry,
and Chandra/AXAF CCD Imaging Spectrometer (ACIS) for counts in the hard and
soft bands. Data reduction is discussed in Davis et al. (2007) for AEGIS and
Salim et al. (2009) for FIDEL.
Photometry was performed on these images as follows, and all measure total
galaxy flux. For the CFHT images, $R$-band magnitudes were measured in
circular apertures of radius $3r_{g}$, where $r_{g}$ is the $\sigma$ of a
Gaussian fit to the image profile. To derive $B$ and $I$-band magnitudes,
$B-R$ and $R-I$ colours were measured in a 1″ radius aperture (Coil et al.,
2004). The resulting $B$ and $I$-band magnitudes differ from magnitudes
measured within $3r_{g}$ only if there are significant colour gradients in the
CFHT images, which there are not (Figure 1). For the WIRC images, photometry
was performed using a Kron-like aperture (Bundy et al., 2006). For the IRAC
images, photometry was performed in a 3″ diameter aperture (Davis et al.,
2007). For MIPS and GALEX images, since the point spread functions (PSFs) are
6″ and 5″ respectively, PSF fluxes were extracted.
K-corrections are applied following Willmer et al. (2006) to obtain rest-frame
$U$ and $B$-band magnitudes, which are given in Table 1. A rest-frame colour-
magnitude diagram in Figure 4 shows all galaxies in AEGIS at $0.65<z<0.85$. It
shows that the Eagle galaxy is extremely bright and blue compared to its
contemporary galaxies.
### 4.1 Single slit kinematics
We look to kinematics derived from the DEEP2 slit spectrum for complementarity
and completeness. The [OIII] emission line in this spectrum is shown in Figure
2. The image FWHM is $\sim 0.7$″. The resolution was $R\sim 5000$ and the slit
width was 1″, which resulted in a spectral resolution of FWHM=$0.56$ Å, or 22
km $s^{-1}$ at the redshift of the Eagle (Gaussian $\sigma=8$ km s-1). The
slit was oriented at the position angle of the photometric major axis which is
$49\degr$ (east of north; upper left panel of Figure 2; measured in §4.5),
which is approximately perpendicular to the gradient of the velocity field in
Figure 1.
As in the IFS data, the spatially extended [OIII] emission line in the galaxy
spectrum is used to measure gas rotation and dispersion profiles, but at 2D
instead of 3D, and under the assumption of a spatially constant $\sigma$. The
ROTCURVE fitting procedure of Weiner et al. (2006a) is used to fit a kinematic
model. Briefly, this procedure fits Gaussians to each row of pixels in the
emission line to obtain profiles of $V$ and $\sigma$ along the slit, and
rejects discrepant values with automatic criteria (Figure 2, bottom panels).
It then measures the light distribution along the slit and fits a Gaussian to
it. Finally, ROTCURVE fits models of the position-velocity distribution along
the slit, taking the seeing into account. The model has two parameters: $V$
(not corrected for inclination) and $\sigma$. The resulting $V$ and $\sigma$
for the Eagle are $0\pm 9$ km s-1 and $45\pm 6$ km s-1, respectively.
The 2D slit spectrum shows no evidence of rotation, whereas the 3D velocity
field shows a rotation gradient of $25\pm 5$ km s-1. This is because the slit
was placed approximately perpendicular to the kinematic major axis. The
$\sigma$ measured from the slit spectrum is shown to be constant along the
slit, as demonstrated by the Gaussian fits to each row of the emission line
(black points in the bottom left panel of Figure 2). The slit spectrum does
not show the $\sigma$ gradient found in the 3D $\sigma$ map because it is
luminosity-weighted and not as deep as the 3D observations. Therefore, it is
biased towards the brightest regions of the galaxy (as depicted in the [OIII]
line map in Figure 1), where the $\sigma$ is 55$\pm 10$ km s-1.
This comparison between 2D and 3D spectroscopy of the same object demonstrates
the power of 3D spectroscopy to reveal the full kinematics of high redshift
galaxies.
Figure 4: The Eagle galaxy (red star) is compared with galaxies at
$0.65<z<0.85$ in the AEGIS survey (points) in colour-magnitude and SFR-$M_{*}$
space. Top: A colour-magnitude diagram (not corrected for dust) shows that the
Eagle is on the very bright and blue end of the distribution of galaxies at
similar redshifts. Uncertainties on magnitudes are 10%. Middle: A SFR-$M_{*}$
diagram where the SFR is derived from line emission for blue galaxies with
F${}_{24um}<60$ $\mu$Jy with and without correcting for extinction (blue and
green points, respectively), and from line emission and far-infrared flux for
galaxies with F${}_{24um}>60$ $\mu$Jy (black points). Uncertainties on SFRs
and $M_{*}$’s are 0.2 dex. The Eagle has a high total SFR for its $M_{*}$.
Bottom: This is the same as the middle plot except only the uncorrected SFRs
derived from emission lines are plotted. The Eagle has a high SFR as derived
from emission lines.
### 4.2 Star-formation rate
Historically, there are two main approaches to determine the star-formation
rate (SFR) of a galaxy. These depend on the wavelength of the data at hand:
optical or UV which trace the output of hot young stars (but which need to be
corrected for interstellar extinction), and far-infrared which acts as a
calorimeter (e.g., Kennicutt, 1998). As multi-wavelength observations become
more commonplace, studies of galaxy SFRs increasingly combine UV/optical and
infrared indicators to obtain better accuracy (e.g., Dale et al., 2007;
Calzetti et al., 2007; Kennicutt et al., 2009). These studies take advantage
of the fact that dust mostly radiates in the infrared and sub-mm, while
intercepting radiation at UV and optical wavelengths. Therefore, the addition
of a UV or optically-determined SFR (not corrected for dust) and an infrared-
determined SFR gives a more accurate account of the total SFR of a system.
Furthermore, a comparison of the two SFRs gives insight into dust content.
Measurements of SFRs from far-infrared data rely on stellar models to predict
the energy output from stars, and assume that most of the radiation from young
stars is extincted and re-processed into the infrared. A SFR is derived from
the $24\mu$m flux of the Eagle following Noeske et al. (2007) using spectral
energy distribution (SED) templates from Chary & Elbaz (2001). A Kroupa (2002)
initial mass function is assumed, which does not differ significantly for the
purposes of this paper from a Chabrier (2003) IMF (Rieke et al., 2009). The
resulting SFR for $F_{24\mu\rm m}=71.8\pm 2.6$ $\mu$Jy is 6 M⊙ yr-1, which has
a factor of $\sim 2$ uncertainty. This is consistent with SFRs derived with
the formulations of Rieke et al. (2009) and Dale & Helou (2002).
SFRs derived from nebular emission lines (like H$\beta$ which is used here)
rely on stellar models to predict the ionizing luminosity of O stars. They
assume that the extinction of the recombination photons is given by the Balmer
decriment (i.e., H$\beta$/H$\alpha$). A calibration from Kennicutt (1998) for
H$\alpha$ line luminosity is typically adopted: SFR (M⊙ yr-1) = $7.9\times
10^{-42}$ $L$(H$\alpha$) (ergs s-1). The resulting SFR is for a Salpeter
(1955) IMF and is converted to a Chabrier (2003) IMF by a multiplicative
factor of 0.66 (Rieke et al., 2009).
An H$\beta$ line luminosity is adopted from the DEEP2 spectrum since the Eagle
is compared with galaxies in DEEP2 for which measurements and calibrations are
performed in the same manner. The H$\beta$ line luminosity is measured
following Weiner et al. (2007), and is $10^{41.9\pm 0.1}$ ergs s-1.
A lower limit to the emission line-determined SFR of the Eagle is derived from
the H$\beta$ line under the assumption of no extinction (i.e., case B
recombination at $10,000$°K: H$\beta$/H$\alpha$ = 0.35, Osterbrock 1989),
namely 11.3 M⊙ yr-1. An upper limit is derived using a lower bound to the
Balmer decriment for a distribution of blue galaxies in DEEP2 over
$0.33<z<0.39$ (the redshift range where both lines can be measured) from
Weiner et al. (2007), namely 0.11. This results in a SFR of 36.0 M⊙ yr-1.
Finally, to estimate the mean SFR of the Eagle from its H$\beta$ line
luminosity, the average Balmer decriment for galaxies in Weiner et al. (2007)
is adopted, namely 0.198. This results in a SFR of 20.3 M⊙ yr-1. In summary,
the SFR of the Eagle derived from H$\beta$ and not corrected further for
extinction is 20.3 M⊙ yr-1 for a Chabrier (2003) IMF, with lower and upper
limits of 11.3 and 36.0 M⊙ yr-1, respectively. In this paper, we adopt the
mean SFR since we know there is some dust present due to its detection at
$24\mu$m.
SFRs derived from UV flux rely on stellar models to predict the UV flux from O
and B stars and assume that the extinction is given by a measurement of the UV
slope. Because intermediate-age stars also contribute to the UV flux, a
simultaneous fit is typically made to the young and older stellar populations,
and a dust model is assumed. Such a fit for the Eagle is adopted from Salim et
al. (2009) and results in a SFR of $9.4\pm 2.0$ M⊙ yr-1 (corrected for
extinction with the UV slope). The uncorrected value is 3.5 M⊙ yr-1 . The
discrepancy between this measurement and the H$\beta$-derived SFR is somewhat
unusual. Along with the low 24$\mu m$-derived SFR, it is evidence that
extinction is low in star-forming regions in the Eagle.
The SFR from the 24$\mu$m flux (6 M⊙ yr-1) is less than even the lower limit
to the SFR derived from H$\beta$ which is not corrected for extinction (11.3
M⊙ yr-1). Even though both these measurements are uncertain to within a factor
of $\sim 2$ due to statistical scatter in the flux-to-SFR conversion, they
differ systematically. The SFR derived from 24$\mu$m is calculated under the
assumption that dust captures all the UV flux (calorimeter assumption), and
the SFR derived from H$\beta$ with zero dust correction (11.3 M⊙ yr-1, lower
limit) assumed that dust captures none of the ionizing or H$\beta$ flux. For
the vast majority of galaxies in the DEEP2 Survey, the SFR derived from
24$\mu$m is greater than the uncorrected line-derived SFR by a factor of $\sim
3$. There are hardly any galaxies where the uncorrected emission line-derived
SFR is greater than the 24$\mu$m-derived SFR. The Eagle is an exception as its
emission-line SFR is greater by at least a factor of $\sim 2$ (11.3 versus 6
M⊙ yr-1). Similarly, Bell et al. (2005) find the 24$\mu$m-derived SFR
dominates the uncorrected UV-derived SFR.
This discrepancy between 24$\mu$m and uncorrected H$\beta$-derived SFRs
suggests that the fraction of line emission escaping the sites of star-
formation is large so that the calorimeter assumption is violated and the
extinction of H$\alpha$ could be lower than normal. In addition, a low dust
content is consistent with the blue colour of the galaxy, uncorrected for dust
(Figure 4, top panel). A total SFR for the Eagle is estimated by adding the
emission line (not corrected for dust) and 24$\mu m$-derived SFRs, and results
in 26.3 M⊙ yr-1, which has a factor of 2–3 uncertainty.
To compare the Eagle with galaxies at similar redshifts, in Figure 4 the SFRs
of galaxies in the AEGIS Survey at $0.65<z<0.85$ are plotted versus stellar
mass (following Noeske et al. 2007), and the Eagle galaxy is highlighted. The
sample is $>80$% complete for $M_{*}\ga 10^{9.6}$ M⊙ and $>95$% complete for
M${}_{*}\ga 10^{10}$ (Noeske et al., 2007). These galaxies are from the same
data set and have SFRs measured in a homogeneous manner. Figure 4 shows that
the Eagle has a high SFR compared to galaxies at similar stellar masses, even
when only the SFR derived from line emission is taken into account.
Figure 5: Multi-wavelength broad band spectral energy distribution of the
Eagle galaxy (black error bars) and the best-fit galaxy stellar population
model to the optical and near-infrared fluxes from Bundy et al. 2006 (red
triangles). The data, from left to right, are $NUV$, $B$, $R$, $I$, $K$, IRAC
channels 1, 2, 3, 4, and MIPS $24\mu m$ fluxes. The fluxes are log $S=-0.34\pm
0.05,0.726\pm 0.09,0.908\pm 0.09,1.111\pm 0.09,1.213\pm 0.02,1.053\pm
0.03,1.018\pm 0.14,0.930\pm 0.14,$ and $1.856\pm 0.04\,\mu$Jy, respectively.
### 4.3 Stellar mass & broad-band spectral energy distribution
The multi-wavelength SED for the Eagle galaxy is shown in Figure 5. Fits of
galaxy stellar population models to the observed $B,R,I,$ and $K_{s}$-band
fluxes were performed by Bundy et al. (2006) to obtain a stellar mass estimate
($M_{*}$) for the Eagle. These are the wavelengths at which stellar population
models are the best-characterised. The galaxy models constituted a grid of
synthetic spectral energy distributions from Bruzual & Charlot (2003) which
spanned a range of exponential star-formation histories, ages (restricted to
be less than the age of the Universe at the redshift of the Eagle),
metallicities, and dust contents. A Chabrier (2003) initial mass function,
which does not differ significantly from the Kroupa (2002) function assumed
for the SFR calculation, was adopted. At each point on the grid of models, the
following is calculated: the $K_{s}$-band stellar mass-to-light ratio, minimum
$\chi^{2}$, and probability that each model accurately describes the galaxy.
The corresponding $M_{*}$ is then determined by scaling the stellar mass-to-
light ratio to the $K_{s}$-band luminosity based on the total $K_{s}$-band
magnitude. The probabilities are then summed (marginalised) across the grid
and binned by model stellar mass, yielding a stellar mass probability
distribution for the galaxy. The median of the distribution is adopted as the
best estimate. The $M_{*}$ measured in this way is robust to degeneracies in
the models, such as those between age and metallicity (Bundy et al., 2006).
The best-fit model has a log $M_{*}$ of $10.0\pm 0.2$ M⊙, where the
uncertainty is taken from the width of the probability distribution. There are
additional systematic uncertainties associated with the IMF and stellar
population synthesis models adopted which can be as large as 0.4 dex (Barro et
al., 2011). At the redshift of the Eagle, taking into account the TP-AGB phase
of stellar evolution does not produce significant changes in stellar mass
estimates (e.g., Barro et al., 2011).
We refrain from including the IRAC channel 1 waveband in the SED fit since
doing so decreases the goodness of the model fit. This is possibly because of
uncertainties in stellar population models at this wavelength. The mid to far-
infrared points are not included in the fit because stellar population models
do not currently include the dust modelling necessary to fit them.
Furthermore, although the best-fit model also provides estimates of age,
metallicity, star-formation history, and dust content, these quantities are
much more affected by degeneracies and are poorly constrained compared with
the stellar mass (Bundy et al., 2006).
### 4.4 Test for AGN contamination
In this section, we test for active galactic nuclei, or AGN, contribution to
the emission lines in the spectrum of the Eagle. This could affect derived
quantities such as kinematics, SFR, and stellar mass. There are two methods
which are regarded as the most effective for the detection of AGN in high
redshift galaxies, where it is generally not possible to isolate the nuclear
regions. These are optical emission line and X-ray selections. Neither
selection is perfect. Emission lines from the AGN can be overpowered by star-
formation, and X-rays can be affected by heavy absorption of gas in close
proximity to the AGN. However, a combination of these two methods is the best
currently available to identify AGN (Yan et al., 2010).
To test for a possible contribution from AGN to the Eagle, we first look to an
emission line diagnostic diagram derived for galaxies in the DEEP2 Survey over
$0.3<z<0.8$ (Yan et al., 2010, Figure 5). This diagram uses integrated rest-
frame $U-B$ colour and [OIII]/H$\beta$ line ratio to separate star-forming
galaxies from AGN. The Eagle galaxy lies in the star-forming region of the
diagram given its [OIII]/H$\beta$ value of $0.53\pm 0.2$ from the DEEP2
spectrum.
Next, we look to the deep $200$ks Chandra data. No counts were detected at the
coordinates of the Eagle in either of the Chandra bands. This places $5\sigma$
upper limits on the X-ray luminosity of AGN in the Eagle of LX = $1.1\times
10^{-16}$ and $8.2\times 10^{-16}$ ergs s-1 cm-2 for the soft and hard bands,
respectively.
Furthermore, we note that the near-infrared through IRAC portion of the SED in
Figure 5 does not show the typical power law shape of an AGN. In conclusion,
the kinematics and emission line luminosities of the Eagle are likely not
strongly affected by AGN.
### 4.5 Metallicity
The oxygen abundances of the Eagle and galaxies in AEGIS at similar redshifts
($0.65<z<0.85$) are estimated. This is done by adopting the relation between
$R_{23}$ and the gas phase oxygen abundance by McGaugh (2001), following
Kobulnicky & Kewley (2004). The upper branch of this calibration is adopted
because the majority of galaxies in Kobulnicky & Kewley (2004) and Kobulnicky
& Zaritsky (1999) with NII/H$\alpha$ measurements fall there. This measurement
relies on equivalent widths of emission lines. It works in part because star-
forming galaxies have a relatively narrow range of continuum shapes between
[OII] and [OIII], and do not have large 4000Å breaks. For the Eagle, rest-
frame equivalent widths of the [OII] $\lambda 3727$, [OIII] $\lambda 5007$,
and H$\beta$ lines ($67.00\pm 1.25$, $77.33\pm 1.82$, and $28.1\pm 0.73$,
respectively and all in Å), result in an oxygen abundance of 12 + log (O/H)
$=8.66\pm 0.05$.
In Figure 6, the oxygen abundances of AEGIS galaxies and the Eagle are plotted
versus rest-frame $B$-band magnitudes and Petrosian radii (measured from
$I$-band $Hubble$ images as described below). This galaxy sample is complete
down to at least $M_{B}$ (AB) $\la 19$ for blue galaxies (no red galaxies are
shown in the figure) in the redshift range plotted (Willmer et al., 2006).
Galaxies from Kobulnicky & Kewley (2004) in the same redshift range are also
compared. The Eagle galaxy is brighter by $\sim 2$ $B$-band magnitudes and
larger by $\sim 0.4$ dex than average for its oxygen abundance. The oxygen
abundance of the Eagle is similar to those of galaxies at similar stellar
masses in the IMAGES Survey at $z\sim 0.6$ (Rodrigues et al., 2008).
Figure 6: Oxygen abundances are compared with rest-frame $B$-band magnitudes
(top) and $I$-band Petrosian radii (bottom) for galaxies in the AEGIS Survey
at $0.65<z<0.85$ (filled circles) and from Kobulnicky & Kewley (2004) (open
circles). Typical uncertainties in the measurements of oxygen abundances,
sizes, and $B$-band magnitudes are 0.05 dex, 0.1 dex, and 10%, respectively.
In addition, for the oxygen abundances, there can be systematic errors in the
calibration which may be $>0.1$ dex (Kobulnicky & Kewley, 2004). Binned
averages for the AEGIS galaxies are shown as open triangles and error bars
show the rms scatter. The Eagle galaxy is plotted as a red star. It is bright
and large for its oxygen abundance.
## 5 A disturbed and highly star-forming galaxy
The SWIFT 3D kinematic fields demonstrate that the Eagle galaxy has a gradient
in $\sigma$ which ranges from $35-75\pm 10$ km s-1 and a $V$ of at least
$25\pm 5$ km s-1. Whatever the process (or processes) is that creates the
abnormally high $\sigma$ values, it is likely active throughout most of the
galaxy. These $\sigma$ values are likely caused by the blurring of velocity
gradients, which may not even have a preferred plane, on scales at or below
the seeing limit (Weiner et al., 2006a; Kassin et al., 2007). The high
$\sigma$’s are consistent with the morphology of the Eagle, which is
quantitatively classified as a major merger, and shows structures unlike those
of a normal disc galaxy.
Table 1 summarises the properties of the Eagle galaxy, and Figures 3–6 compare
it with galaxies at the same epoch. Compared to these galaxies, the Eagle is
extremely bright and blue in the rest-frame optical. It is highly star-forming
with an unusually high SFR as derived from line emission, and a comparably low
SFR as derived from far-infrared luminosity. This suggests that most of the
star-formation in the Eagle occurs in dust-free regions. The Eagle is also
large and bright for its metallicity. These properties are not surprising
given that the Eagle was primarily selected to have extended and bright
emission lines.
The Eagle could be in an early stage of a star-burst brought on by a major
merger. It may have not yet had enough time for metal enrichment, or to have
formed the mass of dust typically found in highly star-forming galaxies. It is
possible that the high SFR could drive winds that expose star-forming regions,
and, given the lower metallicity, may explain the low extinction. (There are
local dwarf galaxies which show similar phenomena, but this galaxy is much
more massive than such systems.) The low metallicity and strong star-burst may
also be explained by a major merger event that brings lower metallicity gas
from the outer parts to the centre of the system (e.g., Kewley, Geller, &
Barton, 2006).
It is likely that the mechanism which causes the star burst is the same, or
initiated by, that which causes the high velocity dispersion and disturbed
morphology. This mechanism is likely a major merger.
## Acknowledgments
The Oxford SWIFT integral field spectrograph is directly supported by a Marie
Curie Excellence Grant from the European Commission (MEXT-CT-2003-002792, Team
Leader: N. Thatte). It is also supported by additional funds from the
University of Oxford Physics Department and the John Fell OUP Research Fund.
Additional funds to host and support SWIFT were provided by Caltech Optical
Observatories.
This paper is based in part on observations obtained at the Hale Telescope at
Palomar Observatory as part of a collaborative agreement between the
California Institute of Technology, its divisions Caltech Optical
Observatories and the Jet Propulsion Laboratory (operated for NASA), and
Cornell University.
Additional support was provided for by the Observational Astrophysics Rolling
Grant at Oxford and the Oxford Astrophysics PATT Linked Grant ST/G004331/1.
The following NSF grants to the DEEP2 Survey are acknowledged: AST00-71198,
AST05-07483, and AST08-08133.
L. Fogarty would like to acknowledge the generous support of the Foley-Béjar
Scholarship through Balliol College, Oxford and the support of the STFC.
This work of LAM was carried out in part at Jet Propulsion Laboratory,
California Institute of Technology, under a contract with NASA. LAM
acknowledges support from the NASA ATFP program.
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|
arxiv-papers
| 2011-07-14T20:00:05 |
2024-09-04T02:49:20.599774
|
{
"license": "Public Domain",
"authors": "Susan A. Kassin, L. Fogarty, T. Goodsall, F. J. Clarke, R. W. C.\n Houghton, G. Salter, N. Thatte, M. Tecza, Roger L. Davies, Benjamin J.\n Weiner, C. N. A. Willmer, Samir Salim, Michael C. Cooper, Jeffrey A. Newman,\n Kevin Bundy, C. J. Conselice, A. M. Koekemoer, Lihwai Lin, Leonidas A.\n Moustakas, Tao Wang",
"submitter": "Susan Kassin",
"url": "https://arxiv.org/abs/1107.2931"
}
|
1107.3106
|
HIN-11-004
# Jet measurements by the CMS experiment in pp and PbPb collisions
Christof Roland for the CMS collaboration
###### Abstract
The energy loss of fast partons traversing the strongly interacting matter
produced in high-energy nuclear collisions is one of the most interesting
observables to probe the nature of the produced medium. The multipurpose
Compact Muon Solenoid (CMS) detector is well designed to measure these hard
scattering processes with its high resolution calorimeters and high precision
silicon tracker. Analyzing data from pp and PbPb collisions at a center-of-
mass energy of 2.76 TeV parton energy loss is observed as a significant
imbalance of dijet transverse momentum. To gain further understanding of the
parton energy loss mechanism the redistribution of the quenched jet energy was
studied using the transverse momentum balance of charged tracks projected onto
the direction of the leading jet. In contrast to pp collisions, a large
fraction the momentum balance for asymmetric jets is found to be carried by
low momentum particles at large angular distance to the jet axis. Further, the
fragmentation functions for leading and subleading jets were reconstructed and
were found to be unmodified compared to measurements in pp collisions. The
results yield a detailed picture of parton propagation in the hot QCD medium.
## 0.1 Introduction
Figure 1: $\Delta\phi_{12}$ distributions for leading jets of
$p_{\mathrm{T},1}>120$ with subleading jets of $p_{\mathrm{T},2}>50$ for 7 TeV
pp collisions (a) and 2.76 TeV PbPb collisions in several centrality bins: (b)
50–100%, (c) 30–50%, (d) 20–30%, (e) 10–20% and (f) 0–10%. Data are shown as
black points, while the histograms show (a) pythia events and (b)-(f) pythia
events embedded into PbPb data. The error bars show the statistical
uncertainties.
Heavy ion collisions at the Large Hadron Collider (LHC) are expected to
produce matter at energy densities exceeding any previously explored in
experiments conducted at particle accelerators. One of the key experimental
signatures suggested to study the quark-gluon plasma(QGP), which is expected
to be formed at these high energy densities, is the energy loss of high-
transverse-momentum partons passing through the medium [1]. This parton energy
loss is often referred to as “jet quenching”. The energy lost by a parton
provides fundamental information on the thermodynamical and transport
properties of the traversed medium.
## 0.2 Data set and experimental method
Figure 2: Dijet asymmetry ratio, $A_{J}$, for leading jets of
$p_{\mathrm{T},1}>$ 120 , subleading jets of $p_{\mathrm{T},2}>$50 and
$\Delta\phi_{12}>2\pi/3$ for 7 TeV pp collisions (a) and 2.76 TeV PbPb
collisions in several centrality bins: (b) 50–100%, (c) 30–50%, (d) 20–30%,
(e) 10–20% and (f) 0–10%. Data are shown as black points, while the histograms
show (a) pythia events and (b)-(f) pythia events embedded into PbPb data. The
error bars show the statistical uncertainities.
The dijet analysis presented in this paper was performed using the data
collected from pp and PbPb collisions at a nucleon-nucleon center-of-mass
energy of $\sqrt{s_{{}_{NN}}}$$=2.76$ TeV at the Compact Muon Solenoid (CMS)
detector. The CMS detector has a solid angle acceptance of nearly 4$\pi$ and
is designed to measure jets and energy flow, an ideal feature for studying
heavy ion collisions. The CMS detector is described in detail elsewhere [2].
This analysis is based on a total integrated PbPb (pp) luminosity of 6.7
$\mu$b-1 (225 nb-1). The jets used for analysis were reconstructed with two
different algorithms. The results related to dijet correlations uses jets
reconstructed from calorimeter energies using an iterative cone algorithm with
a cone radius of 0.5. The fragmentation function results use jets
reconstructed using the CMS particle flow algorithm, which combines
information from all detector systems [3] and the anti-kT algorithm, as
encoded in the FastJet framework, with a resolution parameter of R = 0.3 [4].
For both algorithms the underlying event is subtracted using the method
described in [5] and the reconstructed jet energies are corrected for the
detector response based on pythia [6] simulations. Charged particles are
reconstructed in the CMS Silicon tracking system [2].
## 0.3 Dijet Correlations
For this analysis dijets are selected with a leading jet of
$p_{\mathrm{T},1}>$ 120 , a subleading jet of $p_{\mathrm{T},2}>$50 both
within $|\eta|<$ 2 and with an opening angle of $\Delta\phi_{12}>2\pi/3$,
unless otherwise noted. More details on the analysis presented in this section
can be found in [7].
### 0.3.1 Dijet azimuthal correlations
Figure 3: Left Panel: Average missing transverse momentum,
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$, for tracks with
$>0.5$ , projected onto the leading jet axis (solid circles). The
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ values are shown
as a function of dijet asymmetry $A_{J}$ for 0–30% central events. Middle
Panel: The $\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$
values as a function of $A_{J}$ inside ($\Delta R<0.8$) one of the leading or
subleading jet cones. Right panel:
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ outside ($\Delta
R>0.8$) the leading and subleading jet cones. For the solid circles, vertical
bars and brackets represent the statistical and systematic uncertainties,
respectively. Colored bands show the contribution to
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ for five ranges
of track . For the individual ranges, the statistical uncertainties are shown
as vertical bars.
One possible medium effect on the dijet properties is a change of the back-to-
back alignment of the two partons. This can be studied using the event-
normalized differential dijet distribution, ($1/N$)($dN/d\Delta\phi_{12}$),
versus $\Delta\phi_{12}$. Figure 1 shows distributions of $\Delta\phi_{12}$
between leading and subleading jets which pass the respective selections. In
Figure 1 (a), the dijet $\Delta\phi_{12}$ distributions are plotted for 7 TeV
pp data in comparison to the corresponding pythia simulations using the
anti-$k_{\rm T}$ algorithm for jets based on calorimeter information. pythia
provides a good description of the experimental data, with slightly larger
tails seen in the pythia simulations. Figures 1 (b)-(f) show the dijet
$\Delta\phi_{12}$ distributions for PbPb data in five centrality bins,
compared to pythia+data simulations. For all centrality bins a good agreement
with the pythia+data reference is found for the bulk of the data, especially
for $\Delta\phi$${}_{12}>2$.
### 0.3.2 Dijet momentum balance
To characterize the dijet momentum balance (or imbalance) quantitatively, we
use the asymmetry ratio,
$A_{J}=\frac{p_{\mathrm{T},1}-p_{\mathrm{T},2}}{p_{\mathrm{T},1}+p_{\mathrm{T},2}}~{},$
(1)
where the subscript $1$ always refers to the leading jet, so that $A_{J}$ is
positive by construction. In Fig. 2 (a), the $A_{J}$ dijet asymmetry
observable calculated by pythia is compared to pp data at $\sqrt{s}$ = 7 TeV.
Data and event generator are found to be in excellent agreement. The
centrality dependence of $A_{J}$ for PbPb collisions can be seen in Figs. 2
(b)-(f), in comparison to pythia+data simulations. The dijet momentum balance
exhibits a dramatic change in shape for the most central collisions. In
contrast, the pythia simulations only exhibit a modest broadening, even when
embedded in the highest multiplicity PbPb events. This observation is
consistent with a degradation of the parton energy, or jet quenching, in the
medium produced in central PbPb collisions.
### 0.3.3 Overall momentum balance of dijet events
Information about the overall momentum balance in the dijet events can be
obtained using the projection of missing of reconstructed charged tracks onto
the leading jet axis. For each event, this projection was calculated as
$\displaystyle{\not}p_{\mathrm{T}}^{\parallel}=\sum_{\rm
i}{-p_{\mathrm{T}}^{\rm i}\cos{(\phi_{\rm i}-\phi_{\rm Leading\ Jet})}},$ (2)
where the sum is over all tracks with $>0.5$ and $|\eta|<2.4$. The results
were then averaged over events to obtain
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$. The leading and
subleading jets were required to have $|\eta|<1.6$. The left panel of Fig. 3
shows $\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ as a
function of $A_{J}$ for the 0–30% most central PbPb collisions. Using tracks
with $|\eta|<2.4$ and $>0.5$ , one sees that, even for events with large
observed dijet asymmetry, the momentum balance of the events, shown as solid
circles, is recovered within uncertainties. The figure also shows the
contributions to $\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$
for five transverse momentum ranges from 0.5–1 to $>8$ . The vertical bars for
each range denote statistical uncertainties. A large negative contribution to
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ (i.e., in the
direction of the leading jet) by the $>8$ range is balanced by the combined
contributions of low-momentum tracks from the 0.5–2 regions.
Further insight into the radial dependence of the momentum balance can be
gained by studying
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle$ separately for
tracks inside cones of size $\Delta R=0.8$ around the leading and subleading
jet axes, and for tracks outside of these cones. The results of this study are
shown in Fig. 3 for the in-cone balance (middle panel) and out-of-cone balance
(right panel). As the underlying PbPb event is not $\phi$-symmetric on an
event-by-event basis, the back-to-back requirement was tightened to
$\Delta\phi_{12}>5\pi/6$ for this study.
One observes that the in-cone imbalance of
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle\approx-20$ is
found for the $A_{J}>0.33$ selection, which is balanced by a corresponding
out-of-cone imbalance of
$\langle\displaystyle{\not}p_{\mathrm{T}}^{\parallel}\rangle\approx 20$ . In
the PbPb data the out-of-cone contribution is carried almost entirely by
tracks with $0.5<$$<4$ .
## 0.4 Fragmentation Functions
Figure 4: Fragmentation functions reconstructed in pp, peripheral PbPb, and
central PbPb data, with the leading (open circles) and subleading (solid
points) jets shown in the top row. The middle row is the ratio of each PbPb
fragmentation function to its pp reference. The pp reference is constructed by
convoluting the pp fragmentation with jet resolution effects of PbPb
collisions and reweighting with jet . Error bars are statistical, the hollow
boxes represent the systematic uncertainty for the leading jet, and yellow
boxes are the systematic uncertainty for the subleading jet. The bottom row
shows the jet distributions of the samples (pp, peripheral PbPb, and central
PbPb). It is important to note that jet distributions of leading and
subleading jets are different.
Figure 5: Fragmentation functions in $A_{J}$ bins, reconstructed in central
PbPb and pp reference for the leading (open circles) and subleading (solid
points) jets. The middle row shows the ratio of each fragmentation function to
its convoluted pp reference. The systematic uncertainty is represented by an
hollow box (leading jet) or yellow box (subleading jet). Error bars shown are
statistical. The bottom row shows the jet distributions. The reference
distributions are reweighted to match the jet distributions in PbPb.
Fragmentation functions are measured by correlating tracks within the jet
cones of reconstructed dijets with the jet axis for each jet. For this
analysis dijets are selected with a leading jet of $p_{\mathrm{T},1}>$ 100 , a
subleading jet of $p_{\mathrm{T},2}>$40 within $|\eta|<$ 2 and an opening
angle of $\Delta\phi_{12}>2\pi/3$. The reconstructed jets are corrected to the
generator final state particle level and the reconstructed charged particles
are corrected for tracking efficiency. The fragmentation function is a
function of $\xi$, defined as:
$\xi=-\ln\mathrm{z}=-\ln\frac{p_{T}^{track}}{p_{T}^{jet}},$ (3)
where jet is the transverse momentum of the reconstructed jet, and track is
the transverse momentum of tracks within the jet cone ($\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}=0.3$ around the jet axis). Tracks are
selected with ${}^{track}>4\GeVc$, which restricts this measurement to the low
$\xi$ region of the fragmentation function. For the PbPb analysis, this track
selection minimizes the contribution of the underlying event to the jet
fragmentation function. Figure 4 shows the reconstructed leading and
subleading jet fragmentation functions (top row) in a cone of $\Delta R=0.3$
around the respective jet axis for pp collisions (left panel) peripheral PbPb
(middle panel) and central PbPb (right panel). The corresponding jet
distributions are shown in the bottom row.
For a direct comparison between pp and PbPb, the pp data has to include the
resolution deterioration due to the underlying event fluctuations seen in
PbPb. For this purpose the pp jet has been artificially smeared by the
fluctuations observed in PbPb collisions taking into account the mean and RMS
of the fluctuations as well as the correlation of the background level seen in
back-to-back jet cones and the centrality dependence of the fluctuation
parameters. The ratio between the PbPb and fluctuation convoluted pp reference
fragmentation functions are shown in the middle rows for peripheral and
central PbPb collisions. In both cases, the fragmentation functions of the
leading and subleading jet are in agreement with pp collisions.
### 0.4.1 Fragmentation Functions in Bins of $A_{J}$
To study a potential effect of parton energy loss on the fragmentation
properties of partons in more detail, we divide the data sample into classes
of dijet imbalance. Four $A_{J}$ bins are chosen, which split the data sample
into approximately equal parts: 0 $<A_{J}<$ 0.13, 0.13 $<A_{J}<$ 0.24, 0.24
$<A_{J}<$ 0.35, and 0.35 $<A_{J}<$ 0.70. The fragmentation functions are
reconstructed separately for leading and subleading jets in central events.
Figure 5 shows the fragmentation functions in bins of increasing dijet
imbalance ($A_{J}$) from left to right (top row), the corresponding ratio to
the fluctuation convoluted pp reference (middle row) and the jet distribution
for leading and subleading jet (bottom row). For both leading and subleading
jets, independent of the dijet imbalance bin, the PbPb fragmentation functions
closely resemble those of the pp reference.
## 0.5 Summary
The CMS detector has been used to study jet production in PbPb collisions at
$\sqrt{s_{{}_{NN}}}$$=\,2.76~{}TeV$. No visible modification of the dijet
azimuthal correlation due to the nuclear medium was observed while a strong
increase in the fraction of highly unbalanced jets has been seen in central
PbPb collisions compared with peripheral collisions and model calculations.
This observation is consistent with a high degree of jet quenching in the
produced matter. A large fraction of the momentum balance of the unbalanced
jets is carried by low- particles at large radial distance to the jet axis.
The hard component of fragmentation functions reconstructed in PbPb collisions
for different event centrality and dijet imbalance $A_{J}$ exhibit a universal
behavior closely resembling jets of the same reconstructed energy fragmenting
without having traversed a nuclear medium, as seen in the comparison with pp
collisions. These results provide qualitative constraints on the nature of the
jet modification in PbPb collisions and quantitative input to models of the
transport properties of the medium created in these collisions.
## References
* [1] J. D. Bjorken, “Energy loss of energetic partons in QGP:possible extinction of high $p_{T}$ jets in hadron-hadron collisions”, FERMILAB-PUB-82-059-THY.
* [2] CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004.
* [3] M. Nguyen, “Jet reconstruction with particle flow in heavy-ion collisions with CMS”, These proceedings (2011).
* [4] M. Cacciari, G. P. Salam, and G. Soyez, “The anti-$k_{t}$ jet clustering algorithm”, JHEP 04 (2008) 063.
* [5] O. Kodolova, I. Vardanian, A. Nikitenko et al., “The performance of the jet identification and reconstruction in heavy ions collisions with CMS detector”, Eur. Phys. J. C50 (2007) 117.
* [6] T. Sjöstrand, S. Mrenna, and P. Skands, “PYTHIA 6.4 Physics and Manual”, JHEP 05 (2006) 026 (tune D6T with PDFs CTEQ6L1 used).
* [7] CMS Collaboration, “Observation and studies of jet quenching in PbPb collisions at nucleon-nucleon center-of-mass energy = 2.76 TeV”, arXiv:1102.1957.
|
arxiv-papers
| 2011-07-15T16:57:16 |
2024-09-04T02:49:20.613111
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christof Roland (for the CMS Collaboration)",
"submitter": "Christof Roland",
"url": "https://arxiv.org/abs/1107.3106"
}
|
1107.3211
|
# Stanley Conjecture on intersection of three monomial primary ideals
Andrei Zarojanu Andrei Zarojanu , Faculty of Mathematics and Computer
Sciences, University of Bucharest, Str. Academiei 14, Bucharest, Romania
andrei.zarojanu@yahoo.com
###### Abstract.
We show that the Stanley’s Conjecture holds for an intersection of
three monomial primary ideals of a polynomial algebra S over a field.
Key words : Monomial Ideals, Stanley decompositions, Stanley depth.
2000 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
## Introduction
Let $K$ be a field and $S=K[x_{1},...,x_{n}]$ be the polynomial ring over $K$
in $n$ variables. Let $I\subset S$ be a monomial ideal of $S,u\in I$ a
monomial and $uK[Z],Z\subset\\{x_{1},...,x_{n}\\}$ the linear $K$-subspace of
$I$ of all elements $uf$, $f\in K[Z]$. A presentation of $I$ as a finite
direct sum of spaces $\mathcal{D}:I=\bigoplus_{i=1}^{r}u_{i}K[Z_{i}]$ is
called a Stanley decomposition of $I$. Set
$\operatorname{sdepth}(\mathcal{D})=\operatorname{min}\\{|Z_{i}|:i=1,...,r\\}$
and
$\operatorname{sdepth}\ I:=\operatorname{max}\\{\operatorname{sdepth}\
({\mathcal{D}}):\;{\mathcal{D}}\;\text{is a Stanley decomposition of}\;I\\}.$
The Stanley’s Conjecture [11] says that sdepth $I\geq$ depth $I$. This is
proved if either $I$ is an intersection of four monomial prime ideals by [6,
Theorem 2.6] and [8, Theorem 4.2], or $I$ is the intersection of two monomial
irreducible ideals by [10, Theorem 5.6], or a square free monomial ideal of
$K[x_{1},\ldots,x_{5}]$ by [7] (a short exposition on this subject is given in
[9]). It is the purpose of our paper to show that the Stanley’s Conjecture
holds for intersections of three monomial primary ideals (see Theorem 2.2).
## 1\. Computing depth
Let $I\subset S$ be a monomial ideal and $I=\bigcap_{i=1}^{s}Q_{i}$ an
irredundant primary decompostion of I, where the $Q_{i}$ are monomial primary
ideals. Set $P_{i}=\sqrt{Q_{i}}$. According to Lyubeznik [5]
$\operatorname{size}I$ is the number $v+(n-h)-1$, where $h$ = height
$\sum_{j=1}^{s}Q_{j}$ and $v$ is the minimum number $t$ such that there exist
$1\leq j_{1}<...<j_{t}\leq s$ with
$\sqrt{\sum\limits_{k=1}^{t}Q_{j_{k}}}=\sqrt{\sum\limits_{j=1}^{s}Q_{j}}.$
In [5] it shows that $\operatorname{depth}_{S}I\geq 1+\operatorname{size}I$.
In the study of the Stanley’s Conjecture, we may always assume that $h=n$,
that is $\sum_{i=1}^{s}P_{i}=m=:(x_{1},\ldots,x_{n})$, because each free
variable on $I$ increases depth and sdepth with $1$.
###### Lemma 1.1.
Let $I\subset S$ be a monomial ideal and $I=\bigcap\limits_{i=1}^{3}Q_{i}$ an
irredundant primary decomposition of I, where each $Q_{i}$ is $P_{i}$ \-
primary. Suppose that $P_{i}\not=m$ for all $i\in[3]$. Then
1. (a)
If $Q_{1}\subset Q_{2}+Q_{3}$ and $P_{1}\not\subset P_{i}$ for $i=2,3$, then
$\operatorname{depth}_{S}S/I=1+\operatorname{min}\\{\dim S/(P_{1}+P_{2}),\dim
S/(P_{1}+P_{3})\\}.$
2. (b)
If $Q_{1}\subset Q_{2}+Q_{3}$ and $P_{1}\subset P_{2}$,$P_{1}\not\subset
P_{3}$, then
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\dim S/P_{2},1+\dim
S/(P_{1}+P_{3})\\}.$
3. (c)
If $Q_{1}\subset Q_{2}+Q_{3}$ and $P_{1}\subset P_{i}$ for $i=2,3$ then
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\dim S/P_{2},\dim
S/P_{3}\\}.$
4. (d)
If $Q_{i}\not\subset\sum\limits_{j=1,\ j\neq i}^{3}\ Q_{j},$ for all $i$ then
$\operatorname{depth}_{S}S/I=1$ if and only if $\operatorname{size}I=1$.
5. (e)
If $Q_{i}\not\subset\sum\limits_{j=1,\ j\neq i}^{3}\ Q_{j},$ for all $i$ then
$\operatorname{depth}_{S}S/I=2$ if and only if $\operatorname{size}I=2$.
###### Proof.
As $\operatorname{Ass}_{S}S/I=\\{P_{1},P_{2},P_{3}\\}$ we get
$\operatorname{depth}_{S}S/I>0$ by assumptions. We have the following exact
sequences
1. (1)
$0\rightarrow\frac{S}{I}\rightarrow\frac{S}{Q_{1}\cap
Q_{2}}\oplus\frac{S}{Q_{1}\cap Q_{3}}\rightarrow\frac{S}{Q_{1}}\rightarrow 0,$
2. (2)
$0\rightarrow\frac{S}{Q_{1}\cap
Q_{2}}\rightarrow\frac{S}{Q_{1}}\oplus\frac{S}{Q_{2}}\rightarrow\frac{S}{Q_{1}+Q_{2}}\rightarrow
0,$
3. (3)
$0\rightarrow\frac{S}{Q_{1}\cap
Q_{3}}\rightarrow\frac{S}{Q_{1}}\oplus\frac{S}{Q_{3}}\rightarrow\frac{S}{Q_{1}+Q_{3}}\rightarrow
0.$
Apply Depth Lemma in (2) and (3). If $P_{1}$ is not properly contained in
$P_{2}$ or $P_{3}$ then $\operatorname{depth}\frac{S}{Q_{1}\cap
Q_{3}}=1+\operatorname{depth}\frac{S}{Q_{1}+Q_{3}}$ and
$\operatorname{depth}\frac{S}{Q_{1}\cap
Q_{2}}=1+\operatorname{depth}_{S}\frac{S}{Q_{1}+Q_{2}}$. If $P_{1}\subset
P_{2}$ then $\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}}\geq\operatorname{depth}_{S}\frac{S}{Q_{2}}=\dim\frac{S}{P_{2}}$. But
$\operatorname{depth}_{S}\frac{S}{Q_{1}\cap Q_{2}}\leq\dim\frac{S}{Q_{2}}$,
that is $\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}}=\dim\frac{S}{P_{2}}$. Similarly,
$\operatorname{depth}_{S}\frac{S}{Q_{1}\cap Q_{3}}=\dim\frac{S}{P_{3}}$ if
$P_{1}\subset P_{3}$.
The statements (a),(b), (c) follow if we show that
$\operatorname{depth}_{S}S/I=\operatorname{min}\\{\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}},\operatorname{depth}_{S}\frac{S}{Q_{1}\cap Q_{3}}\\}.$
If
$\operatorname{depth}_{S}\frac{S}{Q_{1}}>\operatorname{min}\\{\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}},\operatorname{depth}_{S}\frac{S}{Q_{1}\cap Q_{3}}\\}$ then by Depth
Lemma applied in (1) we get the above equality. If
$\operatorname{depth}_{S}\frac{S}{Q_{1}}=\operatorname{min}\\{\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}},\operatorname{depth}_{S}\frac{S}{Q_{1}\cap Q_{3}}\\}$ then we get
similarly
$\operatorname{depth}_{S}S/I\geq\operatorname{depth}_{S}S/Q_{1}=\operatorname{depth}_{S}S/P_{1}$.
As $P_{1}\in\operatorname{Ass}S/I$ then $\operatorname{depth}_{S}S/I\leq\dim
S/P_{1}=\operatorname{depth}_{S}S/Q_{1}.$ Thus
$\operatorname{depth}_{S}S/I=\operatorname{depth}_{S}\frac{S}{Q_{1}}$, which
is enough.
(d) If $\operatorname{depth}_{S}S/I=1$ then $2=\operatorname{depth}_{S}I\geq
1+\operatorname{size}I$, that is $1\geq\operatorname{size}I\geq 0$. But
$\operatorname{size}I\not=0$ because the primary decomposition is irredundant.
Conversely, if $\operatorname{size}I=1$ then $v=2$ and we may assume that
$P_{2}+P_{3}=P_{1}+P_{2}+P_{3}=m$. We consider the exact sequences
1. (4)
$0\rightarrow\frac{S}{I}\rightarrow\frac{S}{Q_{1}\cap
Q_{2}}\oplus\frac{S}{Q_{3}}\rightarrow\frac{S}{Q_{3}+(Q_{1}\cap
Q_{2})}\rightarrow 0,$
2. (5)
$0\rightarrow\frac{S}{Q_{1}\cap
Q_{2}}\rightarrow\frac{S}{Q_{1}}\oplus\frac{S}{Q_{2}}\rightarrow\frac{S}{Q_{1}+Q_{2}}\rightarrow
0.$
From (5) we have $\operatorname{depth}_{S}\frac{S}{Q_{1}\cap
Q_{2}}=1+\operatorname{depth}_{S}\frac{S}{Q_{1}+Q_{2}}\geq 1$ by Depth Lemma.
Note that $\operatorname{depth}_{S}S/Q_{3}\geq 1$ and
$\operatorname{depth}_{S}\frac{S}{Q_{3}+(Q_{1}\cap
Q_{2})}=\operatorname{depth}_{S}\frac{S}{(Q_{1}+Q_{3})\cap(Q_{2}+Q_{3})}=0$
because $\sqrt{Q_{2}+Q_{3}}=m$, and $Q_{1}\not\subset Q_{2}+Q_{3}$. Thus Depth
Lemma applied in (4) gives $\operatorname{depth}_{S}S/I=1$.
(e) If $\operatorname{depth}_{S}S/I=2$, then $\operatorname{depth}_{S}I=3\geq
1+\operatorname{size}I$. But $\operatorname{size}I\leq 1$ was the subject of
(d), so $\operatorname{size}I=2$. Conversely, suppose that
$\operatorname{size}I=2$, that is $v=3$. Then
$P_{i}\not\subset\sum\limits_{j=1,\ j\neq i}^{3}\ P_{j},$ for all $i$ and by
[4, Proposition 2.1] we get $\operatorname{depth}_{S}I\leq 3$. As
$\operatorname{depth}_{S}I\geq 1+\operatorname{size}I$ we get
$\operatorname{depth}_{S}S/I=2$.
## 2\. Stanley’s depth
In this section we introduce a new way of splitting, inspired from [4], that
helps us to prove the Stanley Conjecture when
$I=\bigcap\limits_{i=1}^{3}Q_{i}$ is an irredundant primary decomposition of
I.
###### Theorem 2.1.
Let $I$ be a monomial ideal and $I=Q_{1}\cap Q_{2}$ an irredundant primary
decomposition of $I$ , where $Q_{i}$ is $P_{i}$ primary. Then the Stanley
conjecture holds for $I$.
###### Proof.
As usual we my suppose that $P_{1}+P_{2}=m$. Also we may suppose that
$P_{i}\not=m$ for all $i$, because otherwise $\operatorname{depth}_{S}I=1$ and
there exists nothing to show. Applying Depth Lemma in the above exact sequence
(2) we get $\operatorname{depth}_{S}S/I=1$, so
$\operatorname{depth}_{S}I=2=1+\operatorname{size}I$. By [3, Theorem 3.1] we
have $\operatorname{sdepth}_{S}I\geq\operatorname{depth}_{S}I$.
###### Theorem 2.2.
Let $I$ be a monomial ideal and $I=\bigcap\limits_{i=1}^{3}Q_{i}$ an
irredundant primary decomposition of $I$ , where $Q_{i}$ is $P_{i}$ primary.
Then the Stanley conjecture holds for $I$.
###### Proof.
We may suppose as above $P_{1}+P_{2}+P_{3}=m$ and $P_{i}\not=m$ for all $i$.
If $Q_{i}\not\subset\sum\limits_{j=1,\ j\neq i}^{3}\ Q_{j}$, for all $i\in[3]$
we have according to Lemma 1.1 minimal depth that is
$\operatorname{depth}I=1+\operatorname{size}I$. Then by [3, Theorem 3.1] we
get $\operatorname{sdepth}_{S}I\geq\operatorname{depth}_{S}I$. Now suppose
that $Q_{1}\subset Q_{2}+Q_{3}$. It follows that $\operatorname{size}I$ = 1.
If $P_{1}+P_{2}=m$ or $P_{1}+P_{3}=m$ then $\dim\frac{S}{Q_{1}+Q_{2}}=0$ or
$\dim\frac{S}{Q_{1}+Q_{3}}=0$ therefore $\operatorname{depth}_{S}S/I=1$ that
is $\operatorname{depth}_{S}I=2$. Then again we get
$\operatorname{sdepth}_{S}I\geq
1+\operatorname{size}I=2=\operatorname{depth}_{S}I$ by by [3, Theorem 3.1].
Otherwise $P_{1}+P_{2}\neq m\not=P_{1}+P_{3}$. Let $P_{1}=(x_{1},...,x_{r})$
and $P_{3}=(x_{e+1},...,x_{t})$, $2\leq r\leq n-1,e+1\leq r$. If $r=1$ then
$Q_{1}\subset Q_{2}$ or $Q_{1}\subset Q_{3}$ because $Q_{1}\subset
Q_{2}+Q_{3}$. This is false since the primary decomposition is irredundant. If
$r=n$ then $P_{1}=m$, which is not possible. If $e+1>r$ then $Q_{1}\subset
Q_{2}$, also a contradiction. We will prove this case by induction on $n$. If
$n=3$, then $\operatorname{sdepth}_{S}I\geq
1+\operatorname{size}I=2\geq\operatorname{depth}_{S}I$, because $I$ is not
principal. Assume now $n>3$. We set $S^{\prime}=K[x_{1},...,x_{r}]$,
${\bar{S}}:=K[x_{1},...,x_{e},x_{r+1},...,x_{n}]$ and
$J_{3}=\bigoplus\limits_{w}w((I:w)\cap{\bar{S}})$, where $w$ runs in the
finite set of monomials of $K[x_{e+1},...,x_{r}]\setminus Q_{3}$.
We claim that $I=Q_{1}\cap Q_{2}\cap(Q_{3}\cap S^{\prime})S\oplus J_{3}$. It
is enough to see the inclusion $"\subset"$. Let $a\in I$ be a monomial, then
$a=uv$, where $u\in{\bar{S}}$ and $v\in K[x_{e+1},...,x_{r}]$ are monomials.
If $v\not\in Q_{3}$ then $u\in(I:v)\cap{\bar{S}}$, so $a\in J_{3}$. If $v\in
Q_{3}$ then $a\in(Q_{3}\cap S^{\prime})S$. As $a\in I$ we get $a\in Q_{1}\cap
Q_{2}$ therefore a $\in Q_{1}\cap Q_{2}\cap(Q_{3}\cap S^{\prime})S$. The above
sum is direct. Indeed, let $a=uv\in Q_{1}\cap Q_{2}\cap(Q_{3}\cap
S^{\prime})S\cap J_{3}$ be as above. Then $v\not\in Q_{3}$ because $a\in
J_{3}$. But $v$ must be in $(Q_{3}\cap S^{\prime})S$. Contradiction!
The ideal $I^{\prime}:=Q_{1}\cap Q_{2}\cap(Q_{3}\cap S^{\prime})S\subset
P_{1}+P_{2}\not=m$ and so is an extension of an ideal from less than
$n$-variables and we may apply the induction hypothesis for $I^{\prime}$, that
is
$\operatorname{sdepth}_{S}I^{\prime}\geq\operatorname{depth}_{S}I^{\prime}$.
Since
$\operatorname{sdepth}_{S}I\geq\operatorname{min}\\{\operatorname{sdepth}_{S}I^{\prime},\\{\operatorname{sdepth}_{\bar{S}}((I:w)\cap{\bar{S}})\\}_{w}\\}$
it remains to show that
$\operatorname{depth}_{S}I^{\prime}\geq\operatorname{depth}_{S}I$ and
$\operatorname{depth}_{\bar{S}}((I:w)\cap{\bar{S}})\geq\operatorname{depth}_{S}I$,
applying again the induction hypothesis since $\bar{S}$ has less than
$n$-variables. The first inequality follows because $\dim S/(P_{3}\cap
S^{\prime})S\geq\dim S/P_{3}$, $\dim S/(P_{1}+(P_{3}\cap S^{\prime})S)\geq\dim
S/P_{1}+P_{3}$ using Lemma 1.1 (a), (b), (c).
For the second inequality note that for $w\not\in Q_{1}\cup Q_{2}\cup Q_{3}$
we have $(Q_{i}:w)$ primary and so $L_{i}:=(Q_{i}:w)\cap{\bar{S}}$ is
${\bar{P}}_{i}:=P_{i}\cap{\bar{S}}$-primary too. We have
$\dim{\bar{S}}/{\bar{P}_{i}}=\dim S/P_{i}$ for $i=1,3$ because
$(x_{e+1},\ldots,x_{r})\subset P_{1}\cap P_{3}$. Thus
$\dim{\bar{S}}/({\bar{P}_{1}}+{\bar{P}_{i}})=\dim S/(P_{1}+P_{i})$ for all
$i=2,3$. Using Lemma 1.1 we are done because $\dim S/P_{2}$ appears in the
formulas only when $P_{1}\subset P_{2}$, that is when
$\dim{\bar{S}}/{\bar{P}_{2}}=\dim S/P_{2}$.
If $w\in Q_{2}\setminus(Q_{1}\cup Q_{3})$ then
$\operatorname{depth}_{\bar{S}}{\bar{S}}/(L_{1}\cap
L_{3})=1+\dim{\bar{S}}/({\bar{P}}_{1}+{\bar{P}}_{3})=1+\dim
S/(P_{1}+P_{3})\geq\operatorname{depth}_{S}S/I$
by the same lemma, the only problem could appear when $P_{1}\subset P_{3}$,
but in this case
$\dim{\bar{S}}/({\bar{P}}_{1}+{\bar{P}}_{3})=\dim
S/(P_{1}+P_{3})={\bar{S}}/{\bar{P}}_{3}=\dim S/P_{3}$
and it follows
$\operatorname{depth}_{\bar{S}}{\bar{S}}/(L_{1}\cap
L_{3})=1+\dim{\bar{S}}/({\bar{P}}_{1}+{\bar{P}}_{3})>\dim
S/P_{3}\geq\operatorname{depth}_{S}S/I.$
If $w\in(Q_{1}\cap Q_{2})\setminus Q_{3}$ then
$\operatorname{depth}_{\bar{S}}{\bar{S}}/L_{3}=\dim
S/P_{3}\geq\operatorname{depth}_{S}S/I$ by [1].
## References
* [1] W. Bruns and J. Herzog, Cohen-Macaulay rings Revised edition. Cambridge University Press (1998).
* [2] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169.
* [3] J. Herzog, D. Popescu, M. Vladoiu, Stanley depth and size of a monomial ideal, arXiv:AC/1011.6462v1, 2010, to appear in Proceed. AMS.
* [4] M. Ishaq, Values and bounds of the Stanley depth, to appear in Carpathian J. Math., arXiv:AC/1010.4692.
* [5] G. Lyubeznik, On the Arithmetical Rank of Monomial ideals, J. Algebra 112, 86-89 (1988).
* [6] A. Popescu, Special Stanley Decompositions, Bull. Math. Soc. Sc. Math. Roumanie, 53(101), no 4 (2010), arXiv:AC/1008.3680.
* [7] D. Popescu, An inequality between depth and Stanley depth, Bull. Math. Soc. Sc. Math. Roumanie 52(100), (2009), 377-382, arXiv:AC/0905.4597v2.
* [8] D. Popescu, Stanley conjecture on intersections of four monomial prime ideals, arXiv.AC/1009.5646.
* [9] D. Popescu, Bounds of Stanley depth, An. St. Univ. Ovidius. Constanta, 19(2),(2011), 187-194.
* [10] D. Popescu, I. Qureshi, Computing the Stanley depth, J. Algebra, 323 (2010), 2943-2959.
* [11] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193.
|
arxiv-papers
| 2011-07-16T08:56:24 |
2024-09-04T02:49:20.621825
|
{
"license": "Public Domain",
"authors": "Andrei Zarojanu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1107.3211"
}
|
1107.3226
|
# On Measurement and Computation
Huimin Zheng Haixing Hu Nan Wu Fangmin Song
Department of Computer Science & and Technology
Nanjing University
zhenghuimin555@126.com fmsong@nju.edu.cn
###### Abstract
Inspired by the work of Feynman, Deutsch, We formally propose the theory of
physical computation and accordingly, the physical complexity theory. To
achieve this, a framework that could be used to evaluate almost all forms of
computation making use of various physical mechanisms is established. Here, we
focus on applying this to Quantum Computation. As a preliminary study on more
general problems, some examples of other physical mechanisms are also
discussed in this paper.
Keywords: Quantum Computation, Physical Computation, computational complexity
###### Contents
1. 1 Introduction
1. 1.1 Quantum Computation
2. 1.2 Physical Computation
3. 1.3 The structure of the article
2. 2 Models of Quantum Computation
1. 2.1 Quantum Turing Machine
2. 2.2 Quantum Circuit Model
3. 3 The Theory of Physical Computation
1. 3.1 Observer
2. 3.2 Physical States
3. 3.3 Physical processes and the operation $\circ$
4. 3.4 Physical Operator and the operation of operator
5. 3.5 Physical Computability
1. 3.5.1 Deterministic Physical Computation
2. 3.5.2 Non-deterministic Physical Computation
6. 3.6 Estimation of the Complexity of Physical Resource
1. 3.6.1 Framework for the Complexity with respect to General Physical Resource
2. 3.6.2 Some Common Examples
1. 3.6.2.1 Mean of Three Numbers
2. 3.6.2.2 Sorting Without Repeat
3. 3.6.2.3 Volume of irregular shape
4. 3.6.2.4 The centroid of Irregular Shape
3. 3.6.3 Graph Isomorphism, Graph Spectrum and Oscillators
1. 3.6.3.1 Spectrum of Graph
2. 3.6.3.2 Harmonic Oscillator of multi-freedom
3. 3.6.3.3 The characteristic oscillators for a Graph
4. 3.6.3.4 Comments
4. 3.6.4 Steiner Tree Problem
5. 3.6.5 DNA Computation
7. 3.7 Preliminary Discussion of the classic theory of Computation
1. 3.7.1 Turing computable is physical computable
2. 3.7.2 PLATO Machine
3. 3.7.3 Recursive function whose derivative is not recursive
4. 3.7.4 Physical States which is not computable
5. 3.7.5 A few Comments
4. 4 Physical Resource Complexity for Quantum Computation
1. 4.1 Physical Resource Complexity for Quantum Computation
1. 4.1.1 The RCEF for Quantum Computation
2. 4.1.2 Deutsch-Josza Algorithm
3. 4.1.3 Shor’s Algorithm
4. 4.1.4 Grover Algorithm
2. 4.2 Quantum Simulation and Quantum Algorithm
5. 5 Conclusions and Future Works
6. 6 Acknowledgement
## 1 Introduction
### 1.1 Quantum Computation
The research of quantum computation has been lasting for about 30 years since
R.Feynman proposed the concept of so-called ‘quantum computer’ in 1982[2].
Founding out that there do exist some quantum systems which are suspected
cannot be efficiently simulated by classical computers, early researchers
naturally speculated that quantum mechanism itself may provide stunning power
of computation. In order to strictly define what is ‘quantum computation’,
researchers introduced various new computational models, including Quantum
Turing Machine [1] and Quantum Circuit Model[5]. However, at that time, no
convincing evidence was discovered to support the conjecture that quantum
mechanism can really be used to speed up the computation of some hard problems
greatly.
D.Deutsch found the first evidence that quantum computers may surpass the
Turing Machine[9] in query. In fact, he constructed a special scene in which
DTM has to query the oracle for $O(2^{n})$ times to find the correct
answer(for certain) in worst cases while QTM need just once query in all
cases.
One of the most remarkable results is quantum factorization, which is due to
Peter Shor[10, 11]. The best classical algorithm for factorization so far has
to run for $O(\log^{3}L)$ steps. However, Shor showed that one can use a
family of quantum circuit, which contain $O(\log^{3}L)$ gates and needs only
$O(L^{2}\log L\log\log L)$ operations to get the right answer.
Grover’s Algorithm [13] is another successful example of quantum algorithms.
This algorithm can be used to search a database without structure. It is easy
to proof that the time complexity of this problem for Turing Machine is
$O(n)$. However, there does exist a quantum algorithm whose time complexity is
$O(\sqrt{n})$. Since it has been proved that this is the optimal algorithm for
all algorithms that considering quantum mechanics [18] , so the complexity of
Grover’s algorithm can be looked as the quantum complexity of this problem.
One of the most important reason that why Quantum algorithms(especially Shor’s
algorithm) seem so interesting to many computer scientists is that their
existence indicate a huge challenge to strong Church-Turing thesis, which
states that _Any model of computation can be simulated on a probabilistic
Turing machine with at most a polynomial increase in the number of elementary
operations required._
### 1.2 Physical Computation
On the other hand, with the exciting research in quantum computation as well
as other new paradigms of computations(e.g.DNA computation), the idea that we
may just look physical processes as computations(not just the Turing
Machine)was also developed. The seeds of this idea can be traced back to
Feynman[2], Deuthsch[3] and Pitowsky[7] et al.
It is not very hard to understand and appreciate this idea, for at first
glance, this point of view has at least three benefits:
* •
It can include the concept of classical algorithms easily, for an algorithm on
Turing Machine(its physical implementation)can also be thought as a family of
physical processes (and the corresponding measurement).
* •
We can try to solve some special kinds of problems with less time or space
than the lower bound with respect to Turing machines.
* •
Being the ones which could be directly simulated, some physical methods can
also enlighten us to design smart algorithms on Turing machines.
What’s more, currently, it seems that we cannot exclude the possibility that
there does exist a family of physical processes which can help us to calculate
some problems which cannot be solved by a universal Turing Machine in
principle.
However, because of vagueness and extraordinary generality, the theory of so-
called ‘physical computation’ has a significant defeat yet.
* •
In many cases, people cannot decide how to define the resource for a ‘physical
algorithm’. And as a result they cannot proof or even conjecture formally
whether a ‘physical algorithm’ is really superior to any algorithms of TMs
with the same extensionality.
Note that the theory of quantum computation is almost free from such defeat,
for researchers have completed the formal definition of the computational
model of quantum computation in the early years. Roughly speaking, things tend
to go wrong when:
* •
People adopt a design on which the physical postulates it depends is just an
empirical one.
* •
More than one different systems of physical postulates are used.
### 1.3 The structure of the article
The structure of this paper is as follows. In Sec. II we shall introduce two
well-known models of quantum computation and the definition of complexity
respectively. And after that we will formally establish the theoretical
foundation of physical computation and propose the theory of physical
computability in Sec. III. In the beginning of Sec. IV we try to use the
theory of physical computation to reanalyze the quantum algorithms. In the end
of Sec. IV, we focus on the topic about how to construct problems which take
advantage of quantum simulations.
## 2 Models of Quantum Computation
### 2.1 Quantum Turing Machine
Quatnum Turing Machine was first introduced by Benioff[1] in 1980 and was
developed by Deustch and Yao. The modern definitioin was given by Bernstein
and Vazirani in 1997[4].
###### Definition 2.1
(Quantum Turing Machine, Bernstein 1997) Let $\tilde{C}$ be a set of complex
nmber $\alpha$ satisfying: For each $\alpha$, there exists a polynomial time
algorithm to compute the value of $Im(\alpha)$ and $Re(\alpha)$ close to
$2^{-n}$ within the true value.
A Quantum Turing Machine $M$ is defined as the triple $(\Sigma,Q,\delta)$,
where $\Sigma$ is a fintie alphabet with an identified symbol $\\#$ , $Q$ is a
finite set of states with an identified initial state $q_{0}$ and final state
$q_{f}\neq q_{0}$; $\delta$, the quantum transform function
$\delta:Q\times\Sigma\rightarrow\tilde{C}^{\Sigma\times Q\times\\{L,R\\}}$.
The QTM has a two-way infinite tape of cells indexed by $Z$, and a single
read/write tape head that moves along the tape. We define configurations
initial configurations and final configurations exactly as for DTMs. Let $S$
be the inner-product space of finite complex linear combinations of
configurations of $M$ with the Euclidian norm. We call each element $phi\in S$
a superposition of $M$. The QTM $M$ defines a linear operator
$U_{M}:S\rightarrow S$ called the time evolution operator of $M$ as follows:
If $M$ starts in configurations $c$ with current state $p$ and scanned symbol
$\sigma$. The after one step $M$ will be in superposition of configurations
$\psi=\sum_{i}\alpha_{i}c_{i}$, where each non-zero $\alpha_{i}$ corresponds
to a $\delta(p,\sigma,\tau,q,d)$, and $c_{i}$ is the new configuration that
results from applying this transiton to $c$. Extending thi map to the entire
space $S$ through linearity gives the linear time evolution operator $U_{M}$.
###### Definition 2.2
If $U_{M}$ can keep Euclidian norm, then we say $M$ is well deformed.
###### Theorem 2.3
If QTM is in the superposition $\psi=\sum_{i}\alpha_{i}c_{i}$ and is observed,
the probability of the observer gets the configuration $c_{i}$ is
$|\alpha_{i}|^{2}$, and then $M$ is in the state $\psi^{\prime}=c_{i}$.
###### Theorem 2.4
A QTM is well-deformed if and only if its time evolution operator is unitary.
In QTM, the number of the read/write tape head moves during a computation is
the cost of time.
###### Theorem 2.5
There exists a universal QTM, which is polynomially equivalent to any QTMs.
### 2.2 Quantum Circuit Model
The first quantum circuit model was due to Deutsch. Then quantum circuit model
was improved by Yao[5], who also proved that for any QTM, there exists a
uniform family of quantum circuit which is polynomially equivalent to that
QTM.
Not like QTM, quantum circuit model tends to describe an algorithm by using
universal quantum gates and circuits without loops. Quantum circuit model does
not need infinite many quantum gates, but finite many quantum gates which
called the universal quantum gates. It has been proved that Hadamard Gate,
phase gate, C-NOT Gate and $\pi/8$ Gate are universal. For any finite
dimensional $U$ operators, we can always approach it effectively by means of a
universal family of circuits $\mathscr{U}$, which only consists 4 gates above,
i.e.
$\forall\varepsilon(\exists
n\in\mathscr{U}),E(U,\tilde{U}_{n})\equiv\max_{|\psi\rangle}\|(U-\tilde{U}_{n})|\psi\rangle\|<\varepsilon$
The scale of a quantum circuit is defined as the number of the universal gates
and the depth is defined as the longest path from input to output, if the
gates is looked as a vertex.
Both Quantum circuit model and QTM are important models of quantum
computation. But we do not know whether they are the most natural models of
quantum computation or do they fully take the advance of quantum mechanics, no
matter in the theory of quantum computability and quantum computational
complexity.
## 3 The Theory of Physical Computation
### 3.1 Observer
Measurement is in terms of observer. Though there are many differences among
people’s opinions about the exact definition of human beings, we prudently
assume that an observer is classical, that is, the observer will never get
incompatible results during one measurement.
In this article, we will never use the terminology such like ‘a observer of
the observer’, or in other words, by ‘observer’ we always mean the last one
outside the whole experiment.
In order to unify various forms of results, we require that the observer only
accept the symbols on a tape(just something like the one of TM) and also only
use this to initialize an experiment.
So we define the legal inputs and outputs as the elements in set $\Sigma^{+}$,
where
$\Sigma=\\{\leavevmode\nobreak\ 0,1,*,.\leavevmode\nobreak\ \\}$
and $\Sigma^{+}$ the finite string composed by elements in $\Sigma$.
The concept of observer is crucial to our theory.
### 3.2 Physical States
We use (usually finite) attributes which may contribute to the computations to
label the physical states. In addition, though may not be actually concerned
in every computation, three fundamental quantities, namely, space, energy and
mass are always included in a state for the sake of analysis of resource and
complexity.
So we have:
$\Omega\subset\\{x_{1}\\}^{A_{1}}\times\\{x_{2}\\}^{A_{2}}\times\dots\times\\{x_{n}\\}^{A_{n}}\times\\{m\\}^{\mathfrak{M}}\times\\{s\\}^{\mathfrak{S}}\times\\{e\\}^{\mathfrak{E}}$
Or more generally(Quantum)
$\Omega\subset\\{x_{1}\\}^{A_{1}}\times\\{x_{2}\\}^{A_{2}}\times\dots\times\\{x_{n}\\}^{A_{n}}\times\\{\mathbb{C}^{m}\\}^{\mathfrak{M}}\times\\{\mathbb{C}^{s}\\}^{\mathfrak{S}}\times\\{\mathbb{C}^{e}\\}^{\mathfrak{E}}$
For simplicity, fundamental attributes are usually omitted, i.e.
$\Omega\subset\\{x_{1}\\}^{A_{1}}\times\\{x_{2}\\}^{A_{2}}\times\dots\times\\{x_{n}\\}^{A_{n}}$
For a certain attribute $A_{i}$, what really matters is its type which is
constrained by its dimension. Note that dimensionless quantity(e.g.friction
coefficient) can also be assigned to a null type. When a quantity is expressed
by other quantities’ combination, it’s dimension type should be preserved, or
rather, any equations should be dimensional balanced.
For example:
$E^{[D:ML^{2}T^{-2}]}::=m^{[D:M]}g^{[D:LT^{-2}]}h^{[D:L]}=mgh^{[D:ML^{2}T^{-2}]}$
$E^{[D:ML^{2}T^{-2}]}::=\frac{1}{2}(m)^{[D:M]}(v^{2})^{[D:L^{2}T^{-2}]}=\frac{1}{2}(mv^{2})^{[D:ML^{2}T^{-2}]}$
$E^{[D:ML^{2}T^{-2}]}::=(m)^{[D:M]}(c^{2})^{[D:L^{2}T^{-2}]}=(mc^{2})^{[D:ML^{2}T^{-2}]}$
are all dimensional balanced.
### 3.3 Physical processes and the operation $\circ$
Physical process on a state space $\Omega$ is a set of physical state whose
elements are labeled by moment $t(t\in[0,T],T\in\mathbb{R}^{+})$.
$P\equiv\\{\leavevmode\nobreak\ (T,\tilde{P})|\leavevmode\nobreak\
T\in\mathbb{R}^{+},\tilde{P}:[\leavevmode\nobreak\ 0,T\leavevmode\nobreak\
]\to\Omega\leavevmode\nobreak\ \\}$
If two physical processes on $\Omega$ satisfies
$(\pi_{2}P_{1})(\pi_{1}P_{1})=(\pi_{2}P_{2})(0)$
we can define operator $\circ:\\{P\\}\times\\{P\\}\to\\{P\\}$ i.e.
$P_{2}\circ P_{1}=P_{3}$
satisfies:
1. 1.
$\pi_{1}P_{3}=\pi_{1}P_{1}+\pi_{1}P_{2}$
2. 2.
if $0\leq t\leq\pi_{1}P_{1}$ $(\pi_{2}P_{3})(t)=(\pi_{2}P_{1})(t)$
3. 3.
if $\pi_{1}P_{1}\leq t\leq\pi_{1}P_{1}+\pi_{1}P_{2}$
$(\pi_{2}P_{3})(t)=(\pi_{2}P_{2})(t-\pi_{1}P_{1})$
For convenience, we introduce $\rhd P\lhd$ as the initial state of $P$, and
$\lhd P\rhd$ the final state of $P$, i.e.
$\rhd P\lhd\equiv(\pi_{2}P)(0),\lhd P\rhd\equiv(\pi_{2}P)(\pi_{1}P)$
### 3.4 Physical Operator and the operation of operator
Physical operator is a tuple whose first component is a state $x$ in $\Omega$
and the second component is a physical process whose initial state is $x$,
i.e.
$O\equiv\\{(x,P)|x\in\Omega,\rhd P\lhd=x\\}$
In particular, a _deterministic physical operator_ $O$ means:$O$ is a physical
operator, and
$\textrm{if\leavevmode\nobreak\
}\pi_{2}O(x_{1})\neq\pi_{2}O(x_{2}),\textrm{then\leavevmode\nobreak\
}x_{1}\neq x_{2}$
if we only care about the effect the operator do to the initial state, we can
look operator as a mapping in $\Omega$,i.e.$O:\Omega\to\Omega$.
The operation between two deterministic physical operator is defined as
follows(suppose $O_{1},O_{2}$ are productive):
$O_{3}=O_{2}\circ O_{1}$
which satisfies
$\forall x(O_{3}(x)\equiv O_{2}(\lhd O_{1}(x)\rhd)\circ O_{1}(x))$
In some more general cases, it is useful to talk about non-deterministic
physical operators or random physical operators. A random physical operator
$\tilde{O}$ contains the tuples which has the same initial states but
different physical processes, i.e.
$\tilde{O}\equiv\\{(x,P)|x\in\Omega,\rhd P\lhd=x\\}.$
People cannot decide the output $\tilde{O}(x)$ just by the initial state
$x\in\Omega$.
Similarly, if just care about extensionality, we can look operator as a
relationship on $\Omega$ i.e.$\tilde{O}:\Omega\times\Omega$
We can also define operations between two non-deterministic operators, if some
preconditions are satisfied. To do so, we first expand the definition of some
symbols.
$O(x)\equiv\\{P|\rhd P\lhd=x\\}$ $O(X)\equiv\\{P|\rhd P\lhd\in
X\subset\Omega\\}$ $\lhd O(x)\rhd\equiv\\{y|y\in\Omega,\exists P\in
O(x)s.t.\lhd P\rhd=y\\}$
So $O_{2}\circ O_{1}$ (if they are productive) can be defined as
$O_{2}\circ O_{1}(x)\equiv\\{P_{2}\circ P_{1}|P_{1}\in O_{1}(x),P_{2}\in
O_{2}(\lhd O_{1}(x)\rhd)\\}$
Note that of all the processes created by random physical operators, their
last states shall be exposed to outside world in the end by default.
### 3.5 Physical Computability
#### 3.5.1 Deterministic Physical Computation
###### Definition 3.1
(Deterministic Physical System)Deterministic Physical System $\mathscr{P}$ is
a Five-Tuple
$\mathscr{P}\equiv(\Omega,\Sigma,\nabla,\mathcal{H},\Delta)$
where:
* •
$\Sigma=\\{0,1,*,.\\}$ $\Sigma^{+}$ is the collection of finite string formed
by elements in $\Sigma$ , $\Omega_{\Sigma^{+}}$ is the the set of physical
implementation of $\Sigma^{+}$.
* •
$\Omega=\\{\psi_{i},i\in\Lambda\\}$ $\Omega\neq\Omega_{\Sigma^{+}}$ $\Lambda$
is an index set $\Omega$ is a set of distinguishable physical states(labeled
by their attributes).
* •
$\nabla:\Omega_{\Sigma^{+}}\rightarrow\Omega$ initialization operator
* •
$\mathcal{H}:\Omega\rightarrow\Omega$ evolution operator
* •
$\Delta:\Omega\rightarrow\Omega_{\Sigma^{+}}$ Measurement operator
Since Hilbert’s 6th problem has not been solved yet, i.e. the whole theory of
physics has not been axiomatized, we do not know that whether there exist some
additional fundamental mathematical constraints should be included in this
theory, though Beggs et al. have discussed the axioms of measurement based on
Hempel’s axioms[19]. Now, maybe the only restrictions here are the finiteness
of the resource cost by a physical process and the finiteness of the
attributes used to label a physical state set.
As a result this system may looks looser than many classical computational
models and may contains the ability to surpass all these models. We would like
to let physicists to add more necessary restrictions into the system.
Of course, when it comes to a specific branch of the physics, we can always
know what is a legal states and processes. However, we wish to keep some
freedom, i.e. to let the observer combine various axioms in physics so as to
optimize the computations.
###### Definition 3.2
(Partial Physical Computable Arithmetic Functions)For any partial arithmetic
function $f:\mathbb{N}\rightarrow\mathbb{N}$ is said to be _partial physical
computable_ if and only if there exists a Deterministic Physical System
$\mathscr{P}\equiv(\Omega,\Sigma,\nabla,\mathcal{H},\Delta)$
which satisfies
1. If $x\in dom(f)$, then,
$(\Delta\circ\mathcal{H}\circ\nabla)(x)\doteq f(x)$
Similarly, we can define _Total Physically Computable Arithmetic Functions_
###### Definition 3.3
(Total Physically Computable Arithmetic Functions) For any total arithmetic
function $f:\mathbb{N}\rightarrow\mathbb{N}$ is said to be _Total Physically
Computable_ , if and only if there exists a Deterministic Physical System
$(\Omega,\Sigma,\nabla,\mathcal{H},\Delta)$ which satisfies
1. $\forall x\in\mathbb{N}$ we have:
$(\Delta\circ\mathcal{H}\circ\nabla)(x)\doteq f(x)$
In order to extend the definition of physical computability to non-arithmetic
functions, we should take into consideration the precision of the measurement
and computation. Therefore, we need a distance function to measure the
precision of two values and define the computability as the ability of
computing in any desired precision.
###### Definition 3.4
(Partial Physically Computable Functions)Given a partial function
$f:A\rightarrow B$ and a metric $\mathcal{D}:B\times B\rightarrow\mathbb{R}$
$f$ is said to be partial physically computable with respect to the metric
$\mathcal{D}$ , if and only if for any $\epsilon>0$ there exists
$(\Omega,\Sigma,\nabla,\mathcal{H},\Delta)$ s.t. for any $x\in A$ we have
1. if $x\in dom(f)$,
$\mathcal{D}\Big{(}(\Delta\circ\mathcal{H}\circ\nabla)(x),f(x)\Big{)}<\epsilon$
Similarly, we can also define _Total Physical Computable Functions_.
###### Definition 3.5
(Total Physically Computable Functions)Given a total function $f:A\rightarrow
B$ and a metric $\mathcal{D}:B\times B\rightarrow\mathbb{R}$ $f$ is said to be
partial physically computable with respect to the metric $\mathcal{D}$ if and
only if for any $\epsilon>0$ there exists
$(\Omega,\Sigma,\nabla,\mathcal{H},\Delta)$, such that,
1. $\forall x\in A$
$\mathcal{D}\Big{(}(\Delta\circ\mathcal{H}\circ\nabla)(x),f(x)\Big{)}<\epsilon$
#### 3.5.2 Non-deterministic Physical Computation
On the other hand, many physical processes are considered to be non-
deterministic, which enable us to implement so-called ‘randomized algorithms’
and ‘quantum algorithms’. Our _Probabilistic Physical System_ is defined as
follows.
###### Definition 3.6
(Probabilistic Physical System)Probabilistic Physical System $\mathscr{P}^{*}$
is a five-tuple:
$\mathscr{P}^{*}\equiv(\Omega,\Sigma,\nabla,\mathcal{H^{*}},\Delta)$
. where,
* •
$\Sigma=\\{0,1,*,.\\}$.
* •
$\Omega=\\{\psi_{i},i\in\Lambda\\}$.
* •
$\nabla:\Omega_{\Sigma^{+}}\rightarrow\Omega$ which is also called
initialization operator
* •
$\mathcal{H^{*}}:\Omega\times\Omega$ which also called evolution operator,
which is non-deterministic.
* •
$\Delta:\Omega\rightarrow\Omega_{\Sigma^{+}}$, which is also called
measurement operator.
Non-deterministic does not necessarily cause probability, but let’s convention
that in this article we always discussed the randomness which has a
probabilistic distribution.
Definition of the computable functions by means of $\mathscr{P}^{*}$ is an
analog to that of $\mathscr{P}$. As an example, we define Total Non-
deterministic Physical Computable Functions.
###### Definition 3.7
(Total Non-deterministic Physical Computable Functions(Las Vegas)) For any
total function $f:\mathbb{N}\rightarrow\mathbb{N}$ is said to be total non-
deterministic physical computable function, if and only if there exists a
five-tuple
$(\Omega,\Sigma,\nabla,\mathcal{H}^{*},\Delta)$
s.t.
1. $\forall x((\Delta\circ\mathcal{H}^{*}\circ\nabla)(x)\doteq f(x))$
Because of randomness, for any identical inputs $x$, the system may call
different process to compute. The above definition is the counterpart of the
definition of the so called Las Vegas algorithm.
###### Definition 3.8
(Total Non-deterministic Physical Computable Functions(Monte Carlo))For any
total function $f:\mathbb{N}\rightarrow\mathbb{N}$ is said to be total non-
deterministic physical computable, if and only if there exists
$(\Omega,\Sigma,\nabla,\mathcal{H}^{*},\Delta)$
s.t.
1. $\forall x\in\mathbb{N}$
$\Pr\\{Event(x)\text{occurs}\\}>2/3$
where
$Event(x)\equiv\Big{(}(\Delta\circ\mathcal{H}^{*}\circ\nabla)(x)\doteq
f(x)\Big{)}$
### 3.6 Estimation of the Complexity of Physical Resource
For the physical systems defined above, we can even ignore that whether there
exists a physical mechanism in reality to implement it. Any functions which
could be written as the composition of the three operators would be considered
as computable(deterministic version).
$\begin{array}[]{ccc}\Omega&\rightarrow\mathcal{H}&\Omega\\\
\nabla\uparrow&&\Delta\downarrow\\\ \Omega_{\Sigma^{+}}&\rightarrow
f&\Omega_{\Sigma^{+}}\end{array}$
But any experiments which implement a certain system will cost resource. We
will focus on four kinds of resource, namely, time, space, energy and mass.
One can also include 7 fundamental sorts of attributes considering 7
dimensions in SI(Systéme international d’unités). However, actually our
decision is not a totally empirical one. In fact, many differential equations
which are established to describe various phenomena involving these attributes
are always related to energy, time, and space. After properly choosing unit to
make all the constant into 1, we can just rewrite these dimensions in the
forms of the combinations of 4 fundamental dimensions, which enable us to
continue to use the 4 attributes to represent the increase of the resource.
The counterexample may occur only when $m$ or more than $m$ new attributes
appear in $n$ equations and $m>n$, however these cases tend to be unlikely to
happen, if these equations(theory) we discussed are assumed to be ‘enough’ for
some phenomena in nature.
###### Definition 3.9
( Metric for Resource ) The resource of a physical process $\mathfrak{R}$
includes:
* •
$\mathfrak{T}$: The (expectation of the)total time the whole process consumed;
* •
$\mathfrak{S}$: The maximum of (the expectation of)the space the whole process
consumed;
* •
$\mathfrak{M}$: The maximum of (the expectation of)the mass the whole process
consumed;
* •
$\mathfrak{E}$: The maximum of (the expectation of)the energy the whole
process consumed.
and $\mathfrak{R}\equiv(\mathfrak{T},\mathfrak{S},\mathfrak{M},\mathfrak{E})$
In the above definitions, the metric of them could be selected as the common
ones. Today, most physicists tends to believe that mass and energy are not
independent, neither do time and space. But for convenience, we still focus
the primitive forms of resource, for actually we don’t care about the
independence here.
In the above definitions, we don’t talk about the potential possibility that
even time could be reused.
We convention that the resource is with respect to an inertial system, i.e.
the observers obtain their results when they are in an inertial system to the
system running the ‘algorithms’, so as to rule out the paradoxes because of
the theory of relativity.
In many cases, we just cannot get a infinite precise estimation about the
resource, but for our purpose, we actually do not need such things. Of course,
there may exist some cases when we could not get an estimation without any
promise of any precision, however, we will not use such processes to construct
our implementation.
Suppose the projections of the fundamental attributes(resource) are
$\pi_{\mathfrak{M}}(S)$, $\pi_{\mathfrak{S}}(S)$ and $\pi_{\mathfrak{E}}(S)$
Then the resource a physical process consumed is:
$\begin{array}[]{lll}\mathfrak{T}P&\equiv&\pi_{1}P\\\
\mathfrak{M}P&\equiv&\max\\{\pi_{\mathfrak{M}}(S),S\in\mathrm{Ran}(\pi_{2}P)\\}\\\
\mathfrak{S}P&\equiv&\max\\{\pi_{\mathfrak{S}}(S),S\in\mathrm{Ran}(\pi_{2}P)\\}\\\
\mathfrak{E}P&\equiv&\max\\{\pi_{\mathfrak{E}}(S),S\in\mathrm{Ran}(\pi_{2}P)\\}\end{array}$
In general cases, when we have to discuss the process of superposition, the
resource can be defined as:
$\begin{array}[]{lll}\mathfrak{T}P&\equiv&\pi_{1}P\\\
\mathfrak{M}P&\equiv&\max\\{\mathrm{E}[\pi_{\mathfrak{M}}(S)],S\in\mathrm{Ran}(\pi_{2}P)\\}\\\
\mathfrak{S}P&\equiv&\max\\{\mathrm{E}[\pi_{\mathfrak{S}}(S)],S\in\mathrm{Ran}(\pi_{2}P)\\}\\\
\mathfrak{E}P&\equiv&\max\\{\mathrm{E}[\pi_{\mathfrak{E}}(S)],S\in\mathrm{Ran}(\pi_{2}P)\\}\end{array}$
So it is easy to see that
$\begin{array}[]{lll}\mathfrak{T}P_{2}\circ
P_{1}&=&\pi_{1}P_{1}+\pi_{1}P_{2}\\\ \mathfrak{M}P_{2}\circ
P_{1}&=&\max\\{\mathfrak{M}P_{1},\mathfrak{M}P_{2}\\}\\\
\mathfrak{S}P_{2}\circ
P_{1}&=&\max\\{\mathfrak{M}P_{1},\mathfrak{M}P_{2}\\}\\\
\mathfrak{E}P_{2}\circ
P_{1}&=&\max\\{\mathfrak{M}P_{1},\mathfrak{M}P_{2}\\}\end{array}$
According to the definition above, the resource of a non-deterministic
physical operator $O$ which is initialized by $x\in\Omega$ should be defined
as:
$\begin{array}[]{lll}\mathfrak{T}O(x)&\equiv&\mathrm{E}[\mathfrak{T}O_{i}(x)],O_{i}(x)\in
O(x)\\\ \mathfrak{M}O(x)&\equiv&\mathrm{E}[\mathfrak{M}O_{i}(x)],O_{i}(x)\in
O(x)\\\ \mathfrak{S}O(x)&\equiv&\mathrm{E}[\mathfrak{S}O_{i}(x)],O_{i}(x)\in
O(x)\\\ \mathfrak{E}O(x)&\equiv&\mathrm{E}[\mathfrak{E}O_{i}(x)],O_{i}(x)\in
O(x)\\\ \end{array}$
So for operators’ operation, we have:
$\begin{array}[]{lll}\mathfrak{T}O_{2}\circ
O_{1}(x)&=&\mathfrak{T}O_{1}(x)+\mathfrak{T}O_{2}(\lhd O_{1}(x)\rhd)\\\
\mathfrak{M}O_{2}\circ
O_{1}(x)&=&\max\\{\mathfrak{M}O_{1}(x),\mathfrak{M}O_{2}(\lhd
O_{1}(x)\rhd)\\}\\\ \mathfrak{S}O_{2}\circ
O_{1}(x)&=&\max\\{\mathfrak{S}O_{1}(x),\mathfrak{S}O_{2}(\lhd
O_{1}(x)\rhd)\\}\\\ \mathfrak{E}O_{2}\circ
O_{1}(x)&=&\max\\{\mathfrak{E}O_{1}(x),\mathfrak{E}O_{2}(\lhd
O_{1}(x)\rhd)\\}\\\ \end{array}$
#### 3.6.1 Framework for the Complexity with respect to General Physical
Resource
###### Definition 3.10
(Resource(deterministic))A resource the physical process which complete the
whole computation consumed $\mathfrak{R}_{\mathscr{P}}$ including:
$\begin{array}[]{lll}\mathfrak{T}_{\mathscr{P}}&\equiv&\mathfrak{T}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{S}_{\mathscr{P}}&\equiv&\mathfrak{S}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{M}_{\mathscr{P}}&\equiv&\mathfrak{M}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{E}_{\mathscr{P}}&\equiv&\mathfrak{E}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
\end{array}$
i.e.
$\mathfrak{R}_{\mathscr{P}}\equiv(\mathfrak{T_{\mathscr{P}}},\mathfrak{S}_{\mathscr{P}},\mathfrak{M}_{\mathscr{P}},\mathfrak{E}_{\mathscr{P}})$
###### Definition 3.11
(Resource(Las Vegas))A resource the physical process which complete the whole
computation consumed $\mathfrak{R}_{\mathscr{P}}$ including:
$\begin{array}[]{lll}\mathfrak{T}_{\mathscr{P}}&\equiv&\mathfrak{T}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{S}_{\mathscr{P}}&\equiv&\mathfrak{S}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{M}_{\mathscr{P}}&\equiv&\mathfrak{M}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{E}_{\mathscr{P}}&\equiv&\mathfrak{E}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
\end{array}$
i.e.
$\mathfrak{R}_{\mathscr{P}}\equiv(\mathfrak{T_{\mathscr{P}}},\mathfrak{S}_{\mathscr{P}},\mathfrak{M}_{\mathscr{P}},\mathfrak{E}_{\mathscr{P}})$
###### Definition 3.12
(Resource(Monte Carlo))A resource the physical process which complete the
whole computation consumed $\mathfrak{R}_{\mathscr{P}}$ including:
$\begin{array}[]{lll}\mathfrak{T}_{\mathscr{P}}&\equiv&\mathfrak{T}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{S}_{\mathscr{P}}&\equiv&\mathfrak{S}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{M}_{\mathscr{P}}&\equiv&\mathfrak{M}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
&&\\\
\mathfrak{E}_{\mathscr{P}}&\equiv&\mathfrak{E}((\Delta\circ\mathcal{H}\circ\nabla)(x))\\\
\end{array}$
i.e.
$\mathfrak{R}_{\mathscr{P}}\equiv(\mathfrak{T_{\mathscr{P}}},\mathfrak{S}_{\mathscr{P}},\mathfrak{M}_{\mathscr{P}},\mathfrak{E}_{\mathscr{P}})$
The corresponding concept of complexity should be defined as the resource
consumed with respect to the length of the input.
###### Definition 3.13
Complexity The complexity of a kind of resource is a function of the length of
the input $x$ $n=\lceil\log x\rceil$
$\begin{array}[]{lll}\mathrm{Conplexity}_{t}(n)&=&\max\\{\mathfrak{T}_{\mathscr{P}(x)}|n-1\leq\log
x\leq n\\}\\\
\mathrm{Conplexity}_{m}(n)&=&\max\\{\mathfrak{M}_{\mathscr{P}(x)}|n-1\leq\log
x\leq n\\}\\\
\mathrm{Conplexity}_{s}(n)&=&\max\\{\mathfrak{S}_{\mathscr{P}(x)}|n-1\leq\log
x\leq n\\}\\\
\mathrm{Conplexity}_{e}(n)&=&\max\\{\mathfrak{E}_{\mathscr{P}(x)}|n-1\leq\log
x\leq n\\}\end{array}$
Note: _actually, the finiteness of resource cost is a necessary precondition
for all “uniform” physical computation models. Another necessary precondition
is that all the physical attributes should be at some trivial states(very
easily to be constructed, e.g. $0^{\circ}\mathrm{C}$, $0$m/s) before the
experiments. If these requirement are not satisfied, the model will be a non-
uniform one. This case is also discussed in details by Beggs et al.[19],[22]_
#### 3.6.2 Some Common Examples
It is interesting to find some new methods to compute problems without the
help of universal Turing Machines. Through the ages, people have found a lot
of such examples, the most famous of them are:
* •
Measure the volume of an object by putting it into the water;
* •
Obtain the centroid of an object by two suspension method;
* •
Compute function sine by analog circuit;
* •
Decide the path of minimum cost using Fermat’s Principle;
* •
Calculate the mean of numbers by the second Law of Thermodynamics.
Actually, we can give even more similar examples:
* •
By making use of resonance, we can easily find the desired tuning fork from a
heap of them. Otherwise, we have to look up the label of them one by one and
even have to compute the frequency one by one if there is no labels on them.
* •
We can compute the square root of an given number $x$ by the law of free fall.
Prepare a vacuum tube T of length $x$ and let it stand vertically, then let an
object o which is small enough fall. Get the time $t$ when it touch the
bottom, and we have $\sqrt{x}=t/c$, where $c=(2/g)^{1/2}$.
* •
We can sort a series of numbers through dangling poises by strings, where the
strings satisfies Hooke’s law. Given an array of numbers $\\{x_{i}\\}$
construct or find poises whose mass is just $x_{i}$, then dangle them by
strings with the same stiffness coefficient. When the system is stable, the
position of the poises with respect to their weight just indicate the
relationship desired.
However, it is hard for us to estimate the cost of the methods above just
after we describe them informally. So we select a part of them to analyze
next.
Conventions: $x$ is the representation of number in digits, $[x]$ is the value
of $x$, $[x]^{A}$ means the attribute $A$ has the value $x$.
$[x]^{\Sigma^{*}}$ means the representation of quantity $x$ though not on the
tape.
##### 3.6.2.1 Mean of Three Numbers
Given three numbers, compute the mean of them making use of law of
thermodynamics. This idea comes from Pitowsky [7].
The strict description of the problem: Given three numbers $x_{1}$, $x_{2}$,
$x_{3}\in[\leavevmode\nobreak\ 0,100\leavevmode\nobreak\ ]$, compute
$\bar{x}=\frac{(x_{1}+x_{2}+x_{3})}{3}\text{\quad(Accurate to two decimal
places)}.$
Pitowsky suggests that since all of the three numbers less than 100 and bigger
than zero, note that the freezing point of water is 0 $C^{\circ}$ and the
boiling point of water is 100 $C^{\circ}$ under the one standard air pressure,
So for each number $x_{i}$, we can prepare the corresponding water of volume
$V$ and temperature of $x_{i}C^{\circ}$. And then pour the water of three
vessels into a bigger one, whose volume is $V^{\prime}(V^{\prime}>3V)$, and
wait. After the water arrived at the balance point, measure the temperature.
Of course, we assume that during the whole procedure, no calory is lose.
Apparently, the physical state the method above deal with is the temperature
of water, so we have
$\Omega=\\{\vec{t}\leavevmode\nobreak\ |t_{i}\in[0,100],i=1,2,3\\}^{T}.$
on the other hand, we suppose the water is heat up from 0 $C^{\circ}$, i.e.
the initial state of the experiment is $([0]^{T},[0]^{T},[0]^{T})$.
Therefore, the process could be depicted as following:
$\nabla:\Sigma^{+}\rightarrow\Omega$, heat up the water to the desired
temperature
$\nabla(x_{1},x_{2},x_{3})=([x_{1}]^{T},[x_{2}]^{T},[x_{3}]^{T})$
$\mathcal{H}:\Omega\rightarrow\Omega$, admixture the water of different
temperature, the second law of thermodynamics is used
$\mathcal{H}([x_{1}]^{T},[x_{2}]^{T},[x_{3}]^{T})=([\bar{x}]^{T},[\bar{x}]^{T},[\bar{x}]^{T})$
$\Delta:\Omega\rightarrow\Sigma^{+}$, measure the temperature of the water
$\Delta([\bar{x}]^{T},[\bar{x}]^{T},[\bar{x}]^{T})=\bar{x}$
For this problem, since the precision is finite, and there are only constant
(three) numbers and the numbers are bounded, we can easily deduct that
$\mathfrak{R}_{\mathscr{P}}$ is a constant. As a matter of fact, for Turing
Machine, we can also find a constant resource costing algorithm which is just
looking up a finite list to solve the problem.
##### 3.6.2.2 Sorting Without Repeat
Description of the Problem:
Input:
Finite number series of length $n$:
$A=\\{x_{i}|x_{i}\in\mathbb{Z^{+}}\cap[0,M](0\leq i\leq n)\\};$
Output:
Finite number series of length $m$:
$B=\\{x_{j}|x_{j}\in A(0\leq j\leq m)\\},$
s.t. if $j_{1}<j_{2}$ then $x_{j_{1}}<x_{j_{2}}$ .
Our plan is: for the given series, select a series of poises of length $n$,
s.t. the mass of the $i$th poise is equivalent to the $i$th number. Dangling
the poises from right to left by strings, whose restoring coefficient are $k$.
Wait until the system is stationary, open the parallel light source and
measure the projection onto the vertical ruler at the right end. The
measurement could be done by machines and present the results onto the tape
for observer. For Example, we can embed some photoconductive diodes in the
ruler by graduations, diodes who is not triggered should be read.
The physical state the method is primarily concerned with is the mass of poise
$M$, the horizontal positions of the poises $X$ and the vertical ones $Y$, the
projections $Y^{\prime}$ and the boole value $B$ indicating which diodes is
triggered, i.e.
$\Omega\equiv\oplus^{n}_{i}\\{m_{i}\\}^{M}\times\\{x_{i}\\}^{X}\times\\{y_{i}\\}^{Y}\times\oplus_{j}\\{(j,B_{j})\\}^{Y^{\prime}\times
B}$
So we have
$\nabla:\Sigma^{+}\rightarrow\Omega$ (Select poises)
$\nabla(\oplus^{n}_{i}x_{i})=(\oplus_{i}[x_{i}]^{M}[i]^{X}[0]^{Y}\oplus^{M}_{j}[(j,0)]^{Y^{\prime}\times
B})$
$\mathcal{H}:\Omega\rightarrow\Omega$ (Dangle poises)
$\mathcal{H}\left(\begin{array}[]{l}\oplus^{n}_{i}[x_{i}]^{M}\\\ {[i]^{X}}\\\
{[0]^{Y}}\\\ \oplus^{M}_{j}[(j,0)]^{Y^{\prime}\times
B}\end{array}\right)=\left(\begin{array}[]{l}\oplus^{n}_{i}[x_{i}]^{M}\\\
{[i]^{X}}\\\ {[[x_{i}]g/k]^{Y}}\\\ \oplus^{M}_{j}[(j,0)]^{Y^{\prime}\times
B}\\\ \end{array}\right)$
$\mathcal{H^{\prime}}:\Omega\rightarrow\Omega$(Open the parallel light)
$\mathcal{H^{\prime}}\left(\begin{array}[]{l}\oplus^{n}_{i}[x_{i}]^{M}\\\
{[i]^{X}}\\\ {[[x_{i}]g/k]^{Y}}\\\ \oplus_{j}[(j,0)]^{Y^{\prime}\times B}\\\
\end{array}\right)=\left(\begin{array}[]{l}\oplus^{n}_{i}[x_{j}]^{M}\\\
{[j]^{X}}\\\ {[[x_{j}]g/k]^{Y}}\\\
\oplus^{M}_{j}[(j,\epsilon_{A}(j))]^{Y^{\prime}\times B}\\\
\end{array}\right)$
$\Delta:\Omega\rightarrow\Sigma^{+}$(read the projection)
$\Delta\left(\begin{array}[]{c}\oplus^{n}_{i}([x_{j}]^{M}\\\ {[j]^{X}}\\\
{[[x_{j}]g/k]^{Y}}\\\ \oplus^{M}_{j}[(j,\epsilon_{A}(j))]^{Y^{\prime}\times
B})\\\ \end{array}\right)=\oplus_{j^{\prime}}(x_{j^{\prime}})$
satisfies if $j_{1}<j_{2}$ then
$[x_{j_{1}}]<[x_{j_{2}}]$
Considering the ideal implementation, we conclude that the
$\mathfrak{R}_{\mathscr{P}}$ is linear, which is superior to Turing Machines
using comparisons, for the complexity for them is proofed to be $O(n\log n)$.
However, there does exist Turing Machine, which is not based on comparisons,
also has a linear time cost.
Note that if the number series is boundless, the complexity of the method
above will be exponential. This is the common defeat of most analog computers.
##### 3.6.2.3 Volume of irregular shape
For this issue, we shall restrict the range of the saying ‘irregular’ so as to
rule out the objects with infinite length of description. So actually, we tend
to discuss a subset of the set of all cases.
Description of the problem:
Inputs:
point series of length $n$:$(x_{i},y_{i})(1<i<n)$, satisfies $c+r\leq
x_{i}\leq a-c-r,c+r\leq y_{i}\leq b-c-r$
Outputs:
The volume of the box of length $a$ and width $b$ and height $h_{0}$, not
including the series of cylinders(radius: $r$ height: $h_{0}$) which are
induced by the series of points.
Our plan is simple. Assume we have a box of material of dense $\rho$, and a
punch to extract circles from it. Then we measure the mass of the rest then
divide it by its dense or just put it into water.
$\nabla:\Sigma^{+}\rightarrow\Omega$
$\nabla(\oplus^{n}_{i=1}a(x_{i},y_{i}))=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[0]^{\Sigma^{\prime}}$
$\mathcal{H}_{1}:\Omega\rightarrow\Omega$
$\mathcal{H}_{1}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[0]^{\Sigma^{\prime}}$ $=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{\Sigma^{\prime}}$
$\mathcal{H}_{2}:\Omega\rightarrow\Omega$
$\mathcal{H}_{2}[\rho h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{\Sigma^{\prime}}$ $=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{\Sigma^{\prime}}$
$\Delta:\Omega\rightarrow\Sigma^{+}$
$\Delta[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{\Sigma^{\prime}}=h_{0}(A-m(\cup^{n}_{i=1}c_{i}))$
Apparently the resource complexity for this method is linear with respect to
the number of the points. However, because most people think that we cannot do
infinitely measurement during one experiment, this method can only provide the
result of finite precision. This is a good news to Turing Machines because
this implies there exists a Turing Machine which is almost equivalently
efficient.
This may be astonish to someone, who may thought that a TM should at least
solve the equations first. However, because of the finite precision, Turing
Machine can just split the object into lattice and use the so-called scan-line
algorithm to find the answer.
##### 3.6.2.4 The centroid of Irregular Shape
Just as the last example, we restrict our topic into the same subsets of all
cases.
Description of Problem:
Inputs:
point series of length $n$:$(x_{i},y_{i})(1<i<n)$, satisfies $c+r\leq
x_{i}\leq a-c-r,c+r\leq y_{i}\leq b-c-r$
Outputs:
The centroid of the box of length $a$ and width $b$ and height $h_{0}$, not
including the series of cylinders which is induced by the series of points.
The method we suggest is similar to the last one, the difference of them is
that this time we will record some points.
$\nabla:\Sigma^{+}\rightarrow\Omega$
$\nabla(\oplus^{n}_{i=1}a(x_{i},y_{i}))=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[0]^{\Sigma}[0]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$\mathcal{H}_{1}:\Omega\rightarrow\Omega$ (Suspend the box by $V_{0}$)
$\mathcal{H}_{1}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[0]^{\Sigma}[0]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[0]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$\mathcal{H}_{2}:\Omega\rightarrow\Omega$ (Suspend the box by
$V^{\prime}_{0}$)
$\mathcal{H}_{2}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[0]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[\frac{c-V^{\prime}_{0}}{|c-V^{\prime}_{0}|}+V^{\prime}_{0}]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$\mathcal{H}_{3}:\Omega\rightarrow\Omega$(Extend the unit vectors: Get the
point of intersection)
$H_{3}[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[\frac{c-V^{\prime}_{0}}{|c-V^{\prime}_{0}|}+V^{\prime}_{0}]^{\Sigma^{\prime}}[0]^{\Sigma^{\prime\prime}}$
$=[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[\frac{c-V^{\prime}_{0}}{|c-V^{\prime}_{0}|}+V^{\prime}_{0}]^{\Sigma^{\prime}}[c]^{\Sigma^{\prime\prime}}$
$\Delta:\Omega\rightarrow\Sigma^{+}$
$\Delta[\rho
h_{0}(A-m(\cup^{n}_{i=1}c_{i}))]^{M}[\frac{c-V_{0}}{|c-V_{0}|}+V_{0}]^{\Sigma}[\frac{c-V^{\prime}_{0}}{|c-V^{\prime}_{0}|}+V^{\prime}_{0}]^{\Sigma^{\prime}}[c]^{\Sigma^{\prime\prime}}=c$
The time the system cost from oscillating to stillness can be bounded by a
constant. Because of the same reason this method does not break up the lower
bound of Turing Machine. But for some other things, we tend to pay more
attention to it. Some relevant issues will be discussed in Sec-V.
#### 3.6.3 Graph Isomorphism, Graph Spectrum and Oscillators
In this part of the section, we shall talk about a complex example in detail.
We do not mean to show that the method we designed here is superior to all of
the TMs constructed by the people of the same aim. We just want to demonstrate
a new style of computation.
##### 3.6.3.1 Spectrum of Graph
Suppose $X=(V,E)$ is a graph, $A$ is it’s adjacent matrix. We say
$f_{A}(\lambda)$ is the characteristic polynomial of $X$, also denoted by
$f_{X}(\lambda)$. $(\lambda_{1},\dots,\lambda_{n})$, the whole root of
$f(\lambda)$, is called the spectrum of graph $X$.
Actually two different adjacent matrices may represent two isomorphic graphs.
If we alter the permutation of the number of the vertices, $A$ will become
$P^{-1}AP$, where $P$ is the corresponding permutation matrix. However, the
characteristic polynomials of them are the same. Therefore, $f_{X}(\lambda)$
and the spectrum $\textrm{spec}(X)=(\lambda_{1},\dots,\lambda_{n})$ are
uniquely determined by $X$.
For the relationship between spectrum and graph, people conjectured that graph
can be uniquely determined by spectrum, i.e. suppose
$\textrm{spec}(A)=\textrm{spec}(B),$
can we conclude that
$A\backsimeq B?$
Unfortunately, the different graphs of the same spectrum were found soon.
Nonetheless, calculating the spectrum is also important. Because we can know a
lot of crucial properties, such as the extensionality, rapid mixing time of
Markov chains on the graph, by the spectrum of the graph. What’s more, when
two graph have same spectrum, and spectrum is never repeating, we have a
polynomial time algorithm to check whether they are isomorphic.
1. 1.
Input graphs $G_{1}$ $G_{2}$, compute their spectrum, denoted by $\Lambda_{1}$
$\Lambda_{2}$.
2. 2.
Compare the spectrums, if $\Lambda_{1}\neq\Lambda_{2}$, then return NOT
ISOMORPHIC;else, continue;
3. 3.
Check whether the product of the two similar matrices is a permutation matrix,
if it is true return ISOMORPHIC, otherwise return NOT ISOMORPHIC;
Notation: Here by $\Lambda_{1}\neq\Lambda_{2}$ we mean after sorting their
eigenvalue, the two series are not identical to each other. And accordingly
$G_{1}$, $G_{2}$ should also be altered into $\widetilde{G}_{1}$,
$\widetilde{G}_{2}$. But for convenience, we do not differeciate $G_{i}$ and
$\widetilde{G}_{i}$. Proof:
1. If $\Lambda_{1}\neq\Lambda_{2}$, then $G_{1}\ncong G_{2}$. So we only consider the case in which $\Lambda_{1}=\Lambda_{2}=\Lambda$.
i.e.Suppose
$G_{1}=P\Lambda P^{T},\leavevmode\nobreak\ G_{2}=Q\Lambda Q^{T}$
then we have
$P^{T}G_{1}P=\Lambda=Q^{T}G_{2}Q$
thus
$G_{1}=\left(QP^{T}\right)^{T}G_{2}\left(QP^{T}\right)$
by the preconditon,$\Lambda$ is never repeating, so $P$ $Q$ is the unique
orthganol matrices.
the rest is to show that if $G_{1},G_{2}$ is isomorphic, then $QP^{T}$ is the
permutation matrix desired.
In fact, if $G_{1}\cong G_{2}$,then there exists a permutation matrix $S$ s.t.
$G_{1}=S^{T}G_{2}S$
Since $G_{2}=Q\Lambda Q^{T}$, the formula above means
$G_{1}=S^{T}Q\Lambda Q^{T}S=\left(Q^{T}S\right)^{T}\Lambda\left(Q^{T}S\right)$
Because of the uniqueness of $P$, we can conclude that $Q^{T}S=P^{T}$, and by
orthgonality of $Q$, we obtain
$S=QP^{T}.$
$\square$
##### 3.6.3.2 Harmonic Oscillator of multi-freedom
Suppose $s$ is the number of freedom of the system, $q_{\alpha
0}(\alpha=1,2,\dots,s)$ is the general coordinates when the system is in
balance. Without lose of generality, we can always assume that $q_{\alpha 0}$
is just zero, i.e. $q_{\alpha 0}=0(\alpha=1,2,\dots,s)$.
Because we only talk about little vibration, so we only keep several terms in
the Taylor series of the Lagrangians $L$ of the system about $q_{\alpha 0}$.
The potential energy:
$V=V_{0}+\sum^{s}_{\alpha=1}\left(\frac{\partial V}{\partial
q_{\alpha}}\right)_{0}q_{\alpha}+\sum^{s}_{\alpha=1}\sum^{s}_{\beta=1}\frac{1}{2}\left(\frac{\partial^{2}V}{\partial
q_{\alpha}\partial q_{\beta}}\right)_{0}q_{\alpha}q_{\beta}+\cdots.$
Note that $V_{0}$ can be omitted. Introduce the notation $k_{\alpha\beta}$
$k_{\alpha\beta}=k_{\beta\alpha}=\left(\frac{\partial^{2}V}{\partial
q_{\alpha}\partial q_{\beta}}\right)_{0},$
which is called the strength coefficient. According to the formula
$\left(\frac{\partial V}{\partial q_{\alpha}}\right)_{0}=0$, the second order
of the potential energy could be represented as
$V=\frac{1}{2}\sum^{s}_{\alpha=1}\sum^{s}_{\beta=1}k_{\alpha\beta}q_{\alpha}q_{\beta}.$
Then assume $\bm{r}_{i}=\bm{r}_{i}(q)$ is not relevant to time, i.e. the
obligation is constant, so the kinetic energy is:
$T=\frac{1}{2}\sum^{n}_{t=1}m_{i}\dot{\bm{r}}_{i}\cdot\dot{\bm{r}}_{i}=\frac{1}{2}\sum^{n}_{i=1}\sum^{s}_{\alpha=1}\sum^{s}_{\beta=1}m_{i}\frac{\partial\bm{r}_{i}}{\partial
q_{\alpha}}\cdot\frac{\partial\bm{r}_{i}}{\partial
q_{\beta}}\dot{q}_{\alpha}\dot{q}_{\beta}.$
Introduce the symbol $m_{\alpha\beta}$,
$m_{\alpha\beta}=m_{\beta\alpha}=\sum^{n}_{i=1}m_{i}\frac{\partial\bm{r}_{i}}{\partial
q_{\alpha}}\cdot\frac{\partial\bm{r}_{i}}{\partial q_{\beta}},$
then the kinetic energy could be represented as
$T=\frac{1}{2}\sum^{s}_{\alpha=1}\sum^{s}_{\beta=1}m_{\alpha\beta}\dot{q}_{\alpha}\dot{q}_{\beta}.$
Keep the formula above to second order and since
$\dot{q}_{\alpha}\dot{q}_{\beta}$ is second order $m_{\alpha\beta}$ should be
expanded to zeroth order. In other words $m_{\alpha\beta}$ could be looked as
constants, we just take the value of them when the system is in balanced
point.
So the Lagrangian could be written as
$L=\frac{1}{2}\sum^{s}_{\alpha=1}\sum^{s}_{\beta=1}(m_{\alpha\beta}\dot{q}_{\alpha}\dot{q}_{\beta}-k_{\alpha\beta}q_{\alpha}q_{\beta}).$
Thus the Lagrangian equation is
$\frac{d}{dt}\frac{\partial}{\partial\dot{q}_{\alpha}}\left(\frac{1}{2}\sum^{s}_{\beta=1}\sum^{s}_{\gamma=1}m_{\beta\gamma}\dot{q}_{\beta}\dot{q}_{\gamma}\right)-\frac{\partial}{\partial
q_{\alpha}}\left(-\frac{1}{2}\sum^{s}_{\beta=1}\sum^{s}_{\gamma=1}k_{\beta\gamma}q_{\beta}q_{\gamma}\right)=0.$
i.e.
$\frac{d}{dt}\left(\frac{1}{2}\sum^{s}_{\gamma=1}m_{\alpha\gamma}\dot{q}_{\gamma}+\frac{1}{2}\sum^{s}_{\beta=1}m_{\beta\alpha}\dot{q}_{\beta}\right)+\left(\frac{1}{2}\sum^{s}_{\gamma=1}k_{\alpha\gamma}q_{\gamma}+\frac{1}{2}\sum^{s}_{\beta=1}k_{\beta\alpha}q_{\beta}\right)=0.$
therefore
$\sum^{s}_{\beta=1}m_{\alpha\beta}\ddot{q}_{\beta}+\sum^{s}_{\beta=1}k_{\alpha\beta}q_{\beta}=0\quad(\alpha=1,2,\dots,s).$
Let
$q_{\beta}=A_{\beta}e^{\lambda t}\quad(\beta=1,2,\dots,s).$
Take it into the former formula, we get the linear equations for $A_{\beta}$.
$\sum^{s}_{\beta=1}(m_{\alpha\beta}\lambda^{2}+k_{\alpha\beta})A_{\beta}=0\quad(\alpha=1,2,\dots,s).$
If the equations have non-trivial solutions, then following conditions should
be hold:
$\begin{vmatrix}m_{11}\lambda^{2}+k_{11}&m_{12}\lambda^{2}+k_{12}&\cdots&m_{1s}\lambda^{2}+k_{1s}\\\
m_{21}\lambda^{2}+k_{21}&m_{22}\lambda^{2}+k_{22}&\cdots&m_{2s}\lambda^{2}+k_{2s}\\\
\vdots&\vdots&&\vdots\\\
m_{s1}\lambda^{2}+k_{s1}&m_{s2}\lambda^{2}+k_{s2}&\cdots&m_{ss}\lambda^{2}+k_{ss}\\\
\end{vmatrix}=0$
This is the equations of times $s$ of $\lambda^{2}$, and we can get $s$
$\lambda^{2}$, denoted by
$\lambda^{2}_{l}\quad(l=1,2,\cdots,s).$
##### 3.6.3.3 The characteristic oscillators for a Graph
Making use of the conclusions above, we construct a specific oscillators for
any given connected graph.
Denote the vertices of graph by numbers $1\sim n$, according to any order. The
mass of a vertex is set 1g. Connect the vertex $1$ and $n$ to ends by strings
whose $k$ is zero by that direction. For the rest, we connect them according
to the adjacent matrix, i.e. if $A_{ij}=1(\textrm{Note that }A_{ij}=A_{ji})$,
connect vertex $i$ and $j$ by a string whose $k=1$. Let’s study the motion of
the system: First, if two vertices is not connected by string, we have
$k_{\alpha\beta}=k_{\beta\alpha}=0$. Second, the vibration is little, so
string is not an obligation. And we take the general coordinates as the usual
displacement vectors, so
$m_{\alpha\beta}=m_{\beta\alpha}=\delta_{\alpha\beta}$, where
$\delta_{\alpha\beta}$ is the well known Kronecker notation.
At last, we obtain the determinant as follows, which is the characteristic
polynomial of our system.
$\begin{vmatrix}\lambda^{2}+d_{1}&-A_{12}&\cdots&-A_{1s}\\\
-A_{21}&\lambda^{2}+d_{2}&\cdots&-A_{2s}\\\ \vdots&\vdots&&\vdots\\\
-A_{s1}&-A_{s2}&\cdots&\lambda^{2}+d_{s}\\\ \end{vmatrix}=0$
It has been proofed that $\lambda^{2}<0$. So let $-\Lambda=\lambda^{2}$ ,we
can see that the determinant above actually compute the spectrum of
$A^{\prime}$ which is converted from $A$ by adding multi-loops(the number of
degrees). If a vertex is in the characteristic position, it will take part in
the vibrations of all frequencies, if no one is in the characteristic
position, then they just vibrate with respective frequency. In both cases,
we’ll measure the frequency and differentiate them by means of FFT, so as to
get the spectrum of $A^{\prime}$.
Apparently, adding multi-loops is not harmful to the decision of whether $A$
and $B$ are isomorphic, for if $A^{\prime}\neq B^{\prime}$, then
$A\not\backsimeq B$ If $A\backsimeq B$, then $A^{\prime}\backsimeq
B^{\prime}$, which will also be checked by the oscillating system.
##### 3.6.3.4 Comments
Suppose now the problem we want to solve is: try to find the
$n_{0}-$th($n_{0}$ is a constant) value of such matrices. It is easy to see
that the period of the system satisfies
$\frac{1}{\sqrt{n}f_{0}}\leq T\leq\frac{\sqrt{n}}{f_{0}}(\text{or
}\frac{1}{\sqrt{n}}f_{0}\leq f\leq\sqrt{n}f_{0}),$
where $f_{0}$ is the eigenfrequency of a single string, considering the
minimum case occurs when the corresponding graph is totally parallel
connected(two vertices with $n-$multiple edges between them), while the
maximum case occurs when it is just a chain.
The total steps of sampling should be
$N=2BL=2\left(\sqrt{n}f_{0}-\frac{f_{0}}{\sqrt{n}}\right)\left(c_{n_{0}}\frac{\sqrt{n}}{f_{0}}\right)$
where the constant $c_{n_{0}}$ is related to the required precise $n_{0}$. So
we have: $O(N)=O(n)$ and the corresponding complexity for fast fourier
transform should be $O(n\ln n)$.
So we can say that the time complexity of this method should be
$O(n^{2})+O\left(\frac{\sqrt{n}}{f_{0}}\right)+O(n\ln n)+O(n)=O(n^{2})$
where the left $O(n^{2})$ is the cost of constructing the system and $O(n\ln
n)$ is the complexity of FFT. On the other hand, if we use the well-known
algorithm called QR-method to get the answer, it will cost such Turing machine
$O(n^{3})$ steps. However, we know little about wether our method is superior
to any most efficient Turing machines.
However, such analysis may not be enough. We cannot obtain a reliable result
unless some specifications for the materials are considered. The procedure of
constructing the system according to an arbitrary graph is actually quite
tricky. For this system need $n$ oscillators of the same mass and different
length. To achieve this, we have to assume that there exists a kind of ‘ideal’
material which, at least for a large enough range, can be stretched to a
desired length easily(in $O(n)$ time and totally $O(n^{2})$) and, at the same
time, keep rigid.
#### 3.6.4 Steiner Tree Problem
Steiner Tree Problem is a problem in combinatorics. The general version of
Steiner Tree Problem is NP-complete, which implies that this problem is
unlikely be solved in polynomial time.
This problem is similar to the Minimal Spanning Tree Problem in metric space.
The difference is that Steiner Tree Problem allow people to add new points
$v^{\prime}(v^{\prime}\not\in V)$ and new edges $e^{\prime}(e^{\prime}\not\in
E)$ into the original graph $G$, if necessary. When $|G|=3$, the new point (in
this case, at most one point is needed)is called Fermat point.
At a time, some people became to believe that the experiments of soup membrane
can be used to solve the Steiner Tree Problem. In fact, when $|G|$ is small,
say, less than 5, this method really works. However, when the number of
vertices is 10 or more, this experiment just cannot give the right answer. One
can attribute the failure to different reasons and derive various
explanations. Of all these potential explanations, the one which states that
‘it is just the errors during the experiment cause the failure’ made many
people conjecture faithfully that classical mechanics can be used to solve NP-
complete Problems in polynomial time(So they try to proof P$=$NP).
In fact, the foundation of the experiment is the well-known property that the
membrane will stay at a stationary state, where the surface it produces will
be just the minimal surface. Unfortunately, this theory has nothing to do with
the fact that the membrane can arrive at the stationary state _fast_. What’s
more, no one can proof the soundness of such property under the framework of
classical mechanics.
#### 3.6.5 DNA Computation
In 1994, Adleman used a probabilistic DNA algorithm to solve HP problem
(Hamilton Path Problem). HP problem is NP-complete, which implies it is
difficult to find a polynomial algorithm to solve it[14].
In order to understand Adleman’s method, the following knowledge seems
necessary.
1. (1)
DNA contains chains consisted by four types of nucleotides, denoted by A, C, G
and T.
2. (2)
These nucleotides forms complementary couples, i.e. A and T are complementary,
C and G are complementary. If the corresponding positions of two DNA chains
are complementary, they will patch up as the twin-helix structure.
3. (3)
PCR, which proposed by Kary Mullis, is method to reproduce the specific chain
we need.
4. (4)
There is a machine called ’sequencer’ which can be used to read out the series
of a DNA chain.
Adleman’s Algorithm contains five procedures(Suppose $|G|=n$):
1. (1)
Randomly produce the paths in the Graph, encoded by DNA chains.
2. (2)
Keep only those paths which begin with $v_{in}$ and end with $v_{out}$.
3. (3)
Keep only the paths whose length is $n$
4. (4)
Keep only those paths which enter all vertices in G at least once.
5. (5)
If any paths remain, return ’True’, else return ’False’.
Note that the first step of Adleman’s Algorithm, which is usually thought to
be work as an initialization operator $\nabla$, is not polynomial with respect
to the resource mass $\mathfrak{M}$ and space $\mathfrak{S}$ at least.
Considering asymptotically we can only sequentially get the mass the algorithm
need, so actually $O(n!)$ mass can cause $O(n!)$ time $\mathfrak{T}$. As a
matter of fact the other steps of this algorithm, which require exponentially
molecules fully blend by polynomially increasing contacting facades, also cost
a lot of resource $\mathfrak{T}$.
It is not very hard to appreciate the conclusion that we can obtain great
power of computation suppose we are provided with corresponding quantity of
mass, and do not take the cost of preparing such equipment at all. For one
thing, let’s consider the following ideal model.
Suppose we have enough universal Turing Machines, each of them are denoted by
their footnotes. What’s more, by some altering in the definition, these UTMs
have the ability to transmit their results to others. And the condition of two
UTMs $U_{i},U_{j}(i\neq j)$ could communicate to each other is that they are
adjacent to each other, denoted by $Adj(U_{i},U_{j})$.
So the computational model constructed following, called ‘Turing Tree’, can
exponentially speed up the computation of any NP-complete problems.
###### Definition 3.14
(Turing Tree) Suppose we have infinite many UTMs, each of them denoted by
unique footnotes, and
$Adj(U_{i},U_{j})\Leftrightarrow j=2i+1\vee i=2j+1\vee j=2i+2\vee i=2j+2,$
then we call this Turing Tree.
It is easy to see that the following relation holds:
$\begin{array}[]{c}Adj(U_{0},U_{1}),Adj(U_{0},U_{2})\\\
Adj(U_{1},U_{3}),Adj(U_{1},U_{4}),Adj(U_{2},U_{5}),Adj(U_{2},U_{6})\\\
\dots\dots\dots\end{array}$
For example, a TSP problem can be solved as following:
1. a
The Observer input the weighted complete graph $G$ to the $U_{0}$, $U_{0}$
decode 0 to a permutation and compute the sum of the weight, and then transmit
$G$ and flag $F=0$ to $U_{1},U_{2}$.
2. b
For index $i$ After $U_{i}$ get $F=0$ and $G$, it check whether
$i<\lceil\log_{2}n!\rceil$,if the answer if ’yes’ then decode $i$ to a
permutation and get the sum, and transmit $G$ and $F=0$ to $U_{2i+1}$ and
$U_{2i+2}$;else check whether $i=\lceil\log_{2}n!\rceil$, if it is true,
decode $i$ to a permutation and get the sum, then submit the weight sum to the
$U_{\lceil i/2\rceil-1}$. Else, do nothing.
3. c
For index $i$, after $U_{i}$ get $F=1$ and two sum(come from
$U_{2i+1},U_{2i+2}$), if it’s index is not zero, then submit
$\min\\{S_{i},S_{2i+1},S_{2i+2}\\}$ and $F=1$ to $U_{\lceil i/2\rceil-1}$.
Else return $\min\\{S_{0},S_{1},S_{2}\\}$ and write it on to the tape.
It is easy to check that the subprocedure of the algorithm which is used to
decode a natural number to a permutation is polynomial. So the cost of time
the Turing Tree consumed should be $O(2\log_{2}n!)\leq O(2n\log_{2}n)$
(Including once sharing the task and once championship for the minimum). So it
is the time to answer how can we ‘easily’ construct a big enough Turing Tree.
### 3.7 Preliminary Discussion of the classic theory of Computation
#### 3.7.1 Turing computable is physical computable
The topic about the existence of a theoretical physical system which can
provide an implementation of universal Turing Machine has been studied by many
scholars. In addition to the current implementation of computers, scholars
have constructed many other wonderful designs on various axiom systems of
physics(e.g. Classical Mechanics, Quantum Mechanics).
Of course, the results above only imply that it is the ideal mathematic model
for a family of physical phenomenons can be look as equivalent to UTM in terms
of computability. After all, we cannot know for sure that some theory of
physics is completely correct. Because of this, when we talk about the ability
of computation for a certain family of physical system, we always assume
either of the two preconditions:
* •
The ideal mathematic model of some branch of physics is believed to be
absolutely right.
* •
At least in a very large scale, the theory works.
#### 3.7.2 PLATO Machine
For several decades after the Church-Turing Thesis was proposed, people failed
to find a counter-example. This kind of counter-example, if they really exist,
should satisfies the property that most people think they can be effectively
computed in principle, and no Turing machine can compute them.
However, many physicists tend to make efforts in another direction, that is,
they want to find a family of processes in nature, whose functional expression
may not be intuitively computable, nor Turing Computable, but actually it can
be used to ‘compute’ a nonrecursive function by measurement.
Suppose the problem we attempt to deal with now may cost infinite many steps
for some computational model(e.g. Turing Machine), does it necessarily mean
that we have to wait infinitely long time to get the results? This is not
always the case, PLATO Machine, which was proposed by H.Weyl[7], is just a
counter-example. Though it is named after Plato, the designer’s main
inspiration comes from one of Zeno’s Paradoxes.
Specifically, PLATO Machines use $(1/2)^{n}$ seconds to execute the $n-$th
step. For instance, suppose the decision problem we want to solve is $\exists
nP(n)$, where $P$ is a predicate and $P(x)$ is used to describe some
properties of $x$. Then PLATO machine $\mathbb{P}$ will check whether $P(1)=1$
holds in $1/2$ seconds, and check whether $P(2)=1$ in $1/4$ seconds,$\dots$,
and check whether $P(n)$ holds in $2^{-n}$ seconds, and so on. It is easy to
conclude that if $\mathbb{P}$ find an answer, it will return the answer in one
second, otherwise it will return false after a second. Considering the sum of
geometric series, the proof is trivial. So the upperbound of the time for
$\mathbb{P}$ to solve any question is
$T=\left(\frac{1}{2}\right)^{1}+\left(\frac{1}{2}\right)^{2}+\cdots+\left(\frac{1}{2}\right)^{n}+\cdots=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1s$
Apparently, if $\mathbb{P}$ does exist, its power is extraordinarily great,
for it can even solve Turing’s Halting Problem in one second.
So far we have seen two idea to implement the PLATO machine $\mathbb{P}$.
However, unfortunately, neither of them are successful. The first one is to
construct the machine according to the definitions of H.Weyl. Apparently, it
is difficult, for people do not believe that time is infinitely divisible. The
second one is to make use of the theory of general relativity. However, the
computing system will also exhaust the resource of the universe which make the
observer cannot get the answer.
#### 3.7.3 Recursive function whose derivative is not recursive
April 1970, J.Myhill published his astonishing result[6]: There exists a
recursive function, whose derivative is not recursive. In order to understand
the principles of the construction, knowing the following fact about the
recursive functions(whose domain is $\mathbb{R}$) should be helpful.
###### Theorem 3.15
Suppose $f$ is a real-valued function, $\\{f_{n}\\}$ is a series of recursive
functions, if there exists a recursive function
$e:\mathbb{N}\rightarrow\mathbb{N}$ s.t. $\forall x\in I$ $k\geq e(n)$
$|f_{k}(x)-f(x)|\leq\frac{1}{2^{n}}$, then $f$ is recursively computable.
J.Myhill’s idea is to build a non-trivial structure(slope or bump) in the
neighborhood of $2^{-n}$ in interval $[0,1]$, where $n\in\mathscr{A}$, and
$\mathscr{A}$ is a recursively enumerable, nonrecursive set. Otherwise
$f(x)=0$. However, in order to make the function computable, the scale of the
structure should shrink as the $n$ is enumerated recursively, or rather,
should be smaller than the bound in the theorem above. As a result, the
derivative of the function is intuitively hard to compute, and on the other
hand we can proof that it is not recursive, because if we could compute it we
can use the result to decide whether $\lceil x\rceil$ is an element of
$\mathscr{A}$ generally, contradicting the nonrecursiveness of $\mathscr{A}$.
Specifically, suppose
$\theta(x)\equiv\left\\{\begin{array}[]{ll}x(x^{2}-1)^{2},&\hbox{if\leavevmode\nobreak\
$-1\leq x\leq 1$;}\\\ 0,&\hbox{if\leavevmode\nobreak\
$|x|>1$.}\end{array}\right.$
It is easy to verify that $\theta(-1)=\theta(0)=\theta(1)=0$
$\theta^{\prime}(-1)=\theta^{\prime}(1)=0$ $\theta^{\prime}(0)=1$ and
$\theta_{min}=\theta(-1/\sqrt{5})\equiv-\lambda$
$\theta_{max}=\theta(+1\sqrt{5})\equiv+\lambda$. We call $\theta$ a $bump$ of
length 2 and height $\lambda$. Then the function
$\theta_{\alpha\beta}(x)\equiv(\beta/\lambda)\theta(x/\alpha)$ satisfies the
following conditions:
$\theta_{\alpha\beta}(-\alpha)=\theta_{\alpha\beta}(0)=\theta_{\alpha\beta}(\alpha)=0,\quad\theta^{\prime}_{\alpha\beta}(-\alpha)=\theta^{\prime}_{\alpha\beta}(\alpha)=0,\quad\theta^{\prime}_{\alpha\beta}(0)=\theta/\lambda\alpha,$
$-\beta\leq\theta_{\alpha\beta}(x)\leq\beta\quad(\alpha\leq x\leq\alpha.)$
For each $n\in\mathscr{A}$ we shall construct a$bump$:
$\theta_{\alpha_{n}\beta_{n}}$ at $2^{-n}$ i.e.
if $n\in\mathscr{A}$, $\delta\in[-\alpha_{n},+\alpha_{n}]$,
$f(2^{-n}+\delta)\equiv\theta_{\alpha_{n}\beta_{n}}(\delta)$, otherwise
$f(x)\equiv 0$. To make $f$ well-defined, parameters
$\alpha_{n},\beta_{n},n\in\mathscr{A}$ is defined as
$\alpha\equiv 2^{-k-2n-2},\quad\beta_{n}\equiv 2^{-k-n-2},$
where $n=h(k)$ and $h$ is a function enumerating $\mathscr{A}$ without
repetitions(It is easy to proof that if there exists a recursive function
enumerating $\mathscr{A}$, then there exists such function with no
repetitions).
For physicists, does J.Myhill’s results imply that if an object move under the
condition that the displacement and the time satisfies the following relations
$\textbf{\emph{r}}(t)=f(t)=\left\\{\begin{array}[]{ll}\theta_{\alpha_{n}\beta_{n}}(\delta)(n\in\mathscr{A}),&\hbox{if\leavevmode\nobreak\
$t=2^{-n}+\delta$,\quad$\delta\in[\\-\alpha_{n},+\alpha_{n}]$;}\\\
0,&\hbox{o.w.}\end{array}\right.$
The speed $\textbf{\emph{v}}\equiv\textbf{\emph{r}}^{\prime}(t)$ will be a
physical quantity which is not computable?
#### 3.7.4 Physical States which is not computable
Pour-El et al published their results in 1997: for a differential equation,
one can design a specific initial state to make the solution after $t$ ($t$
could be take some computable value)seconds is nowhere computable[8].
Consider the IVP of the following wave equation:
$\left\\{\begin{array}[]{ll}\frac{\partial^{2}u}{\partial
t^{2}}=\frac{\partial^{2}x}{\partial x^{2}}+\frac{\partial^{2}y}{\partial
y^{2}}+\frac{\partial^{2}z}{\partial z^{2}},&\hbox{}\\\
u(x,y,z,0)=f(x,y,z),\frac{\partial u}{\partial
t}(x,y,z,0)=0&\hbox{.}\end{array}\right.$
where $(x,y,z)\in\mathbb{R}^{3},\quad t\in[\leavevmode\nobreak\
0,+\infty\leavevmode\nobreak\ )$ for all $f\in\mathscr{C}^{1}$ this IVP has a
form of solution known as Kirchhoff’s formula:
$u(\overrightarrow{x},t)=\iint_{S^{2}}[f(\overrightarrow{x}+tn)+t\nabla
f(\overrightarrow{x}+t\overrightarrow{n})\cdot\overrightarrow{n}]d\sigma(\overrightarrow{n})$
The conclusion Pour-El get is the following theorem:
###### Theorem 3.16
For all compact set $D\subset\mathbb{R}^{3}\times[\leavevmode\nobreak\
0,\infty\leavevmode\nobreak\ )$, there exists a computable function
$f(x,y,z)\in\mathscr{C}^{1}$, s.t. the corresponding solution
$u(\overrightarrow{x},t)$ is not computable in the neighborhood of any point
in $D$.
Pour-El et al construct the initial value through the uncomputable real number
$\sum^{\infty}_{i=0}\frac{1}{2^{a(i)}},\quad a(i)\in\mathscr{A}$.
Apparently, one can conclude that in this wave equation, the initial state is
computable but the state $u(0,0,0,1)$ is a state which can not be compute.
For us, can we safely conclude that
$\\{\text{Turing Computable}\\}\subset\\{\text{Physical Computable}\\}$
but
$\\{\text{Turing Computable}\\}\not\supset\\{\text{Physical Computable}\\}?$
#### 3.7.5 A few Comments
In the above scenario, the use of (actual) infinity is their common theme.
They ask the system to run for infinite steps or just encode the solutions
into real numbers. It is easy to find out that adding either of these two
assumes into a physical system will make the original system extraordinarily
powerful.
For example, we can throw a particle onto a plane $[0,1]\times[0,1]$ at
random(obey the uniform distribution), then we can proof that with high
probability, the $x$-coordinate(or $y$-coordinate) of the center of the
particle will indicate a non-recursive real number. In fact, in cell
$[0,1]\times[0,1]$, the Lebesgue measurement for the recursive real numbers is
0, while the rest is 1, i.e.
$m([0,1]\times[0,1]\cap\mathbb{R}_{r})=0,m([0,1]\times[0,1]\cap\mathbb{R}^{c}_{r})=1$
This is geometric probability and consider the uniform distribution, the
probability of the either event of the two are just their measurement.
Therefore we can look the $x-$coordinate as a function with respect to the
digits. According to Beggs et al, a theoretical machine called SME may help us
to get the value of the position coordinates[20][21][22].
However, does the strict plane really exist in the physical world? We just do
not know.
We propose the some levels for $f$ which is not computable. Suppose $o$ is an
operator, $\hat{o}$ is a physical implementation of $o$ and operators $\Delta$
, $\nabla$ always exist. For $f\in F$ where all elements in $F$ are non-
recursive functions, we have the level of existence as follows.
* •
Existence-I
$\exists p(\Delta\circ p\circ\nabla\doteq f)$
* •
Existence-I*
$\exists\hat{p}\exists\hat{\nabla}\exists F((f\in
F)\wedge(\Pr\\{\Delta\circ\hat{p}\circ\hat{\nabla}\doteq
f^{\prime}|f^{\prime}\in F\\}>0))$
* •
Existence-II
$\exists\hat{\Delta}\exists p(\hat{\Delta}\circ p\circ\nabla\doteq f)$
* •
Existence-II*
$\exists\hat{p}\exists\hat{\nabla}(\Delta\circ\hat{p}\circ\hat{\nabla}\doteq
f)$
* •
Existence-III
$\exists\hat{p}\exists\hat{\Delta}\exists\hat{\nabla}(\hat{\Delta}\circ\hat{p}\circ\hat{\nabla}\doteq
f)$
According to the levels we proposed above, assume the space is continuous, we
can find out that $\mathbb{P}\in\text{Existence-II}$, J.Myhill’s function
$f\in\text{Existence-I}$, Pour-El’s construction $\phi\in\text{Existence-II}$,
our example $x\in\text{Existence-I*}$. Apparently, we wish to get the examples
in Existence-III.
## 4 Physical Resource Complexity for Quantum Computation
### 4.1 Physical Resource Complexity for Quantum Computation
For general quantum computation, we only need to explain the definition of the
physical state set and the required evolution operators. More over, we only
talk about Monte Carlo styled quantum algorithms.
According to von Neumann’s four postulates for quantum mechanics, we require
that the state of any representation should be vectors in Hilbert space,i.e.
$\Omega\subset\mathbb{H}$
and the evolution operators should be unitary, i.e.
$\mathcal{H}\in U(n)$
Without loss of generality, we can assume that the measurement operators is
projection operators(POVM could be substituted by projection operators through
adding more auxiliary qubits)
#### 4.1.1 The RCEF for Quantum Computation
The resource cost by a computation is
$\begin{array}[]{ccc}\mathfrak{R}_{\mathscr{P}}(\Delta\circ\mathcal{H}\circ\nabla)&=&(\mathfrak{T}(\Delta\circ\mathcal{H}\circ\nabla),\\\
&&\mathfrak{S}(\Delta\circ\mathcal{H}\circ\nabla),\\\
&&\mathfrak{E}(\Delta\circ\mathcal{H}\circ\nabla),\\\
&&\mathfrak{G}(\Delta\circ\mathcal{H}\circ\nabla))\end{array}$
Our definition here is special a case of the one in Sec-III. Suppose our
discussion is restricted to QCM, i.e. we have finite kinds of universal
quantum operators, then the number of gates used and the depth of the whole
circuit will be the main parameter which should be took into account. It is
easy to find out that this definition is similar to that of quantum circuit
model. One of the difference between them is that we will also take the cost
of design(usually this costs time) of a new circuit into account. Though in
most cases, this will not cause great difference from the result given by QCM,
however, we don’t think we can safely ignore the potential exceptions just
because it is usually easy to expand the scale of some circuits.
So far, people always assume that qubit is relatively easy to prepared. At
least in the asymptotic sense, no matter how difficult to prepared a quantum
bit, the cost should be bounded by a constant. We will also do this.
#### 4.1.2 Deutsch-Josza Algorithm
Deutsch-Josza algorithm is one of the most successful algorithms in the early
years. The corresponding problem of the algorithm is: consider two sets of
functions:
1. A:
$\bigg{\\{}\varphi|\varphi:\\{0,\dots,2^{n}-1\\}\rightarrow\\{0,1\\}\textrm{,\leavevmode\nobreak\
\leavevmode\nobreak\ }\forall x(\varphi(x)=0)\bigg{\\}}$
2. B:
$\bigg{\\{}\varphi|\varphi:\\{0,\dots,2^{n}-1\\}\rightarrow\\{0,1\\}\textrm{,\leavevmode\nobreak\
\leavevmode\nobreak\
}\Big{|}\\{x|\varphi(x)=0\\}\Big{|}=\Big{|}\\{x|\varphi(x)=1\\}\Big{|}\bigg{\\}}$
Apparently we have $A\cap B=\emptyset$, now suppose $f\in A\cup B$ and there
is an oracle to compute $f$. We are required to decide whether $f\in A$ or
not. It is no doubt that people wish to reduce the times of query the oracle
as much as possible.
Note that the cost of implement the oracle is not taken into account, because
we assume we have implemented it.
The algorithm needs a trivial input $\psi_{0}=|0\rangle^{\otimes n}|1\rangle$,
and used the gate $H^{\otimes n}\otimes H$ onto the state$\psi_{0}$ a nd get
$\psi_{1}$, i.e.
$\psi_{1}=\Big{(}H^{\otimes n}\otimes H\Big{)}\left(|0\rangle^{\otimes
n}|1\rangle\right)$
Note that $H=\frac{1}{\sqrt{2}}\Big{(}(|0\rangle+|1\rangle)\langle
0|+(|0\rangle-|1\rangle)\langle 1|\Big{)}$. By induction we have
$H^{\otimes n}=\frac{1}{\sqrt{2^{n}}}\sum_{x,y}(-1)^{x\cdot y}|x\rangle\langle
y|$
where i.e. $x\cdot y\equiv\bigoplus\limits_{i}x_{i}\wedge y_{i}$. So we get:
$\begin{array}[]{lll}\psi_{1}&=&\Big{(}H^{\otimes n}\otimes
H\Big{)}\left(|0\rangle^{\otimes n}|1\rangle\right)\\\
&=&\frac{1}{\sqrt{2^{n}}}\bigg{(}\sum\limits_{x,y}(-1)^{x\cdot
y}|x\rangle\langle y|\bigg{)}|0\rangle^{\otimes
n}\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\text{(by
orthogonality)}\\\
&=&\frac{1}{\sqrt{2^{n}}}\sum\limits_{x}(-1)^{0}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\\\
&=&\frac{1}{\sqrt{2^{n}}}\sum\limits_{x=0}^{2^{n}-1}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\\\
\end{array}$
Now use the oracle $U_{f}:|x,y\rangle\rightarrow|x,y\oplus f(x)\rangle$ onto
the state $\psi_{1}$ to get $\psi_{2}$
$\begin{array}[]{lll}\psi_{2}&=&U_{f}\Bigg{(}\frac{1}{\sqrt{2^{n}}}\sum\limits_{x=0}^{2^{n}-1}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\Bigg{)}\\\
&=&\frac{1}{\sqrt{2^{n}}}\sum\limits_{x=0}^{2^{n}-1}|x\rangle\bigg{(}f(x)\oplus\frac{|0\rangle-|1\rangle}{\sqrt{2}}\bigg{)}\\\
&&\\\
&=&\frac{1}{\sqrt{2^{n}}}\sum\limits_{x=0}^{2^{n}-1}(-1)^{f(x)}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\\\
\end{array}$
At last we use $H^{\otimes n}\otimes I$ onto $\psi_{2}$to get$\psi_{3}$:
$\begin{array}[]{lll}\psi_{3}&=&\Big{(}H^{\otimes n}\otimes
I\Big{)}\Bigg{(}\frac{1}{\sqrt{2^{n}}}\sum\limits_{x=0}^{2^{n}-1}(-1)^{f(x)}|x\rangle\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\Bigg{)}\\\
&=&\sum\limits_{z}\sum\limits_{x}\frac{(-1)^{x\cdot
z+f(x)}|z\rangle}{2^{n}}\left[\frac{|0\rangle-|1\rangle}{\sqrt{2}}\right]\\\
\end{array}$
The observer is supposed to check the first $n$ qubits, note that the
amplitude of $|0\rangle^{\otimes n}$ is $\sum_{x}(-1)^{f(x)}/2^{n}$. If $f\in
A$, $f(x)$ is constant and the amplitude of $|0\rangle^{\otimes n}$ is $+1$ or
$-1$. So the amplitude of another cases should be zero and the observer will
get $|0\rangle^{\otimes n}$. On the other hand, if $f\in B$, the amplitude of
$|0\rangle^{\otimes n}$ will be zero. So the observer will always get a non-
zero vector.
In our opinion, the procedure could be written as follows.
$\begin{array}[]{lll}\nabla&\equiv&\text{Initialize the
state}|0\rangle^{\otimes n}\otimes|1\rangle\\\ &&\text{Generate the whole
circuit}\\\ \mathcal{H}_{1}&\equiv&H^{\otimes n}\otimes H\\\
\mathcal{H}_{2}&\equiv&U_{f}\\\ \mathcal{H}_{3}&\equiv&H^{\otimes n}\otimes
I\\\ \Delta&\equiv&\sum\limits_{i}|P_{i}\rangle\langle P_{i}|\end{array}$
Let $\mathcal{H}=\mathcal{H}_{3}\circ\mathcal{H}_{2}\circ\mathcal{H}_{1}$
$\left(\Delta\circ\mathcal{H}\circ\nabla\right)(|x\rangle)=\Big{(}\Delta\circ\mathcal{H}_{3}\circ\mathcal{H}_{2}\circ\mathcal{H}_{1}\circ\nabla\Big{)}(|x\rangle)=P(f\in
B)$
Though Deutsch-Jozsa Algorithm is great, someone still think it is not very
useful. In addition to the fact that the problem they studied is not very
important, there does exist an efficient classical probabilistic algorithm to
solve the problem with high probability.
#### 4.1.3 Shor’s Algorithm
Shor’s Algorithms for prime factorization and discrete logarithms[10, 11] is
so far the most exciting quantum algorithms. The appearance of Shor’s
Algorithms is the greatest challenge to strong Church Turing Thesis.
Shor’s Algorithms depends on a technique of so called ”quantum Fourier
Transform”. But of course QFT is not enough. Shor’s Algorithm is totally non-
trivial and marvelous, and few people can produce any algorithms like that
easily.
In order to understand Shor’s Algorithm, it may be enough to gain a clear idea
of quantum ordering algorithm. This is the only subprogram in the Shor’s
Algorithm which has to be implemented by quantum computers so far, and it is
really the most important subprogram.
First, note that
$\sum^{r-1}_{s=0}\exp(-2\pi isk/r)=r\delta_{k0}$
and define $|u_{s}\rangle$ as follows
$|u_{s}\rangle\triangleq\frac{1}{\sqrt{r}}\sum^{r-1}_{k^{\prime}=0}e^{-2\pi
isk^{\prime}/r}\Big{|}x^{k^{\prime}}\textrm{mod}N\Big{\rangle}$
According to the fact above, we can get
$\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}e^{2\pi
isk/r}|u_{s}\rangle=\Big{|}x^{k}\text{mod }N\Big{\rangle}$
In fact
$\displaystyle\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}e^{2\pi isk/r}|u_{s}\rangle=$
$\displaystyle\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}\left(e^{2\pi
isk/r}\frac{1}{\sqrt{r}}\sum^{r-1}_{k^{\prime}=0}e^{-2\pi
isk^{\prime}/r}\Big{|}x^{k^{\prime}}\textrm{mod}N\Big{\rangle}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{r}\sum^{r-1}_{s=0}\left(e^{2\pi
isk/r}\sum^{r-1}_{k^{\prime}=0}e^{-2\pi
isk^{\prime}/r}\Big{|}x^{k^{\prime}}\textrm{mod}N\Big{\rangle}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{r}\sum^{r-1}_{k^{\prime}=0}\sum^{r-1}_{s=0}\textrm{exp}\left({\frac{2\pi
is(k-k^{\prime})}{r}}\right)\Big{|}x^{k^{\prime}}\textrm{mod}N\Big{\rangle}$
$\displaystyle=$
$\displaystyle\frac{1}{r}\sum^{r-1}_{k^{\prime}=0}r\delta_{kk^{\prime}}\Big{|}x^{k^{\prime}}\textrm{mod}N\Big{\rangle}$
$\displaystyle=$ $\displaystyle\Big{|}x^{k}\text{mod }N\Big{\rangle}$
In particular, when $k=0$, we have
$\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}|u_{s}\rangle=|1\rangle^{\otimes L}$
where $L\equiv\lceil\log(N)\rceil$.
Suppose $U_{x,N}$ satisfies
$U_{x,N}|y\rangle\triangleq|xy(\text{mod}N)\rangle$. Considering
$\mathbb{Z}^{*}_{N}$ and the fact that the permutation on orthnormal basis can
be represented as a unitary operator, one can know for sure that $U_{x,N}$ is
unitary. What’s more $u_{s}$ is a eigenvector of $U_{x,N}$ , the corresponding
eigenvalue is $e^{\frac{2\pi is}{r}}$ since
$U_{x,N}|u_{s}\rangle=\frac{1}{\sqrt{r}}\sum^{r-1}_{k=0}e^{\frac{-2\pi
isk}{r}}\Big{|}x^{k+1}\text{mod}N\Big{\rangle}=e^{\frac{2\pi
is}{r}}|u_{s}\rangle$
Reverse the results above, we get the first half of the quantum ordering
Algorithm, which complete the following task:
$|1\rangle^{\otimes
L}=\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}|u_{s}\rangle\xrightarrow[\text{Modular
exponentiation}]{U_{x,N}^{z_{t}2^{t-1}}\cdots
U_{x,N}^{z_{1}2^{0}}}\boxed{\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}e^{2\pi
isk/r}|u_{s}\rangle}=\Big{|}x^{k}\text{mod }N\Big{\rangle}$
where the state $\frac{1}{\sqrt{r}}\sum^{r-1}_{s=0}e^{2\pi
isk/r}|u_{s}\rangle$ is the one we desire. Apparently the eigenvalue contains
the information of $r$. So as to extract the information, we need a sub-
progress named ”quantum phase estimation” which based on inverse quantum
fourier transformation. One can verify that if $t$ is large enough, such like
$t=2L+1+\lceil\log\left(2+\frac{1}{2\varepsilon}\right)\rceil$, for each
$s\in\\{0,\dots,r-1\\}$, we will obtain the estimation of $\varphi\approx s/r$
accurate to $2L+1$ bits with probability at least $(1-\varepsilon)/r$. Through
the continued fractions algorithm, we will get $r$ with high
probability(According to PNT).
In our opinion, the procedure above could be written as:
$\begin{array}[]{lll}\nabla&\equiv&\text{Initialize the
state}|0\rangle^{\otimes t}\otimes|1\rangle^{\otimes L}\\\ &&\text{Generate
the whole circuit}\\\ \mathcal{H}_{1}&\equiv&H^{\otimes n}\otimes H\\\
\mathcal{H}_{2}&\equiv&CU_{x,N}\\\
\mathcal{H}_{3}&\equiv&FT^{{\dagger}}\otimes I^{\otimes L}\\\
\mathcal{H}_{4}&\equiv&CF\otimes I^{\otimes L}\\\
\Delta&\equiv&\sum\limits_{i}|P_{i}\rangle\langle P_{i}|\end{array}$
It is easy to check that except the $\mathcal{H}_{3}$, all operators cost
polynomial time with respect to $\log N$. The complexity of operator modular
exponentiation and continued fraction are both $O(L^{3})$, which are two most
time-consuming subprocedure of the whole algorithm except the
$\mathcal{H}_{3}$(inverse quantum fourier transform).
Note that $\mathcal{H}_{3}$ is indeed not an operator which could be
implemented by polynomial universal gates. Consider a family of gates used in
$\mathcal{H}_{3}$ which is usually noted by $R_{k}$($k\in\\{2,\dots,L\\}$)
$R_{k}=\begin{pmatrix}1&0\\\ 0&e^{2\pi i/2^{k}}\\\ \end{pmatrix}$
In other words, the original Shor’s Algorithm is not a algorithm with super-
polynomial acceleration. In order to overcome this, Coppersmith created a new
algorithm called the AFFT(Approximate Fast Fourier transform) [12] which can
substitute for the procedure QFT.
#### 4.1.4 Grover Algorithm
Quantum Search Algorithm[13], also known as Grover’s Algorithm, is another
quite successful quantum algorithm. Though this algorithm is not faster than
the fastest classical search algorithms super-polynomially, one can proof it
is the fastest one considering quantum mechanics. Therefore, the complexity of
the algorithm is the complexity of the problem it deals with.
The crucial subroutine of Grover’s Algorithm is the Grover iteration, often
denoted by $G$:
* •
Apply Oracle $O:|x\rangle|-\rangle\rightarrow(-1)^{f(x)}|x\rangle|-\rangle$
* •
Apply Hadamard Gates:$H^{\otimes n}$
* •
Perform a conditional phase shift($2|0\rangle\langle 0|-I$) on the computer,
with every non-zero bases receiving a phase shift of $-1$.
* •
Perform Hadamard transformation $H^{\otimes n}$.
Note that $H^{\otimes n}(2|0\rangle\langle 0|-I)H^{\otimes
n}=2|\psi\rangle\langle\psi|-I$ One can proof that Grover iteration can be
looked as a rotation in the plane spanned vectors which denoted the right
answers and the wrong answers.
Let $\Sigma^{\prime}_{x}$ be the sum of all the vectors which indicate a
solution to the search problem, $\Sigma^{\prime\prime}_{x}$ the rest. Define
normalized states:
$\begin{array}[]{ccc}|\alpha\rangle&\equiv&\frac{1}{\sqrt{N-M}}\Sigma^{\prime\prime}_{x}|x\rangle\\\
&&\\\
|\beta\rangle&\equiv&\frac{1}{\sqrt{M}}\Sigma^{\prime}_{x}|x\rangle\end{array}$
thus the initial state
$|\psi\rangle=\frac{1}{N^{1/2}}\Sigma^{N-1}_{x=0}|x\rangle$ could be
represented as
$|\psi\rangle=\sqrt{\frac{N-M}{N}}|\alpha\rangle+\sqrt{\frac{M}{N}}|\beta\rangle$
The action of Operator $O$ is
$O(a|\alpha\rangle+b|\beta\rangle)=a|\alpha\rangle-b|\beta\rangle$, which
could be looked as perform a reflection in $\alpha\beta-$ plane. Similarly
Operator $2|\psi\rangle\langle\psi|-I$ also performs a reflection in
$\alpha\beta-$plane. Thus both two reflections which could be looked as a
rotation occur in the $\alpha\beta-$plane. Let $\cos\theta/2=\sqrt{(N-M)/N}$,
s.t.$|\psi\rangle=\cos\theta/2|\alpha\rangle+\sin\theta/2|\beta\rangle$, apply
the iteration once makes $|\psi\rangle$ become
$G|\psi\rangle=\cos\frac{3\theta}{2}|\alpha\rangle+\sin\frac{3\theta}{2}|\beta\rangle$
$k$ times use of Grover’s Iteration will lead to the following result:
$G^{k}|\psi\rangle=\cos\left(\frac{2k+1}{2}\theta\right)|\alpha\rangle+\sin\left(\frac{2k+1}{2}\theta\right)|\beta\rangle$
Since $|\psi\rangle=\sqrt{(N-M)/N}|\alpha\rangle+\sqrt{M/N}|\beta\rangle$, we
just need to rotate $|\psi\rangle$ $\arccos\sqrt{M/N}$ radians to the one
which is parallel to vector $|\beta\rangle$. So repeating $G$ for
$R=[\frac{\arccos\sqrt{M/N}}{\theta}]$ times will get $|\psi\rangle$ to within
an angle $\theta/2\leq\pi/4$ of $|\beta\rangle$. This is a ’good’ state, for
people only have to repeat the experiment for expected constant times to get
the solution to the problem(Consider geometric probability distribution:
$\mathrm{E}[X]=1/(1/2)=2$).
Apparently $R\leq\lceil\pi/2\theta\rceil$, suppose $M\leq N/2$ then we have
$\frac{\theta}{2}\geq\sin\frac{\theta}{2}=\sqrt{\frac{M}{N}}$. Thus, we
obtain:
$R\leq\left\lceil\frac{\pi}{4}\sqrt{\frac{N}{M}}\right\rceil$
in other words we need repeat $G$ for $R=O(\sqrt{N/M})$ times.
### 4.2 Quantum Simulation and Quantum Algorithm
Quantum Lattice Celluar Automata(QLCA) and Quantum Gas Automata(QG-A) are two
familiar ideal models in the research of quantum simulation[15] . Meyer,
Boghosian[15, 16, 17] have obtained their results respectively by using these
models, that is, they construct some quantum algorithms which demonstrate
exponentially speedup in such models. For Bohosian, the object they tried to
simulated is a QGA which obey lattice Boltzman distribution, where arbitrary
fields can be concerned with. They have proofed that the complexity of
simulation is only related to the dimension of the lattice, but almost has
nothing to do with the number of the particles. However, the number of
particle always cause exponentially hardness on a classical computer. In fact,
Boghosian’s results imply that it is almost impossible for a classical
computer to simulate one evolution step of a quantum system including dozens
of particles.
We’ve mentioned that it is the difficulty of quantum simulation that makes
people believe quantum mechanics can provide enormous power of computation in
the early years.
Note that in this article we do not care about the hardness of simulations.
Generally speaking, the hardness of simulation has nothing to do with the one
of computation. For instance, people may find it difficult to simulate some
classical celluar automaton according to the given regulations, however once
the tedious work has been completed there often exists some more simple
methods to produce the series. A typical example is that the regulations of an
automata actually cause a circle with a finite period in the series. The same
thing can happens to quantum simulations too.
However, it is important to know that there must exists some cases in which
simulations and computations are equivalent. These extreme cases often appears
when the length of regulations is near the Kolmogorov complexity(lower bound
of description) of a series. Still, strictly speaking, at present no one can
proof that polynomially universal unitary operators really cause exponentially
difficulty in classical computation. To understand this, just consider an easy
but helpful fact that almost all the problems we want to efficiently solved on
a quantum computer are in the class BQP, and we have BQP$\subseteq$PSPACE.
Unfortunately PSPACE$=$P is not totally impossible. Of course most people
don’t believe this is true, since this would imply that Shor’s Algorithms can
be polynomially simulated on a classical computers.
Now we discuss how to extract a corresponding quantum algorithm from a method
of quantum simulation, which is believed to be exponentially faster than any
classical one of the same target.
On a high level, we should do following things:
* •
Find a family of experiments of quantum mechanics which can be efficiently
simulated by quantum computers but are believed to be hardly to simulate and
compute by classical computers
* •
Design a ’good’ problem about some non-trivial properties of the last state of
the system, which makes quantum computers able to present the answer to the
observer quickly.
Designing the problem is a crucial step. In most cases, though we may have
quickly obtained the probabilistic distribution very close to the real
experiments, we can not know the whole information in short time. So first we
have to ask a question which can be easily verified by any quantum computers
containing the whole quantum information of the system.
For example, we can ask a question such like:
* •
What the number of the $n_{0}-$th digit of the probability of a certain system
arriving in $\Omega^{\prime}(\Omega^{\prime}\subset\Omega)$?
The problem of this method is that in high dimensional spaces, it is very
likely that the probability of the set $\Omega^{\prime}$ is exponentially
close to zero, which actually enables a classical computers to guess zero
without running and get the right answer in most cases.
Now we propose our version: Suppose $\phi$ is the wave function of the system
we’ve simulated and $|\phi(X)|^{2},X\subset\Omega$ is the probability of
$\vec{x}$ appear in $X$. Try to find two subsets $A,B\subset\Omega$ s.t.
$\frac{3}{7}\leq\frac{|\phi(A)|^{2}}{|\phi(B)|^{2}}\leq 1$
and determine the value of the $n_{0}-$th digit of $\phi(A)$.
For the systems which (probabilistic)Turing Machine cannot simulate in
polynomial time, the question above is intuitively hard to answer, though up
till now no one can proof or disproof it.
On the other hand, if these systems can be efficiently simulated by quantum
computers, repeating following procedure will ensure us to find the answer
relatively much faster than any probabilistic Turing Machine of the same aim.
###### Definition 4.1
(Vector of normal vectors $\vec{x}$)
$\vec{x}\equiv\left(\left(\begin{array}[]{c}0\\\ 0\\\ \vdots\\\
1\end{array}\right),\left(\begin{array}[]{c}0\\\ \vdots\\\ 1\\\
0\end{array}\right),\dots,\left(\begin{array}[]{c}1\\\ \vdots\\\ 0\\\
0\end{array}\right)\right)$
###### Definition 4.2
(Procedure $P_{Q}$) $P_{Q}$(In pseudo-code):
while(find the answer)
{
. Mid-cut the space $\Omega$ by super-plane whose normal vector is $x_{i}$.
. Suppose the two spaces is $\Omega_{1}$ and $\Omega_{2}$
. if(the condition is satisfied(verifies by testing))
. {
. halt
. }
. else
. {
. $\Omega=\mathrm{min}_{|\phi|}\\{\Omega_{1},\Omega_{2}\\}$
. $i++$
. }
}
## 5 Conclusions and Future Works
We formally proposed the theory of physical computation, define the concepts
of resource and complexity. Several examples, including classic mechanics and
quantum mechanics, were discussed and analyzed under the framework of physical
computation. A technique, which is used to converse a method of quantum
simulation into a quantum algorithm, is discussed.
This is a exciting field, we believe there is more exciting topic to
discussed. A very interesting question is: can we find a physical mechanism as
the fastest implementation of an arbitrary functions?
In Sec-III, we talk about the question of calculating the centroid of an
object. We thought it is the limitation of dimensions(only three dimensions)
hide the advance of the method we mentioned. We conjecture that this method
has a excellent counterpart in high dimensional cases. We’ll have a try in the
(quantum)statistics mechanism.
In Sec-IV, we talked about quantum simulations and how to construct a clever
problem to induced a quantum algorithm. Actually, we conjecture that the
problem we construct is a hard one in class $\\#P$, for these questions have a
counting style. However, we are not sure about whether the designed questions
could be in $\\#P-hard$ under some specific statistical models. We shall try
to work on this in the future.
We’ve mentioned that we assume that polynomial qubits is polynomially hard to
prepare. However, it is harder to control the qubits as the number of them
increase[18] so far. So one can still conjecture that preparing qubits itself
is a ”complicated computing”, and the results up till now can be explained as
someone displace the resource consuming procedure, just like DNA Algorithms.
## 6 Acknowledgement
The authors are grateful to Haixing Hu, Nan Wu and Jiasen Wu for their useful
remarks and suggestions.
## References
* [1] P.Benioff. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing Machines J. Stat. Phys.,22(5):563-591,1980
* [2] Richard P.Feynman. Simulating Physics with Computers Int. J. Theor. Phys. 21(1982)467-888
* [3] D.Deutsch. Quantum theory, the Church-Turing Princple and the universal quantum computer. Proc. R. Soc. Lond. A,400:97,1985
* [4] E.Bernstein and U.Vazirani. Quantum complexity theory. SIAM J. Comput., 26(5): 1411-1473, 1997.
* [5] A.C.Yao. Quantum circuit complexity. Proc. of the 34th Ann. IEEE Symp. on Foundations of Computer Science,pages 352-361,1993
* [6] J.Myhill. A Recursive Function,Defined On a Compact Interval and Having a Continuous Derivative that is Not Recursive. Michigan Math.J.18(1971)
* [7] Itamar Pitowsky. The Physical Church Thesis and Physical Computational Complexity A Jerusalem Philosophical Quarterly 39(January 1990)
* [8] Marian B.Pour-El and Ning Zhong. The Wave Equation with Computable Initial Data Whose Unique Solution Is Nowhere Computable Mathematical Logic Quarterly ©Johann Ambrosius Barth 1997
* [9] D.Deutsch and R.Jozsa. Rapid solution of problems by quantum computation. Proc. R.Soc. London A, 439:553, 1992
* [10] P.W.Shor. Algorithms for quantum computation:discrete logarithms and factoring. In Prodeedings, 35th Annual Symp. on Foundations of Computer Sciece, IEEE press, Los Alamitos, CA, 1994
* [11] P.W.Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.SIAM. J. Comp.,26(5):1484-1509,1997
* [12] D.Coppersmith. An approximate Fourier transform usefulin quantum factoring. IBM Research Report RC 1994
* [13] L.K.Grover. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett., 79(2):325,1997
* [14] L.M.Adleman. Molecular computation of solutions to combinatorial problems. Science, 266:1021, 1994
* [15] D.A.Meyer, J.Stat.Phys.85,551(1996)
* [16] B.M.Boghosian and W. Taylor, Phys. Rev. E 57, 54(1998)
* [17] B.M.Boghosian and W.Taylor, Intl. J.Mod. Phys. C8, 705(1997).
* [18] M.A.Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press (2000)
* [19] Edwin Beggs, José Félix Costa, and John V. Tucker _Computational Models of Measurement and Hempel’s Axiomatization_
* [20] Edwin Beggs, José Félix Costa, and John V. Tucker _Physical Oracles: The Turing Machine and the Wheatstone Bridge_
* [21] Edwin Beggs, José Félix Costa, Bruno Loff, and John V. Tucker _Oracles and Advice as Measurements_
* [22] Edwin Beggs, José Félix Costa, Bruno Loff, and John V. Tucker _On the Complexity of Measurement in Classical Physics_
* [23] Edwin Beggs, José Félix Costa, Bruno Loff, and John V. Tucker _Computational complexity with experiments as oracles. II. Upper bounds_
|
arxiv-papers
| 2011-07-16T12:52:20 |
2024-09-04T02:49:20.632622
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huimin Zheng, HaiXing Hu, Nan Wu and Fangmin Song",
"submitter": "HuiMin Zheng",
"url": "https://arxiv.org/abs/1107.3226"
}
|
1107.3271
|
# On the Simulation of Adaptive Measurements via Postselection
Vikram Dhillon dhillonv10@gmail.com
###### Abstract.
In this note we address the question of whether any any quantum computational
model that allows adaptive measurements can be simulated by a model that
allows postselected measurements. We argue in the favor of this question and
prove that adaptive measurements can be simulated by postselection. We also
discuss some potentially stunning consequences of this result such as the
ability to solve #P problems.
## 1\. Introduction
In [3] Aaronson introduced a complexity class ${\bf\sf PostBQP}$, which is is
a complexity class consisting of all of the computational problems solvable in
polynomial time on a quantum Turing machine with postselection and bounded
error. It was also shown equivalent to ${\bf\sf PP}$ which is the class of
decision problems solvable by a probabilistic Turing machine in polynomial
time, with an error probability of less than 1/2 for all instances. Aaronson
then raised an interesting question which asks whether adaptive measurements
made by a quantum computational model be simulated with postselected
measurements. In this note we address this question by asserting that it is
possible to simulate adaptive measurements by postselection on the quantum
circuit model of computation. We also explore the consequences of being able
to atleast theoritically perform this simulation, it is known that ${\bf\sf
P^{PP}}={\bf\sf P^{\\#P}}$ which implies that the complexity of ${\bf\sf PP}$
is equivalent to that of ${\bf\sf P^{\\#P}}$ which is an ${\bf\sf NP}$.Sso if
an adaptive (non-projective) measurement such as a weak measurement can be
simulated, following the work of Lloyd $\mathit{et.al}$ logical gates can be
constructed that allow us to solve ${\bf\sf P^{\\#P}}$ problems.
## 2\. Proof
Before we show how the simulation would work, we want to establish some
definitions to make an easier transition to the proof itself.
###### Definition 1.
An adaptive measurement is an incomplete measurement is made on the system,
and its result used to choose the nature of the second measurement made on the
system, and so on (until the measurement is complete). A complete measurement
is one which leaves the system in a state independent of its initial state,
and hence containing no further information of use. [4]
###### Definition 2.
Postselection is the power of discarding all runs of a computation in which a
given event does not occur. [3]
We will be using the quantum circuit model which is the standard model in
quantum computation theory and most other computational models have been shown
to be equivalent to it. The equivalence also allows us to simulate those
models on the circuit model.
###### Lemma 3.
Measurement based quantum computation (MBQC) employs adaptive local
measurements on a resource state.
###### Proof.
See [4] for this. ∎
###### Lemma 4.
Measurement based models can be simulated on the quantum circuit model.
###### Proof.
Any one-way computation can be made into a quantum circuit by using quantum
gates to prepare the resource state [5]. ∎
###### Axiom 5.
From Lemma 3 and Lemma 4 we can deduce that the qunatum circuit model can
simulate measurement based computation which is a model that allows for
adaptive measurements.
The above mentioned axiom completes the first part of the corrospondence, we
now have to show that the same model that can simulate postselected
measurements to complete the corrospondance. Postselected measurements fall
under the complexity class ${\bf\sf PostBQP}$ and we will also use the
equivalence of ${\bf\sf PostBQP}$ and ${\bf\sf PP}$ shown by Aaronson in [3].
This switch between complexity classes makes this proof simplistic.
###### Axiom 6.
${\bf\sf BQP}\subset{\bf\sf PP}$
###### Lemma 7.
${\bf\sf PP}\cap{\bf\sf BQP}\notin\\{\phi\\}$
###### Proof.
Let us assume that no problem exist at the intersection of ${\bf\sf PP}$ and
${\bf\sf BQP}$ However, we know the aforementioned axiom to be true so there
must atleast be one problem that exist at the intersection of the complexity
classes. That particular problem, by the virtue of being at the intersection
will be both ${\bf\sf PP}$ and ${\bf\sf BQP}$ which is self-evident. We will
represent the problems present at the intersection of the two complexity
classes by the set $\tau$. ∎
###### Lemma 8.
From Lemma 7 we can deduce that elements of $\tau$ can be simulated on a
quantum computer
###### Proof.
The elements of $\tau$ fall in the clas ${\bf\sf BQP}$ which can be simulated
on a quantum computer therefore the elements of that set can also be simulated
by a quantum computation model. ∎
###### Lemma 9.
Elements of $\tau$ can exibit postselected measurements
###### Proof.
The members of $\tau$ are both ${\bf\sf PP}$ and ${\bf\sf BQP}$ where ${\bf\sf
BQP}$ can be simulated on a quantum computer and since ${\bf\sf PP}={\bf\sf
PostBQP}$, elements of $\tau$ can be simulated through the use of postselected
measurements. ∎
###### Axiom 10.
From the preceding proof and Axiom 5, we see the corrospondence is complete.
We can indeed take a quantum computational model (in our case, it is the
standard quantum circuit model) that allows for adaptive measurements (Axiom
5) and simulate it with a model that allows for postselected measurements
(again the quantum circuit model)
The preceeding axiom presents the completed proof, in the following section we
will discuss some speculative consequences of this result.
## 3\. Consequences
Although machines capable of postselectd measurements are implausable, the
ability to simulate a particular type of adaptive measurement called weak
measurement has some very interesting consequences. Lloyd $\mathit{et.al}$ [1]
showed that when weak measurements are made on a set of identical quantum
systems, the single-system density matrix can be determined to a high degree
of accuracy while affecting each system only slightly. If this information can
then be fed back into the system by coherent operations, the single-sytem
density matrix can be made to undergo arbitrary nonlinear dynamics such as
dynamics governed by a nonlinear Schrödinger equation. Nonlinear corrections
to quantum mechanical evolution can then be used to construct nonlinear
quantum gates which can solve $\\#{\bf\sf P}$ and ${\bf\sf NP}$-Complete
problems as shown by Lloyd and Abrams [2].
## References
* [1] Lloyd, Seth and Slotine, Jean-Jacques E. Quantum feedback with weak measurements. Phys. Rev. A (2000).
* [2] Abrams, Daniel S. and Lloyd, Seth. Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for ${\bf\sf NP}$-Complete and $\\#{\bf\sf P}$ Problems. Phys. Rev. Lett. (1998).
* [3] Taylor, P.L., Heinonen, O.: A Quantum Approach to Condensed
* [4] R. Raussendorf, D. E. Browne, and H. J. Briegel. Measurement based Quantum Computation on Cluster States. Phys. Rev. A (2003).
* [5] H. M. Wiseman, D. W. Berry, S. D. Bartlett, B. L. Higgins, and G. J. Pryde. Adaptive Measurements in the Optical Quantum Information Laboratory. IEEE Journal of Selected Topics in Quantum Electronics (2009).
|
arxiv-papers
| 2011-07-17T02:49:01 |
2024-09-04T02:49:20.647310
|
{
"license": "Public Domain",
"authors": "Vikram Dhillon",
"submitter": "Vikram Dhillon",
"url": "https://arxiv.org/abs/1107.3271"
}
|
1107.3278
|
Composite structural motifs of binding sites for delineating biological
functions of proteins
Akira R. Kinjo1∗, Haruki Nakamura1,
1 Institute for Protein Research, Osaka University, Suita, Osaka 565-0871,
Japan.
$\ast$ E-mail: akinjo@protein.osaka-u.ac.jp
Short title: Composite motifs of binding sites
## Abstract
Most biological processes are described as a series of interactions between
proteins and other molecules, and interactions are in turn described in terms
of atomic structures. To annotate protein functions as sets of interaction
states at atomic resolution, and thereby to better understand the relation
between protein interactions and biological functions, we conducted exhaustive
all-against-all atomic structure comparisons of all known binding sites for
ligands including small molecules, proteins and nucleic acids, and identified
recurring elementary motifs. By integrating the elementary motifs associated
with each subunit, we defined composite motifs which represent context-
dependent combinations of elementary motifs. It is demonstrated that function
similarity can be better inferred from composite motif similarity compared to
the similarity of protein sequences or of individual binding sites. By
integrating the composite motifs associated with each protein function, we
define meta-composite motifs each of which is regarded as a time-independent
diagrammatic representation of a biological process. It is shown that meta-
composite motifs provide richer annotations of biological processes than
sequence clusters. The present results serve as a basis for bridging atomic
structures to higher-order biological phenomena by classification and
integration of binding site structures.
## Introduction
Virtually every biological process is realized, at the atomic level, through a
series of interactions between proteins and other molecules. Accordingly, most
proteins, if not all, synchronously or asynchronously interact with multiple
molecules ranging from single atom ions, small (non-polymer) molecules to
proteins, nucleic acids and other macromolecules. The types and combinations
of interactions in proteins are known to modulate their functions. For
example, depending on their ligand-bound or ligand-free forms as well as
interactions with corepressor or coactivator proteins, nuclear receptors can
perform intricate transcriptional regulations [1]. The activity of coronavirus
3C-like proteases is controlled by dimerization through their C-terminal
domain which is absent from picornavirus 3C proteases [2]. Furthermore,
certain homologous proteins can catalyze completely different enzymatic
reactions, namely transferase or hydrolase activities, by adopting different
oligomerization states [3]. Therefore, it is important to identify not only
individual interactions, but also possible combinations of the interactions
that can be accommodated by a protein to fully describe its molecular
functions as well as to distinguish different functions among homologous
proteins.
The advance in genome sequence technologies is making it more and more
imperative to develop effective techniques for inferring protein functions
from sequence information. To date, the most widely used approach for protein
function prediction is the annotation transfer, which is based on the
assumption that protein functions are similar if their sequences are similar
[4, 5, 6]. It has been gradually recognized, however, that such annotation
transfer approaches may be unreliable in many cases [7, 8]. It has also been
shown that function similarity is not a simple function of sequence similarity
[9]. These observations prompt us to have more detailed examination of the
determinants of protein functions.
Structural information has been proved valuable for precisely understanding
protein functions [10]. Thanks to the structural genomics efforts [11, 12], we
now have a great wealth of structural information available for close
examination of sequence-structure-function relationships of proteins. However,
when global topologies or folds of protein structures are considered, it is
often even more difficult to assign a specific function to a particular fold,
for some folds include an extremely diverse set of proteins with diverse
functions [3, 13]. The use of structural information is not limited to finding
global fold similarity and distant evolutionary relationship. In particular,
physical interactions between protein molecules and their ligands observed in
experimentally solved protein structures allow more direct approaches to
elucidate the relationship between protein structures and functions [14, 15].
To date, there have been many methods for detecting potential ligand binding
sites based on structural similarity of proteins [16, 17, 18, 19, 20, 21, 22,
14]. Most of these methods are targeted at predicting protein functions at the
level of ligand binding and catalytic activity. There have also been many
studies on protein-protein interaction interfaces to understand biological
functions of proteins in cellular contexts [23, 24, 25, 26, 27, 28, 15, 29,
30, 31, 32, 33, 34]. However, apart from a few works [35, 36, 37], most of
these studies are focused on particular types of interactions _per se_ and do
not explicitly address how the combination of interactions with small
molecules and macromolecules modulates with biological function of proteins.
To understand the relationship between the patterns of interactions at atomic
level and biological functions of proteins, we herein performed exhaustive
all-against-all structural comparisons of binding site structures at atomic
level using all structures available in the Protein Data Bank (PDB) [38], and
identified recurring structural patterns of ligand binding sites to define
_elementary motifs_. We then defined _composite motifs_ by integrating the
elementary motifs associated with individual subunits. In other words, protein
subunits with the same combination of elementary motifs are said to share an
identical composite motif. We then examined how such composite motifs
correlated with protein functions as defined by the UniProt database [39]. It
is demonstrated that the similarity between composite motifs better
corresponds with the similarity between functions compared to the similarity
between protein sequences or between individual binding sites. Finally, by
integrating all the composite motifs associated with particular functions, we
define _meta-composite motifs_. It is shown that meta-composite motifs are
useful to elucidate the rich internal structures of biological processes
compared to sets of homologous sequence clusters.
## Results
### Identification of elementary and composite motifs
We first generated all biological units as annotated in the PDBML [40] files,
and then extracted 197,690 protein subunits which contained at least one
ligand (non-polymer, protein or nucleic acid) binding site. Here, a ligand
binding site of a subunit is defined as a set of atoms of the subunit that are
in contact with some atoms of the ligand within 5Å. While we do not use any
pre-defined non-redundant data set based on sequence similarity, the
redundancy is taken care of after clustering similar structures (see below).
In this manner, the structural diversity of proteins with highly homologous or
identical amino acid sequences can be preserved in the following analyses
while the structural redundancy is removed.
All-against-all structure comparisons of 410,254 non-polymer binding sites,
346,288 protein binding sites and 20,338 nucleic acid binding sites using the
GIRAF structure search and alignment program [41] followed by complete linkage
clustering yielded 5,869, 7,678 and 398 clusters (with at least 10 members) of
non-polymer, protein and nucleic acid binding sites, respectively. (We did not
use in the following analyses small clusters with less than 10 members because
some small clusters exhibited spurious similarities.) We refer to these
clusters as _elementary motifs_ in the following. An elementary motif can be
regarded as a bundle of mutually similar atomic dispositions of binding sites
(Fig. 1A). It should be noted that the elementary motifs are solely based on
the binding site structures, and they do not directly include the identity of
the binding partners. We have previously performed comprehensive analyses of
elementary motifs [14, 15]. It was found that most elementary motifs were
confined within homologous families. In some exceptional cases, motifs were
shared across non-homologous families with different folds, which included
motifs for metal, mononucleotide or dinucleotide binding for non-polymer
binding sites [14] and coiled-coil motifs for protein binding sites [15].
The set of all elementary motifs contained in a protein subunit is called the
_composite motif_ of the subunit (Fig. 1B,C). Thus, two subunits sharing the
same set of elementary motifs are said to have the same composite motif. In
total, 5,738 composite motifs, each of which is shared by at least 10
subunits, were identified. Our hypothesis is that thus defined composite
motifs exhibit good correspondence with protein functions. In the example in
Fig 1, while the three proteins (LAAO [42], KDM1 [43] and PAO [44]) share the
same elementary motif (N2) for FAD binding and they share the same domain
folds (FAD/NAD(P)-binding domain and FAD-linked reductases C-terminal domain
[45]), their biological functions are similar but different; and these
differences correspond to the differences in their composite motifs.
### Characterization of composite motifs
The number of elementary motifs that comprise a composite motif ranges from 1
to 20 (Fig. 2A). Approximately one third of the composite motifs (1975 out of
5738) consist of only one elementary motif and more than 90% of the composite
motifs are composed of less than or equal to 5 elementary motifs. The number
of composite motifs appears exponentially decreasing as the number of
constitutive elementary motifs increases.
To characterize the diversity of composite motifs, the average and minimum
sequence identities were calculated for pairs of subunits sharing the same
composite motifs (Fig. 2B). Although the majority of composite motifs are
shared between close homologs on average, many of them contain distantly
related subunits. In particular, 118 composite motifs were shared between
subunits whose sequence similarity could not be detected by BLAST [46].
However, only three out of these 118 composite motifs consisted of more than
one and at most two elementary motifs. Thus, if a composite motif consists of
more than one elementary motif, it is most likely to comprise only homologous
proteins.
By defining the similarity between two composite motifs as the fraction of
shared elementary motifs (Eq. 4), we also examined the similarity between
different composite motifs as a function of minimum sequence identity between
them (Fig. 2C). While many composite motifs share no elementary motifs for the
entire range of sequence identities, some do share a significant fraction of
their constitutive elementary motifs in spite of weak sequence similarities.
It is also noted that the composite motif similarities widely vary for high
sequence identities. Thus, while each composite motif comprises homologous
proteins in most cases, the converse does not hold in general so that
composite motif similarity hardly correlates with sequence similarity. This
observation clearly demonstrates that it is not possible to take into account
the structural diversity of binding sites and their combinations by using a
representative set of proteins based on sequence similarity.
### Association of composite motif similarity with function similarity
In order to study the functional relevance of the composite motifs, we next
examined the association between composite motif similarity and function
similarity. Here, the function of a protein is defined as a set of controlled
keywords provided in UniProt [39] and the similarities for composite motifs
and UniProt functions are defined by the Jaccard index (see Materials and
Methods, Eq. 4). For comparison, we also checked sequence identity as well as
binding site similarity (Eq. 3) as measures of subunit similarities in place
of composite motif similarity (Fig. 3A). In order to reduce the bias due to
the redundant data set, we randomly picked one representative from each
composite or elementary motif, or sequence cluster (with 100% sequence
identity cutoff) for this comparison. It is evident that the function
similarity persists even for low composite motif similarities although the
function similarity is not always 100% for 100% composite motif similarity. To
the contrary, we can only infer high function similarities for high sequence
or binding site similarities.
Since many UniProt function annotations, especially those of ligand binding
activities, have been actually derived from the PDB entries, the high
correlation between composite motifs and UniProt functions may appear trivial.
However, the current elementary motifs that constitute composite motifs do not
directly include the information of their ligands, but are solely based on
their binding site structures. The bare binding site similarity does not
correspond with the function similarity as strongly as the composite motif
similarity. In addition, when we used only the UniProt functions under the
Biological process category which are less directly related to molecular
functions, we still observed the highest function similarity for a wide range
of composite motif similarity compared to sequence or binding site
similarities (Fig. 3B). These results demonstrate that composite motifs
sharing a small fraction of elementary motifs imply more function similarity
compared to bare sequence or binding site similarities.
When we examined the correspondence between composite motifs and UniProt
functions excluding those composite motifs that consisted of only one
elementary motif, the correspondence was found to be slightly better (Figs.
3C,D). This indicates that combinations of multiple elementary motifs may
enhance accurate inference of specific protein functions.
Although the similarity between composite motifs implies similar functions, 15
composite motifs were found to be shared by completely different functions.
11, 3, and 1 of these composite motifs consisted of 1, 2, and 3 elementary
motifs, respectively. 7 of them were due to improper annotations for
artificially engineered proteins, to incomplete annotations in the UniProt, or
to a wrong annotation in the PDB, and 3 were due to coiled-coil structures.
Among the remaining 5 composite motifs, 2 composite motifs were actually found
in the same dimeric complexes, and each of them consisted of only 1 elementary
motif shared between remotely homologous proteins.
### Examples of composite motifs sharing the same elementary motif and fold
but with different functions
We have already presented in Introduction an example that demonstrated
different combinations of elementary motifs (i.e., composite motifs) might
modulate function specificity (Fig. 1). The analysis in the previous section
showed that composite motif similarity is a good indicator of function
similarity. In this section, we provide several examples of proteins that
share the same elementary motif and the same fold, but have different
composite motifs and different functions (Fig. 4). These examples show that
the difference in functions can be associated with the difference in composite
motifs within the same family of proteins.
#### Glycine oxidase (GO) and glycerol-3-phosphate dehydrogenase (GlpD)
GO from _Bacillus subtilis_ (PDB 1RYI [47], chain A) and GlpD from
_Escherichia coli_ (PDB 2QCU [48], chain A) share the same elementary motif
for binding the FAD cofactor, and despite the low sequence similarity ($\sim$
14% sequence identity), they share the same fold (FAD/NAD(P)-binding domain
[45]) according to the Matras fold comparison program [49, 50] (Fig. 4A).
While GO forms a homotetramer and has 3 elementary motifs for protein binding,
GlpD is monomeric (however, the latter may form a dimer [48]). In addition,
they have their own elementary motif for binding the respective ligands
(glycolic acid, GOA, in PDB 1RYI and phosphoenolpyruvate, phosphate ion, PO4,
in PDB 2R46). Thus, they have different composite motifs. Although the shared
elementary motif for FAD binding and the shared fold, they exhibit different
enzymatic activities, EC 1.4.3.19 for GO and EC 1.1.5.3 for GlpD, and function
in different contexts, thiamine biosynthesis and glycerol metabolism,
respectively.
#### D-3-phosphoglycerate dehydrogenase (PGDH) and C-terminal-binding protein
3 (CtBP3)
PGDH from _E. coli_ (PDB 1PSD [51], chain A, EC 1.1.1.95) and CtBP3 (also
called CtBP1) from rat (PDB 1HKU [52], chain A, EC 1.1.1.-) share the same
elementary motif for binding the NAD cofactor and the same folds
(NAD(P)-binding Rossmann-fold domain and Flavodoxin-like fold [45]) with 25 %
sequence identity (Fig. 4B). PGDH forms a homotetramer with one of its
protein-protein interface located at its additional ACT domain [53], and is
involved in L-serine biosynthesis. CtBP3, forming a homodimer or heterodimer
with CtBP2, is involved in controlling the structure of the Golgi complex and
acts as a corepressor targeting various transcription regulators [52]. While
these proteins may catalyze very similar reactions, their biological roles are
clearly different.
#### $\beta$-trypsin and coagulation factor VII
Bovine $\beta$-trypsin (PDB 1G3C [54], chain A, EC 3.4.21.4) and human
coagulation factor VII heavy chain (PDB 1WQV [55], chain H, EC 3.4.21.21) are
both serine proteases with 40 % sequence identity. In these structures, they
share the same elementary motif for protease inhibitors at the catalytic sites
in addition to similar calcium ion binding sites although the latter do not
belong to the same elementary motif (Fig. 4C). Factor VII heavy chain is in
complex with its light chain counter part as well as with tissue factor, which
shapes its functional form. On the other hand, $\beta$-trypsin is not known to
form a similar complex structure. Thus, the difference in their complex
structures can be associated with the difference in their functions: digestion
for $\beta$-trypsin and blood coagulation for Factor VII.
#### Cytochrome $b_{2}$ and glycolate oxidase (GOX)
Mitochondrial cytochrome $b_{2}$, also known as L-lactate dehydrogenase, from
_Saccharomyces cerevisiae_ (PDB 1FCB [56], chain A, EC 1.1.2.3) and glycolate
oxidase (GOX) from spinach (PDB 1AL7 [57], chain A, EC 1.1.3.15) share the
TIM-barrel fold with 40 % sequence identity, and have the same elementary
motif for flavin mononucleotide (FMN) (Fig. 4D). Although they have roughly
equivalent homotetrameric complexes, the number of interacting subunits are
different: a subunit of cytochrome $b_{2}$ interacts with all 3 other subunits
whereas that of GOX interacts with only 2 out of 3 other subunits. In
addition, cytochrome $b_{2}$ also has an elementary motif for heme binding in
its additional heme-binding domain which is utilized for transferring
electrons to cytochrome $c$ following oxidation of lactate [56]; such function
is not associated with GOX.
### Meta-composite motifs for annotating functions
While each composite motif describes a particular state of a protein subunit,
any biological process is realized as a series of interaction patterns. In
this sense, composite motifs only represent snapshots of biological processes.
To have a more integrative view of biological processes, we define _meta-
composite motifs_ by grouping all the composite motifs associated with
particular functions (Fig. 5A,B). For 3,359 UniProt functions, 2,760 meta-
composite motifs were identified. The number of composite motifs associated
with meta-composite motifs ranged from 1 to 157, with the average of 2.39 (S.D
4.62). While the same UniProt function implies the same meta-composite motif
by definition, the converse does not hold in general as there are more
functions than meta-composite motifs. Meta-composite motifs thus allow us to
understand protein functions as an ensemble of snapshots of ligand-bound
states of proteins. For comparison, we analogously defined meta-sequence
motifs by associating each function with corresponding sequence clusters
(complete linkage). We defined two types of sequence clusters, the one (type-1
sequence cluster) is based solely on BLAST E-value cutoff of 0.05, the other
(type-2 sequence cluster) is based on sequence identity cutoff of 100%. Thus,
the former sequence clusters include a wide range of homologous sequences
while the latter include only (almost) identical sequences.
We then compared the meta-composite motif or meta-sequence motif similarities
with function similarity (Fig. 5C). It is not surprising that the function
similarity appears lower for the meta-composite motif similarity than for
composite motif similarity because, by definition, different meta-composite
motifs always have different functions while different composite motifs may
have identical functions. Although the differences are small, we can still
observe that similar meta-composite motifs imply more similarity in functions
than either type-1 or type-2 meta-sequence motifs (Fig. 5C).
It is also noted that the average size of meta-composite motifs
(2.39$\pm$4.62) is statistically significantly greater than those of meta-
sequence motifs (1.88$\pm$4.42 for type-1, 1.86$\pm$3.43 for type-2). This
indicates that the composite motifs more finely dissect protein functions than
the sequence clusters.
### Network structure of meta-composite motifs in biological processes
Since the meta-composite motifs are defined by grouping together all composite
motifs associated with particular functions, they are more suitable for
analyzing, rather than predicting, protein functions in terms of interaction
states of proteins. For example, we can identify a meta-composite motif for
the UniProt keyword “Transcription” (Fig. 6A), and subsequently connect the
constituent composite motifs (nodes) based on relations such as common
elementary motifs or common sequences. When a protein in one composite motif
interacts with another protein in another (possibly the same) composite motif,
an edge representing protein-protein interaction can be also drawn. In the
case of composite motifs, nodes may be also characterized according to their
constituent elementary motifs (i.e., interaction states). We can observe a
variety of interaction states of nodes and relations between nodes. If two
nodes share identical sequences, it reflects a transition between different
interaction states, possibly changing their atomic structures. For example,
there are PDB entries of human cellular tumor antigen p53 with or without
bound DNA (e.g., PDB 1UOL [58] and 2AC0 [59]) which share the same elementary
motif for zinc binding but have different composite motifs depending on the
presence or absence of the elementary motif for DNA binding. Similarly, there
are PDB entries of yeast RNA polymerase II with or without bound DNA/RNA in
which the subunit RPB2 (e.g., PDB 1I3Q [60], chain B and 1Y1W [61], chain B)
share some elementary motifs for protein binding, but other corresponding
protein binding sites belong to different elementary motifs due to slight
conformational changes (not shown), and an elementary motif for binding DNA is
present in only one of the entries; thus these subunits identical in amino
acid sequence have different composite motifs which are connected by edges of
the common protein binding motifs and of the common sequence. Such description
is not possible with meta-sequence motifs (Fig. 6B) because sequence
similarity alone cannot discriminate different interaction states.
To evaluate the properties of networks of meta motifs more generally and more
quantitatively, we identified the meta motif for each upper-most keyword in
the hierarchy of the UniProt Biological process category, and compared various
network characteristics of meta-composite motifs against those of meta-
sequence motifs (Fig. 7). On average (Fig. 7A), meta-composite motifs include
more nodes (i.e., composite motifs), more connected components, as well as
more connections between nodes representing common sequences (identified by
the UniProt accession) and protein-protein interactions, compared to both
type-1 and type-2 meta-sequence motifs. In particular, the increased number of
edges representing common sequences indicates that many identical proteins are
split into different composite motifs. The same trend is also observed for a
particular meta-composite motif obtained for the keyword “Transcription” (Fig.
7B). As expected, the type-1 meta-sequence motifs exhibit rather poor
characteristics in most aspects because many homologs are grouped into large
clusters so that differences in interaction states of proteins cannot be
differentiated. While the type-2 meta-sequence motifs sometimes contain more
edges for common elementary motifs, this is simply because many elementary
motifs shared among homologous proteins are split into different sequence
clusters irrespective of interaction states, which is reflected in the lower
number of edges representing common sequences. Thus, the classification of
proteins in terms of composite motifs allows us to inspect the organization of
proteins involved in individual biological processes.
In summary, the observation that meta-composite motifs have more counts in
nodes, connected components, common sequences and protein-protein interactions
implies that meta-composite motifs discriminate the subtle differences in the
interaction states or conformations of the proteins involved in the biological
processes and such discrimination is not possible with meta-sequence motifs.
## Discussion
Structural classifications of proteins have been traditionally targeted at
elucidating the universality of protein architectures based on the notion of
structural domains. As such, it is not necessarily suitable for analyzing
specific functions of particular proteins [62]. In other words, the current
protein structure classifications, for a good reason, ignore the differences
among protein structures within the same families or folds. The examples shown
in Figs. 1 and 4 clearly show that although those proteins share the same
folds, they have varied functions. Such limitations of fold classifications
with respect to specific assignment of protein functions have been known for
some time [13]. Recently, seemingly minute differences within protein folds
are being recognized as determinants of functional specificity as exemplified
by the concept of “embellishments” proposed by Orengo and coworkers [63, 64].
Although it is often assumed that domains are the units of functions, there
are inherent limitations in this assumption. For example, it has been known
that the combination of domains generates new functions [65], therefore it is
questionable to assign one function to one domain. Furthermore, the very
definition of domains is problematic as there exists no universally accepted
definition of domains [66].
In this study, we avoided the complications regarding the definition of
domains, and directly analyzed the atomic structures of binding sites
irrespective of overall topology or homology of proteins. Nevertheless, it has
been previously shown that thus identified elementary structural motifs are
mostly confined within homologous protein families [14, 67], especially for
protein binding sites [15]. In this sense, the classification of binding site
structures are effectively not very different from the traditional protein
classifications. However, by combining the elementary motifs found in
individual subunit structures solved under different experimental conditions,
it becomes possible to specify a particular interaction state for a particular
subunit. Thus, the classification of proteins based on composite motifs
differs from the traditional classification schemes in that the notion of the
composite motif allows us to explicate the universality of binding site
structures and the diversity of their combinations at the same time. It should
be stressed that the redundancy of the current PDB is essential for
identifying elementary and composite motifs since the diversity of atomic
structures is not negligible even for highly homologous or identical proteins
[14, 68]. In addition, different interaction states of the same protein are
also useful for characterizing conformational transitions [69, 70, 71].
We have demonstrated that the similarity between composite motifs of proteins
well indicates the similarity between their functions (Figs. 3A,B). A recent
study also indicates that the integration of non-polymer and protein binding
sites enhances the detection of functional specificity [37]. These results
manifest the importance of the context-dependent combination of ligand binding
motifs for understanding protein functions. The application of composite
motifs to function prediction, however, requires some caveats. In case when we
know a protein structure with bound ligands, we first need to identify the
elementary motifs to which the binding sites belong to. But it may not be
always possible to identify all the necessary elementary motifs. In case when
we only have a protein structure in its ligand-free form, it is necessary to
predict its binding sites if any should exist. In this case, we need to rely
on prediction based on prediction, which necessarily leads to low accuracy.
While this limitation is inherent in any annotation transfer approaches, it is
more stringent on the one based on composite motifs because it requires more
interaction states to be solved for similar proteins. In any case, it is
preferable to accumulate more structures in the PDB, not only those of
completely novel folds, but also those of known folds but in new ligand-bound
forms. It is worth noting that the function prediction by composite motif
similarity is not based on supervised learning or parameter fitting so that
the results obtained here should hold mostly valid for newly solved structures
to the extent that the distribution of functionally characterized proteins in
the PDB stays the same.
By grouping the composite motifs associated with particular functions, we
defined meta-composite motifs. It was demonstrated that the description based
on meta-composite motifs provided us with a detailed annotation of biological
processes (Figs. 5,6). By describing biological processes in terms of
composite motifs rather than individual structures, we can abstract the
pattern of interactions so that the commonality and specificity of the
interactions in different contexts, such as species or pathways, for example,
can be delineated. Although there are currently some limitations in this
description, such as the absence of temporal relation between composite motifs
or the lack of experimental structures for some possible transient complexes,
these limitations may be overcome in the future by complementing meta-
composite motifs with other experimental information such as gene/protein
expression or interactome analyses.
In summary, we have introduced composite motifs that well describe protein
functions based on the context-dependent combinations of structural patterns
of binding sites, and provide a useful means to describe the atomic details of
biological processes.
## Materials and Methods
### Data set
We have used all the PDB entries as of December 29, 2010 (70,231 entries). All
the biological units were generated for each entry as annotated in the PDBML
files [40], except for those with icosahedral, helical, or point symmetries
(mostly viruses). For the latter, only the corresponding (icosahedral, etc.)
asymmetric units were used. Entries without annotated biological units were
treated as they are given. Some PDB entries contain more than one biological
unit all of which were used in the present study since alternative oligomeric
states may (or may not) be biologically relevant. The biological units in the
PDB are defined by authors and/or software (PQS[72] and/or PISA[73]). In
total, 197,690 subunits in 79,826 biological units contained at least one
ligand binding site.
A ligand binding site of a subunit is defined as a set of at least 10 atoms in
the subunit that are in contact with some atoms of a ligand within 5Å radius.
In this study, ligands include non-polymers, proteins, and nucleic acids. The
non-polymer ligands are those annotated as such in the PDBML [40] files, but
water molecules were discarded. The protein ligands are those annotated as
“polypeptide(L)” with at least 25 amino acid residues. The nucleic acid
ligands are those annotated as “polydeoxyribonucleotide,” “polyribonucleotide”
or “polydeoxyribonucleotide/polyribonucleotide hybrid.”
### Similarity between binding site structures
To compare binding site structures, we used the GIRAF structural search and
alignment program [41] with some modifications to enable faster database
search and flexible alignments (unpublished). GIRAF produces an atom-wise
alignment for a pair of binding sites. After all-against-all comparisons of
binding sites, elementary motifs were defined as complete-linkage clusters
with a cutoff GIRAF score [41] of 15, as in our previous studies [14, 15]. The
cutoff value was chosen so that the largest cluster did not predominate all
the other clusters due to the “phase transition” of the similarity networks
[74, 14]. The GIRAF score is defined as
$G(A,B)=\frac{N_{A,B}\sum_{a}w(\mathbf{x}^{A}_{a},\mathbf{x}^{B}_{a})}{\min\left[N_{A},N_{B}\right]}$
(1)
where $N_{A}$ and $N_{B}$ are the number of atoms of the binding sites $A$ and
$B$ respectively, and $N_{A,B}$ is the number of aligned atom pairs. The
weight $w(\mathbf{x}^{A}_{a},\mathbf{x}^{B}_{a})$ for the aligned atom pairs
$\mathbf{x}^{A}_{a}$ and $\mathbf{x}^{B}_{a}$ ($a=1,\cdots,N_{A,B}$) is
defined as
$w(\mathbf{x}^{A}_{a},\mathbf{x}^{B}_{a})=\max\left[1-d(\mathbf{x}^{A}_{a},\mathbf{x}^{B}_{a})/d_{c},0\right]$
(2)
where $d(\mathbf{x}^{A}_{a},\mathbf{x}^{B}_{a})$ is the distance between two
atoms in a superimposed coordinate system and the cutoff distance $d_{c}$ is
set to 2.5 Å.
Clusters with less than 10 members were excluded in this study because
structural similarity in small clusters may be coincidental. In fact, when
there were protein pairs not detected by BLAST within a cluster, the fraction
of such pairs was 79% on average for clusters with less than 10 members while
that for clusters with at least 10 members was 36%. Although motifs shared
between remote homologs or non-homologs may provide interesting examples, we
expect many of them are not biologically relevant.
The raw GIRAF score largely depends on the size of binding sites. Therefore,
when comparing binding site similarity with function similarity, we used a
normalized similarity measure so that binding sites of varying sizes can be
compared on the same scale. Let $N_{A}$, $N_{B}$ and $N_{A,B}$ be defined as
above, then the normalized similarity $S(A,B)$ between the binding sites $A$
and $B$ is defined as
$S(A,B)=100\times\frac{2N_{A,B}}{N_{A}+N_{B}}~{}~{}(\%).$ (3)
### Functions defined by UniProt keywords
For each subunit in the data set, the corresponding UniProt [39] accession
identifier was obtained from the struct_ref category of the PDBML file. In
total, 186,791 subunits with at least 1 ligand binding site in the PDB were
annotated by UniProt. For thus identified UniProt entries, their keywords were
extracted. The UniProt keywords are a set of controlled vocabulary to describe
the properties of proteins and they are organized in a hierarchical order. In
most cases, these keywords are manually assigned by curators, hence they are
expected to be more reliable. This is in contrast to the Gene Ontology
annotations (http://geneontology.org) for the PDB which are mostly
automatically annotated and are likely to contain a large number of erroneous
annotations.
For each subunit, all the keywords annotated either explicitly or implicitly
via the keyword hierarchy, were extracted except for those belonging to the
Technical term, Disease, or Domain categories. We define the function of a
subunit as the set of the UniProt keywords associated with it. In other words,
two subunits whose associated sets of keywords are exactly identical are
defined to have the same function. In total, 7,991 UniProt functions were
defined. The similarity between two UniProt functions are defined by the
Jaccard index between the sets of keywords associated with the functions (see
below, Eq. 4)
### Similarity between two sets
The similarity measures for composite motifs, functions or meta motifs are
based on comparison between two sets. Given the sets $A$ and $B$, their
similarity is defined by the Jaccard index $J(A,B)$:
$J(A,B)=100\times\frac{|A\cap B|}{|A\cup B|}~{}~{}(\%).$ (4)
For a given composite motif, function, meta-composite motif or meta-sequence
motif, the set consists of elementary motifs, UniProt keywords, composite
motifs, or sequence clusters, respectively.
### Sequence clusters
To define meta-sequence motifs, complete-linkage clustering was applied to the
result of an all-against-all BLAST [46] comparison with two different
criteria. In one case, all pairs of sequences in a cluster must have BLAST
E-value of at most 0.05. This resulted in 3,327 clusters with at least 10
members. These clusters are referred to as type-1 sequence clusters. In the
other case, all pairs of sequences in a cluster must have 100% sequence
identity as well as E-value of at most 0.05. This resulted in 4,594 clusters
with at least 10 members, which are referred to as type-2 sequence clusters.
When BLAST produces more than one alignment for a pair of sequences, the
alignments were integrated into one alignment as long as they were mutually
consistent.
### Comparison between motif similarity and function similarity
Although we did not use any representative set for defining elementary and
composite motifs based on sequence similarity, we did use representatives of
motifs and sequences when their similarities were compared with function
similarity (c.f., Figs. 3 and 5C) in order to reduce the bias due to different
sizes of clusters. For composite motifs, a representative was randomly
selected from each composite motif. For binding sites, a representative was
randomly selected from each elementary motif. For protein sequences, a
representative was randomly selected from each type-2 sequence cluster.
Average function similarities for a given range of motif, binding site or
sequence similarity (Fig. 3) were calculated for 10 sets of randomly selected
representatives and the standard deviations of the average function similarity
are shown as error bars. Only those points with at least 500 (50 for nucleic
acid binding sites) samples on average are shown in Figs. 3A,B.
For meta-composite and meta-sequence motifs, 50 % of the all observed pairs of
meta motifs were randomly selected and the average function similarity was
calculated. This procedure was iterated 10 times, and data points with at
least 10 samples are reported with the standard deviation of the average
values in Fig. 5C.
We have confirmed that selecting different random sets of representatives in
all the above cases did not alter the results significantly.
### Downloadable data
The results of all-against-all comparison of binding sites and classifications
are made available for download at http://pdbj.org/giraf/cmotif/.
## Acknowledgments
We thank Toshiaki Katayama and Hideki Hatanaka for helping with uploading data
to TogoDB.
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## Figure Legends
Figure 1: Examples of elementary and composite motifs. A: Concrete examples
of elementary motifs (corresponding to B). Several binding sites belonging to
each elementary motif are superimposed. The binding site atoms that constitute
the elementary motif are shown in ball-and-stick representation with CPK
coloring and ligands are shown in green wireframes (non-polymers) or tubes
(proteins). These binding sites include subunits shown in C. Non-polymer
ligands are phenylalanine and its analogs (N1), FAD (N2), and polyamines (N3).
B: In this example, the combinations of 3 non-polymer binding elementary
motifs (cyan triangles labeled N1, N2 and N3) and 3 protein binding elementary
motif (orange rectangles labeled P1, P2 and P3) found in various protein
subunits (black dots) define 3 distinct composite motifs (hexagons in magenta
labeled C1, C2, and C3). Examples of each elementary motif are shown in
molecular figures (A) right above the triangles or rectangles, and those of
each composite motif are shown in molecular figures (C) right below the
hexagons. Direct correspondence between elementary and composite motifs is
indicated by thick edges in pale magenta. C: Concrete examples of composite
motifs (corresponding to B). These 3 composite motifs share the same
elementary motif for FAD binding (labeled N2 in B). Subunits (colored pink)
containing the composite motifs (C1, C2, C3) are shown with elementary motifs
in ball-and-stick representations (protein binding sites in orange, non-
polymer binding sites in cyan) and with ligands in green (spacefill for non-
polymers, cartoon for proteins). From left to right: L-amino acid oxidase
(LAAO) from _Calloselasma rhodostoma_ in homo-dimeric form (PDB ID: 1F8S [42],
chain A); human lysine-specific histone demethylase 1 (KDM1) (PDB ID: 2IW5
[43], chain A); polyamine oxidase (PAO) from _Zea mays_ in putative homo-
dimeric form (PDB ID: 3KU9 [44], chain A, pdbx_struct_assembly.id 3). The
protein figures were created using jV [75]. The network diagrams (also in
Figs. 5 and 6) were created using Cytoscape [76]. Figure 2: Characterization
of composite motifs. A: Histogram of the number of elementary motifs
comprising composite motifs. B: Histograms of the average and minimum sequence
identities (%) between pairs of subunits within each composite motif. C:
Composite motif similarity as a function of minimum sequence identity between
pairs of composite motifs. Sequence identity between two composite motifs is
defined as the sequence identity between two protein sequences, one belonging
to the one motif, the other to the other motif. Figure 3: Correspondence
between composite motifs and protein functions. A: Average UniProt function
similarity as a function of similarity between subunits based on composite
motifs, individual binding sites or sequence identity. Data points with
insufficient number of samples were discarded (see Materials and Methods).
Error bars indicate the standard deviation of the average function similarity
based on 10 bootstrap samplings. B: Same as A, except that only the UniProt
functions of the Biological process category were used. C: Composite motifs
with more than one elementary motif (n$>$1) are compared with those with at
least one elementary motif (n$>$0), the latter are the same as in A. D: Same
as C, except that only the UniProt functions of the Biological process
category were used. Figure 4: Examples of differences in composite motifs and
functions. Left column: superposition of common elementary motifs (pink and
cyan) and their ligands (magenta and blue). Center column: the biological unit
containing the subunit with the elementary motif shown in the left column in
pink, with interacting molecules (other than that in the left column) in green
and non-interacting molecules in grey. Right column: the biological unit
containing the subunit with the elementary motif shown in the left column in
cyan, with interacting molecules (other than that in the left column) in green
and non-interacting molecules in grey. A: Glycine oxidase (center) and
glycerol-3-phosphate dehydrogenase (right), sharing FAD binding motif (left).
B: D-3-phosphoglycerate dehydrogenase (center) and C-terminal binding protein
3 (right) sharing NAD binding motif (left). C: $\beta$-trypsin (center) and
coagulation factor VII (right) sharing protease inhibitor binding motif
(left). D: Cytochrome $b_{2}$ (center) and glycolate oxidase (right) sharing
FMN binding motif (left). Figure 5: Meta-composite motifs. A: A meta-
composite motif is defined as a set of all composite motifs (hexagons in
magenta) associated with particular UniProt functions (green circles). The
associations are defined through individual protein subunits (black dots); see
text for the detailed definitions. Each composite motifs are associated with
elementary motifs for non-polymer (triangles in cyan), protein (rectangles in
orange), or nucleic acid (diamonds in blue) binding sites (c.f. Fig. 1). B: A
simplified representation of the diagram shown in A. C: Average function
similarity as a function of meta-composite motif similarity or meta-sequence
motif (type-1 and type-2) similarity. Figure 6: Network structure of the
meta motif for biological process. Examples of a meta-composite motif (A) and
a type-1 meta-sequence motif (B) for the UniProt biological process
“Transcription.” A: The meta-composite motif, i.e., the set of composite
motifs (colored hexagons) associated with Transcription. B: type-1 meta-
sequence motif, i.e., the set of type-1 sequence clusters associated with the
same keyword. Figure 7: Characteristics of meta motif networks. A: Average
counts of composite motifs or sequence clusters (denoted CM/SC), connected
components (CC) as well as edges representing sharing of common elementary
motifs (CEM) for non-polymer, protein and nucleic acid binding sites, common
sequences (CS) and protein-protein interactions (PPI). B: The same counts for
nodes and various edges, but only for the meta motifs for the UniProt keyword
“Transcription” (corresponding to the diagrams in Fig. 6).
|
arxiv-papers
| 2011-07-17T05:56:04 |
2024-09-04T02:49:20.653368
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Akira R. Kinjo and Haruki Nakamura",
"submitter": "Akira Kinjo",
"url": "https://arxiv.org/abs/1107.3278"
}
|
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